Average Error: 0.1 → 0.1
Time: 4.6s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r10449 = e;
        double r10450 = v;
        double r10451 = sin(r10450);
        double r10452 = r10449 * r10451;
        double r10453 = 1.0;
        double r10454 = cos(r10450);
        double r10455 = r10449 * r10454;
        double r10456 = r10453 + r10455;
        double r10457 = r10452 / r10456;
        return r10457;
}
↓
double f(double e, double v) {
        double r10458 = e;
        double r10459 = v;
        double r10460 = sin(r10459);
        double r10461 = r10458 * r10460;
        double r10462 = 1.0;
        double r10463 = r10462 * r10462;
        double r10464 = cos(r10459);
        double r10465 = r10458 * r10464;
        double r10466 = r10465 * r10465;
        double r10467 = r10463 - r10466;
        double r10468 = r10461 / r10467;
        double r10469 = r10462 - r10465;
        double r10470 = r10468 * r10469;
        return r10470;
}

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 flip-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))