Average Error: 0.1 → 0.1
Time: 15.0s
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 r19396 = e;
        double r19397 = v;
        double r19398 = sin(r19397);
        double r19399 = r19396 * r19398;
        double r19400 = 1.0;
        double r19401 = cos(r19397);
        double r19402 = r19396 * r19401;
        double r19403 = r19400 + r19402;
        double r19404 = r19399 / r19403;
        return r19404;
}
↓
double f(double e, double v) {
        double r19405 = e;
        double r19406 = v;
        double r19407 = sin(r19406);
        double r19408 = r19405 * r19407;
        double r19409 = 1.0;
        double r19410 = r19409 * r19409;
        double r19411 = r19405 * r19405;
        double r19412 = cos(r19406);
        double r19413 = 2.0;
        double r19414 = pow(r19412, r19413);
        double r19415 = r19411 * r19414;
        double r19416 = r19410 - r19415;
        double r19417 = r19408 / r19416;
        double r19418 = r19405 * r19412;
        double r19419 = r19409 - r19418;
        double r19420 = r19417 * r19419;
        return r19420;
}

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. 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 2019208 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))