# coding: utf-8

# # Conditioning of evaluating tan()

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import numpy as np
import matplotlib.pyplot as pt


# Let us estimate the sensitivity of evaluating the $\tan$ function:

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x = np.linspace(-5, 5, 1000)
pt.ylim([-10, 10])
pt.plot(x, np.tan(x))


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x = np.pi/2 - 0.0001
#x = 0.1
x


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np.tan(x)


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dx = 0.00005
np.tan(x+dx)


# ## Condition number estimates

# ### From evaluation data
# 

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np.abs(np.tan(x+dx) - np.tan(x))/np.abs(np.tan(x)) / (np.abs(dx) / np.abs(x))


# ### Using the derivative estimate

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import sympy as sp

xsym = sp.Symbol("x")

f = sp.tan(xsym)
df = f.diff(xsym)
df


# Evaluate the derivative estimate. Use `.subs(xsym, x)` to substitute in the value of `x`.

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(xsym*df/f).subs(xsym, x)


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