Average Error: 0.1 → 0.1
Time: 14.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + \left(e \cdot \left(\sqrt[3]{\cos v} \cdot \sqrt[3]{\cos v}\right)\right) \cdot \sqrt[3]{\cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 + \left(e \cdot \left(\sqrt[3]{\cos v} \cdot \sqrt[3]{\cos v}\right)\right) \cdot \sqrt[3]{\cos v}}
double f(double e, double v) {
        double r22348 = e;
        double r22349 = v;
        double r22350 = sin(r22349);
        double r22351 = r22348 * r22350;
        double r22352 = 1.0;
        double r22353 = cos(r22349);
        double r22354 = r22348 * r22353;
        double r22355 = r22352 + r22354;
        double r22356 = r22351 / r22355;
        return r22356;
}
↓
double f(double e, double v) {
        double r22357 = e;
        double r22358 = v;
        double r22359 = sin(r22358);
        double r22360 = r22357 * r22359;
        double r22361 = 1.0;
        double r22362 = cos(r22358);
        double r22363 = cbrt(r22362);
        double r22364 = r22363 * r22363;
        double r22365 = r22357 * r22364;
        double r22366 = r22365 * r22363;
        double r22367 = r22361 + r22366;
        double r22368 = r22360 / r22367;
        return r22368;
}

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-cbrt0.1

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

  4. Applied associate-*r*0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019294 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))