Average Error: 0.1 → 0.2
Time: 5.1s
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 r13394 = e;
        double r13395 = v;
        double r13396 = sin(r13395);
        double r13397 = r13394 * r13396;
        double r13398 = 1.0;
        double r13399 = cos(r13395);
        double r13400 = r13394 * r13399;
        double r13401 = r13398 + r13400;
        double r13402 = r13397 / r13401;
        return r13402;
}
↓
double f(double e, double v) {
        double r13403 = e;
        double r13404 = 1.0;
        double r13405 = v;
        double r13406 = cos(r13405);
        double r13407 = r13403 * r13406;
        double r13408 = r13404 + r13407;
        double r13409 = sqrt(r13408);
        double r13410 = r13403 / r13409;
        double r13411 = sin(r13405);
        double r13412 = r13411 / r13409;
        double r13413 = r13410 * r13412;
        return r13413;
}

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-frac0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020049 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))