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 r15624 = e;
        double r15625 = v;
        double r15626 = sin(r15625);
        double r15627 = r15624 * r15626;
        double r15628 = 1.0;
        double r15629 = cos(r15625);
        double r15630 = r15624 * r15629;
        double r15631 = r15628 + r15630;
        double r15632 = r15627 / r15631;
        return r15632;
}
↓
double f(double e, double v) {
        double r15633 = e;
        double r15634 = v;
        double r15635 = cos(r15634);
        double r15636 = 1.0;
        double r15637 = fma(r15635, r15633, r15636);
        double r15638 = r15633 / r15637;
        double r15639 = sin(r15634);
        double r15640 = r15638 * r15639;
        return r15640;
}

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 2020033 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))