Average Error: 30.2 → 0.0
Time: 11.1s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\tan \left(\frac{x}{2}\right)\]
double f(double x) {
        double r2841154 = 1.0;
        double r2841155 = x;
        double r2841156 = cos(r2841155);
        double r2841157 = r2841154 - r2841156;
        double r2841158 = sin(r2841155);
        double r2841159 = r2841157 / r2841158;
        return r2841159;
}
↓
double f(double x) {
        double r2841160 = x;
        double r2841161 = 2.0;
        double r2841162 = r2841160 / r2841161;
        double r2841163 = tan(r2841162);
        return r2841163;
}


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

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 30.2

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

  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 2019101 +o rules:numerics
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

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

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