Average Error: 0.1 → 0.1
Time: 6.1s
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 r9347 = e;
        double r9348 = v;
        double r9349 = sin(r9348);
        double r9350 = r9347 * r9349;
        double r9351 = 1.0;
        double r9352 = cos(r9348);
        double r9353 = r9347 * r9352;
        double r9354 = r9351 + r9353;
        double r9355 = r9350 / r9354;
        return r9355;
}
↓
double f(double e, double v) {
        double r9356 = e;
        double r9357 = v;
        double r9358 = cos(r9357);
        double r9359 = 1.0;
        double r9360 = fma(r9358, r9356, r9359);
        double r9361 = r9356 / r9360;
        double r9362 = sin(r9357);
        double r9363 = r9361 * r9362;
        return r9363;
}

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)))))