Average Error: 0.1 → 0.1
Time: 20.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r23824 = e;
        double r23825 = v;
        double r23826 = sin(r23825);
        double r23827 = r23824 * r23826;
        double r23828 = 1.0;
        double r23829 = cos(r23825);
        double r23830 = r23824 * r23829;
        double r23831 = r23828 + r23830;
        double r23832 = r23827 / r23831;
        return r23832;
}
↓
double f(double e, double v) {
        double r23833 = e;
        double r23834 = v;
        double r23835 = sin(r23834);
        double r23836 = r23833 * r23835;
        double r23837 = cos(r23834);
        double r23838 = 1.0;
        double r23839 = fma(r23837, r23833, r23838);
        double r23840 = r23836 / r23839;
        return r23840;
}

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}}\]

  3. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))