Average Error: 30.6 → 0.0
Time: 9.5s
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 r2094505 = 1.0;
        double r2094506 = x;
        double r2094507 = cos(r2094506);
        double r2094508 = r2094505 - r2094507;
        double r2094509 = sin(r2094506);
        double r2094510 = r2094508 / r2094509;
        return r2094510;
}
↓
double f(double x) {
        double r2094511 = x;
        double r2094512 = 2.0;
        double r2094513 = r2094511 / r2094512;
        double r2094514 = tan(r2094513);
        return r2094514;
}

Error

Bits error versus x

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Results

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Target

Original30.6
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.6

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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