Average Error: 0.1 → 0.1
Time: 20.7s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)}\]
\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 r23875 = e;
        double r23876 = v;
        double r23877 = sin(r23876);
        double r23878 = r23875 * r23877;
        double r23879 = 1.0;
        double r23880 = cos(r23876);
        double r23881 = r23875 * r23880;
        double r23882 = r23879 + r23881;
        double r23883 = r23878 / r23882;
        return r23883;
}
↓
double f(double e, double v) {
        double r23884 = e;
        double r23885 = v;
        double r23886 = sin(r23885);
        double r23887 = r23884 * r23886;
        double r23888 = cos(r23885);
        double r23889 = 1.0;
        double r23890 = fma(r23888, r23884, r23889);
        double r23891 = r23887 / r23890;
        return r23891;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))