Average Error: 29.4 → 0.0
Time: 10.8s
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 r1521552 = 1.0;
        double r1521553 = x;
        double r1521554 = cos(r1521553);
        double r1521555 = r1521552 - r1521554;
        double r1521556 = sin(r1521553);
        double r1521557 = r1521555 / r1521556;
        return r1521557;
}
↓
double f(double x) {
        double r1521558 = x;
        double r1521559 = 2.0;
        double r1521560 = r1521558 / r1521559;
        double r1521561 = tan(r1521560);
        return r1521561;
}

Error

Bits error versus x

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Results

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Target

Original29.4
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 29.4

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

  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 2019143 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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