Average Error: 0.1 → 0.1
Time: 23.4s
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 r22722 = e;
        double r22723 = v;
        double r22724 = sin(r22723);
        double r22725 = r22722 * r22724;
        double r22726 = 1.0;
        double r22727 = cos(r22723);
        double r22728 = r22722 * r22727;
        double r22729 = r22726 + r22728;
        double r22730 = r22725 / r22729;
        return r22730;
}
↓
double f(double e, double v) {
        double r22731 = e;
        double r22732 = v;
        double r22733 = sin(r22732);
        double r22734 = r22731 * r22733;
        double r22735 = cos(r22732);
        double r22736 = 1.0;
        double r22737 = fma(r22735, r22731, r22736);
        double r22738 = r22734 / r22737;
        return r22738;
}

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 2019323 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))