Average Error: 36.7 → 12.7
Time: 1.9m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[-\left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)} + \left(\frac{\sin x}{\cos x} + \frac{\sin x}{\cos x \cdot \left(\frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x} - 1\right)}\right)\right)\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.2
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

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

  3. Applied tan-sum21.3

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

  5. Applied frac-2neg21.3

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

  6. Applied simplify21.3

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

  7. Taylor expanded around -inf 12.7

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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)))