Average Error: 30.1 → 0.0
Time: 11.1s
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 r6738814 = 1.0;
        double r6738815 = x;
        double r6738816 = cos(r6738815);
        double r6738817 = r6738814 - r6738816;
        double r6738818 = sin(r6738815);
        double r6738819 = r6738817 / r6738818;
        return r6738819;
}
↓
double f(double x) {
        double r6738820 = x;
        double r6738821 = 2.0;
        double r6738822 = r6738820 / r6738821;
        double r6738823 = tan(r6738822);
        return r6738823;
}

Error

Bits error versus x

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Results

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Target

Original30.1
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.1

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

  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 2019124 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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