Average Error: 30.3 → 1.6
Time: 39.9s
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\left(\sqrt[3]{\frac{\sin x}{1 + \cos x}} \cdot \sqrt[3]{\frac{\sin x}{1 + \cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{1 + \cos x}}\]

Error

Bits error versus x

Target

Original30.3
Target0.0
Herbie1.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.3

    \[\frac{1 - \cos x}{\sin x}\]
  2. Using strategy rm

  3. Applied flip--30.5

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

  4. Applied simplify14.8

    \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{\sin x}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt15.5

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

  7. Applied simplify15.4

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

  8. Applied simplify1.6

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

Runtime

Time bar (total: 39.9s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))