Error due to subtraction¶

In [11]:

from math import sin,cos

Example 1¶

Consider computing $$ x=10^{-8}, \qquad y = 1 - \cos(x) $$

In [12]:

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 $$ x = 10^{-8}, \qquad z = 2 \sin^2(x/2) $$ which does not involve subtraction of nearly equal quantities.

In [13]:

z = 2.0*sin(0.5*x)**2
print("%24.14e" % z)

    5.00000000000000e-17

Example 2¶

$$ x=10^{-8}, \qquad y = \frac{1 - \cos(x)}{x^2} $$

In [14]:

y = (1.0 - cos(x))/x**2
print("%24.14e" % y)

    0.00000000000000e+00

$$ x=10^{-8}, \qquad z = \frac{2 \sin^2(x/2)}{x^2} $$

In [15]:

z = 2.0*sin(0.5*x)**2 / x**2
print("%24.14e" % z)

    5.00000000000000e-01