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 r18474 = e;
        double r18475 = v;
        double r18476 = sin(r18475);
        double r18477 = r18474 * r18476;
        double r18478 = 1.0;
        double r18479 = cos(r18475);
        double r18480 = r18474 * r18479;
        double r18481 = r18478 + r18480;
        double r18482 = r18477 / r18481;
        return r18482;
}
↓
double f(double e, double v) {
        double r18483 = e;
        double r18484 = 1.0;
        double r18485 = v;
        double r18486 = cos(r18485);
        double r18487 = r18483 * r18486;
        double r18488 = r18484 + r18487;
        double r18489 = sqrt(r18488);
        double r18490 = r18483 / r18489;
        double r18491 = sin(r18485);
        double r18492 = r18491 / r18489;
        double r18493 = r18490 * r18492;
        return r18493;
}

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-sqrt0.1

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

  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 2020027 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))