Average Error: 30.1 → 0.0
Time: 11.4s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\tan \left(\frac{x}{2}\right)\]
double f(double x) {
        double r2300892 = 1.0;
        double r2300893 = x;
        double r2300894 = cos(r2300893);
        double r2300895 = r2300892 - r2300894;
        double r2300896 = sin(r2300893);
        double r2300897 = r2300895 / r2300896;
        return r2300897;
}
↓
double f(double x) {
        double r2300898 = x;
        double r2300899 = 2.0;
        double r2300900 = r2300898 / r2300899;
        double r2300901 = tan(r2300900);
        return r2300901;
}


\frac{1 - \cos x}{\sin x}
↓
\tan \left(\frac{x}{2}\right)

Error

Bits error versus x

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 2019102 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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