Average Error: 0.1 → 0.1
Time: 6.3s
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 r9322 = e;
        double r9323 = v;
        double r9324 = sin(r9323);
        double r9325 = r9322 * r9324;
        double r9326 = 1.0;
        double r9327 = cos(r9323);
        double r9328 = r9322 * r9327;
        double r9329 = r9326 + r9328;
        double r9330 = r9325 / r9329;
        return r9330;
}
↓
double f(double e, double v) {
        double r9331 = e;
        double r9332 = v;
        double r9333 = cos(r9332);
        double r9334 = 1.0;
        double r9335 = fma(r9333, r9331, r9334);
        double r9336 = r9331 / r9335;
        double r9337 = sin(r9332);
        double r9338 = r9336 * r9337;
        return r9338;
}

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. Using strategy rm

  4. Applied associate-/l*0.3

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

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))