Average Error: 37.7 → 13.9
Time: 56.0s
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0642737776012086 \cdot 10^{-61} \lor \neg \left(\varepsilon \le 1.2125662404792727 \cdot 10^{-114}\right):\\ \;\;\;\;\left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_* \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \left(\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.7
Target14.7
Herbie13.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.0642737776012086e-61 or 1.2125662404792727e-114 < eps

    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-sum8.0

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

    6. Applied flip3--8.0

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

    7. Applied associate-/r/8.0

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

    8. Simplified8.0

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

    10. Applied fma-udef8.0

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

    11. Applied distribute-lft-in8.0

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

    12. Applied associate--l+6.9

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

    if -1.0642737776012086e-61 < eps < 1.2125662404792727e-114

    1. Initial program 48.5

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

    2. Initial simplification48.5

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

    4. Applied tan-sum48.5

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

    5. Taylor expanded around 0 26.0

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

    6. Simplified26.0

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.0642737776012086 \cdot 10^{-61} \lor \neg \left(\varepsilon \le 1.2125662404792727 \cdot 10^{-114}\right):\\ \;\;\;\;\left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_* \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \left(\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} - \tan x\right)\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 56.0s)Debug logProfile

herbie shell --seed 2018251 +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)))