Average Error: 0.1 → 0.1
Time: 24.1s
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 r21582 = e;
        double r21583 = v;
        double r21584 = sin(r21583);
        double r21585 = r21582 * r21584;
        double r21586 = 1.0;
        double r21587 = cos(r21583);
        double r21588 = r21582 * r21587;
        double r21589 = r21586 + r21588;
        double r21590 = r21585 / r21589;
        return r21590;
}
↓
double f(double e, double v) {
        double r21591 = e;
        double r21592 = 1.0;
        double r21593 = v;
        double r21594 = cos(r21593);
        double r21595 = r21591 * r21594;
        double r21596 = r21592 + r21595;
        double r21597 = sqrt(r21596);
        double r21598 = r21591 / r21597;
        double r21599 = sin(r21593);
        double r21600 = r21599 / r21597;
        double r21601 = r21598 * r21600;
        return r21601;
}

Error

Bits error versus e

Bits error versus v

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Your Program's Arguments

Results

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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 2019199 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))