Average Error: 0.1 → 0.1
Time: 20.2s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\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 r716687 = e;
        double r716688 = v;
        double r716689 = sin(r716688);
        double r716690 = r716687 * r716689;
        double r716691 = 1.0;
        double r716692 = cos(r716688);
        double r716693 = r716687 * r716692;
        double r716694 = r716691 + r716693;
        double r716695 = r716690 / r716694;
        return r716695;
}
↓
double f(double e, double v) {
        double r716696 = e;
        double r716697 = v;
        double r716698 = sin(r716697);
        double r716699 = r716696 * r716698;
        double r716700 = cos(r716697);
        double r716701 = 1.0;
        double r716702 = fma(r716700, r716696, r716701);
        double r716703 = r716699 / r716702;
        return r716703;
}

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

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

  4. Applied associate-*l/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))