Conditioning of evaluating tan()

import numpy as np
import matplotlib.pyplot as pt

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

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

[<matplotlib.lines.Line2D at 0x7f448ae83ba8>]

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

1.5706963267948966

np.tan(x)

9999.9999666616441

dx = 0.00005
np.tan(x+dx)

19999.99998335545

Condition number estimates

From evaluation data

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

31413.926693068603

Using the derivative estimate

import sympy as sp

xsym = sp.Symbol("x")

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

tan(x)**2 + 1

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

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

15706.9633726542