Average Error: 0.1 → 0.1
Time: 4.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r11261 = e;
        double r11262 = v;
        double r11263 = sin(r11262);
        double r11264 = r11261 * r11263;
        double r11265 = 1.0;
        double r11266 = cos(r11262);
        double r11267 = r11261 * r11266;
        double r11268 = r11265 + r11267;
        double r11269 = r11264 / r11268;
        return r11269;
}
↓
double f(double e, double v) {
        double r11270 = e;
        double r11271 = v;
        double r11272 = sin(r11271);
        double r11273 = r11270 * r11272;
        double r11274 = 1.0;
        double r11275 = r11274 * r11274;
        double r11276 = cos(r11271);
        double r11277 = r11270 * r11276;
        double r11278 = r11277 * r11277;
        double r11279 = r11275 - r11278;
        double r11280 = r11273 / r11279;
        double r11281 = r11274 - r11277;
        double r11282 = r11280 * r11281;
        return r11282;
}

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 flip-+0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}{1 - e \cdot \cos v}}}\]

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)}\]

  5. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]

Reproduce

herbie shell --seed 2020034 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))