Average Error: 36.5 → 13.2
Time: 1.3m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.591317359301214 \cdot 10^{-37} \lor \neg \left(\varepsilon \le 1.392132940226014 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.5
Target15.0
Herbie13.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -9.591317359301214e-37 or 1.392132940226014e-31 < eps

    1. Initial program 29.3

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

    2. Initial simplification29.3

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

    4. Applied tan-sum2.3

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

    6. Applied tan-quot2.4

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

    7. Applied frac-sub2.4

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

    8. Simplified2.4

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

    if -9.591317359301214e-37 < eps < 1.392132940226014e-31

    1. Initial program 45.8

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

    2. Initial simplification45.8

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

    4. Applied tan-sum45.8

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

    5. Taylor expanded around 0 27.1

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

    6. Simplified27.1

      \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.591317359301214 \cdot 10^{-37} \lor \neg \left(\varepsilon \le 1.392132940226014 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018274 +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)))