Average Error: 0.1 → 0.2
Time: 16.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{1}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{\sin v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \frac{1}{\frac{\mathsf{fma}\left(\cos v, e, 1\right)}{\sin v}}
double f(double e, double v) {
        double r408846 = e;
        double r408847 = v;
        double r408848 = sin(r408847);
        double r408849 = r408846 * r408848;
        double r408850 = 1.0;
        double r408851 = cos(r408847);
        double r408852 = r408846 * r408851;
        double r408853 = r408850 + r408852;
        double r408854 = r408849 / r408853;
        return r408854;
}
↓
double f(double e, double v) {
        double r408855 = e;
        double r408856 = 1.0;
        double r408857 = v;
        double r408858 = cos(r408857);
        double r408859 = fma(r408858, r408855, r408856);
        double r408860 = sin(r408857);
        double r408861 = r408859 / r408860;
        double r408862 = r408856 / r408861;
        double r408863 = r408855 * r408862;
        return r408863;
}

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 clear-num0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))