Average Error: 0.1 → 0.1
Time: 6.1s
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 r15288 = e;
        double r15289 = v;
        double r15290 = sin(r15289);
        double r15291 = r15288 * r15290;
        double r15292 = 1.0;
        double r15293 = cos(r15289);
        double r15294 = r15288 * r15293;
        double r15295 = r15292 + r15294;
        double r15296 = r15291 / r15295;
        return r15296;
}
↓
double f(double e, double v) {
        double r15297 = e;
        double r15298 = 1.0;
        double r15299 = v;
        double r15300 = cos(r15299);
        double r15301 = r15297 * r15300;
        double r15302 = r15298 + r15301;
        double r15303 = sqrt(r15302);
        double r15304 = r15297 / r15303;
        double r15305 = sin(r15299);
        double r15306 = r15305 / r15303;
        double r15307 = r15304 * r15306;
        return r15307;
}

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