Average Error: 0.1 → 0.1
Time: 15.2s
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 r22799 = e;
        double r22800 = v;
        double r22801 = sin(r22800);
        double r22802 = r22799 * r22801;
        double r22803 = 1.0;
        double r22804 = cos(r22800);
        double r22805 = r22799 * r22804;
        double r22806 = r22803 + r22805;
        double r22807 = r22802 / r22806;
        return r22807;
}
↓
double f(double e, double v) {
        double r22808 = e;
        double r22809 = v;
        double r22810 = sin(r22809);
        double r22811 = r22808 * r22810;
        double r22812 = cos(r22809);
        double r22813 = 1.0;
        double r22814 = fma(r22812, r22808, r22813);
        double r22815 = r22811 / r22814;
        return r22815;
}

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 2019209 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))