Average Error: 0.1 → 0.1
Time: 16.8s
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 r532693 = e;
        double r532694 = v;
        double r532695 = sin(r532694);
        double r532696 = r532693 * r532695;
        double r532697 = 1.0;
        double r532698 = cos(r532694);
        double r532699 = r532693 * r532698;
        double r532700 = r532697 + r532699;
        double r532701 = r532696 / r532700;
        return r532701;
}
↓
double f(double e, double v) {
        double r532702 = e;
        double r532703 = v;
        double r532704 = sin(r532703);
        double r532705 = r532702 * r532704;
        double r532706 = cos(r532703);
        double r532707 = 1.0;
        double r532708 = fma(r532706, r532702, r532707);
        double r532709 = r532705 / r532708;
        return r532709;
}

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 2019135 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))