Average Error: 36.6 → 0.4

Time: 1.4m

Precision: 64

Internal Precision: 2368

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\frac{\left(\cos x + \sin x \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x - \frac{\sin x \cdot \cos x}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

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Out

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Target

Original	36.6
Target	14.8
Herbie	0.4

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

Initial program 36.6
\[\tan \left(x + \varepsilon\right) - \tan x\]
Using strategy rm
Applied tan-quot36.7
\[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied tan-sum21.8
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
Applied frac-sub21.8
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon \cdot {\left(\sin x\right)}^{2}}{\cos \varepsilon \cdot \cos x} + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x}}\]
Applied simplify0.4
\[\leadsto \color{blue}{\frac{\left(\cos x + \sin x \cdot \frac{\sin x}{\cos x}\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x - \frac{\sin x \cdot \cos x}{\frac{\cos x \cdot \cos \varepsilon}{\sin \varepsilon}}}}\]

Runtime

herbie shell --seed 2018167 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))