Average Error: 30.4 → 0.0
Time: 11.5s
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 r1357603 = 1.0;
        double r1357604 = x;
        double r1357605 = cos(r1357604);
        double r1357606 = r1357603 - r1357605;
        double r1357607 = sin(r1357604);
        double r1357608 = r1357606 / r1357607;
        return r1357608;
}
↓
double f(double x) {
        double r1357609 = x;
        double r1357610 = 2.0;
        double r1357611 = r1357609 / r1357610;
        double r1357612 = tan(r1357611);
        return r1357612;
}

Error

Bits error versus x

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Results

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Target

Original30.4
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.4

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

  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 2019130 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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