Average Error: 30.3 → 1.6
Time: 17.3s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}} \cdot \frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{\sqrt[3]{\sin x}}\]

Error

Bits error versus x

Target

Original30.3
Comparison0.0
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-- 30.6

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

  4. Applied simplify 15.2

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

  6. Applied add-cube-cbrt 15.8

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

  7. Applied add-sqr-sqrt 15.8

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

  8. Applied times-frac 15.9

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

  9. Applied times-frac 1.6

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}} \cdot \frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{\sqrt[3]{\sin x}}}\]
  10. Removed slow pow expressions

Runtime

Time bar (total: 17.3s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067901057 3396600083 3715501224 3126139233 3908045574 1593683916)'
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 1

  :target
  (tan (/ x 2))

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