Average Error: 0.1 → 0.3
Time: 17.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{\sin v}{\frac{\sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)}}{e}}}{\sqrt[3]{1 + e \cdot \cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\frac{\sin v}{\frac{\sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)}}{e}}}{\sqrt[3]{1 + e \cdot \cos v}}
double f(double e, double v) {
        double r21360 = e;
        double r21361 = v;
        double r21362 = sin(r21361);
        double r21363 = r21360 * r21362;
        double r21364 = 1.0;
        double r21365 = cos(r21361);
        double r21366 = r21360 * r21365;
        double r21367 = r21364 + r21366;
        double r21368 = r21363 / r21367;
        return r21368;
}
↓
double f(double e, double v) {
        double r21369 = v;
        double r21370 = sin(r21369);
        double r21371 = cos(r21369);
        double r21372 = e;
        double r21373 = 1.0;
        double r21374 = fma(r21371, r21372, r21373);
        double r21375 = cbrt(r21374);
        double r21376 = r21375 * r21375;
        double r21377 = r21376 / r21372;
        double r21378 = r21370 / r21377;
        double r21379 = r21372 * r21371;
        double r21380 = r21373 + r21379;
        double r21381 = cbrt(r21380);
        double r21382 = r21378 / r21381;
        return r21382;
}

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 add-cube-cbrt0.2

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\left(\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}\right) \cdot \sqrt[3]{1 + e \cdot \cos v}}}\]

  4. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{e \cdot \sin v}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}}{\sqrt[3]{1 + e \cdot \cos v}}}\]

  5. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{\sin v}{\frac{\sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)}}{e}}}}{\sqrt[3]{1 + e \cdot \cos v}}\]

  6. Final simplification0.3

    \[\leadsto \frac{\frac{\sin v}{\frac{\sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\cos v, e, 1\right)}}{e}}}{\sqrt[3]{1 + e \cdot \cos v}}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))