Average Error: 0.1 → 0.1
Time: 4.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}
double f(double e, double v) {
        double r12391 = e;
        double r12392 = v;
        double r12393 = sin(r12392);
        double r12394 = r12391 * r12393;
        double r12395 = 1.0;
        double r12396 = cos(r12392);
        double r12397 = r12391 * r12396;
        double r12398 = r12395 + r12397;
        double r12399 = r12394 / r12398;
        return r12399;
}
↓
double f(double e, double v) {
        double r12400 = e;
        double r12401 = 1.0;
        double r12402 = v;
        double r12403 = cos(r12402);
        double r12404 = r12400 * r12403;
        double r12405 = r12401 + r12404;
        double r12406 = sqrt(r12405);
        double r12407 = r12400 / r12406;
        double r12408 = sin(r12402);
        double r12409 = 3.0;
        double r12410 = pow(r12401, r12409);
        double r12411 = pow(r12404, r12409);
        double r12412 = r12410 + r12411;
        double r12413 = sqrt(r12412);
        double r12414 = r12408 / r12413;
        double r12415 = r12407 * r12414;
        double r12416 = r12401 * r12401;
        double r12417 = r12404 * r12404;
        double r12418 = r12401 * r12404;
        double r12419 = r12417 - r12418;
        double r12420 = r12416 + r12419;
        double r12421 = sqrt(r12420);
        double r12422 = r12415 * r12421;
        return r12422;
}

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. Using strategy rm

  6. Applied flip3-+0.1

    \[\leadsto \frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}}\]

  7. Applied sqrt-div0.1

    \[\leadsto \frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\color{blue}{\frac{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}}\]

  8. Applied associate-/r/0.1

    \[\leadsto \frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \color{blue}{\left(\frac{\sin v}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}} \cdot \sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}\right)}\]

  9. Applied associate-*r*0.1

    \[\leadsto \color{blue}{\left(\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}\]

  10. Final simplification0.1

    \[\leadsto \left(\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))