Consider computing $$ x=10^{-8}, \qquad y = 1 - \cos(x) $$
from numpy import sin,cos
x = 1.0e-8
y = 1.0 - cos(x)
print "%24.14e" % y
0.00000000000000e+00
Because of roundoff error, the result is zero. But mathematically we have the equivalent expression $$ z = 2 \sin^2(x/2) $$ which does not involve subtraction of nearly equal quantities.
z = 2.0*sin(0.5*x)**2
print "%24.14e" % z
5.00000000000000e-17