Average Error: 30.2 → 0.0
Time: 16.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\tan \left(\frac{x}{2}\right)\]
\frac{1 - \cos x}{\sin x}
↓
\tan \left(\frac{x}{2}\right)
double f(double x) {
        double r77401 = 1.0;
        double r77402 = x;
        double r77403 = cos(r77402);
        double r77404 = r77401 - r77403;
        double r77405 = sin(r77402);
        double r77406 = r77404 / r77405;
        return r77406;
}
↓
double f(double x) {
        double r77407 = x;
        double r77408 = 2.0;
        double r77409 = r77407 / r77408;
        double r77410 = tan(r77409);
        return r77410;
}

Error

Bits error versus x

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Results

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Target

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

Derivation

  1. Initial program 30.2

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

  3. Applied add-exp-log30.2

    \[\leadsto \frac{\color{blue}{e^{\log \left(1 - \cos x\right)}}}{\sin x}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u30.2

    \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(1 - \cos x\right)\right)\right)}}}{\sin x}\]

  6. Taylor expanded around inf 30.2

    \[\leadsto \frac{\color{blue}{1 - \cos x}}{\sin x}\]
  7. Using strategy rm

  8. Applied hang-p0-tan0.0

    \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)}\]

  9. Final simplification0.0

    \[\leadsto \tan \left(\frac{x}{2}\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))