Average Error: 29.7 → 0.5
Time: 18.2s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -3.348635337413482 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \mathbf{if}\;x \le 0.20464312961310469:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \sin x}{\frac{\sin x}{\frac{1}{1 + \cos x}}}\\ \end{array}\]

Error

Bits error versus x

Target

Original29.7
Comparison0.0
Herbie0.5
\[ \tan \left(\frac{x}{2}\right) \]

Derivation

  1. Split input into 3 regimes.

  2. if x < -3.348635337413482e-19

    1. Initial program 3.2

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

    3. Applied flip-- 3.6

      \[\leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]

    4. Applied simplify 0.9

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

    if -3.348635337413482e-19 < x < 0.20464312961310469

    1. Initial program 60.2

      \[\frac{1 - \cos x}{\sin x}\]

    2. Applied taylor 0.0

      \[\leadsto \frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\]

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

    if 0.20464312961310469 < x

    1. Initial program 0.9

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

    3. Applied flip-- 1.4

      \[\leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\cos x\right)}^2}{1 + \cos x}}}{\sin x}\]

    4. Applied simplify 1.0

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

    6. Applied div-inv 1.0

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

    7. Applied associate-/l* 1.0

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\frac{\sin x}{\frac{1}{1 + \cos x}}}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 18.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 1

  :target
  (tan (/ x 2))

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