#!/usr/bin/env python
# coding: utf-8

# # Error due to subtraction

# Consider computing
# $$
# x = 10^{-8}, \qquad y = \sqrt{1+x^2} - 1
# $$

# In[1]:


from math import sqrt
x = 1.0e-8
y = sqrt(1.0 + x**2) - 1.0
print("%20.10e" % y)


# The result is zero due to round-off error. An equivalent expression is
# $$
# y = \frac{x^2}{\sqrt{1 + x^2} + 1}
# $$

# In[2]:


y = x**2/(sqrt(1.0+x**2) + 1.0)
print("%20.10e" % y)