Average Error: 0.1 → 0.2
Time: 5.1s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}
double f(double e, double v) {
        double r12404 = e;
        double r12405 = v;
        double r12406 = sin(r12405);
        double r12407 = r12404 * r12406;
        double r12408 = 1.0;
        double r12409 = cos(r12405);
        double r12410 = r12404 * r12409;
        double r12411 = r12408 + r12410;
        double r12412 = r12407 / r12411;
        return r12412;
}
↓
double f(double e, double v) {
        double r12413 = e;
        double r12414 = 1.0;
        double r12415 = v;
        double r12416 = cos(r12415);
        double r12417 = r12413 * r12416;
        double r12418 = r12414 + r12417;
        double r12419 = cbrt(r12418);
        double r12420 = r12419 * r12419;
        double r12421 = r12413 / r12420;
        double r12422 = sin(r12415);
        double r12423 = r12422 / r12419;
        double r12424 = r12421 * r12423;
        return r12424;
}

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 times-frac0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))