Average Error: 0.1 → 0.1
Time: 18.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{-\sin v}{-\mathsf{fma}\left(\cos v, e, 1\right)}\]
\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 r18691 = e;
        double r18692 = v;
        double r18693 = sin(r18692);
        double r18694 = r18691 * r18693;
        double r18695 = 1.0;
        double r18696 = cos(r18692);
        double r18697 = r18691 * r18696;
        double r18698 = r18695 + r18697;
        double r18699 = r18694 / r18698;
        return r18699;
}
↓
double f(double e, double v) {
        double r18700 = e;
        double r18701 = v;
        double r18702 = sin(r18701);
        double r18703 = -r18702;
        double r18704 = cos(r18701);
        double r18705 = 1.0;
        double r18706 = fma(r18704, r18700, r18705);
        double r18707 = -r18706;
        double r18708 = r18703 / r18707;
        double r18709 = r18700 * r18708;
        return r18709;
}

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{e}{1} \cdot \frac{\sin v}{1 + e \cdot \cos v}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{e} \cdot \frac{\sin v}{1 + e \cdot \cos v}\]

  6. Simplified0.1

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

  8. Applied frac-2neg0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))