Average Error: 0.1 → 0.1
Time: 8.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v
double f(double e, double v) {
        double r10514 = e;
        double r10515 = v;
        double r10516 = sin(r10515);
        double r10517 = r10514 * r10516;
        double r10518 = 1.0;
        double r10519 = cos(r10515);
        double r10520 = r10514 * r10519;
        double r10521 = r10518 + r10520;
        double r10522 = r10517 / r10521;
        return r10522;
}
↓
double f(double e, double v) {
        double r10523 = e;
        double r10524 = v;
        double r10525 = cos(r10524);
        double r10526 = 1.0;
        double r10527 = fma(r10525, r10523, r10526);
        double r10528 = r10523 / r10527;
        double r10529 = sin(r10524);
        double r10530 = r10528 * r10529;
        return r10530;
}

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. Using strategy rm

  4. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{\sin v}}}\]
  5. Using strategy rm

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))