Average Error: 36.9 → 0.4

Time: 1.6m

Precision: 64

Internal Precision: 2368

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

↓

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

Error

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

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Target

Original	36.9
Target	15.1
Herbie	0.4

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

Derivation

Initial program 36.9
\[\tan \left(x + \varepsilon\right) - \tan x\]
Using strategy rm
Applied tan-quot36.9
\[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
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-sub21.9
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
Taylor expanded around inf 0.4
\[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon \cdot {\left(\sin x\right)}^{2}}{\cos \varepsilon \cdot \cos x} + \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
Applied simplify0.4
\[\leadsto \color{blue}{\frac{\left(\frac{\sin x \cdot \sin x}{\cos x} + \cos x\right) \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x - \tan x \cdot \left(\cos x \cdot \tan \varepsilon\right)}}\]

Runtime

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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