Average Error: 30.8 → 0.2
Time: 30.2s
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;(e^{\log_* (1 + \frac{\sin x}{1 + \cos x})} - 1)^* \le 1.4823632321274158:\\ \;\;\;\;(e^{\log_* (1 + \frac{\sin x}{1 + \cos x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Error

Bits error versus x

Target

Original30.8
Target0.0
Herbie0.2
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (expm1 (log1p (/ (sin x) (+ 1 (cos x))))) < 1.4823632321274158

    1. Initial program 38.7

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

    3. Applied flip--38.7

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

    4. Applied simplify18.7

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

    6. Applied expm1-log1p-u18.7

      \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x})} - 1)^*}\]

    7. Applied simplify0.2

      \[\leadsto (e^{\color{blue}{\log_* (1 + \frac{\sin x}{1 + \cos x})}} - 1)^*\]

    if 1.4823632321274158 < (expm1 (log1p (/ (sin x) (+ 1 (cos x)))))

    1. Initial program 0.4

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

    3. Applied div-sub0.4

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 30.2s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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