Average Error: 37.5 → 0.5

Time: 1.4m

Precision: 64

Internal Precision: 2368

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

↓

\[\frac{\sin \varepsilon}{\cos x} \cdot \frac{\frac{\cos x}{\cos \varepsilon} + \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}}\]

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	37.5
Target	15.2
Herbie	0.5

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

Derivation

Initial program 37.5
\[\tan \left(x + \varepsilon\right) - \tan x\]
Using strategy rm
Applied tan-quot37.5
\[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied tan-sum22.4
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
Applied frac-sub22.4
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{\frac{\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{\cos x \cdot \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right)}}\]
Applied simplify0.5
\[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos x} \cdot \frac{\frac{\cos x}{\cos \varepsilon} + \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos x}}}\]

Runtime

herbie shell --seed 2018296 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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