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tl;dr: A method to compute mod( $x,2\pi$ ) (and mod( $x,\pi$ ), mod( $x,\pi/2)$ ) in Julia (using the ieee754_rem_pio2 in OpenLibm) is described, implemented, and tested. It appears to be as accurate as possible, and twice as slow as the naive alternative. This is in response to this Julia issue.

Modpi - what is the issue, and why should we care?¶

Why would one be interested in the remainder of x modulo pi? Well, for trigonometry for starters. How to compute the $sin$ of a large number, for example?

In [1]:

sin(111111111111111)

Out[1]:

0.7321216741892959

Well, we know that we can compute $\sin$ using $\sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...$ , or some other series. There are two important properties of $sin$ (and other trigonometric functions) here:

the series approximation converges well for small values of $x$
the function is periodic with period $2\pi$ .

Now, $111111111111111 = 1768388256576 * 2\pi + 0.82......$ , and therefore

$\sin(111111111111111) =$

$\sin(111111111111111 - floor(111111111111111 / 2\pi) * 2\pi ) =$

$\sin(mod(111111111111111,2\pi)) =$

$\sin(0.82......)$

Let's do it:

In [2]:

mod(111111111111111,2pi)

Out[2]:

0.8257627936194325

In [3]:

sin(mod(111111111111111,2pi))

Out[3]:

0.7350651674203722

Oops... that's off the result computed above already in the 3rd significant digit:

In [4]:

sin(111111111111111.0) - sin(mod(111111111111111.0,2pi))

Out[4]:

-0.002943493231076344

What's the problem?

The problem is the naive way of computing $\mod(x,2\pi)$ , by subtracting two huge numbers that are very close to each other, leading to cancellation (most digits are zero, and very few significant digits are left, and the rest is just random rubbish).

This problem has been recognized a long time ago, and essentially been tackled, under the name "range reduction". Here is a paper with an algorithm and a few references. Fortunately, the OpenLibm that comes with Julia already contains a routine to tackle this problem, ieee754_rem_pio2 ("remainder with respect to pi over 2"). Let's wrap it up and test it.

In [23]:

function ieee754_rem_pio2(x::Float64)
    # rem_pio2 essentially computes x mod pi/2 (ie within a quarter circle)
    # and returns the result as 
    # y between + and - pi/4 (for maximal accuracy (as the sign bit is exploited))
    # and n, where n specifies the integer part of the division, or, at any rate, 
    # in which quadrant we are.
    # The invariant fulfilled by the returned values seems to be
    #  x = y + n*pi/2 (where y = y1+y2 is a double-double and y2 is the "tail" of y).
    # Note: for very large x (thus n), the invariant might hold only modulo 2pi 

    y = [0.0,0.0]
    n = ccall((:__ieee754_rem_pio2,libm), Cint, (Float64,Ptr{Float64}),x,y)

    return (n,y)
end

Out[23]:

ieee754_rem_pio2 (generic function with 1 method)

Let's test it with a few values:

In [6]:

for k = [-20:20]
  println(rpad(lpad("$k",3," ")*"*pi/8 + 0.2 = $(k/8*pi+0.2)",40," "), ieee754_rem_pio2(k/8*pi+0.2))
end

-20*pi/8 + 0.2 = -7.653981633974483     (-5,[0.20000000000000048,8.503679843877493e-19])
-19*pi/8 + 0.2 = -7.261282552275758     (-5,[0.5926990816987251,-2.6905207631241164e-17])
-18*pi/8 + 0.2 = -6.868583470577034     (-4,[-0.5853981633974479,2.2884754890353088e-17])
-17*pi/8 + 0.2 = -6.475884388878311     (-4,[-0.19269908169872416,-4.8708207252758256e-18])
-16*pi/8 + 0.2 = -6.083185307179586     (-4,[0.20000000000000043,-4.8708207252758256e-18])
-15*pi/8 + 0.2 = -5.6904862254808615    (-4,[0.592699081698725,2.2884754890353088e-17])
-14*pi/8 + 0.2 = -5.297787143782138     (-3,[-0.5853981633974479,-3.8347585050568314e-17])
-13*pi/8 + 0.2 = -4.905088062083414     (-3,[-0.1926990816987242,-1.05920094349394e-17])
-12*pi/8 + 0.2 = -4.5123889803846895    (-3,[0.20000000000000037,-1.05920094349394e-17])
-11*pi/8 + 0.2 = -4.119689898685965     (-3,[0.592699081698725,-3.8347585050568314e-17])
-10*pi/8 + 0.2 = -3.726990816987241     (-2,[-0.585398163397448,1.1442377445176544e-17])
 -9*pi/8 + 0.2 = -3.334291735288517     (-2,[-0.19269908169872385,1.1442377445176544e-17])
 -8*pi/8 + 0.2 = -2.941592653589793     (-2,[0.2000000000000003,1.1442377445176544e-17])
 -7*pi/8 + 0.2 = -2.548893571891069     (-2,[0.5926990816987244,1.1442377445176544e-17])
 -6*pi/8 + 0.2 = -2.1561944901923447    (-1,[-0.585398163397448,-4.9789962508669555e-17])
 -5*pi/8 + 0.2 = -1.7634954084936207    (-1,[-0.19269908169872413,5.721188722588272e-18])
 -4*pi/8 + 0.2 = -1.3707963267948966    (-1,[0.2,5.721188722588272e-18])
 -3*pi/8 + 0.2 = -0.9780972450961725    (-1,[0.5926990816987242,-4.9789962508669555e-17])
 -2*pi/8 + 0.2 = -0.5853981633974483    (-1,[0.9853981633974483,-4.9789962508669555e-17])
 -1*pi/8 + 0.2 = -0.19269908169872413   (-1,[1.3780972450961724,6.12323399538461e-17])
  0*pi/8 + 0.2 = 0.2                    (1,[-1.3707963267948966,-6.12323399538461e-17])
  1*pi/8 + 0.2 = 0.5926990816987241     (1,[-0.9780972450961726,4.9789962508669555e-17])
  2*pi/8 + 0.2 = 0.9853981633974482     (1,[-0.5853981633974484,4.9789962508669555e-17])
  3*pi/8 + 0.2 = 1.3780972450961724     (1,[-0.19269908169872424,-5.721188722588272e-18])
  4*pi/8 + 0.2 = 1.7707963267948965     (1,[0.1999999999999999,-5.721188722588272e-18])
  5*pi/8 + 0.2 = 2.163495408493621      (1,[0.5926990816987242,4.9789962508669555e-17])
  6*pi/8 + 0.2 = 2.556194490192345      (2,[-0.5853981633974482,-1.1442377445176544e-17])
  7*pi/8 + 0.2 = 2.948893571891069      (2,[-0.19269908169872407,-1.1442377445176544e-17])
  8*pi/8 + 0.2 = 3.3415926535897933     (2,[0.20000000000000007,-1.1442377445176544e-17])
  9*pi/8 + 0.2 = 3.7342917352885174     (2,[0.5926990816987242,-1.1442377445176544e-17])
 10*pi/8 + 0.2 = 4.126990816987242      (3,[-0.5853981633974483,3.8347585050568314e-17])
 11*pi/8 + 0.2 = 4.519689898685965      (3,[-0.1926990816987246,1.05920094349394e-17])
 12*pi/8 + 0.2 = 4.91238898038469       (3,[0.19999999999999998,1.05920094349394e-17])
 13*pi/8 + 0.2 = 5.305088062083414      (3,[0.5926990816987245,3.8347585050568314e-17])
 14*pi/8 + 0.2 = 5.697787143782138      (4,[-0.5853981633974483,-2.2884754890353088e-17])
 15*pi/8 + 0.2 = 6.090486225480862      (4,[-0.19269908169872466,4.8708207252758256e-18])
 16*pi/8 + 0.2 = 6.483185307179586      (4,[0.19999999999999993,4.8708207252758256e-18])
 17*pi/8 + 0.2 = 6.875884388878311      (4,[0.5926990816987245,-2.2884754890353088e-17])
 18*pi/8 + 0.2 = 7.268583470577035      (5,[-0.5853981633974484,2.6905207631241164e-17])
 19*pi/8 + 0.2 = 7.661282552275758      (5,[-0.1926990816987247,-8.503679843877493e-19])
 20*pi/8 + 0.2 = 8.053981633974482      (5,[0.19999999999999898,-8.503679843877493e-19])

So, to implement mod2pi, modpi, and modpio2, we need to add $\mod(n,4)*\pi/2$ , $\mod(n,2)*\pi/2$ , and $\mod(n,1)*\pi/2 = 0$ , respectively. However, because the underlying ieee754_rem_pio2 function returns values in $(-\pi/4,\pi/4)$ , we need to look at the sign of y, and if negative, bump it into the positive realm by adding $2\pi, \pi, \pi/2$ respectively. Thus, here's our first implementation:

In [7]:

function mod2pi_v0(x::Float64) # or modtau(x)
# with r = mod2pi(x)
# a) 0 <= r < 2π  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*2π with k integer

# note: mod(n,4) is 0,1,2,3; while mod1(n) is 4,1,2,3.
# We use the latter to push negative y in quadrant 0 into the positive (one revolution, + 4*pi/2)
    (n,y) = ieee754_rem_pio2(x)

    if y[1] > 0
        r = y[1] + mod(n,4) * π/2
    else
        r = y[1] + mod1(n,4) * π/2 
    end
    return r
end


function modpi_v0(x::Float64)
# with r = modpi(x)
# a) 0 <= r < π  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*π with k integer
    (n,y) = ieee754_rem_pio2(x)

    if y[1] > 0.0
        r = y[1] + mod(n,2) * π/2
    else
        r = y[1] + mod1(n,2) * π/2
    end
    return r
end


function modpio2_v0(x::Float64)
# with r = modpio2(x)
# a) 0 <= r < π/2  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*π/2 with k integer
    (n,y) = ieee754_rem_pio2(x)
    if y[1] > 0.0
        r = y[1] # + mod(n,1) * π/2
    else
        r = y[1] + π/2 
    end
    return r
end

Out[7]:

modpio2_v0 (generic function with 1 method)

Let's see whether these do better at the original problem:

In [8]:

sin(111111111111111.0) - sin(mod(111111111111111.0,2pi))  # bad...

Out[8]:

-0.002943493231076344

In [9]:

sin(111111111111111.0) - sin(mod2pi_v0(111111111111111.0))  # better!

Out[9]:

1.1102230246251565e-16

Much better. Let's test it systematically. We want to cover numbers near $\pi$ , or multiples thereof, because then the cancellation will be particularly severe. We also want to cover all quadrants, so let's take multiples of $\pi/16 \pm \delta\pi$ , where $\delta\pi$ is small, so we end up on both sides of zero. Next, the paper on range reduction from above has, in section 1.2, a float $\Gamma = 6411027962775774 / 2^{48}$ that is very close to a multiple of $\pi/4$ . As we are looking for $\pi/2$ , we take $\Gamma\Gamma = 6411027962775774 / 2^{47}$ . Finally, to find integers $x$ close to a multiple of $\pi$ , ie for which $x = k\pi + \epsilon$ with small $\epsilon$ , we obviously have $x/k \approx \pi$ , thus we choose the numerators of the convergents of the continuous fraction approximation to $\pi$ . Thus, here are our test cases:

In [10]:

function allTestCases() # for modpi
    deltaPi = 0.00001
    pips = [π/16*k + deltaPi for k in [-20:20]]  # multiples of pi/16 plus a bit
    pims = [π/16*k - deltaPi for k in [-20:20]]  # multiples of pi/16 minus a bit

    # numerators of continuous fraction approximations to pi
    # see http://oeis.org/A002485
    # (reason: for max cancellation, we want x = k*pi + eps for small eps, so x/k ≈ pi)
    contFractionNumerators = [22, 333, 355, 103993, 104348, 208341, 312689, 833719, 1146408, 
    4272943, 5419351, 80143857, 165707065, 245850922, 411557987, 1068966896, 2549491779, 6167950454, 
    14885392687, 21053343141, 1783366216531, 3587785776203, 5371151992734, 8958937768937, 139755218526789, 
    428224593349304, 5706674932067741, 6134899525417045 ] 
    # note: these numerators below are too big to represent as a Float64 exactly
    # 30246273033735921, 66627445592888887, 430010946591069243, 2646693125139304345
    # (all natural numbers <= 9007199254740992 can be represented as a Float64 exactly)
    closeToPi = 3.14159265359

    ΓΓ = 6411027962775774 / 2^47  
    # from [2], section 1.2: 
    # the Float64 greater than 8, and less than 2**63 − 1 closest to a multiple of π/4 is
    # Γ = 6411027962775774 / 2^48 
    # Gamma mod pi/4 should be ε ≈ 3.9405531196482 × 10^−19 
    # Wolfram Alpha: mod( 6411027962775774 * 2^(-48) , pi/4) 
    # = 3.094903182941788500075335732804827979316517057683310544... × 10^-19
    # we take twice Γ, as we want to be close to pi/2, not pi/4
    # ΓΓ = 2*Γ 
 
    return vcat([-pi/2, pi/2, -pi, pi, 2pi, -2pi, ΓΓ,closeToPi,-closeToPi],pips,pims,contFractionNumerators)
end

Out[10]:

allTestCases (generic function with 1 method)

Now, we test all of these against Mathematica WolframAlpha. However, before we test modpi with all these numbers, there is one more little issue to address. If we print a Float64 in Julia, we get the closest decimal representation that, if parsed, resolves back to the original Float64. However, WolframAlpha will interpret the same sequence of digits differently. For example, "0.1" is approximated in Julia by a Float64 that is exactly $(1 + 2702159776422298 / 2^{52}) / 2^4$ , but WolframAlpha interprets it as $1/10$ .

In [11]:

(1+2702159776422298/2^52)/2^4

Out[11]:

0.1

So, below, with a little helper function that outputs the exact representation of a Float64, and WolframAlpha, we have our list of test cases:

In [12]:

modpiSolns = [
"mod(-1.5707963267948966,pi) = mod(-(1+2570638124657944/2^52)*2.0^(0),pi) = 1.57079632679489668046366164900741030340188131343760582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(1.5707963267948966,pi) = mod(+(1+2570638124657944/2^52)*2.0^(0),pi) = 1.5707963267948965579989817342720925807952880859375",
"mod(-3.141592653589793,pi) = mod(-(1+2570638124657944/2^52)*2.0^(1),pi) = 1.22464679914735317722606593227500105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933... × 10^-16",
"mod(3.141592653589793,pi) = mod(+(1+2570638124657944/2^52)*2.0^(1),pi) = 3.141592653589793115997963468544185161590576171875",
"mod(6.283185307179586,pi) = mod(+(1+2570638124657944/2^52)*2.0^(2),pi) = 3.14159265358979299353328355380886743898398294437489417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(-6.283185307179586,pi) = mod(-(1+2570638124657944/2^52)*2.0^(2),pi) = 2.44929359829470635445213186455000211641949889184615632812572417997256069650684234135964296173026564613294187689219101164463450718816256962234900568205403877042211119289245897909860763928857621951331866... × 10^-16",
"mod(45.553093477052,pi) = mod(+(1+1907428335405278/2^52)*2.0^(5),pi) = 1.57079632679489661985030232822810914211365184624851850635077570769057031199307401920751244521036104824992678881404770694068617546629184875584496828620126435569602256217286070452216497527871463097465249799664...",
"mod(3.14159265359,pi) = mod(+(1+2570638124658410/2^52)*2.0^(1),pi) = 2.06823107110214443817242336338901406144179025055407692183593713791001371965174657882932017851913486717693352906155390449417768274640591871518882549715897298061478894440355377051045069618035571189024334... × 10^-13",
"mod(-3.14159265359,pi) = mod(-(1+2570638124658410/2^52)*2.0^(1),pi) = 3.14159265358958641535553316883568564186083049796896164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.9269808169872418,pi) = mod(-(1+4339175044666918/2^52)*2.0^(1),pi) = 2.35620449019234470335066281789739528427110881828146164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.73063127613788,pi) = mod(-(1+3897035185165141/2^52)*2.0^(1),pi) = 2.55255403104170655105593060965009877214418621085958664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.5342817352885176,pi) = mod(-(1+3454895325663363/2^52)*2.0^(1),pi) = 2.74890357189106884285040825146541842946993083976583664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.337932194439156,pi) = mod(-(1+3012755466161586/2^52)*2.0^(1),pi) = 2.94525311274043069055567604321812191734300823234396164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.1415826535897935,pi) = mod(-(1+2570615606659808/2^52)*2.0^(1),pi) = 9.9999999997438875103017539386904715834618751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288... × 10^-6",
"mod(-2.9452331127404316,pi) = mod(-(1+2128475747158031/2^52)*2.0^(1),pi) = 0.19635954084936159159277809350664217834466085445323082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.7488835718910694,pi) = mod(-(1+1686335887656253/2^52)*2.0^(1),pi) = 0.39270908169872388338725573532196183567040548335948082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.5525340310417075,pi) = mod(-(1+1244196028154476/2^52)*2.0^(1),pi) = 0.58905862254808573109252352707466532354348287593760582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.356184490192345,pi) = mod(-(1+802056168652698/2^52)*2.0^(1),pi) = 0.78540816339744802288700116888998498086922750484385582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.1598349493429834,pi) = mod(-(1+359916309150921/2^52)*2.0^(1),pi) = 0.98175770424680987059226896064268846874230489742198082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.9634854084936209,pi) = mod(-(1+4339152526668781/2^52)*2.0^(0),pi) = 1.17810724509617238443135152748931621079438314449229332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.7671358676442588,pi) = mod(-(1+3454872807665226/2^52)*2.0^(0),pi) = 1.37445678594553445418122424427332778339379415523448082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.5707863267948967,pi) = mod(-(1+2570593088661671/2^52)*2.0^(0),pi) = 1.57080632679489652393109696105733935599320516597666832097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.3744367859455346,pi) = mod(-(1+1686313369658116/2^52)*2.0^(0),pi) = 1.76715586764425859368096967784135092859261617671885582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.1780872450961726,pi) = mod(-(1+802033650654561/2^52)*2.0^(0),pi) = 1.96350540849362066343084239462536250119202718746104332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.9817377042468104,pi) = mod(-(1+4339107490672507/2^52)*2.0^(-1),pi) = 2.15985494934298284420301757392502811615460500728526207097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.7853881633974483,pi) = mod(-(1+2570548052665397/2^52)*2.0^(-1),pi) = 2.35620449019234491395289029070903968875401601802744957097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.5890386225480863,pi) = mod(-(1+801988614658287/2^52)*2.0^(-1),pi) = 2.55255403104170698370276300749305126135342702876963707097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.3926890816987242,pi) = mod(-(1+2570457980672850/2^52)*2.0^(-2),pi) = 2.74890357189106905345263572427706283395283803951182457097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.1963395408493621,pi) = mod(-(1+2570277836687755/2^52)*2.0^(-3),pi) = 2.94525311274043115095808405668998791714304075252451988347494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(1.0e-5,pi) = mod(+(1+1399358476216561/2^52)*2.0^(-17),pi) = 0.000010000000000000000818030539140313095458623138256371021270751953125",
"mod(0.19635954084936205,pi) = mod(+(1+2570998412628133/2^52)*2.0^(-3),pi) = 0.1963595408493620519951861069785081781446933746337890625",
"mod(0.3927090816987241,pi) = mod(+(1+2570818268643038/2^52)*2.0^(-2),pi) = 0.39270908169872409398948320813360624015331268310546875",
"mod(0.5890586225480862,pi) = mod(+(1+802168758643381/2^52)*2.0^(-1),pi) = 0.58905862254808616373935592491761781275272369384765625",
"mod(0.7854081633974482,pi) = mod(+(1+2570728196650491/2^52)*2.0^(-1),pi) = 0.78540816339744823348922864170162938535213470458984375",
"mod(0.9817577042468103,pi) = mod(+(1+4339287634657601/2^52)*2.0^(-1),pi) = 0.98175770424681030323910135848564095795154571533203125",
"mod(1.1781072450961723,pi) = mod(+(1+802123722647107/2^52)*2.0^(0),pi) = 1.1781072450961722619666716127539984881877899169921875",
"mod(1.3744567859455343,pi) = mod(+(1+1686403441650662/2^52)*2.0^(0),pi) = 1.374456785945534331716544329538010060787200927734375",
"mod(1.5708063267948964,pi) = mod(+(1+2570683160654217/2^52)*2.0^(0),pi) = 1.5708063267948964014664170463220216333866119384765625",
"mod(1.7671558676442585,pi) = mod(+(1+3454962879657772/2^52)*2.0^(0),pi) = 1.76715586764425847121628976310603320598602294921875",
"mod(1.9635054084936205,pi) = mod(+(1+4339242598661327/2^52)*2.0^(0),pi) = 1.9635054084936205409661624798900447785854339599609375",
"mod(2.159854949342982,pi) = mod(+(1+359961345147192/2^52)*2.0^(1),pi) = 2.159854949342982166626825346611440181732177734375",
"mod(2.3562044901923445,pi) = mod(+(1+802101204648970/2^52)*2.0^(1),pi) = 2.35620449019234445842130298842675983905792236328125",
"mod(2.5525540310417063,pi) = mod(+(1+1244241064150747/2^52)*2.0^(1),pi) = 2.552554031041706306126570780179463326930999755859375",
"mod(2.7489035718910686,pi) = mod(+(1+1686380923652525/2^52)*2.0^(1),pi) = 2.748903571891068597921048421994782984256744384765625",
"mod(2.9452531127404304,pi) = mod(+(1+2128520783154302/2^52)*2.0^(1),pi) = 2.94525311274043044562631621374748647212982177734375",
"mod(3.1416026535897927,pi) = mod(+(1+2570660642656080/2^52)*2.0^(1),pi) = 9.9999999994989581504722833032452583970068748941790250554076921835937137910013719651746578829320178519134867176933529061553904494177682746405918715188825497158972980614788944403553770510450696180355712... × 10^-6",
"mod(3.3379521944391546,pi) = mod(+(1+3012800502157857/2^52)*2.0^(1),pi) = 0.19635954084936134666341826403600673313147439945301917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.534301735288517,pi) = mod(+(1+3454940361659635/2^52)*2.0^(1),pi) = 0.39270908169872363845789590585132639045721902835926917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.7306512761378787,pi) = mod(+(1+3897080221161412/2^52)*2.0^(1),pi) = 0.58905862254808548616316369760402987833029642093739417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.927000816987241,pi) = mod(+(1+4339220080663190/2^52)*2.0^(1),pi) = 0.78540816339744777795764133941934953565604104984364417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(-3.9270008169872415,pi) = mod(-(1+4339220080663191/2^52)*2.0^(1),pi) = 2.35618449019234501641579219379753717908846111320333664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.7306512761378796,pi) = mod(-(1+3897080221161414/2^52)*2.0^(1),pi) = 2.55253403104170686412105998555024066696153850578146164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.5343017352885173,pi) = mod(-(1+3454940361659636/2^52)*2.0^(1),pi) = 2.74888357189106915591553762736556032428728313468771164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.3379521944391555,pi) = mod(-(1+3012800502157859/2^52)*2.0^(1),pi) = 2.94523311274043100362080541911826381216036052726583664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.141602653589793,pi) = mod(-(1+2570660642656081/2^52)*2.0^(1),pi) = 3.14158265358979329541528306093358346948610515617208664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.9452531127404313,pi) = mod(-(1+2128520783154304/2^52)*2.0^(1),pi) = 0.19633954084936190465790746940678407316201314937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.748903571891069,pi) = mod(-(1+1686380923652526/2^52)*2.0^(1),pi) = 0.39268908169872419645238511122210373048775777828135582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.552554031041707,pi) = mod(-(1+1244241064150749/2^52)*2.0^(1),pi) = 0.58903862254808604415765290297480721836083517085948082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.356204490192345,pi) = mod(-(1+802101204648971/2^52)*2.0^(1),pi) = 0.78538816339744833595213054479012687568657979976573082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.159854949342983,pi) = mod(-(1+359961345147194/2^52)*2.0^(1),pi) = 0.98173770424681018365739833654283036355965719234385582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.9635054084936208,pi) = mod(-(1+4339242598661328/2^52)*2.0^(0),pi) = 1.17808724509617247545187597835815002088540182125010582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.7671558676442587,pi) = mod(-(1+3454962879657773/2^52)*2.0^(0),pi) = 1.37443678594553454520174869514216159348481283199229332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.5708063267948966,pi) = mod(-(1+2570683160654218/2^52)*2.0^(0),pi) = 1.57078632679489661495162141192617316608422384273448082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.3744567859455346,pi) = mod(-(1+1686403441650663/2^52)*2.0^(0),pi) = 1.76713586764425868470149412871018473868363485347666832097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.1781072450961725,pi) = mod(-(1+802123722647108/2^52)*2.0^(0),pi) = 1.96348540849362075445136684549419631128304586421885582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.9817577042468104,pi) = mod(-(1+4339287634657602/2^52)*2.0^(-1),pi) = 2.15983494934298282420123956227820788388245687496104332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.7854081633974483,pi) = mod(-(1+2570728196650492/2^52)*2.0^(-1),pi) = 2.35618449019234489395111227906221945648186788570323082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.5890586225480863,pi) = mod(-(1+802168758643382/2^52)*2.0^(-1),pi) = 2.55253403104170696370098499584623102908127889644541832097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.39270908169872415,pi) = mod(-(1+2570818268643039/2^52)*2.0^(-2),pi) = 2.74888357189106908896200894388806962286227331172862144597494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-0.19635954084936208,pi) = mod(-(1+2570998412628134/2^52)*2.0^(-3),pi) = 2.94523311274043115871188166067208119546168432247080894597494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.0e-5,pi) = mod(-(1+1399358476216561/2^52)*2.0^(-17),pi) = 3.14158265358979323846182535274036257110171077623684944995367384035469140628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(0.19633954084936206,pi) = mod(+(1+2570277836687754/2^52)*2.0^(-3),pi) = 0.196339540849362059748983710960601456463336944580078125",
"mod(0.39268908169872413,pi) = mod(+(1+2570457980672849/2^52)*2.0^(-2),pi) = 0.392689081698724129498856427744613029062747955322265625",
"mod(0.5890386225480861,pi) = mod(+(1+801988614658286/2^52)*2.0^(-1),pi) = 0.5890386225480861437375779132707975804805755615234375",
"mod(0.7853881633974482,pi) = mod(+(1+2570548052665396/2^52)*2.0^(-1),pi) = 0.785388163397448213487450630054809153079986572265625",
"mod(0.9817377042468103,pi) = mod(+(1+4339107490672506/2^52)*2.0^(-1),pi) = 0.9817377042468102832373233468388207256793975830078125",
"mod(1.1780872450961724,pi) = mod(+(1+802033650654560/2^52)*2.0^(0),pi) = 1.17808724509617235298719606362283229827880859375",
"mod(1.3744367859455344,pi) = mod(+(1+1686313369658115/2^52)*2.0^(0),pi) = 1.3744367859455344227370687804068438708782196044921875",
"mod(1.5707863267948965,pi) = mod(+(1+2570593088661670/2^52)*2.0^(0),pi) = 1.570786326794896492486941497190855443477630615234375",
"mod(1.7671358676442586,pi) = mod(+(1+3454872807665225/2^52)*2.0^(0),pi) = 1.7671358676442585622368142139748670160770416259765625",
"mod(1.9634854084936206,pi) = mod(+(1+4339152526668780/2^52)*2.0^(0),pi) = 1.96348540849362063198668693075887858867645263671875",
"mod(2.1598349493429825,pi) = mod(+(1+359916309150919/2^52)*2.0^(1),pi) = 2.159834949342982479691954722511582076549530029296875",
"mod(2.3561844901923448,pi) = mod(+(1+802056168652697/2^52)*2.0^(1),pi) = 2.356184490192344771486432364326901733875274658203125",
"mod(2.5525340310417066,pi) = mod(+(1+1244196028154474/2^52)*2.0^(1),pi) = 2.55253403104170661919170015607960522174835205078125",
"mod(2.748883571891069,pi) = mod(+(1+1686335887656252/2^52)*2.0^(1),pi) = 2.7488835718910689109861777978949248790740966796875",
"mod(2.9452331127404308,pi) = mod(+(1+2128475747158029/2^52)*2.0^(1),pi) = 2.945233112740430758691445589647628366947174072265625",
"mod(3.141582653589793,pi) = mod(+(1+2570615606659807/2^52)*2.0^(1),pi) = 3.141582653589793050485923231462948024272918701171875",
"mod(3.337932194439155,pi) = mod(+(1+3012755466161584/2^52)*2.0^(1),pi) = 0.19633954084936165972854763993614862794882669437489417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.534281735288517,pi) = mod(+(1+3454895325663362/2^52)*2.0^(1),pi) = 0.39268908169872395152302528175146828527457132328114417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.730631276137879,pi) = mod(+(1+3897035185165139/2^52)*2.0^(1),pi) = 0.58903862254808579922829307350417177314764871585926917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.9269808169872413,pi) = mod(+(1+4339175044666917/2^52)*2.0^(1),pi) = 0.78538816339744809102277071531949143047339334476551917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(22.0,pi) = mod(+(1+1688849860263936/2^52)*2.0^(4),pi) = 0.00885142487144733076149631704347981061981420437425925317538785384528515599653700960375622260518052412496339440702385347034308773314592437792248414310063217784801128108643035226108248763935731548732624899832...",
"mod(333.0,pi) = mod(+(1+1354598325420032/2^52)*2.0^(8),pi) = 3.13277137307170996142244475565219715929721306561388879763081780767927733994805514405634333907770786187445091610535780205514631599718886566883726214650948266772016921629645528391623731459035973230989373497484...",
"mod(355.0,pi) = mod(+(1+1741626418397184/2^52)*2.0^(8),pi) = 0.00003014435336405372129768941617408571985787061304222983126106921674608965838315503206473634077131801726622399909934887839555912078420781503438688148163372811789639468094711507176015760676809286683801954435...",
"mod(103993.0,pi) = mod(+(1+2642744916836352/2^52)*2.0^(16),pi) = 3.14157352425401364804137006517503018949571128462221990835905001896913552019593641341924635058293272291618832384236767454664957926617754765887823153914653127814591646313301288487020333576664284942659544192789...",
"mod(104348.0,pi) = mod(+(1+2667140331077632/2^52)*2.0^(16),pi) = 0.00001101501758446330002437131170139101839975586015631721536649587806520356811056982327626158158697295130646132818471677795129377741117324218725901249968388881352875511202147883640384875046198736305149704344...",
"mod(208341.0,pi) = mod(+(1+2654942623956992/2^52)*2.0^(17),pi) = 3.14158453927159811134139443648673158051411104048237622557441651484720072376404698324252261216451969586749478517055239132460087304358872090106549055164621516695944521824503436370660718451710483678964693897133...",
"mod(312689.0,pi) = mod(+(1+868356487905280/2^52)*2.0^(18),pi) = 2.90069938933617877542451893008733534139696742672181483841841744952104594855443776404840398960083665315998545480145545832221144931191152739015601741793832268987065511732143745138864461786922231647158597... × 10^-6",
"mod(833719.0,pi) = mod(+(1+2657992050737152/2^52)*2.0^(19),pi) = 3.14159034067037678369894528552459175518479383441722966920409335168209976585594409211805070897249889754080110514146199423551751746648734472412027086368105104360482495955526900658150996180634057523427988214327...",
"mod(1.146408e6,pi) = mod(+(1+420185240502272/2^52)*2.0^(20),pi) = 5.87779972881415077326764018958322965832009550570043987177791732880615683647927779932034371430395306178613634489043881995068386074403922301611569987864477230727508447806913401705828009489526214389300436... × 10^-7",
"mod(4.272943e6,pi) = mod(+(1+84437983297536/2^52)*2.0^(22),pi) = 3.14159210401029542794417726581664863015369133044588137933605488505729840770299503590139050507561318872671964098236546136716350267164556793588717569839101463703651714208061242732171507929036904381292305004458...",
"mod(5.419351e6,pi) = mod(+(1+1315384200265728/2^52)*2.0^(22),pi) = 3.82004750708966112093011647042794877630803261284050974705412148820324696852011356117678675511398777330827176437639516531304810601080841179018325213840634637668871217131295571404954295784087554749162140... × 10^-8",
"mod(8.0143857e7,pi) = mod(+(1+874763572477952/2^52)*2.0^(26),pi) = 3.14159263881694642049673419603295449006652001357044717700741947263430675615757062871728906982575890468500790414041247406248664649838040944906482632404631401392500987850031641113551504622638314153549969887158...",
"mod(1.65707065e8,pi) = mod(+(1+1056606817091584/2^52)*2.0^(27),pi) = 8.65478143496479283480806791601818899147100884047004723119419558177519294537964410073515122454559736833697797859473725690814071454276305173366818717701291531848387749318946794370229795161899094380176246... × 10^-9",
"mod(2.45850922e8,pi) = mod(+(1+3745788417015808/2^52)*2.0^(27),pi) = 3.14159264747172785546152703084102240608470900504145601747746670382850233793276357409693317056091012923060527247739045265722390340652112399182787805771450119093792519698419390432498298992868109315449064267334...",
"mod(4.11557987e8,pi) = mod(+(1+2401197617053696/2^52)*2.0^(28),pi) = 2.53671605196367648236958743790572859713735903697256934271488151342174752084854244595394428579405455430108612460486731570511125630286557038325420725050055641276613287640889128900803008984309962204629913... × 10^-9",
"mod(1.068966896e9,pi) = mod(+(1+4463544628150272/2^52)*2.0^(29),pi) = 3.14159265254515995938887999558019728189616619931617409142260538925826536477625861579401806246879870081871438107956270186695853481674363659755901882422291569193903802251645965714276556794474127284068988676594...",
"mod(2.549491779e9,pi) = mod(+(1+843072155942912/2^52)*2.0^(31),pi) = 4.47449493816149706970976233303722197019495577867890936615779430401846755180508920207307551467187143433646915044596696119497365034532889215443076399478064252395175148483303135651614725663715466720569020... × 10^-10",
"mod(6.167950454e9,pi) = mod(+(1+1963965187883008/2^52)*2.0^(32),pi) = 3.14159265344005894702117940952214974850361059335516524715838726248982422557995212615503590288341380375308866794685653195615192705573836666662479725510906849089516652730680995410937183924797072416812082020708...",
"mod(1.4885392687e10,pi) = mod(+(1+3300633133711360/2^52)*2.0^(33),pi) = 1.47980910933221759456269961916604584979614430234776276979795068989333010234511075289901023009068306300795365662712861011872933904331764909404251146367829101604918014490927524901658264139193178277114987... × 10^-10",
"mod(2.1053343141e10,pi) = mod(+(1+1015407956983808/2^52)*2.0^(34),pi) = 3.14159265358803985795440116897841971042021517833477967739316353946961929456928513638954697817331482676215697424765189761886478806761130057095656216451331963726299562891172796860029936414962898830731399848419...",
"mod(1.783366216531e12,pi) = mod(+(1+2801068395540480/2^52)*2.0^(40),pi) = 6.96948240875758165283364652450017592218369365167838571237684767728582201531913110512760529814875987841007291472111488973274399752796445730686810635597303227231604049063965142781067801484038929704503126... × 10^-13",
"mod(3.587785776203e12,pi) = mod(+(1+2844185642293248/2^52)*2.0^(41),pi) = 3.14159265358943375443615268530898643972511521351921641612349921661209466410474230079261080439434034782178672622333391220180901104555784937046215505597469325853419023536619117669842729443519112391028207634360...",
"mod(5.371151992734e12,pi) = mod(+(1+996460013189120/2^52)*2.0^(42),pi) = 3.37464214385060194766920180395831736328964513722462875515942586261884366107892162736040369453515697892612846187277924980541538489592093576898951719503435891484259641931614955023236781384150844501479806... × 10^-13",
"mod(8.958937768937e12,pi) = mod(+(1+83376510325248/2^52)*2.0^(43),pi) = 3.14159265358977121865053774550375335990551104525554538063722167948761060669100418515871869655707638819124024192122652504799628897053839090895174714955159221025369367125767543634035890939021436069166622718810...",
"mod(1.39755218526789e14,pi) = mod(+(1+4440734358344000/2^52)*2.0^(46),pi) = 7.16703280049355852405580552051994292235944787877057242894865818968232636596038712584350583584681705588885972394333979767169688540821492356534377064696422753798816392058646400432320064809223397832709837... × 10^-15",
"mod(4.28224593349304e14,pi) = mod(+(1+2347993866218368/2^52)*2.0^(48),pi) = 3.14159265358979271974893922617932552732207260508431245898085799120489745266557323213781657771845391870874778237239419162716811898993140599960797179432228824156563456394028940083212066878222733029361050389004...",
"mod(5.706674932067741e15,pi) = mod(+(1+1203075304697245/2^52)*2.0^(52),pi) = 4.23754646451256218416429262194162608653524752956234482551589924717953528741279504902951631892985510393600689510284748380679359369235443057957270202960326099424609213900534623202922619105230013018255129... × 10^-16",
"mod(6.134899525417045e15,pi) = mod(+(1+1631299898046549/2^52)*2.0^(52),pi) = 3.14159265358979314350358567743554394375133479924692111250561094743938000425549795009134531899795882166037967535790458522785762927467978667896734102976534619883583752426638882544133456931685053321622960912005..." ]



## 2*pi
mod2piSolns = [
"mod(-1.5707963267948966,2*pi) = mod(-(1+2570638124657944/2^52)*2^(0),2*pi) = 4.71238898038468991892630503228691318759905071281271164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(1.5707963267948966,2*pi) = mod(+(1+2570638124657944/2^52)*2^(0),2*pi) = 1.5707963267948965579989817342720925807952880859375",
"mod(-3.141592653589793,2*pi) = mod(-(1+2570638124657944/2^52)*2^(1),2*pi) = 3.14159265358979336092732329801482060680376262687521164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(3.141592653589793,2*pi) = mod(+(1+2570638124657944/2^52)*2^(1),2*pi) = 3.141592653589793115997963468544185161590576171875",
"mod(6.283185307179586,2*pi) = mod(+(1+2570638124657944/2^52)*2^(2),2*pi) = 6.28318530717958623199592693708837032318115234375",
"mod(-6.283185307179586,2*pi) = mod(-(1+2570638124657944/2^52)*2^(2),2*pi) = 2.44929359829470635445213186455000211641949889184615632812572417997256069650684234135964296173026564613294187689219101164463450718816256962234900568205403877042211119289245897909860763928857621951331866... × 10^-16",
"mod(45.553093477052,2*pi) = mod(+(1+1907428335405278/2^52)*2.0^(5),2*pi) = 1.57079632679489661985030232822810914211365184624851850635077570769057031199307401920751244521036104824992678881404770694068617546629184875584496828620126435569602256217286070452216497527871463097465249799664...",
"mod(3.14159265359,2*pi) = mod(+(1+2570638124658410/2^52)*2.0^(1),2*pi) = 3.14159265359000006156975359772332012653350830078125",
"mod(-3.14159265359,2*pi) = mod(-(1+2570638124658410/2^52)*2.0^(1),2*pi) = 3.14159265358958641535553316883568564186083049796896164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.9269808169872418,2*pi) = mod(-(1+4339175044666918/2^52)*2.0^(1),2*pi) = 2.35620449019234470335066281789739528427110881828146164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.73063127613788,2*pi) = mod(-(1+3897035185165141/2^52)*2.0^(1),2*pi) = 2.55255403104170655105593060965009877214418621085958664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.5342817352885176,2*pi) = mod(-(1+3454895325663363/2^52)*2.0^(1),2*pi) = 2.74890357189106884285040825146541842946993083976583664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.337932194439156,2*pi) = mod(-(1+3012755466161586/2^52)*2.0^(1),2*pi) = 2.94525311274043069055567604321812191734300823234396164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.1415826535897935,2*pi) = mod(-(1+2570615606659808/2^52)*2.0^(1),2*pi) = 3.14160265358979298235015368503344157466875286125021164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.9452331127404316,2*pi) = mod(-(1+2128475747158031/2^52)*2.0^(1),2*pi) = 3.33795219443915483005542147678614506254183025382833664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.7488835718910694,2*pi) = mod(-(1+1686335887656253/2^52)*2.0^(1),2*pi) = 3.53430173528851712184989911860146471986757488273458664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.5525340310417075,2*pi) = mod(-(1+1244196028154476/2^52)*2.0^(1),2*pi) = 3.73065127613787896955516691035416820774065227531271164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.356184490192345,2*pi) = mod(-(1+802056168652698/2^52)*2.0^(1),2*pi) = 3.92700081698724126134964455216948786506639690421896164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.1598349493429834,2*pi) = mod(-(1+359916309150921/2^52)*2.0^(1),2*pi) = 4.12335035783660310905491234392219135293947429679708664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.9634854084936209,2*pi) = mod(-(1+4339152526668781/2^52)*2.0^(0),2*pi) = 4.31969989868596562289399491076881909499155254386739914194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.7671358676442588,2*pi) = mod(-(1+3454872807665226/2^52)*2.0^(0),2*pi) = 4.51604943953532769264386762755283066759096355460958664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.5707863267948967,2*pi) = mod(-(1+2570593088661671/2^52)*2.0^(0),2*pi) = 4.71239898038468976239374034433684224019037456535177414194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.3744367859455346,2*pi) = mod(-(1+1686313369658116/2^52)*2.0^(0),2*pi) = 4.90874852123405183214361306112085381278978557609396164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.1780872450961726,2*pi) = mod(-(1+802033650654561/2^52)*2.0^(0),2*pi) = 5.10509806208341390189348577790486538538919658683614914194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.9817377042468104,2*pi) = mod(-(1+4339107490672507/2^52)*2.0^(-1),2*pi) = 5.30144760293277608266566095720453100035177440666036789194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.7853881633974483,2*pi) = mod(-(1+2570548052665397/2^52)*2.0^(-1),2*pi) = 5.49779714378213815241553367398854257295118541740255539194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.5890386225480863,2*pi) = mod(-(1+801988614658287/2^52)*2.0^(-1),2*pi) = 5.69414668463150022216540639077255414555059642814474289194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.3926890816987242,2*pi) = mod(-(1+2570457980672850/2^52)*2.0^(-2),2*pi) = 5.89049622548086229191527910755656571815000743888693039194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.1963395408493621,2*pi) = mod(-(1+2570277836687755/2^52)*2.0^(-3),2*pi) = 6.08684576633022438942072743996949080134021015189962570444988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(1.0e-5,2*pi) = mod(+(1+1399358476216561/2^52)*2.0^(-17),2*pi) = 0.000010000000000000000818030539140313095458623138256371021270751953125",
"mod(0.19635954084936205,2*pi) = mod(+(1+2570998412628133/2^52)*2.0^(-3),2*pi) = 0.1963595408493620519951861069785081781446933746337890625",
"mod(0.3927090816987241,2*pi) = mod(+(1+2570818268643038/2^52)*2.0^(-2),2*pi) = 0.39270908169872409398948320813360624015331268310546875",
"mod(0.5890586225480862,2*pi) = mod(+(1+802168758643381/2^52)*2.0^(-1),2*pi) = 0.58905862254808616373935592491761781275272369384765625",
"mod(0.7854081633974482,2*pi) = mod(+(1+2570728196650491/2^52)*2.0^(-1),2*pi) = 0.78540816339744823348922864170162938535213470458984375",
"mod(0.9817577042468103,2*pi) = mod(+(1+4339287634657601/2^52)*2.0^(-1),2*pi) = 0.98175770424681030323910135848564095795154571533203125",
"mod(1.1781072450961723,2*pi) = mod(+(1+802123722647107/2^52)*2.0^(0),2*pi) = 1.1781072450961722619666716127539984881877899169921875",
"mod(1.3744567859455343,2*pi) = mod(+(1+1686403441650662/2^52)*2.0^(0),2*pi) = 1.374456785945534331716544329538010060787200927734375",
"mod(1.5708063267948964,2*pi) = mod(+(1+2570683160654217/2^52)*2.0^(0),2*pi) = 1.5708063267948964014664170463220216333866119384765625",
"mod(1.7671558676442585,2*pi) = mod(+(1+3454962879657772/2^52)*2.0^(0),2*pi) = 1.76715586764425847121628976310603320598602294921875",
"mod(1.9635054084936205,2*pi) = mod(+(1+4339242598661327/2^52)*2.0^(0),2*pi) = 1.9635054084936205409661624798900447785854339599609375",
"mod(2.159854949342982,2*pi) = mod(+(1+359961345147192/2^52)*2.0^(1),2*pi) = 2.159854949342982166626825346611440181732177734375",
"mod(2.3562044901923445,2*pi) = mod(+(1+802101204648970/2^52)*2.0^(1),2*pi) = 2.35620449019234445842130298842675983905792236328125",
"mod(2.5525540310417063,2*pi) = mod(+(1+1244241064150747/2^52)*2.0^(1),2*pi) = 2.552554031041706306126570780179463326930999755859375",
"mod(2.7489035718910686,2*pi) = mod(+(1+1686380923652525/2^52)*2.0^(1),2*pi) = 2.748903571891068597921048421994782984256744384765625",
"mod(2.9452531127404304,2*pi) = mod(+(1+2128520783154302/2^52)*2.0^(1),2*pi) = 2.94525311274043044562631621374748647212982177734375",
"mod(3.1416026535897927,2*pi) = mod(+(1+2570660642656080/2^52)*2.0^(1),2*pi) = 3.14160265358979273742079385556280612945556640625",
"mod(3.3379521944391546,2*pi) = mod(+(1+3012800502157857/2^52)*2.0^(1),2*pi) = 3.337952194439154585126061647315509617328643798828125",
"mod(3.534301735288517,2*pi) = mod(+(1+3454940361659635/2^52)*2.0^(1),2*pi) = 3.534301735288516876920539289130829274654388427734375",
"mod(3.7306512761378787,2*pi) = mod(+(1+3897080221161412/2^52)*2.0^(1),2*pi) = 3.7306512761378787246258070808835327625274658203125",
"mod(3.927000816987241,2*pi) = mod(+(1+4339220080663190/2^52)*2.0^(1),2*pi) = 3.92700081698724101642028472269885241985321044921875",
"mod(-3.9270008169872415,2*pi) = mod(-(1+4339220080663191/2^52)*2.0^(1),2*pi) = 2.35618449019234501641579219379753717908846111320333664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.7306512761378796,2*pi) = mod(-(1+3897080221161414/2^52)*2.0^(1),2*pi) = 2.55253403104170686412105998555024066696153850578146164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.5343017352885173,2*pi) = mod(-(1+3454940361659636/2^52)*2.0^(1),2*pi) = 2.74888357189106915591553762736556032428728313468771164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.3379521944391555,2*pi) = mod(-(1+3012800502157859/2^52)*2.0^(1),2*pi) = 2.94523311274043100362080541911826381216036052726583664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-3.141602653589793,2*pi) = mod(-(1+2570660642656081/2^52)*2.0^(1),2*pi) = 3.14158265358979329541528306093358346948610515617208664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.9452531127404313,2*pi) = mod(-(1+2128520783154304/2^52)*2.0^(1),2*pi) = 3.33793219443915514312055085268628695735918254875021164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.748903571891069,2*pi) = mod(-(1+1686380923652526/2^52)*2.0^(1),2*pi) = 3.53428173528851743491502849450160661468492717765646164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.552554031041707,2*pi) = mod(-(1+1244241064150749/2^52)*2.0^(1),2*pi) = 3.73063127613787928262029628625431010255800457023458664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.356204490192345,2*pi) = mod(-(1+802101204648971/2^52)*2.0^(1),2*pi) = 3.92698081698724157441477392806962975988374919914083664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-2.159854949342983,2*pi) = mod(-(1+359961345147194/2^52)*2.0^(1),2*pi) = 4.12333035783660342212004171982233324775682659171896164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.9635054084936208,2*pi) = mod(-(1+4339242598661328/2^52)*2.0^(0),2*pi) = 4.31967989868596571391451936163765290508257122062521164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.7671558676442587,2*pi) = mod(-(1+3454962879657773/2^52)*2.0^(0),2*pi) = 4.51602943953532778366439207842166447768198223136739914194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.5708063267948966,2*pi) = mod(-(1+2570683160654218/2^52)*2.0^(0),2*pi) = 4.71237898038468985341426479520567605028139324210958664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.3744567859455346,2*pi) = mod(-(1+1686403441650663/2^52)*2.0^(0),2*pi) = 4.90872852123405192316413751198968762288080425285177414194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.1781072450961725,2*pi) = mod(-(1+802123722647108/2^52)*2.0^(0),2*pi) = 5.10507806208341399291401022877369919548021526359396164194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.9817577042468104,2*pi) = mod(-(1+4339287634657602/2^52)*2.0^(-1),2*pi) = 5.30142760293277606266388294555771076807962627433614914194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.7854081633974483,2*pi) = mod(-(1+2570728196650492/2^52)*2.0^(-1),2*pi) = 5.49777714378213813241375566234172234067903728507833664194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.5890586225480863,2*pi) = mod(-(1+802168758643382/2^52)*2.0^(-1),2*pi) = 5.69412668463150020216362837912573391327844829582052414194988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.39270908169872415,2*pi) = mod(-(1+2570818268643039/2^52)*2.0^(-2),2*pi) = 5.89047622548086232742465232716757250705944271110372726694988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-0.19635954084936208,2*pi) = mod(-(1+2570998412628134/2^52)*2.0^(-3),2*pi) = 6.08682576633022439717452504395158407965885372184591476694988918461563281257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(-1.0e-5,2*pi) = mod(-(1+1399358476216561/2^52)*2.0^(-17),2*pi) = 6.28317530717958647692446873601986545529888017561195527092861843266250781257241799725606965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392885762...",
"mod(0.19633954084936206,2*pi) = mod(+(1+2570277836687754/2^52)*2.0^(-3),2*pi) = 0.196339540849362059748983710960601456463336944580078125",
"mod(0.39268908169872413,2*pi) = mod(+(1+2570457980672849/2^52)*2.0^(-2),2*pi) = 0.392689081698724129498856427744613029062747955322265625",
"mod(0.5890386225480861,2*pi) = mod(+(1+801988614658286/2^52)*2.0^(-1),2*pi) = 0.5890386225480861437375779132707975804805755615234375",
"mod(0.7853881633974482,2*pi) = mod(+(1+2570548052665396/2^52)*2.0^(-1),2*pi) = 0.785388163397448213487450630054809153079986572265625",
"mod(0.9817377042468103,2*pi) = mod(+(1+4339107490672506/2^52)*2.0^(-1),2*pi) = 0.9817377042468102832373233468388207256793975830078125",
"mod(1.1780872450961724,2*pi) = mod(+(1+802033650654560/2^52)*2.0^(0),2*pi) = 1.17808724509617235298719606362283229827880859375",
"mod(1.3744367859455344,2*pi) = mod(+(1+1686313369658115/2^52)*2.0^(0),2*pi) = 1.3744367859455344227370687804068438708782196044921875",
"mod(1.5707863267948965,2*pi) = mod(+(1+2570593088661670/2^52)*2.0^(0),2*pi) = 1.570786326794896492486941497190855443477630615234375",
"mod(1.7671358676442586,2*pi) = mod(+(1+3454872807665225/2^52)*2.0^(0),2*pi) = 1.7671358676442585622368142139748670160770416259765625",
"mod(1.9634854084936206,2*pi) = mod(+(1+4339152526668780/2^52)*2.0^(0),2*pi) = 1.96348540849362063198668693075887858867645263671875",
"mod(2.1598349493429825,2*pi) = mod(+(1+359916309150919/2^52)*2.0^(1),2*pi) = 2.159834949342982479691954722511582076549530029296875",
"mod(2.3561844901923448,2*pi) = mod(+(1+802056168652697/2^52)*2.0^(1),2*pi) = 2.356184490192344771486432364326901733875274658203125",
"mod(2.5525340310417066,2*pi) = mod(+(1+1244196028154474/2^52)*2.0^(1),2*pi) = 2.55253403104170661919170015607960522174835205078125",
"mod(2.748883571891069,2*pi) = mod(+(1+1686335887656252/2^52)*2.0^(1),2*pi) = 2.7488835718910689109861777978949248790740966796875",
"mod(2.9452331127404308,2*pi) = mod(+(1+2128475747158029/2^52)*2.0^(1),2*pi) = 2.945233112740430758691445589647628366947174072265625",
"mod(3.141582653589793,2*pi) = mod(+(1+2570615606659807/2^52)*2.0^(1),2*pi) = 3.141582653589793050485923231462948024272918701171875",
"mod(3.337932194439155,2*pi) = mod(+(1+3012755466161584/2^52)*2.0^(1),2*pi) = 3.33793219443915489819119102321565151214599609375",
"mod(3.534281735288517,2*pi) = mod(+(1+3454895325663362/2^52)*2.0^(1),2*pi) = 3.53428173528851718998566866503097116947174072265625",
"mod(3.730631276137879,2*pi) = mod(+(1+3897035185165139/2^52)*2.0^(1),2*pi) = 3.730631276137879037690936456783674657344818115234375",
"mod(3.9269808169872413,2*pi) = mod(+(1+4339175044666917/2^52)*2.0^(1),2*pi) = 3.926980816987241329485414098598994314670562744140625",
"mod(22.0,2*pi) = mod(+(1+1688849860263936/2^52)*2.0^(4),2*pi) = 3.15044407846124056922413970032298269481698360374936507415033244615310156228274600823179104794729759210711148092030616011743693234269650660964784355122911329529829538378836887336664213226230627041770821342713...",
"mod(333.0,2*pi) = mod(+(1+1354598325420032/2^52)*2.0^(8),2*pi) = 6.27436402666150319988508813893170004349438246498899461860576239998709374623426414268437816441982492985659900261864010870224016060673944790056262155463796378517045331899839380502179695921330868724027569940365...",
"mod(355.0,2*pi) = mod(+(1+1741626418397184/2^52)*2.0^(8),2*pi) = 3.14162279794315729218394107269567696991702726998814805080620566152456249594459215366009956168288838599941431051238165552548940373033479004675974628961011484556818049738288563617731980222971704779721998397317...",
"mod(103993.0,2*pi) = mod(+(1+2642744916836352/2^52)*2.0^(16),2*pi) = 6.28316617784380688650401344845453307369288068399732572933399461127695192648214541204728117592504979089833641035564998119374342387572812989060359094727501239559620056583495140597576298038959180435697740635670...",
"mod(104348.0,2*pi) = mod(+(1+2667140331077632/2^52)*2.0^(16),2*pi) = 3.14160366860737770176266775459120427521556915523526213819031108818588160985431956845131108692370404093345454784146702342504513838696175547391261842062816500626381285781395999994196349337341094229343346147225...",
"mod(208341.0,2*pi) = mod(+(1+2654942623956992/2^52)*2.0^(17),2*pi) = 3.14158453927159811134139443648673158051411104048237622557441651484720072376404698324252261216451969586749478517055239132460087304358872090106549055164621516695944521824503436370660718451710483678964693897133...",
"mod(312689.0,2*pi) = mod(+(1+868356487905280/2^52)*2.0^(18),2*pi) = 2.90069938933617877542451893008733534139696742672181483841841744952104594855443776404840398960083665315998545480145545832221144931191152739015601741793832268987065511732143745138864461786922231647158597... × 10^-6",
"mod(833719.0,2*pi) = mod(+(1+2657992050737152/2^52)*2.0^(19),2*pi) = 3.14159034067037678369894528552459175518479383441722966920409335168209976585594409211805070897249889754080110514146199423551751746648734472412027086368105104360482495955526900658150996180634057523427988214327...",
"mod(1.146408e6,2*pi) = mod(+(1+420185240502272/2^52)*2.0^(20),2*pi) = 3.14159324136976611987772071004352184252013523138465639101893177009954928690189264655581475737648849837745426512691679569097583967793665663564766101969846898192751483021038632801896135045095844445659635372924...",
"mod(4.272943e6,2*pi) = mod(+(1+84437983297536/2^52)*2.0^(22),2*pi) = 6.28318475760008866640682064909615151435086072982098720031099947736511481398920403452942533041773025670886772749564776801425734728119615016761253510651949575448680124478255094842727472391331799874330501447339...",
"mod(5.419351e6,2*pi) = mod(+(1+1315384200265728/2^52)*2.0^(22),2*pi) = 3.14159269179026830935925459258066758847665716245543194938004206284903128831867868382917043710998461912202581959599995041104549774003164233980947730996100250151374786958906023423511678511837853333913743934502...",
"mod(8.0143857e7,2*pi) = mod(+(1+874763572477952/2^52)*2.0^(26),2*pi) = 6.28318529240673965895937757931245737426368941294555299798236406494212316244377962734532389516787597266715599065369478070958049110793099168079018573217479513137529398120225493224107469084933209646588166330039...",
"mod(1.65707065e8,2*pi) = mod(+(1+1056606817091584/2^52)*2.0^(27),2*pi) = 3.14159266224457467342743621808757080021535839084611466144499182350201198806140194400767892607726829252774545485026028524183110151769129677448841114179666829446319942118581601429502758832524690654937290823057...",
"mod(2.45850922e8,2*pi) = mod(+(1+3745788417015808/2^52)*2.0^(27),2*pi) = 3.14159264747172785546152703084102240608470900504145601747746670382850233793276357409693317056091012923060527247739045265722390340652112399182787805771450119093792519698419390432498298992868109315449064267334...",
"mod(4.11557987e8,2*pi) = mod(+(1+2401197617053696/2^52)*2.0^(28),2*pi) = 2.53671605196367648236958743790572859713735903697256934271488151342174752084854244595394428579405455430108612460486731570511125630286557038325420725050055641276613287640889128900803008984309962204629913... × 10^-9",
"mod(1.068966896e9,2*pi) = mod(+(1+4463544628150272/2^52)*2.0^(29),2*pi) = 3.14159265254515995938887999558019728189616619931617409142260538925826536477625861579401806246879870081871438107956270186695853481674363659755901882422291569193903802251645965714276556794474127284068988676594...",
"mod(2.549491779e9,2*pi) = mod(+(1+843072155942912/2^52)*2.0^(31),2*pi) = 4.47449493816149706970976233303722197019495577867890936615779430401846755180508920207307551467187143433646915044596696119497365034532889215443076399478064252395175148483303135651614725663715466720569020... × 10^-10",
"mod(6.167950454e9,2*pi) = mod(+(1+1963965187883008/2^52)*2.0^(32),2*pi) = 3.14159265344005894702117940952214974850361059335516524715838726248982422557995212615503590288341380375308866794685653195615192705573836666662479725510906849089516652730680995410937183924797072416812082020708...",
"mod(1.4885392687e10,2*pi) = mod(+(1+3300633133711360/2^52)*2.0^(33),2*pi) = 1.47980910933221759456269961916604584979614430234776276979795068989333010234511075289901023009068306300795365662712861011872933904331764909404251146367829101604918014490927524901658264139193178277114987... × 10^-10",
"mod(2.1053343141e10,2*pi) = mod(+(1+1015407956983808/2^52)*2.0^(34),2*pi) = 3.14159265358803985795440116897841971042021517833477967739316353946961929456928513638954697817331482676215697424765189761886478806761130057095656216451331963726299562891172796860029936414962898830731399848419...",
"mod(1.783366216531e12,2*pi) = mod(+(1+2801068395540480/2^52)*2.0^(40),2*pi) = 6.96948240875758165283364652450017592218369365167838571237684767728582201531913110512760529814875987841007291472111488973274399752796445730686810635597303227231604049063965142781067801484038929704503126... × 10^-13",
"mod(3.587785776203e12,2*pi) = mod(+(1+2844185642293248/2^52)*2.0^(41),2*pi) = 3.14159265358943375443615268530898643972511521351921641612349921661209466410474230079261080439434034782178672622333391220180901104555784937046215505597469325853419023536619117669842729443519112391028207634360...",
"mod(5.371151992734e12,2*pi) = mod(+(1+996460013189120/2^52)*2.0^(42),2*pi) = 3.14159265359013070267702844347426980437756523111143478548866705518333234887247088299414271750485310835160160221117491949328112253453112377021495150170538006916978753859342278074749125957797219171176611527331...",
"mod(8.958937768937e12,2*pi) = mod(+(1+83376510325248/2^52)*2.0^(43),2*pi) = 6.28318530717956445711318112878325624410268044463065120161216627179542701297721318378675352189919345617338832843450883169509013358008897314067710655768007332770397777395961395744591855401316331562204819161692...",
"mod(1.39755218526789e14,2*pi) = mod(+(1+4440734358344000/2^52)*2.0^(46),2*pi) = 3.14159265358980040549544387683802694000268991931802818042282336288024535494439868095440078572924291148798393333033819550681778794934825392861076762305204646122093106692947650926948023108695327813103005666278...",
"mod(4.28224593349304e14,2*pi) = mod(+(1+2347993866218368/2^52)*2.0^(48),2*pi) = 3.14159265358979271974893922617932552732207260508431245898085799120489745266557323213781657771845391870874778237239419162716811898993140599960797179432228824156563456394028940083212066878222733029361050389004...",
"mod(5.706674932067741e15,2*pi) = mod(+(1+1203075304697245/2^52)*2.0^(52),2*pi) = 4.23754646451256218416429262194162608653524752956234482551589924717953528741279504902951631892985510393600689510284748380679359369235443057957270202960326099424609213900534623202922619105230013018255129... × 10^-16",
"mod(6.134899525417045e15,2*pi) = mod(+(1+1631299898046549/2^52)*2.0^(52),2*pi) = 3.14159265358979314350358567743554394375133479924692111250561094743938000425549795009134531899795882166037967535790458522785762927467978667896734102976534619883583752426638882544133456931685053321622960912005..."
]




## pi/2
modpio2Solns = [
"mod(-1.5707963267948966,pi/2) = mod(-(1+2570638124657944/2^52)*2^(0),pi/2) = 6.12323399573676588613032966137500529104874722961539082031431044993140174126710585339910740432566411533235469223047752911158626797040642405587251420513509692605527798223114744774651909822144054878329667... × 10^-17",
"mod(1.5707963267948966,pi/2) = mod(+(1+2570638124657944/2^52)*2^(0),pi/2) = 1.5707963267948965579989817342720925807952880859375",
"mod(-3.141592653589793,pi/2) = mod(-(1+2570638124657944/2^52)*2^(1),pi/2) = 1.22464679914735317722606593227500105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933... × 10^-16",
"mod(3.141592653589793,pi/2) = mod(+(1+2570638124657944/2^52)*2^(1),pi/2) = 1.57079632679489649676664177690443371949199147218744708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(6.283185307179586,pi/2) = mod(+(1+2570638124657944/2^52)*2^(2),pi/2) = 1.57079632679489637430196186216911599688539824468734126853758311153827539057068650205794776198682439802677787023007654002935923308567412665241196088780727832382457384594709221834166053306557656760442705335678...",
"mod(-6.283185307179586,pi/2) = mod(-(1+2570638124657944/2^52)*2^(2),pi/2) = 2.44929359829470635445213186455000211641949889184615632812572417997256069650684234135964296173026564613294187689219101164463450718816256962234900568205403877042211119289245897909860763928857621951331866... × 10^-16",
"mod(45.553093477052,pi/2) = mod(+(1+1907428335405278/2^52)*2.0^(5),pi/2) = 6.18980636588357700015067146560965595863303411536662108849969519893495032539302514258852745557406553617139253161516557639982288582137023796970880510821891443969385152967240153509461515782240852843965031... × 10^-19",
"mod(3.14159265359,pi/2) = mod(+(1+2570638124658410/2^52)*2.0^(1),pi/2) = 2.06823107110214443817242336338901406144179025055407692183593713791001371965174657882932017851913486717693352906155390449417768274640591871518882549715897298061478894440355377051045069618035571189024334... × 10^-13",
"mod(-3.14159265359,pi/2) = mod(-(1+2570638124658410/2^52)*2.0^(1),pi/2) = 1.57079632679468979612421147719593419976224579828140873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.9269808169872418,pi/2) = mod(-(1+4339175044666918/2^52)*2.0^(1),pi/2) = 0.78540816339744808411934112625764384217252411859390873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.73063127613788,pi/2) = mod(-(1+3897035185165141/2^52)*2.0^(1),pi/2) = 0.98175770424680993182460891801034733004560151117203373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.5342817352885176,pi/2) = mod(-(1+3454895325663363/2^52)*2.0^(1),pi/2) = 1.17810724509617222361908655982566698737134614007828373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.337932194439156,pi/2) = mod(-(1+3012755466161586/2^52)*2.0^(1),pi/2) = 1.37445678594553407132435435157837047524442353265640873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.1415826535897935,pi/2) = mod(-(1+2570615606659808/2^52)*2.0^(1),pi/2) = 9.9999999997438875103017539386904715834618751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288... × 10^-6",
"mod(-2.9452331127404316,pi/2) = mod(-(1+2128475747158031/2^52)*2.0^(1),pi/2) = 0.19635954084936159159277809350664217834466085445323082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.7488835718910694,pi/2) = mod(-(1+1686335887656253/2^52)*2.0^(1),pi/2) = 0.39270908169872388338725573532196183567040548335948082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.5525340310417075,pi/2) = mod(-(1+1244196028154476/2^52)*2.0^(1),pi/2) = 0.58905862254808573109252352707466532354348287593760582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.356184490192345,pi/2) = mod(-(1+802056168652698/2^52)*2.0^(1),pi/2) = 0.78540816339744802288700116888998498086922750484385582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.1598349493429834,pi/2) = mod(-(1+359916309150921/2^52)*2.0^(1),pi/2) = 0.98175770424680987059226896064268846874230489742198082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.9634854084936209,pi/2) = mod(-(1+4339152526668781/2^52)*2.0^(0),pi/2) = 1.17810724509617238443135152748931621079438314449229332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.7671358676442588,pi/2) = mod(-(1+3454872807665226/2^52)*2.0^(0),pi/2) = 1.37445678594553445418122424427332778339379415523448082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.5707863267948967,pi/2) = mod(-(1+2570593088661671/2^52)*2.0^(0),pi/2) = 9.9999999999046997752694175879138946204662891154104874722961539082031431044993140174126710585339910740432566411533235469223047752911158626797040642405587251420513509692605527798223114744774651909822144... × 10^-6",
"mod(-1.3744367859455346,pi/2) = mod(-(1+1686313369658116/2^52)*2.0^(0),pi/2) = 0.19635954084936197444964798620159948649403147703130291048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-1.1780872450961726,pi/2) = mod(-(1+802033650654561/2^52)*2.0^(0),pi/2) = 0.39270908169872404419952070298561105909344248777349041048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.9817377042468104,pi/2) = mod(-(1+4339107490672507/2^52)*2.0^(-1),pi/2) = 0.58905862254808622497169588228527667405602030759770916048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.7853881633974483,pi/2) = mod(-(1+2570548052665397/2^52)*2.0^(-1),pi/2) = 0.78540816339744829472156859906928824665543131833989666048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.5890386225480863,pi/2) = mod(-(1+801988614658287/2^52)*2.0^(-1),pi/2) = 0.98175770424681036447144131585329981925484232908208416048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.3926890816987242,pi/2) = mod(-(1+2570457980672850/2^52)*2.0^(-2),pi/2) = 1.17810724509617243422131403263731139185425333982427166048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.1963395408493621,pi/2) = mod(-(1+2570277836687755/2^52)*2.0^(-3),pi/2) = 1.37445678594553453172676236505023647504445605283696697298747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(1.0e-5,pi/2) = mod(+(1+1399358476216561/2^52)*2.0^(-17),pi/2) = 0.000010000000000000000818030539140313095458623138256371021270751953125",
"mod(0.19635954084936205,pi/2) = mod(+(1+2570998412628133/2^52)*2.0^(-3),pi/2) = 0.1963595408493620519951861069785081781446933746337890625",
"mod(0.3927090816987241,pi/2) = mod(+(1+2570818268643038/2^52)*2.0^(-2),pi/2) = 0.39270908169872409398948320813360624015331268310546875",
"mod(0.5890586225480862,pi/2) = mod(+(1+802168758643381/2^52)*2.0^(-1),pi/2) = 0.58905862254808616373935592491761781275272369384765625",
"mod(0.7854081633974482,pi/2) = mod(+(1+2570728196650491/2^52)*2.0^(-1),pi/2) = 0.78540816339744823348922864170162938535213470458984375",
"mod(0.9817577042468103,pi/2) = mod(+(1+4339287634657601/2^52)*2.0^(-1),pi/2) = 0.98175770424681030323910135848564095795154571533203125",
"mod(1.1781072450961723,pi/2) = mod(+(1+802123722647107/2^52)*2.0^(0),pi/2) = 1.1781072450961722619666716127539984881877899169921875",
"mod(1.3744567859455343,pi/2) = mod(+(1+1686403441650662/2^52)*2.0^(0),pi/2) = 1.374456785945534331716544329538010060787200927734375",
"mod(1.5708063267948964,pi/2) = mod(+(1+2570683160654217/2^52)*2.0^(0),pi/2) = 9.9999999997822350953546822701912880272387890095895125277038460917968568955006859825873289414660089259567433588466764530776952247088841373202959357594412748579486490307394472201776885255225348090177856... × 10^-6",
"mod(1.7671558676442585,pi/2) = mod(+(1+3454962879657772/2^52)*2.0^(0),pi/2) = 0.19635954084936185198496807146628176388743824953119708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(1.9635054084936205,pi/2) = mod(+(1+4339242598661327/2^52)*2.0^(0),pi/2) = 0.39270908169872392173484078825029333648684926027338458951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.159854949342982,pi/2) = mod(+(1+359961345147192/2^52)*2.0^(1),pi/2) = 0.58905862254808554739550365497168873963359303468744708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.3562044901923445,pi/2) = mod(+(1+802101204648970/2^52)*2.0^(1),pi/2) = 0.78540816339744783918998129678700839695933766359369708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.5525540310417063,pi/2) = mod(+(1+1244241064150747/2^52)*2.0^(1),pi/2) = 0.98175770424680968689524908853971188483241505617182208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.7489035718910686,pi/2) = mod(+(1+1686380923652525/2^52)*2.0^(1),pi/2) = 1.17810724509617197868972673035503154215815968507807208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.9452531127404304,pi/2) = mod(+(1+2128520783154302/2^52)*2.0^(1),pi/2) = 1.37445678594553382639499452210773503003123707765619708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(3.1416026535897927,pi/2) = mod(+(1+2570660642656080/2^52)*2.0^(1),pi/2) = 9.9999999994989581504722833032452583970068748941790250554076921835937137910013719651746578829320178519134867176933529061553904494177682746405918715188825497158972980614788944403553770510450696180355712... × 10^-6",
"mod(3.3379521944391546,pi/2) = mod(+(1+3012800502157857/2^52)*2.0^(1),pi/2) = 0.19635954084936134666341826403600673313147439945301917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.534301735288517,pi/2) = mod(+(1+3454940361659635/2^52)*2.0^(1),pi/2) = 0.39270908169872363845789590585132639045721902835926917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.7306512761378787,pi/2) = mod(+(1+3897080221161412/2^52)*2.0^(1),pi/2) = 0.58905862254808548616316369760402987833029642093739417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.927000816987241,pi/2) = mod(+(1+4339220080663190/2^52)*2.0^(1),pi/2) = 0.78540816339744777795764133941934953565604104984364417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(-3.9270008169872415,pi/2) = mod(-(1+4339220080663191/2^52)*2.0^(1),pi/2) = 0.78538816339744839718447050215778573698987641351578373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.7306512761378796,pi/2) = mod(-(1+3897080221161414/2^52)*2.0^(1),pi/2) = 0.98173770424681024488973829391048922486295380609390873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.5343017352885173,pi/2) = mod(-(1+3454940361659636/2^52)*2.0^(1),pi/2) = 1.17808724509617253668421593572580888218869843500015873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.3379521944391555,pi/2) = mod(-(1+3012800502157859/2^52)*2.0^(1),pi/2) = 1.37443678594553438438948372747851237006177582757828373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-3.141602653589793,pi/2) = mod(-(1+2570660642656081/2^52)*2.0^(1),pi/2) = 1.57078632679489667618396136929383202738752045648453373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321...",
"mod(-2.9452531127404313,pi/2) = mod(-(1+2128520783154304/2^52)*2.0^(1),pi/2) = 0.19633954084936190465790746940678407316201314937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.748903571891069,pi/2) = mod(-(1+1686380923652526/2^52)*2.0^(1),pi/2) = 0.39268908169872419645238511122210373048775777828135582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.552554031041707,pi/2) = mod(-(1+1244241064150749/2^52)*2.0^(1),pi/2) = 0.58903862254808604415765290297480721836083517085948082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.356204490192345,pi/2) = mod(-(1+802101204648971/2^52)*2.0^(1),pi/2) = 0.78538816339744833595213054479012687568657979976573082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-2.159854949342983,pi/2) = mod(-(1+359961345147194/2^52)*2.0^(1),pi/2) = 0.98173770424681018365739833654283036355965719234385582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.9635054084936208,pi/2) = mod(-(1+4339242598661328/2^52)*2.0^(0),pi/2) = 1.17808724509617247545187597835815002088540182125010582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.7671558676442587,pi/2) = mod(-(1+3454962879657773/2^52)*2.0^(0),pi/2) = 1.37443678594553454520174869514216159348481283199229332097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.5708063267948966,pi/2) = mod(-(1+2570683160654218/2^52)*2.0^(0),pi/2) = 1.57078632679489661495162141192617316608422384273448082097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881...",
"mod(-1.3744567859455346,pi/2) = mod(-(1+1686403441650663/2^52)*2.0^(0),pi/2) = 0.19633954084936206547017243707043329658505015378911541048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-1.1781072450961725,pi/2) = mod(-(1+802123722647108/2^52)*2.0^(0),pi/2) = 0.39268908169872413522004515385444486918446116453130291048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.9817577042468104,pi/2) = mod(-(1+4339287634657602/2^52)*2.0^(-1),pi/2) = 0.58903862254808620496991787063845644178387217527349041048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.7854081633974483,pi/2) = mod(-(1+2570728196650492/2^52)*2.0^(-1),pi/2) = 0.78538816339744827471979058742246801438328318601567791048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.5890586225480863,pi/2) = mod(-(1+802168758643382/2^52)*2.0^(-1),pi/2) = 0.98173770424681034446966330420647958698269419675786541048747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.39270908169872415,pi/2) = mod(-(1+2570818268643039/2^52)*2.0^(-2),pi/2) = 1.17808724509617246973068725224831818076368861204106853548747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-0.19635954084936208,pi/2) = mod(-(1+2570998412628134/2^52)*2.0^(-3),pi/2) = 1.37443678594553453948055996903232975336309962278325603548747229615390820314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(-1.0e-5,pi/2) = mod(-(1+1399358476216561/2^52)*2.0^(-17),pi/2) = 1.57078632679489661923050366110061112900312607654929653946620154420078320314310449931401741267105853399107404325664115332354692230477529111586267970406424055872514205135096926055277982231147447746519098221440...",
"mod(0.19633954084936206,pi/2) = mod(+(1+2570277836687754/2^52)*2.0^(-3),pi/2) = 0.196339540849362059748983710960601456463336944580078125",
"mod(0.39268908169872413,pi/2) = mod(+(1+2570457980672849/2^52)*2.0^(-2),pi/2) = 0.392689081698724129498856427744613029062747955322265625",
"mod(0.5890386225480861,pi/2) = mod(+(1+801988614658286/2^52)*2.0^(-1),pi/2) = 0.5890386225480861437375779132707975804805755615234375",
"mod(0.7853881633974482,pi/2) = mod(+(1+2570548052665396/2^52)*2.0^(-1),pi/2) = 0.785388163397448213487450630054809153079986572265625",
"mod(0.9817377042468103,pi/2) = mod(+(1+4339107490672506/2^52)*2.0^(-1),pi/2) = 0.9817377042468102832373233468388207256793975830078125",
"mod(1.1780872450961724,pi/2) = mod(+(1+802033650654560/2^52)*2.0^(0),pi/2) = 1.17808724509617235298719606362283229827880859375",
"mod(1.3744367859455344,pi/2) = mod(+(1+1686313369658115/2^52)*2.0^(0),pi/2) = 1.3744367859455344227370687804068438708782196044921875",
"mod(1.5707863267948965,pi/2) = mod(+(1+2570593088661670/2^52)*2.0^(0),pi/2) = 1.570786326794896492486941497190855443477630615234375",
"mod(1.7671358676442586,pi/2) = mod(+(1+3454872807665225/2^52)*2.0^(0),pi/2) = 0.19633954084936194300549252233511557397845692628900958951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(1.9634854084936206,pi/2) = mod(+(1+4339152526668780/2^52)*2.0^(0),pi/2) = 0.39268908169872401275536523911912714657786793703119708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.1598349493429825,pi/2) = mod(+(1+359916309150919/2^52)*2.0^(1),pi/2) = 0.58903862254808586046063303087183063445094532960932208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.3561844901923448,pi/2) = mod(+(1+802056168652697/2^52)*2.0^(1),pi/2) = 0.78538816339744815225511067268715029177668995851557208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.5525340310417066,pi/2) = mod(+(1+1244196028154474/2^52)*2.0^(1),pi/2) = 0.98173770424680999996037846443985377964976735109369708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.748883571891069,pi/2) = mod(+(1+1686335887656252/2^52)*2.0^(1),pi/2) = 1.17808724509617229175485610625517343697551197999994708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(2.9452331127404308,pi/2) = mod(+(1+2128475747158029/2^52)*2.0^(1),pi/2) = 1.37443678594553413946012389800787692484858937257807208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(3.141582653589793,pi/2) = mod(+(1+2570615606659807/2^52)*2.0^(1),pi/2) = 1.57078632679489643125460153982319658217433400148432208951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559...",
"mod(3.337932194439155,pi/2) = mod(+(1+3012755466161584/2^52)*2.0^(1),pi/2) = 0.19633954084936165972854763993614862794882669437489417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.534281735288517,pi/2) = mod(+(1+3454895325663362/2^52)*2.0^(1),pi/2) = 0.39268908169872395152302528175146828527457132328114417902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.730631276137879,pi/2) = mod(+(1+3897035185165139/2^52)*2.0^(1),pi/2) = 0.58903862254808579922829307350417177314764871585926917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(3.9269808169872413,pi/2) = mod(+(1+4339175044666917/2^52)*2.0^(1),pi/2) = 0.78538816339744809102277071531949143047339334476551917902505540769218359371379100137196517465788293201785191348671769335290615539044941776827464059187151888254971589729806147889444035537705104506961803557118...",
"mod(22.0,pi/2) = mod(+(1+1688849860263936/2^52)*2.0^(4),pi/2) = 0.00885142487144733076149631704347981061981420437425925317538785384528515599653700960375622260518052412496339440702385347034308773314592437792248414310063217784801128108643035226108248763935731548732624899832...",
"mod(333.0,pi/2) = mod(+(1+1354598325420032/2^52)*2.0^(8),pi/2) = 1.56197504627681334219112306401244571719862836592633588714334551152536913680495064474232592640664932788337687284871664873159939369241357455297458244244524210899502716494548602336345749227888525484470275276044...",
"mod(355.0,pi/2) = mod(+(1+1741626418397184/2^52)*2.0^(8),pi/2) = 0.00003014435336405372129768941617408571985787061304222983126106921674608965838315503206473634077131801726622399909934887839555912078420781503438688148163372811789639468094711507176015760676809286683801954435...",
"mod(103993.0,pi/2) = mod(+(1+2642744916836352/2^52)*2.0^(16),pi/2) = 1.57077719745911702881004837353527874739712658493466699787157772281522731705283191410522893791187418892511428058572652122310265696140225654301555183508229071942077441178204362431742351345516837196140445971348...",
"mod(104348.0,pi/2) = mod(+(1+2667140331077632/2^52)*2.0^(16),pi/2) = 0.00001101501758446330002437131170139101839975586015631721536649587806520356811056982327626158158697295130646132818471677795129377741117324218725901249968388881352875511202147883640384875046198736305149704344...",
"mod(208341.0,pi/2) = mod(+(1+2654942623956992/2^52)*2.0^(17),pi/2) = 1.57078821247670149211007274484698013841552634079482331508694421869329252062094248392850519949346116187642074191391123800105395073881342978520281084758197460823430316689406510315382736220563035932445595675693...",
"mod(312689.0,pi/2) = mod(+(1+868356487905280/2^52)*2.0^(18),pi/2) = 2.90069938933617877542451893008733534139696742672181483841841744952104594855443776404840398960083665315998545480145545832221144931191152739015601741793832268987065511732143745138864461786922231647158597... × 10^-6",
"mod(833719.0,pi/2) = mod(+(1+2657992050737152/2^52)*2.0^(19),pi/2) = 1.57079401387548016446762359388484031308620913472967675871662105552819156271283959280403329630144036354972706188482084091197059516171205360825759115961681048487968290820429974602873013949486609776908889992887...",
"mod(1.146408e6,pi/2) = mod(+(1+420185240502272/2^52)*2.0^(20),pi/2) = 5.87779972881415077326764018958322965832009550570043987177791732880615683647927779932034371430395306178613634489043881995068386074403922301611569987864477230727508447806913401705828009489526214389300436... × 10^-7",
"mod(4.272943e6,pi/2) = mod(+(1+84437983297536/2^52)*2.0^(22),pi/2) = 1.57079577721539880871285557417689718805510663075832846884858258890339020455989053658737309240455465473564559772572430804361658036687027682002449599432677407831137509072964316676893525697889456634773206783018...",
"mod(5.419351e6,pi/2) = mod(+(1+1315384200265728/2^52)*2.0^(22),pi/2) = 3.82004750708966112093011647042794877630803261284050974705412148820324696852011356117678675511398777330827176437639516531304810601080841179018325213840634637668871217131295571404954295784087554749162140... × 10^-8",
"mod(8.0143857e7,pi/2) = mod(+(1+874763572477952/2^52)*2.0^(26),pi/2) = 1.57079631202204980126541250439320304796793531388289426651994717648039855301446612940327165715470037069393386088377132073893972419360511833320214661998207345519986782714934715058273522391490866407030871665717...",
"mod(1.65707065e8,pi/2) = mod(+(1+1056606817091584/2^52)*2.0^(27),pi/2) = 8.65478143496479283480806791601818899147100884047004723119419558177519294537964410073515122454559736833697797859473725690814071454276305173366818717701291531848387749318946794370229795161899094380176246... × 10^-9",
"mod(2.45850922e8,pi/2) = mod(+(1+3745788417015808/2^52)*2.0^(27),pi/2) = 1.57079632067683123623020533920127096398612430535390310698999440767459413478965907478291575788985159523953122922074929933367698110174583287596519835365026063221278314563322464377220316761720661568929966045894...",
"mod(4.11557987e8,pi/2) = mod(+(1+2401197617053696/2^52)*2.0^(28),pi/2) = 2.53671605196367648236958743790572859713735903697256934271488151342174752084854244595394428579405455430108612460486731570511125630286557038325420725050055641276613287640889128900803008984309962204629913... × 10^-9",
"mod(1.068966896e9,pi/2) = mod(+(1+4463544628150272/2^52)*2.0^(29),pi/2) = 1.57079632575026334015755830394044583979758149962862118093513309310435716163315411648000064979774016682764033782292154854341161251196834548169633912015867513321389597116549039658998574563326679537549890455154...",
"mod(2.549491779e9,pi/2) = mod(+(1+843072155942912/2^52)*2.0^(31),pi/2) = 4.47449493816149706970976233303722197019495577867890936615779430401846755180508920207307551467187143433646915044596696119497365034532889215443076399478064252395175148483303135651614725663715466720569020... × 10^-10",
"mod(6.167950454e9,pi/2) = mod(+(1+1963965187883008/2^52)*2.0^(32),pi/2) = 1.57079632664516232778985771788239830640502589366761233667091496633591602243684762684101849021235526976201462469021537863260500475096307555076211755104482793217002447595584069355659201693649624670292983799267...",
"mod(1.4885392687e10,pi/2) = mod(+(1+3300633133711360/2^52)*2.0^(33),pi/2) = 1.47980910933221759456269961916604584979614430234776276979795068989333010234511075289901023009068306300795365662712861011872933904331764909404251146367829101604918014490927524901658264139193178277114987... × 10^-10",
"mod(2.1053343141e10,pi/2) = mod(+(1+1015407956983808/2^52)*2.0^(34),pi/2) = 1.57079632679314323872307947733866826832163047864722676690569124331571109142618063707552956550225629277108293099101074429531786576283600945509388246044907907853785357756075870804751954183815451084212301626979...",
"mod(1.783366216531e12,pi/2) = mod(+(1+2801068395540480/2^52)*2.0^(40),pi/2) = 6.96948240875758165283364652450017592218369365167838571237684767728582201531913110512760529814875987841007291472111488973274399752796445730686810635597303227231604049063965142781067801484038929704503126... × 10^-13",
"mod(3.587785776203e12,pi/2) = mod(+(1+2844185642293248/2^52)*2.0^(41),pi/2) = 1.57079632679453713520483099366923499762653051383166350563602692045818646096163780147859339172328181383071268296669275887826208874078255825459947535191045269980904818401522191614564747212371664644509109412920...",
"mod(5.371151992734e12,pi/2) = mod(+(1+996460013189120/2^52)*2.0^(42),pi/2) = 3.37464214385060194766920180395831736328964513722462875515942586261884366107892162736040369453515697892612846187277924980541538489592093576898951719503435891484259641931614955023236781384150844501479806... × 10^-13",
"mod(8.958937768937e12,pi/2) = mod(+(1+83376510325248/2^52)*2.0^(43),pi/2) = 1.57079632679487459941921605386400191780692634556799247014974938333370240354789968584470128388601785420016619866458537172444936666576309979308906744548735165152855161990670617578757908707873988322647524497370...",
"mod(1.39755218526789e14,pi/2) = mod(+(1+4440734358344000/2^52)*2.0^(46),pi/2) = 7.16703280049355852405580552051994292235944787877057242894865818968232636596038712584350583584681705588885972394333979767169688540821492356534377064696422753798816392058646400432320064809223397832709837... × 10^-15",
"mod(4.28224593349304e14,pi/2) = mod(+(1+2347993866218368/2^52)*2.0^(48),pi/2) = 1.57079632679489610051761753453957408522348790539675954849338569505098924952246873282379916504739538471767373911575303830362119668515611488374529209025804768284049251258932014027934084647075285282841952167563...",
"mod(5.706674932067741e15,pi/2) = mod(+(1+1203075304697245/2^52)*2.0^(52),pi/2) = 4.23754646451256218416429262194162608653524752956234482551589924717953528741279504902951631892985510393600689510284748380679359369235443057957270202960326099424609213900534623202922619105230013018255129... × 10^-16",
"mod(6.134899525417045e15,pi/2) = mod(+(1+1631299898046549/2^52)*2.0^(52),pi/2) = 1.57079632679489652427226398579579250165275009955936820201813865128547180111239345077732790632690028766930563210126343190431070696990449556310466132570110564011069547291541956488855474700537605575103862690565..."
]

Out[12]:

119-element Array{UTF8String,1}:
 "mod(-1.5707963267948966,pi/2) = mod(-(1+2570638124657944/2^52)*2^(0),pi/2) = 6.12323399573676588613032966137500529104874722961539082031431044993140174126710585339910740432566411533235469223047752911158626797040642405587251420513509692605527798223114744774651909822144054878329667... × 10^-17"    
 "mod(1.5707963267948966,pi/2) = mod(+(1+2570638124657944/2^52)*2^(0),pi/2) = 1.5707963267948965579989817342720925807952880859375"                                                                                                                                                                        
 "mod(-3.141592653589793,pi/2) = mod(-(1+2570638124657944/2^52)*2^(1),pi/2) = 1.22464679914735317722606593227500105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933... × 10^-16"     
 "mod(3.141592653589793,pi/2) = mod(+(1+2570638124657944/2^52)*2^(1),pi/2) = 1.57079632679489649676664177690443371949199147218744708951252770384609179685689550068598258732894146600892595674335884667645307769522470888413732029593575944127485794864903073944722017768852552253480901778559..."         
 "mod(6.283185307179586,pi/2) = mod(+(1+2570638124657944/2^52)*2^(2),pi/2) = 1.57079632679489637430196186216911599688539824468734126853758311153827539057068650205794776198682439802677787023007654002935923308567412665241196088780727832382457384594709221834166053306557656760442705335678..."         
 "mod(-6.283185307179586,pi/2) = mod(-(1+2570638124657944/2^52)*2^(2),pi/2) = 2.44929359829470635445213186455000211641949889184615632812572417997256069650684234135964296173026564613294187689219101164463450718816256962234900568205403877042211119289245897909860763928857621951331866... × 10^-16"     
 "mod(45.553093477052,pi/2) = mod(+(1+1907428335405278/2^52)*2.0^(5),pi/2) = 6.18980636588357700015067146560965595863303411536662108849969519893495032539302514258852745557406553617139253161516557639982288582137023796970880510821891443969385152967240153509461515782240852843965031... × 10^-19"      
 "mod(3.14159265359,pi/2) = mod(+(1+2570638124658410/2^52)*2.0^(1),pi/2) = 2.06823107110214443817242336338901406144179025055407692183593713791001371965174657882932017851913486717693352906155390449417768274640591871518882549715897298061478894440355377051045069618035571189024334... × 10^-13"        
 "mod(-3.14159265359,pi/2) = mod(-(1+2570638124658410/2^52)*2.0^(1),pi/2) = 1.57079632679468979612421147719593419976224579828140873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321..."          
 "mod(-3.9269808169872418,pi/2) = mod(-(1+4339175044666918/2^52)*2.0^(1),pi/2) = 0.78540816339744808411934112625764384217252411859390873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321..."     
 "mod(-3.73063127613788,pi/2) = mod(-(1+3897035185165141/2^52)*2.0^(1),pi/2) = 0.98175770424680993182460891801034733004560151117203373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321..."       
 "mod(-3.5342817352885176,pi/2) = mod(-(1+3454895325663363/2^52)*2.0^(1),pi/2) = 1.17810724509617222361908655982566698737134614007828373146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321..."     
 "mod(-3.337932194439156,pi/2) = mod(-(1+3012755466161586/2^52)*2.0^(1),pi/2) = 1.37445678594553407132435435157837047524442353265640873146241688846172460942931349794205223801317560197322212976992345997064076691432587334758803911219272167617542615405290778165833946693442343239557294664321..."      
 ⋮                                                                                                                                                                                                                                                                                                        
 "mod(2.549491779e9,pi/2) = mod(+(1+843072155942912/2^52)*2.0^(31),pi/2) = 4.47449493816149706970976233303722197019495577867890936615779430401846755180508920207307551467187143433646915044596696119497365034532889215443076399478064252395175148483303135651614725663715466720569020... × 10^-10"        
 "mod(6.167950454e9,pi/2) = mod(+(1+1963965187883008/2^52)*2.0^(32),pi/2) = 1.57079632664516232778985771788239830640502589366761233667091496633591602243684762684101849021235526976201462469021537863260500475096307555076211755104482793217002447595584069355659201693649624670292983799267..."          
 "mod(1.4885392687e10,pi/2) = mod(+(1+3300633133711360/2^52)*2.0^(33),pi/2) = 1.47980910933221759456269961916604584979614430234776276979795068989333010234511075289901023009068306300795365662712861011872933904331764909404251146367829101604918014490927524901658264139193178277114987... × 10^-10"     
 "mod(2.1053343141e10,pi/2) = mod(+(1+1015407956983808/2^52)*2.0^(34),pi/2) = 1.57079632679314323872307947733866826832163047864722676690569124331571109142618063707552956550225629277108293099101074429531786576283600945509388246044907907853785357756075870804751954183815451084212301626979..."        
 "mod(1.783366216531e12,pi/2) = mod(+(1+2801068395540480/2^52)*2.0^(40),pi/2) = 6.96948240875758165283364652450017592218369365167838571237684767728582201531913110512760529814875987841007291472111488973274399752796445730686810635597303227231604049063965142781067801484038929704503126... × 10^-13"   
 "mod(3.587785776203e12,pi/2) = mod(+(1+2844185642293248/2^52)*2.0^(41),pi/2) = 1.57079632679453713520483099366923499762653051383166350563602692045818646096163780147859339172328181383071268296669275887826208874078255825459947535191045269980904818401522191614564747212371664644509109412920..."      
 "mod(5.371151992734e12,pi/2) = mod(+(1+996460013189120/2^52)*2.0^(42),pi/2) = 3.37464214385060194766920180395831736328964513722462875515942586261884366107892162736040369453515697892612846187277924980541538489592093576898951719503435891484259641931614955023236781384150844501479806... × 10^-13"    
 "mod(8.958937768937e12,pi/2) = mod(+(1+83376510325248/2^52)*2.0^(43),pi/2) = 1.57079632679487459941921605386400191780692634556799247014974938333370240354789968584470128388601785420016619866458537172444936666576309979308906744548735165152855161990670617578757908707873988322647524497370..."        
 "mod(1.39755218526789e14,pi/2) = mod(+(1+4440734358344000/2^52)*2.0^(46),pi/2) = 7.16703280049355852405580552051994292235944787877057242894865818968232636596038712584350583584681705588885972394333979767169688540821492356534377064696422753798816392058646400432320064809223397832709837... × 10^-15" 
 "mod(4.28224593349304e14,pi/2) = mod(+(1+2347993866218368/2^52)*2.0^(48),pi/2) = 1.57079632679489610051761753453957408522348790539675954849338569505098924952246873282379916504739538471767373911575303830362119668515611488374529209025804768284049251258932014027934084647075285282841952167563..."    
 "mod(5.706674932067741e15,pi/2) = mod(+(1+1203075304697245/2^52)*2.0^(52),pi/2) = 4.23754646451256218416429262194162608653524752956234482551589924717953528741279504902951631892985510393600689510284748380679359369235443057957270202960326099424609213900534623202922619105230013018255129... × 10^-16"
 "mod(6.134899525417045e15,pi/2) = mod(+(1+1631299898046549/2^52)*2.0^(52),pi/2) = 1.57079632679489652427226398579579250165275009955936820201813865128547180111239345077732790632690028766930563210126343190431070696990449556310466132570110564011069547291541956488855474700537605575103862690565..."

Now we can start testing. In the function below, xDec is the Float64 extracted from the decimal representation on the left of the first equal sign. xExact is the Float64 extracted from the exact representation from the formula in the middle. xSoln is the number on the right, namely the solution of $\mod(x,pi)$ etc., as taken from WolframAlpha. We compute the Julia answer as xNew = modpi(x), and xOld = mod(x,pi), and compute the difference newDiff, oldDiff respectively to the correct solution xSoln. One small complication here: if the true answer is, say, 0.01, and we're off by 0.02, we could reach 0.03, or, in the other direction, 3.1315926... etc. Thus, to be generous, we take oldDiff = min(oldDiff, abs(pi - oldDiff)). Here's the code:

In [13]:

function testModPi_v0()
    # for each divisor (2pi, pi, pi/2), we're testing that:
    # the number x in exact representation 
    for divisorStr in ["2pi","pi","pi/2"]
#        numbersRegEx = Regex(string("mod\\(([-.0-9]*),",regexStr,"\\) = mod\\(([-.+()^/*0-9]*),",regexStr,"\\) = ([.0-9]*)\\.{3,3}?(...10\\^(-?[0-9]*))?"))
        numbersRegEx = r"[^(]*\(([^,]*),[^(]*\(([^,]*),[^=]*= ([0-9]*.[0-9]+)[^^]*\^?(-?[0-9]+)?"
        xDivisor = eval(parse(divisorStr))
        tol = eps(Float64) * xDivisor

        normalizedDivisorStr = replace(divisorStr,"/","o")              # 2pi, pi, pio2
        solnList = eval(parse("mod" * normalizedDivisorStr * "Solns"))  # see below: mod2piSolns, modpiSolns, modpio2Solns
        modFn = eval(parse("mod" * normalizedDivisorStr*"_v0"))

        println("\nTesting mod"*normalizedDivisorStr, " with ", length(solnList), " instances.")
        errsNew, errsOld = Array(Float64,0), Array(Float64,0)
        for testCase in solnList
            decStr, exactStr, solnSignificand, solnExponent = match(numbersRegEx,testCase).captures
            solnStr = solnExponent == nothing ? solnSignificand : string(solnSignificand, "e", solnExponent)
            print(lpad(decStr,20," "),": ")
            xDec = eval(parse(convert(ASCIIString,decStr)))
            xExact = eval(parse(convert(ASCIIString,exactStr)))
#            print(solnStr)
            xSoln = eval(parse(convert(ASCIIString,solnStr)))
            # 1. want: xDec ≈ xExact  (dfference of the original x in different representations (decimal vs exact))
            # (might be off because for some x::Float64, if we have eval(parse(repr(x))) != x)
            # 2. want: xNew := modFn(xExact)  ≈  xSoln  <--- this is the crucial bit, xNew close to xSoln
            # 3. know: xOld := mod(xExact,xDivisor) might be quite a bit off from xSoln - that's expected
            xOld = mod(xExact,xDivisor)
            xNew = modFn(xExact)

            reprDiff = abs(xDec - xExact) # should be zero
            newDiff  = abs(xNew - xSoln)  # should be zero, ideally (our new function)
            oldDiff  = abs(xOld - xSoln)  # should be zero in a perfect world, but often bigger due to cancellation
            oldDiff  = min(oldDiff, abs(xDivisor - oldDiff)) # we are being generous here:
            # if xOld happens to end up "on the wrong side of 0", ie 
            # if xSoln = 3.14 (correct), but xOld reports 0.01, 
            # we don't take the long way around the circle of 3.14 - 0.01, but the short way of 3.1415.. - (3.14 - 0.1) 
            if reprDiff == zero(reprDiff)
                 print("Repr ok  ")
            else print("Repr ERR ") end
            if abs(newDiff) <= tol && abs(newDiff) <= abs(oldDiff)
                 print("new ok  ")
            else print("new ERR ") end
            push!(errsNew,abs(newDiff))
            push!(errsOld,abs(oldDiff))
            println("RepErr $(reprDiff)  NewErr $(newDiff)  OldErr $(oldDiff)")
        end
        sort!(errsNew)
        sort!(errsOld)
        println("Total err = $(sum(errsNew)) (new), $(sum(errsOld)) (old).")
        sort!(errsNew, rev = true)
        sort!(errsOld, rev = true)
        println("Total err = $(sum(errsNew)) (new), $(sum(errsOld)) (old).")

        N, runs = 10000, 10
        newTimes, oldTimes = Array(Float64,0), Array(Float64,0)
        println("\nSpeed: ")
        for k = 1:runs
            testCases = vcat(repmat(allTestCases(),N),linspace(-1e30,1e30,100*N) .* randn(100*N))
            oldTime = @elapsed for tc in testCases x = mod(tc,pi) end
            newTime = @elapsed for tc in testCases x = modFn(tc) end
            push!(oldTimes, oldTime)
            push!(newTimes, newTime)
            println("Items: $(length(testCases))  NewTime: $newTime, OldTime: $oldTime, Ratio: $(newTime/oldTime)")
        end
        println("Averages:")
        println("Items: $(runs)  NewTime: $(mean(newTimes)), OldTime: $(mean(oldTimes)), Ratio: $(sum(newTimes)/sum(oldTimes))")
    end
end

Out[13]:

testModPi_v0 (generic function with 1 method)

Let's run it. On the left is the number x we test. "Repr ok" indicates that the x derived from the decimal representation lines up with the formulaic representation (used for WolframAlpha). "new ok" indicates that the new v0 modpi from above has a small error, and smaller than the naive mod(x,pi). Next, we print the error of the new (modpi) and old (mod(x,pi)) solutions against the WolframAlpha correct solution.

In [14]:

testModPi_v0()

Testing mod2pi with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
   3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.7763568394002505e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 8.881784197001252e-16  OldErr 8.881784197001252e-16
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 6.55112222065421e-17  OldErr 3.785779876435205e-16
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 8.326672684688674e-17  OldErr 1.3877787807814457e-16
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 3.3306690738754696e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 3.3306690738754696e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 8.881784197001252e-16  OldErr 8.881784197001252e-16
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 8.881784197001252e-16  OldErr 8.881784197001252e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 2.7755575615628914e-16
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 4.440892098500626e-16
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.141582653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 8.881784197001252e-16  OldErr 1.2434497875801753e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.3766765505351941e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0536463075113716e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.0674130730167235e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.46886971872118e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.6656631629530239e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125545757172404e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.124145386834698e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.45954623124112e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691173986608e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 2.4043730695311183e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453886613
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406251633
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00034923478860182655
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00544789856308725
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 1.4359632664255433e-14 (new), 0.4845155554649995 (old).
Total err = 1.4359632664255433e-14 (new), 0.4845155554649995 (old).

Speed: 
Items: 2190000  NewTime: 1.043648358, OldTime: 0.329823616, Ratio: 3.1642620703060875
Items: 2190000  NewTime: 0.912111966, OldTime: 0.395030342, Ratio: 2.3089668539942183
Items: 2190000  NewTime: 0.936323767, OldTime: 0.333052285, Ratio: 2.8113416696720757
Items: 2190000  NewTime: 1.119435337, OldTime: 0.416654736, Ratio: 2.686721739315595
Items: 2190000  NewTime: 0.966345107, OldTime: 0.398716975, Ratio: 2.423636733800963
Items: 2190000  NewTime: 0.924069663, OldTime: 0.350590045, Ratio: 2.6357555674463033
Items: 2190000  NewTime: 0.937606333, OldTime: 0.319389125, Ratio: 2.9356238506868384
Items: 2190000  NewTime: 0.885002133, OldTime: 0.336631333, Ratio: 2.6289951238733917
Items: 2190000  NewTime: 0.908559927, OldTime: 0.316386159, Ratio: 2.871680385360979
Items: 2190000  NewTime: 0.890804363, OldTime: 0.362796966, Ratio: 2.4553798583861366
Averages:
Items: 10  NewTime: 0.9523906954000001, OldTime: 0.35590715819999996, Ratio: 2.6759526282548376

Testing modpi with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473532e-16
   3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   6.283185307179586: Repr ok  new ERR RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 0.0
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.7763568394002505e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473515e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 6.55112222065421e-17  OldErr 6.55112222065421e-17
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 8.326672684688674e-17  OldErr 1.3877787807814457e-16
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 4.440892098500626e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 1.6653345369377348e-16
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ERR RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 0.0
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.141582653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.569533971325427e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2878587085651816e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.383850919598438e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0536463075113716e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0676643436464606e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.468895374173894e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 1.6656631629530239e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125561418735498e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 3.124144942745488e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.459546132324573e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691173986608e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 2.4043730695311183e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453886613
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406255992
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 4.440892098500626e-16  OldErr 0.00034923478860138246
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0054478985630871885
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 1.0251807473142353e-14 (new), 0.4845155554649873 (old).
Total err = 1.0251807473142353e-14 (new), 0.48451555546498726 (old).

Speed: 
Items: 2190000  NewTime: 0.85899531, OldTime: 0.312095208, Ratio: 2.752350205902552
Items: 2190000  NewTime: 0.924686098, OldTime: 0.351355138, Ratio: 2.6317705307044634
Items: 2190000  NewTime: 0.934326479, OldTime: 0.304161849, Ratio: 3.071806941178872
Items: 2190000  NewTime: 0.919958421, OldTime: 0.36893931, Ratio: 2.493522365507758
Items: 2190000  NewTime: 0.916990902, OldTime: 0.329800087, Ratio: 2.7804446940609813
Items: 2190000  NewTime: 0.906802094, OldTime: 0.350689161, Ratio: 2.5857716600485405
Items: 2190000  NewTime: 0.942499937, OldTime: 0.341421289, Ratio: 2.760518946432775
Items: 2190000  NewTime: 0.910477693, OldTime: 0.362204418, Ratio: 2.51371227890434
Items: 2190000  NewTime: 1.028523904, OldTime: 0.327362985, Ratio: 3.141845447187623
Items: 2190000  NewTime: 0.886481422, OldTime: 0.350269101, Ratio: 2.530858187231308
Averages:
Items: 10  NewTime: 0.9229742259999998, OldTime: 0.3398298546, Ratio: 2.7159892325716806

Testing modpio2 with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123233995736766e-17
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473532e-16
   3.141592653589793: Repr ok  new ERR RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 0.0
   6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.775737858763662e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473515e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123201175701337e-17
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 6.55112222065421e-17  OldErr 6.55112222065421e-17
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 8.326672684688674e-17  OldErr 8.326672684688674e-17
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123201175701337e-17
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 1.1102230246251565e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 2.220446049250313e-16
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 5.551115123125783e-17
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 5.551115123125783e-17  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ERR RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 0.0
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 1.1102230246251565e-16  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.141582653589793: Repr ok  new ERR RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 0.0
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.569533971325427e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2878587085651816e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.383850919598438e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0538683521162966e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0676643436464606e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.468895374173894e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 1.665665383399073e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125561418735498e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.124145164790093e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.459546132324573e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691396031213e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4043730717515643e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 2.220446049250313e-16  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453908817
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406255992
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0003492347886016045
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0054478985630871885
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 5.033759257404118e-15 (new), 0.484515555464982 (old).
Total err = 5.033759257404118e-15 (new), 0.48451555546498193 (old).

Speed: 
Items: 2190000  NewTime: 1.145805877, OldTime: 0.309216702, Ratio: 3.7055109558732693
Items: 2190000  NewTime: 0.901169008, OldTime: 0.41662444, Ratio: 2.1630248287882488
Items: 2190000  NewTime: 0.887643365, OldTime: 0.375393682, Ratio: 2.364566607170549
Items: 2190000  NewTime: 0.878303605, OldTime: 0.341441745, Ratio: 2.5723380865453342
Items: 2190000  NewTime: 0.941794494, OldTime: 0.331873837, Ratio: 2.8378087966000165
Items: 2190000  NewTime: 0.958109597, OldTime: 0.346970082, Ratio: 2.761360839751019
Items: 2190000  NewTime: 0.959344308, OldTime: 0.306538346, Ratio: 3.129606199414934
Items: 2190000  NewTime: 1.017380036, OldTime: 0.581026338, Ratio: 1.7510050224263671
Items: 2190000  NewTime: 0.977133609, OldTime: 0.406962439, Ratio: 2.4010412641546015
Items: 2190000  NewTime: 0.948788345, OldTime: 0.352802529, Ratio: 2.689290090094564
Averages:
Items: 10  NewTime: 0.9615472244, OldTime: 0.376885014, Ratio: 2.551301295307008

We get: Total err = 1.4359632664255433e-14 (new), 0.4845155554649995 (old). Thus, the new approximation is better. However, the error is not zero. Some investigation reveals:

occasionally, one is a bit or two off due to the rounding after adding the required multiple of $\pi$ .
the wrapped function ieee754_rem_pio2 behaves somewhat unusual for values x that are already in $(-\pi/4, \pi/4)$ , in that it moves them outside that quarter circle (instead of just bouncing the same number back with $n==0$ ).

Thus, we modify the functions in two ways:

we add the required multiple of $\pi$ in double precision.
we intercept values that are already in range, and handle them separately by bouncing them right back (possibly after adding a multiple of $\pi$ to get it into the required range).

First, we compute $k\pi/4$ for $k=1,2,3,4$ in double-double precision.

In [15]:

const pi1o2_bf = pi * BigFloat(1/2)
const pi1o2_h  = convert(Float64, pi1o2_bf)
const pi1o2_l  = convert(Float64, pi1o2_bf - pi1o2_h)

const pi2o2_bf = pi * BigFloat(1)
const pi2o2_h  = convert(Float64, pi2o2_bf)
const pi2o2_l  = convert(Float64, pi2o2_bf - pi2o2_h)

const pi3o2_bf = pi * BigFloat(3/2)
const pi3o2_h  = convert(Float64, pi3o2_bf)
const pi3o2_l  = convert(Float64, pi3o2_bf - pi3o2_h)

const pi4o2_bf = pi * BigFloat(2)
const pi4o2_h  = convert(Float64, pi4o2_bf)
const pi4o2_l  = convert(Float64, pi4o2_bf - pi4o2_h)

Out[15]:

2.4492935982947064e-16

Next, we define a double-double addition routine, as outlined in this paper:

In [16]:

function add22Cond(xh::Float64, xl::Float64, yh::Float64, yl::Float64)
    # This algorithm,  due to Dekker [1], computes the sum of 
    # two double-double numbers as a double-double, with a relative error smaller than 2^−103
    # [1] http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN362160546_0018&DMDID=DMDLOG_0023&LOGID=LOG_0023&PHYSID=PHYS_0232
    # [2] http://ftp.nluug.nl/pub/os/BSD/FreeBSD/distfiles/crlibm/crlibm-1.0beta3.pdf
    r = xh+yh
    s = (abs(xh) > abs(yh)) ? (xh-r+yh+yl+xl) : (yh-r+xh+xl+yl)
    zh = r+s
    zl = r-zh+s
    return (zh,zl)
end

function add22Cond_h(xh::Float64, xl::Float64, yh::Float64, yl::Float64)
    # as above, but only compute and return high double
    r = xh+yh
    s = (abs(xh) > abs(yh)) ? (xh-r+yh+yl+xl) : (yh-r+xh+xl+yl)
    zh = r+s
    return zh
end

Out[16]:

add22Cond_h (generic function with 1 method)

In [17]:

function mod2pi(x::Float64) # or modtau(x)
# with r = mod2pi(x)
# a) 0 <= r < 2π  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*2π with k integer

# note: mod(n,4) is 0,1,2,3; while mod(n-1,4)+1 is 1,2,3,4.
# We use the latter to push negative y in quadrant 0 into the positive (one revolution, + 4*pi/2)

    if x < pi4o2_h
        if 0.0 <= x return x end
        if x > -pi4o2_h 
            return add22Cond_h(x,0.0,pi4o2_h,pi4o2_l)
        end
    end

    (n,y) = ieee754_rem_pio2(x)

    if iseven(n)
        if n & 2 == 2 # add pi
            return add22Cond_h(y[1],y[2],pi2o2_h,pi2o2_l)
        else # add 0 or 2pi
            if y[1] > 0.0
                return y[1]
            else # else add 2pi
                return add22Cond_h(y[1],y[2],pi4o2_h,pi4o2_l)
            end
        end
    else # add pi/2 or 3pi/2
        if n & 2 == 2 # add 3pi/2
            return add22Cond_h(y[1],y[2],pi3o2_h,pi3o2_l) 
        else # add pi/2
            return add22Cond_h(y[1],y[2],pi1o2_h,pi1o2_l) 
        end
    end
end

function modpi(x::Float64)
# with r = modpi(x)
# a) 0 <= r < π  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*π with k integer
    if x < pi2o2_h
        if 0.0 <= x return x end
        if x > -pi2o2_h 
            return add22Cond_h(x,0.0,pi2o2_h,pi2o2_l)
        end
    end
    (n,y) = ieee754_rem_pio2(x)
    if iseven(n)
        if y[1] > 0.0
            return y[1]
        else # else add pi
            return add22Cond_h(y[1],y[2],pi2o2_h,pi2o2_l)
        end
    else # add pi/2
        return add22Cond_h(y[1],y[2],pi1o2_h,pi1o2_l)
    end
end


function modpio2(x::Float64)
# with r = modpio2(x)
# a) 0 <= r < π/2  (note: boundary open or closed - a bit fuzzy, due to rem_pio2 implementation)
# b) r-x = k*π/2 with k integer

# Note: we explicitly test for 0 <= values < pi/2, because 
# ieee754_rem_pio2 behaves weirdly for arguments that are already
# within -pi/4, pi/4 e.g. 
# ieee754_rem_pio2(0.19633954084936206)  returns
# (1,[-1.3744567859455346,-6.12323399538461e-17])
# which does not add up to 0.19633954084936206
    if x < pi1o2_h
        if x >= 0.0 return x end
        if x > -pi1o2_h
            zh = add22Cond_h(x,0.0,pi1o2_h,pi1o2_l)
            return zh
        end
    end
    (n,y) = ieee754_rem_pio2(x)
    if y[1] > 0.0
        return y[1]
    else
        zh = add22Cond_h(y[1],y[2],pi1o2_h,pi1o2_l)
        return zh
    end
end

Out[17]:

modpio2 (generic function with 1 method)

Let's test it.

In [18]:

function testModPi()
    # for each divisor (2pi, pi, pi/2), we're testing that:
    # the number x in exact representation 
    for divisorStr in ["2pi","pi","pi/2"]
        numbersRegEx = r"[^(]*\(([^,]*),[^(]*\(([^,]*),[^=]*= ([0-9]*.[0-9]+)[^^]*\^?(-?[0-9]+)?"
        xDivisor = eval(parse(divisorStr))
        tol = eps(Float64) * xDivisor

        normalizedDivisorStr = replace(divisorStr,"/","o")              # 2pi, pi, pio2
        solnList = eval(parse("mod" * normalizedDivisorStr * "Solns"))  # see below: mod2piSolns, modpiSolns, modpio2Solns
        modFn = eval(parse("mod" * normalizedDivisorStr))

        println("\nTesting mod"*normalizedDivisorStr, " with ", length(solnList), " instances.")
        errsNew, errsOld = Array(Float64,0), Array(Float64,0)
        for testCase in solnList
            decStr, exactStr, solnSignificand, solnExponent = match(numbersRegEx,testCase).captures
            solnStr = solnExponent == nothing ? solnSignificand : string(solnSignificand, "e", solnExponent)
            print(lpad(decStr,20," "),": ")
            xDec = eval(parse(convert(ASCIIString,decStr)))
            xExact = eval(parse(convert(ASCIIString,exactStr)))
            xSoln = eval(parse(convert(ASCIIString,solnStr)))
            # 1. want: xDec ≈ xExact  (dfference of the original x in different representations (decimal vs exact))
            # (might be off because for some x::Float64, if we have eval(parse(repr(x))) != x)
            # 2. want: xNew := modFn(xExact)  ≈  xSoln  <--- this is the crucial bit, xNew close to xSoln
            # 3. know: xOld := mod(xExact,xDivisor) might be quite a bit off from xSoln - that's expected
            xOld = mod(xExact,xDivisor)
            xNew = modFn(xExact)

            reprDiff = abs(xDec - xExact) # should be zero
            newDiff  = abs(xNew - xSoln)  # should be zero, ideally (our new function)
            oldDiff  = abs(xOld - xSoln)  # should be zero in a perfect world, but often bigger due to cancellation
            oldDiff  = min(oldDiff, abs(xDivisor - oldDiff)) # we are being generous here:
            # if xOld happens to end up "on the wrong side of 0", ie 
            # if xSoln = 3.14 (correct), but xOld reports 0.01, 
            # we don't take the long way around the circle of 3.14 - 0.01, but the short way of 3.1415.. - (3.14 - 0.1) 
            if reprDiff == zero(reprDiff)
                 print("Repr ok  ")
            else print("Repr ERR ") end
            if abs(newDiff) <= tol && abs(newDiff) <= abs(oldDiff)
                 print("new ok  ")
            else print("new ERR ") end
            push!(errsNew,abs(newDiff))
            push!(errsOld,abs(oldDiff))
            println("RepErr $(reprDiff)  NewErr $(newDiff)  OldErr $(oldDiff)")
        end
        sort!(errsNew)
        sort!(errsOld)
        println("Total err = $(sum(errsNew)) (new), $(sum(errsOld)) (old).")
        sort!(errsNew, rev = true)
        sort!(errsOld, rev = true)
        println("Total err = $(sum(errsNew)) (new), $(sum(errsOld)) (old).")

        N, runs = 10000, 10
        newTimes, oldTimes = Array(Float64,0), Array(Float64,0)
        println("\nSpeed: ")
        for k = 1:runs
            testCases = vcat(repmat(allTestCases(),N),linspace(-1e30,1e30,100*N) .* randn(100*N))
            oldTime = @elapsed for tc in testCases x = mod(tc,pi) end
            newTime = @elapsed for tc in testCases x = modFn(tc) end
            push!(oldTimes, oldTime)
            push!(newTimes, newTime)
            println("Items: $(length(testCases))  NewTime: $newTime, OldTime: $oldTime, Ratio: $(newTime/oldTime)")
        end
        println("Averages:")
        println("Items: $(runs)  NewTime: $(mean(newTimes)), OldTime: $(mean(oldTimes)), Ratio: $(sum(newTimes)/sum(oldTimes))")
    end
end

Out[18]:

testModPi (generic function with 1 method)

In [19]:

testModPi()

Testing mod2pi with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.7763568394002505e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.785779876435205e-16
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.3877787807814457e-16
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.3306690738754696e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.3306690738754696e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.7755575615628914e-16
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.141582653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.881784197001252e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2434497875801753e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.3766765505351941e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0536463075113716e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0674130730167235e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.46886971872118e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.6656631629530239e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125545757172404e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.124145386834698e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.45954623124112e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691173986608e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4043730695311183e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453886613
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406251633
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00034923478860182655
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00544789856308725
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 0.0 (new), 0.4845155554649995 (old).
Total err = 0.0 (new), 0.4845155554649995 (old).

Speed: 
Items: 2190000  NewTime: 0.858867848, OldTime: 0.456972053, Ratio: 1.8794756536238335
Items: 2190000  NewTime: 0.961220303, OldTime: 0.501049624, Ratio: 1.9184133805477122
Items: 2190000  NewTime: 0.853484811, OldTime: 0.402626978, Ratio: 2.119790420501827
Items: 2190000  NewTime: 0.859911539, OldTime: 0.414182918, Ratio: 2.0761636987646117
Items: 2190000  NewTime: 0.924028018, OldTime: 0.347771712, Ratio: 2.6569959146073385
Items: 2190000  NewTime: 0.880319305, OldTime: 0.393905948, Ratio: 2.234846438520903
Items: 2190000  NewTime: 0.881358813, OldTime: 0.388341169, Ratio: 2.2695477156582387
Items: 2190000  NewTime: 0.893572548, OldTime: 0.387251896, Ratio: 2.3074710730402725
Items: 2190000  NewTime: 0.731399435, OldTime: 0.367134668, Ratio: 1.9921829746680313
Items: 2190000  NewTime: 0.726011154, OldTime: 0.345401089, Ratio: 2.101936493894494
Averages:
Items: 10  NewTime: 0.8570173774000001, OldTime: 0.40046380549999994, Ratio: 2.1400620121710356

Testing modpi with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473532e-16
   3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.7763568394002505e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473515e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.55112222065421e-17
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.3877787807814457e-16
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.6653345369377348e-16
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.141582653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.440892098500626e-16
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.569533971325427e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2878587085651816e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.383850919598438e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0536463075113716e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0676643436464606e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.468895374173894e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.6656631629530239e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125561418735498e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.124144942745488e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.459546132324573e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691173986608e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4043730695311183e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453886613
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406255992
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00034923478860138246
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0054478985630871885
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 0.0 (new), 0.4845155554649873 (old).
Total err = 0.0 (new), 0.48451555546498726 (old).

Speed: 
Items: 2190000  NewTime: 0.766528536, OldTime: 0.346521943, Ratio: 2.212063482513718
Items: 2190000  NewTime: 0.766918337, OldTime: 0.349841166, Ratio: 2.192190089487639
Items: 2190000  NewTime: 0.799230459, OldTime: 0.345382747, Ratio: 2.3140428001749607
Items: 2190000  NewTime: 0.794279021, OldTime: 0.349050548, Ratio: 2.275541538470812
Items: 2190000  NewTime: 0.9882292, OldTime: 0.399884659, Ratio: 2.4712856013813727
Items: 2190000  NewTime: 0.916824455, OldTime: 0.38328332, Ratio: 2.392028056425727
Items: 2190000  NewTime: 0.874914328, OldTime: 0.380125763, Ratio: 2.301644384992658
Items: 2190000  NewTime: 0.91351045, OldTime: 0.399918904, Ratio: 2.284239231661827
Items: 2190000  NewTime: 0.857834711, OldTime: 0.427136629, Ratio: 2.0083379714082072
Items: 2190000  NewTime: 0.818110721, OldTime: 0.355423843, Ratio: 2.301789081156269
Averages:
Items: 10  NewTime: 0.8496380217999999, OldTime: 0.37365695220000006, Ratio: 2.273845078480517

Testing modpio2 with 119 instances.
 -1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123233995736766e-17
  1.5707963267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  -3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473532e-16
   3.141592653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  -6.283185307179586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4492935982947064e-16
     45.553093477052: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.775737858763662e-15
       3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246467991473515e-16
      -3.14159265359: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.9269808169872418: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
   -3.73063127613788: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.5342817352885176: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  -3.337932194439156: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.1415826535897935: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
 -2.9452331127404316: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.7488835718910694: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.5525340310417075: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356184490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -2.1598349493429834: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9634854084936209: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.7671358676442588: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.5707863267948967: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123201175701337e-17
 -1.3744367859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -1.1780872450961726: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -0.9817377042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.7853881633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.5890386225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.3926890816987242: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
 -0.1963395408493621: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
              1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.55112222065421e-17
 0.19635954084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.326672684688674e-17
  0.3927090816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.5890586225480862: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.7854081633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  0.9817577042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  1.1781072450961723: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.3744567859455343: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.5708063267948964: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.123201175701337e-17
  1.7671558676442585: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  1.9635054084936205: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
   2.159854949342982: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.3562044901923445: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.5525540310417063: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.7489035718910686: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452531127404304: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  3.1416026535897927: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2246402351402674e-16
  3.3379521944391546: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534301735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.7306512761378787: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.927000816987241: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -3.9270008169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.7306512761378796: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.5343017352885173: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -3.3379521944391555: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  -3.141602653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -2.9452531127404313: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.748903571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.552554031041707: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.356204490192345: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  -2.159854949342983: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -1.9635054084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.7671558676442587: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.5708063267948966: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 -1.3744567859455346: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -1.1781072450961725: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 -0.9817577042468104: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.7854081633974483: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
 -0.5890586225480863: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
-0.39270908169872415: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
-0.19635954084936208: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
             -1.0e-5: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
 0.19633954084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
 0.39268908169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  0.5890386225480861: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  0.7853881633974482: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  0.9817377042468103: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.1780872450961724: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.3744367859455344: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.220446049250313e-16
  1.5707863267948965: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  1.7671358676442586: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  1.9634854084936206: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.551115123125783e-17
  2.1598349493429825: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.3561844901923448: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  2.5525340310417066: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   2.748883571891069: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
  2.9452331127404308: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.141582653589793: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0
   3.337932194439155: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.534281735288517: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
   3.730631276137879: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
  3.9269808169872413: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.1102230246251565e-16
                22.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.569533971325427e-16
               333.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2878587085651816e-14
               355.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.383850919598438e-14
            103993.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0538683521162966e-12
            104348.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.0676643436464606e-12
            208341.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.121503469737945e-12
            312689.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.2189154521319881e-11
            833719.0: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.249978064445713e-11
          1.146408e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.468895374173894e-11
          4.272943e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.665665383399073e-10
          5.419351e6: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.1125561418735498e-10
         8.0143857e7: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 3.124145164790093e-9
        1.65707065e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.459546132324573e-9
        2.45850922e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.583691396031213e-9
        4.11557987e8: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 1.604323752371775e-8
       1.068966896e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 4.1670166428531275e-8
       2.549491779e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 9.938357040137513e-8
       6.167950454e9: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 2.4043730717515643e-7
     1.4885392687e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 5.80258184884533e-7
     2.1053343141e10: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 8.206954920098042e-7
   1.783366216531e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 6.951867952347794e-5
   3.587785776203e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00013985805453908817
   5.371151992734e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.00020937673406255992
   8.958937768937e12: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0003492347886016045
 1.39755218526789e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.0054478985630871885
 4.28224593349304e14: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.016692930477863133
5.706674932067741e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.2224559947753089
6.134899525417045e15: Repr ok  new ok  RepErr 0.0  NewErr 0.0  OldErr 0.239148925253172
Total err = 0.0 (new), 0.484515555464982 (old).
Total err = 0.0 (new), 0.48451555546498193 (old).

Speed: 
Items: 2190000  NewTime: 0.863646387, OldTime: 0.362025741, Ratio: 2.3855938658240325
Items: 2190000  NewTime: 0.88491586, OldTime: 0.366783726, Ratio: 2.4126366500786354
Items: 2190000  NewTime: 0.8127882, OldTime: 0.380150835, Ratio: 2.1380676435973105
Items: 2190000  NewTime: 0.815705414, OldTime: 0.356042439, Ratio: 2.2910342269619157
Items: 2190000  NewTime: 0.80539267, OldTime: 0.353045084, Ratio: 2.281274280539196
Items: 2190000  NewTime: 0.809362052, OldTime: 0.346495817, Ratio: 2.3358494166179216
Items: 2190000  NewTime: 0.82259848, OldTime: 0.360018574, Ratio: 2.28487789077238
Items: 2190000  NewTime: 0.839431475, OldTime: 0.360737058, Ratio: 2.326989856972222
Items: 2190000  NewTime: 0.940298983, OldTime: 0.359745725, Ratio: 2.613787788583172
Items: 2190000  NewTime: 0.960367203, OldTime: 0.366171548, Ratio: 2.622724808209293
Averages:
Items: 10  NewTime: 0.8554506724, OldTime: 0.3611216547, Ratio: 2.3688711581438153

To note: Total err = 0.0 (new), 0.484515555464982 (old). The error of the new modpi is zero (compared to the WolframAlpha solution). So, we get every single bit right. Also, the new modpi etc. take about 2.5 times longer than the naive mod(x,pi).

Let's conclude with a final example:

In [20]:

sin(333.0) - sin(mod(333.0,2pi))    # naive way

Out[20]:

-1.2734605037145741e-14

In [21]:

sin(333.0) - sin(mod2pi_v0(333.0))  # better

Out[21]:

5.863365348801608e-16

In [22]:

sin(333.0) - sin(mod2pi(333.0))     # best

Out[22]:

-3.0184188481996443e-16

Note that using sin directly is still more accurate, as it uses the ieee_754_rem_pio2 function, which returns x between $(-\pi/4,\pi/4)$ , ie within a quarter circle, and dispatches to an internal sin or cos function depending on the quadrant of the returned solution, and then fixes the sign. Using the mod2pi function returns a number in the whole circle (4 times larger), thus we lose 2 bits of accuracy (and, indeed, the decision whether to use sin or cos, and whether to flip the sign of the result, requires exactly two bits.)

Summary:

an implementation of mod2pi, modpi, modpio2 (for Float64 arguments) is proposed for Julia with the usual semantics (ie, the result is between 0 and 2pi, pi, pi/2, respectively).
this implementation essentially wraps up and exposes the OpenLibm range reduction routine "ieee754_rem_pio2"
the accuracy is as good as possible (to the last bit) on the (hard) values tested, while the "naive" method can be wildly off
the runtime appears to be about twice the runtime of the naive method
if you want to use trigonometric functions directly, however, there is no need to pass the argument through mod2pi, as the built-in trig functions do range reduction internally (and by passing it through mod2pi, one actually loses 2 bits of accuracy)