Average Error: 37.3 → 13.6
Time: 1.6m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.4997145788344413 \cdot 10^{-73}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{elif}\;\varepsilon \le 2.0746387791450197 \cdot 10^{-20}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}\right)} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.7
Herbie13.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.4997145788344413e-73

    1. Initial program 31.4

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

    2. Initial simplification31.4

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

    4. Applied tan-sum6.0

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

    6. Applied tan-quot6.0

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

    7. Applied frac-sub6.1

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

    8. Simplified6.0

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

    if -2.4997145788344413e-73 < eps < 2.0746387791450197e-20

    1. Initial program 46.3

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

    2. Initial simplification46.3

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

    4. Applied tan-sum46.3

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

    6. Applied add-cube-cbrt46.3

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \color{blue}{\left(\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\right)}} - \tan x\]

    7. Applied associate-*r*46.3

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{\left(\tan \varepsilon \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right) \cdot \sqrt[3]{\tan x}}} - \tan x\]
    8. Using strategy rm

    9. Applied tan-quot46.3

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\color{blue}{\frac{\sin x}{\cos x}}}\right)\right) \cdot \sqrt[3]{\tan x}} - \tan x\]

    10. Applied cbrt-div46.3

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \left(\sqrt[3]{\tan x} \cdot \color{blue}{\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}}\right)\right) \cdot \sqrt[3]{\tan x}} - \tan x\]

    11. Taylor expanded around 0 27.3

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

    12. Simplified27.3

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

    if 2.0746387791450197e-20 < eps

    1. Initial program 30.2

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

    2. Initial simplification30.2

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

    4. Applied tan-sum1.1

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

    6. Applied add-cbrt-cube1.5

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

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.4997145788344413 \cdot 10^{-73}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{elif}\;\varepsilon \le 2.0746387791450197 \cdot 10^{-20}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} \cdot \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x}\right)} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018257 +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)))