Average Error: 0.1 → 0.1
Time: 20.4s
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 r1268899 = e;
        double r1268900 = v;
        double r1268901 = sin(r1268900);
        double r1268902 = r1268899 * r1268901;
        double r1268903 = 1.0;
        double r1268904 = cos(r1268900);
        double r1268905 = r1268899 * r1268904;
        double r1268906 = r1268903 + r1268905;
        double r1268907 = r1268902 / r1268906;
        return r1268907;
}
↓
double f(double e, double v) {
        double r1268908 = e;
        double r1268909 = v;
        double r1268910 = sin(r1268909);
        double r1268911 = cos(r1268909);
        double r1268912 = 1.0;
        double r1268913 = fma(r1268911, r1268908, r1268912);
        double r1268914 = r1268910 / r1268913;
        double r1268915 = r1268908 * r1268914;
        return r1268915;
}

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 2019168 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))