Average Error: 0.1 → 0.1
Time: 11.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r11277 = e;
        double r11278 = v;
        double r11279 = sin(r11278);
        double r11280 = r11277 * r11279;
        double r11281 = 1.0;
        double r11282 = cos(r11278);
        double r11283 = r11277 * r11282;
        double r11284 = r11281 + r11283;
        double r11285 = r11280 / r11284;
        return r11285;
}
↓
double f(double e, double v) {
        double r11286 = e;
        double r11287 = v;
        double r11288 = sin(r11287);
        double r11289 = r11286 * r11288;
        double r11290 = 1.0;
        double r11291 = r11290 * r11290;
        double r11292 = r11286 * r11286;
        double r11293 = cos(r11287);
        double r11294 = 2.0;
        double r11295 = pow(r11293, r11294);
        double r11296 = r11292 * r11295;
        double r11297 = r11291 - r11296;
        double r11298 = r11289 / r11297;
        double r11299 = r11286 * r11293;
        double r11300 = r11290 - r11299;
        double r11301 = r11298 * r11300;
        return r11301;
}

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 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. Simplified0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}}} \cdot \left(1 - e \cdot \cos v\right)\]

  6. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))