Average Error: 0.1 → 0.1
Time: 39.0s
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 r904506 = e;
        double r904507 = v;
        double r904508 = sin(r904507);
        double r904509 = r904506 * r904508;
        double r904510 = 1.0;
        double r904511 = cos(r904507);
        double r904512 = r904506 * r904511;
        double r904513 = r904510 + r904512;
        double r904514 = r904509 / r904513;
        return r904514;
}
↓
double f(double e, double v) {
        double r904515 = e;
        double r904516 = v;
        double r904517 = sin(r904516);
        double r904518 = cos(r904516);
        double r904519 = 1.0;
        double r904520 = fma(r904518, r904515, r904519);
        double r904521 = r904517 / r904520;
        double r904522 = r904515 * r904521;
        return r904522;
}

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 2019152 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))