Average Error: 0.1 → 0.1
Time: 15.9s
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 r954663 = e;
        double r954664 = v;
        double r954665 = sin(r954664);
        double r954666 = r954663 * r954665;
        double r954667 = 1.0;
        double r954668 = cos(r954664);
        double r954669 = r954663 * r954668;
        double r954670 = r954667 + r954669;
        double r954671 = r954666 / r954670;
        return r954671;
}
↓
double f(double e, double v) {
        double r954672 = e;
        double r954673 = v;
        double r954674 = sin(r954673);
        double r954675 = r954672 * r954674;
        double r954676 = cos(r954673);
        double r954677 = 1.0;
        double r954678 = fma(r954676, r954672, r954677);
        double r954679 = r954675 / r954678;
        return r954679;
}

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

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

  4. Applied associate-*r/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))