Average Error: 0.1 → 0.2
Time: 5.1s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{e \cdot \sin v}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}}{\sqrt[3]{1 + e \cdot \cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\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}}
double f(double e, double v) {
        double r12607 = e;
        double r12608 = v;
        double r12609 = sin(r12608);
        double r12610 = r12607 * r12609;
        double r12611 = 1.0;
        double r12612 = cos(r12608);
        double r12613 = r12607 * r12612;
        double r12614 = r12611 + r12613;
        double r12615 = r12610 / r12614;
        return r12615;
}
↓
double f(double e, double v) {
        double r12616 = e;
        double r12617 = v;
        double r12618 = sin(r12617);
        double r12619 = r12616 * r12618;
        double r12620 = 1.0;
        double r12621 = cos(r12617);
        double r12622 = r12616 * r12621;
        double r12623 = r12620 + r12622;
        double r12624 = cbrt(r12623);
        double r12625 = r12624 * r12624;
        double r12626 = r12619 / r12625;
        double r12627 = r12626 / r12624;
        return r12627;
}

Error

Bits error versus e

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))