Average Error: 0.1 → 0.1
Time: 20.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e
double f(double e, double v) {
        double r865718 = e;
        double r865719 = v;
        double r865720 = sin(r865719);
        double r865721 = r865718 * r865720;
        double r865722 = 1.0;
        double r865723 = cos(r865719);
        double r865724 = r865718 * r865723;
        double r865725 = r865722 + r865724;
        double r865726 = r865721 / r865725;
        return r865726;
}
↓
double f(double e, double v) {
        double r865727 = v;
        double r865728 = sin(r865727);
        double r865729 = e;
        double r865730 = cos(r865727);
        double r865731 = 1.0;
        double r865732 = fma(r865729, r865730, r865731);
        double r865733 = r865728 / r865732;
        double r865734 = r865733 * r865729;
        return r865734;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.1

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

  5. Applied associate-*r*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))