Average Error: 0.1 → 0.1
Time: 14.5s
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 r18498 = e;
        double r18499 = v;
        double r18500 = sin(r18499);
        double r18501 = r18498 * r18500;
        double r18502 = 1.0;
        double r18503 = cos(r18499);
        double r18504 = r18498 * r18503;
        double r18505 = r18502 + r18504;
        double r18506 = r18501 / r18505;
        return r18506;
}
↓
double f(double e, double v) {
        double r18507 = e;
        double r18508 = v;
        double r18509 = sin(r18508);
        double r18510 = r18507 * r18509;
        double r18511 = cos(r18508);
        double r18512 = 1.0;
        double r18513 = fma(r18511, r18507, r18512);
        double r18514 = r18510 / r18513;
        return r18514;
}

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 2019195 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))