Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e}{\sqrt{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}} \cdot \left(\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)} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}\right)
double f(double e, double v) {
        double r13338 = e;
        double r13339 = v;
        double r13340 = sin(r13339);
        double r13341 = r13338 * r13340;
        double r13342 = 1.0;
        double r13343 = cos(r13339);
        double r13344 = r13338 * r13343;
        double r13345 = r13342 + r13344;
        double r13346 = r13341 / r13345;
        return r13346;
}
↓
double f(double e, double v) {
        double r13347 = e;
        double r13348 = 1.0;
        double r13349 = 3.0;
        double r13350 = pow(r13348, r13349);
        double r13351 = v;
        double r13352 = cos(r13351);
        double r13353 = r13347 * r13352;
        double r13354 = pow(r13353, r13349);
        double r13355 = r13350 + r13354;
        double r13356 = sqrt(r13355);
        double r13357 = r13347 / r13356;
        double r13358 = r13348 * r13348;
        double r13359 = r13353 * r13353;
        double r13360 = r13348 * r13353;
        double r13361 = r13359 - r13360;
        double r13362 = r13358 + r13361;
        double r13363 = sqrt(r13362);
        double r13364 = sin(r13351);
        double r13365 = r13348 + r13353;
        double r13366 = sqrt(r13365);
        double r13367 = r13364 / r13366;
        double r13368 = r13363 * r13367;
        double r13369 = r13357 * r13368;
        return r13369;
}

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

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

  7. Applied sqrt-div0.1

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

  8. Applied associate-/r/0.1

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

  9. Applied associate-*l*0.1

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

  10. Final simplification0.1

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

Reproduce

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