Average Error: 37.5 → 14.0
Time: 1.2m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.057872224018183 \cdot 10^{-21}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right) \cdot \left(1 + \tan \varepsilon \cdot \tan x\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 1.035628088159927 \cdot 10^{-71}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.5
Target15.2
Herbie14.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -8.057872224018183e-21

    1. Initial program 30.8

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

    2. Initial simplification30.8

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

    4. Applied tan-sum1.5

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

    6. Applied flip--1.6

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

    7. Applied associate-/r/1.6

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

    8. Applied fma-neg1.6

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

    9. Simplified1.6

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

    if -8.057872224018183e-21 < eps < 1.035628088159927e-71

    1. Initial program 46.2

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

    2. Initial simplification46.2

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

    4. Applied tan-sum46.2

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

    5. Taylor expanded around 0 27.3

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    6. Simplified27.3

      \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*}\]

    if 1.035628088159927e-71 < eps

    1. Initial program 31.2

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

    2. Initial simplification31.2

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

    4. Applied tan-sum5.9

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

    6. Applied div-inv6.0

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

    7. Applied fma-neg6.0

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

    8. Taylor expanded around inf 6.1

      \[\leadsto (\color{blue}{\left(\frac{\sin \varepsilon}{\cos \varepsilon} + \frac{\sin x}{\cos x}\right)} \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.057872224018183 \cdot 10^{-21}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right) \cdot \left(1 + \tan \varepsilon \cdot \tan x\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 1.035628088159927 \cdot 10^{-71}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\sin x}{\cos x} + \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018273 +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)))