Average Error: 0.1 → 0.1
Time: 6.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}
double f(double e, double v) {
        double r17658 = e;
        double r17659 = v;
        double r17660 = sin(r17659);
        double r17661 = r17658 * r17660;
        double r17662 = 1.0;
        double r17663 = cos(r17659);
        double r17664 = r17658 * r17663;
        double r17665 = r17662 + r17664;
        double r17666 = r17661 / r17665;
        return r17666;
}
↓
double f(double e, double v) {
        double r17667 = e;
        double r17668 = 1.0;
        double r17669 = v;
        double r17670 = cos(r17669);
        double r17671 = r17667 * r17670;
        double r17672 = r17668 + r17671;
        double r17673 = sqrt(r17672);
        double r17674 = r17667 / r17673;
        double r17675 = sin(r17669);
        double r17676 = r17675 / r17673;
        double r17677 = r17674 * r17676;
        return r17677;
}

Error

Bits error versus e

Bits error versus v

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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-sqr-sqrt0.2

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

  4. Applied times-frac0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019356 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))