Average Error: 30.8 → 0.0
Time: 11.4s
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 r1972353 = 1.0;
        double r1972354 = x;
        double r1972355 = cos(r1972354);
        double r1972356 = r1972353 - r1972355;
        double r1972357 = sin(r1972354);
        double r1972358 = r1972356 / r1972357;
        return r1972358;
}
↓
double f(double x) {
        double r1972359 = x;
        double r1972360 = 2.0;
        double r1972361 = r1972359 / r1972360;
        double r1972362 = tan(r1972361);
        return r1972362;
}

Error

Bits error versus x

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Results

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Target

Original30.8
Target0.0
Herbie0.0
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Initial program 30.8

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

  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 2019158 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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