Average Error: 0.1 → 0.1
Time: 32.7s
Precision: 64
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{1 + \left(\sqrt[3]{\cos v \cdot e} \cdot \sqrt[3]{\cos v \cdot e}\right) \cdot \sqrt[3]{\cos v \cdot e}}\]
double f(double e, double v) {
        double r1234486 = e;
        double r1234487 = v;
        double r1234488 = sin(r1234487);
        double r1234489 = r1234486 * r1234488;
        double r1234490 = 1.0;
        double r1234491 = cos(r1234487);
        double r1234492 = r1234486 * r1234491;
        double r1234493 = r1234490 + r1234492;
        double r1234494 = r1234489 / r1234493;
        return r1234494;
}
↓
double f(double e, double v) {
        double r1234495 = e;
        double r1234496 = v;
        double r1234497 = sin(r1234496);
        double r1234498 = 1.0;
        double r1234499 = cos(r1234496);
        double r1234500 = r1234499 * r1234495;
        double r1234501 = cbrt(r1234500);
        double r1234502 = r1234501 * r1234501;
        double r1234503 = r1234502 * r1234501;
        double r1234504 = r1234498 + r1234503;
        double r1234505 = r1234497 / r1234504;
        double r1234506 = r1234495 * r1234505;
        return r1234506;
}


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

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. Using strategy rm

  7. Applied add-cube-cbrt0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019102 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))