Average Error: 36.7 → 12.9
Time: 2.0m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \frac{(\left(\cos x \cdot \sin x\right) \cdot \left((\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(-1\right))_*\right) + \left(\cos x \cdot \sin x\right))_*}{\cos x \cdot \left(\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)\right)}\right)\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.7
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  2. Using strategy rm

  3. Applied tan-sum21.9

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
  4. Using strategy rm

  5. Applied frac-2neg21.9

    \[\leadsto \color{blue}{\frac{-\left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x\]

  6. Applied simplify21.9

    \[\leadsto \frac{-\left(\tan x + \tan \varepsilon\right)}{\color{blue}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*}} - \tan x\]

  7. Taylor expanded around -inf 12.9

    \[\leadsto \color{blue}{-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)}\]
  8. Using strategy rm

  9. Applied frac-add12.9

    \[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \color{blue}{\frac{\sin x \cdot \left(\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)\right) + \cos x \cdot \sin x}{\cos x \cdot \left(\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)\right)}}\right)\]

  10. Applied simplify12.9

    \[\leadsto -\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \frac{\color{blue}{(\left(\cos x \cdot \sin x\right) \cdot \left((\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(-1\right))_*\right) + \left(\cos x \cdot \sin x\right))_*}}{\cos x \cdot \left(\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)\right)}\right)\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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