Average Error: 0.1 → 0.2
Time: 5.0s
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 r11483 = e;
        double r11484 = v;
        double r11485 = sin(r11484);
        double r11486 = r11483 * r11485;
        double r11487 = 1.0;
        double r11488 = cos(r11484);
        double r11489 = r11483 * r11488;
        double r11490 = r11487 + r11489;
        double r11491 = r11486 / r11490;
        return r11491;
}
↓
double f(double e, double v) {
        double r11492 = e;
        double r11493 = 1.0;
        double r11494 = v;
        double r11495 = cos(r11494);
        double r11496 = r11492 * r11495;
        double r11497 = r11493 + r11496;
        double r11498 = cbrt(r11497);
        double r11499 = r11498 * r11498;
        double r11500 = r11492 / r11499;
        double r11501 = sin(r11494);
        double r11502 = r11501 / r11498;
        double r11503 = r11500 * r11502;
        return r11503;
}

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