(1.1-0.8) == 0.3

sum([0.1]*8)

# associativity failure
(1.0e-8+1.0)-1.0 == 1.0e-8+(1.0-1.0)

# commutativity failure
(0.1+0.2+0.3) == (0.3+0.2+0.1)

int('0b1101',2)

int('0b1110101',2)

bin(1742)

0.3

1.1-0.8

from numpy import abs
def near(a, b, rtol=1e-5, atol=1e-8):
    try:
        return abs(a-b)<(atol+rtol*abs(b))
    except TypeError:
        return False

from numpy import arange

dx = 0.1
xhi = 2.0
xlo = 1.0
x = arange(xlo, xhi+dx, dx) #same array as above

print('x[9] from arange =', repr(x[9]))
print(1.9)
print('x[9] == 1.9?', (x[9] == 1.9)) #numerical drift is too great
print('near(x[9], 1.9)?', near(x[1], 1.1))

print (1.0e-8+1.0)-1.0
print 1.0e-8+(1.0-1.0)

x = 5.0/9.0
y = 100*x
print y, int(y), round(y)

1.7976931348623157e+308 + 1

1.7976931348623157e+308 + 1e250

1.7976931348623157e+308 + 1e300

# Find the point at which addition of the unit value can no longer be captured accurately by the addition operation.
x = 1.0
p = 0
while x != x + 1:
    x *= 2
    p += 1
print "x =", x
print "p =", p

import sys
print sys.float_info

float('inf')/float('inf')

# actual multiplier on Patriot interceptor hardware
dt = 2**-4+2**-5 + 2**-8+2**-9 + 2**-12+2**-13 + 2**-16+2**-17 + 2**-20+2**-21

# "real life" tenth of a second (or in this case, double precision)
dT = 0.1

uptime = 100*60*60*10 #ds (deciseconds ... a little odd)

assumed_diff = (uptime*dt)
real_diff = (uptime*dT)

real_diff - assumed_diff

# difference in Scud missile position as a result
v = 1676 # m/s
v*(real_diff-assumed_diff) # m

sum([0.1]*8)

from math import fsum
fsum([0.1]*8)

corrected_diff = fsum([dt]*uptime)
real_diff - corrected_diff

sum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])

# fsum will solve issues related to drift in repeated operations.
from math import fsum
fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1])

# fsum will not solve fundamental precision errors---
print 'fsum([1.1, -0.8]) == 0.3 is', fsum([1.1, -0.8]) == 0.3
#---but will mitigate accumulated error.
print 'fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) == 1.0 is', fsum([.1, .1, .1, .1, .1, .1, .1, .1, .1, .1]) == 1.0