Average Error: 0.1 → 0.1
Time: 17.1s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{\sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}
double f(double e, double v) {
        double r531459 = e;
        double r531460 = v;
        double r531461 = sin(r531460);
        double r531462 = r531459 * r531461;
        double r531463 = 1.0;
        double r531464 = cos(r531460);
        double r531465 = r531459 * r531464;
        double r531466 = r531463 + r531465;
        double r531467 = r531462 / r531466;
        return r531467;
}
↓
double f(double e, double v) {
        double r531468 = e;
        double r531469 = v;
        double r531470 = sin(r531469);
        double r531471 = cos(r531469);
        double r531472 = 1.0;
        double r531473 = fma(r531471, r531468, r531472);
        double r531474 = r531470 / r531473;
        double r531475 = r531468 * r531474;
        return r531475;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))