Average Error: 0.1 → 0.1
Time: 20.0s
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 r514786 = e;
        double r514787 = v;
        double r514788 = sin(r514787);
        double r514789 = r514786 * r514788;
        double r514790 = 1.0;
        double r514791 = cos(r514787);
        double r514792 = r514786 * r514791;
        double r514793 = r514790 + r514792;
        double r514794 = r514789 / r514793;
        return r514794;
}
↓
double f(double e, double v) {
        double r514795 = e;
        double r514796 = v;
        double r514797 = sin(r514796);
        double r514798 = cos(r514796);
        double r514799 = 1.0;
        double r514800 = fma(r514798, r514795, r514799);
        double r514801 = r514797 / r514800;
        double r514802 = r514795 * r514801;
        return r514802;
}

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 2019152 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))