Average Error: 0.1 → 0.1
Time: 15.3s
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 r23534 = e;
        double r23535 = v;
        double r23536 = sin(r23535);
        double r23537 = r23534 * r23536;
        double r23538 = 1.0;
        double r23539 = cos(r23535);
        double r23540 = r23534 * r23539;
        double r23541 = r23538 + r23540;
        double r23542 = r23537 / r23541;
        return r23542;
}
↓
double f(double e, double v) {
        double r23543 = e;
        double r23544 = v;
        double r23545 = sin(r23544);
        double r23546 = r23543 * r23545;
        double r23547 = cos(r23544);
        double r23548 = 1.0;
        double r23549 = fma(r23547, r23543, r23548);
        double r23550 = r23546 / r23549;
        return r23550;
}

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 2019235 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))