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

# # 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)


# 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)


# ## 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)


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

# In[15]:


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