Average Error: 30.3 → 0.0
Time: 11.2s
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 r1289149 = 1.0;
        double r1289150 = x;
        double r1289151 = cos(r1289150);
        double r1289152 = r1289149 - r1289151;
        double r1289153 = sin(r1289150);
        double r1289154 = r1289152 / r1289153;
        return r1289154;
}
↓
double f(double x) {
        double r1289155 = x;
        double r1289156 = 2.0;
        double r1289157 = r1289155 / r1289156;
        double r1289158 = tan(r1289157);
        return r1289158;
}

Error

Bits error versus x

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Results

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Target

Original30.3
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.3

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

  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 2019146 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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