Average Error: 0.1 → 0.1
Time: 4.6s
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 r10747 = e;
        double r10748 = v;
        double r10749 = sin(r10748);
        double r10750 = r10747 * r10749;
        double r10751 = 1.0;
        double r10752 = cos(r10748);
        double r10753 = r10747 * r10752;
        double r10754 = r10751 + r10753;
        double r10755 = r10750 / r10754;
        return r10755;
}
↓
double f(double e, double v) {
        double r10756 = e;
        double r10757 = v;
        double r10758 = cos(r10757);
        double r10759 = 1.0;
        double r10760 = fma(r10758, r10756, r10759);
        double r10761 = r10756 / r10760;
        double r10762 = sin(r10757);
        double r10763 = r10761 * r10762;
        return r10763;
}

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. Using strategy rm

  5. Applied associate-/r/0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))