Average Error: 29.9 → 0.0
Time: 10.7s
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 r1664922 = 1.0;
        double r1664923 = x;
        double r1664924 = cos(r1664923);
        double r1664925 = r1664922 - r1664924;
        double r1664926 = sin(r1664923);
        double r1664927 = r1664925 / r1664926;
        return r1664927;
}
↓
double f(double x) {
        double r1664928 = x;
        double r1664929 = 2.0;
        double r1664930 = r1664928 / r1664929;
        double r1664931 = tan(r1664930);
        return r1664931;
}

Error

Bits error versus x

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Results

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Target

Original29.9
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 29.9

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

  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 2019135 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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