Average Error: 0.1 → 0.1
Time: 19.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sin v
double f(double e, double v) {
        double r17898 = e;
        double r17899 = v;
        double r17900 = sin(r17899);
        double r17901 = r17898 * r17900;
        double r17902 = 1.0;
        double r17903 = cos(r17899);
        double r17904 = r17898 * r17903;
        double r17905 = r17902 + r17904;
        double r17906 = r17901 / r17905;
        return r17906;
}
↓
double f(double e, double v) {
        double r17907 = e;
        double r17908 = v;
        double r17909 = cos(r17908);
        double r17910 = 1.0;
        double r17911 = fma(r17909, r17907, r17910);
        double r17912 = r17907 / r17911;
        double r17913 = sin(r17908);
        double r17914 = r17912 * r17913;
        return r17914;
}

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. Using strategy rm

  3. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}}\]

  4. Simplified0.3

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

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))