Average Error: 0.1 → 0.1
Time: 24.1s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\sin v \cdot \sqrt{\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) + \left(1 - \cos v \cdot e\right)}}{\sqrt{\cos v \cdot e + 1}} \cdot \frac{e}{\sqrt{1 + {\left(\cos v \cdot e\right)}^{3}}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{\sin v \cdot \sqrt{\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) + \left(1 - \cos v \cdot e\right)}}{\sqrt{\cos v \cdot e + 1}} \cdot \frac{e}{\sqrt{1 + {\left(\cos v \cdot e\right)}^{3}}}
double f(double e, double v) {
        double r880903 = e;
        double r880904 = v;
        double r880905 = sin(r880904);
        double r880906 = r880903 * r880905;
        double r880907 = 1.0;
        double r880908 = cos(r880904);
        double r880909 = r880903 * r880908;
        double r880910 = r880907 + r880909;
        double r880911 = r880906 / r880910;
        return r880911;
}
↓
double f(double e, double v) {
        double r880912 = v;
        double r880913 = sin(r880912);
        double r880914 = cos(r880912);
        double r880915 = e;
        double r880916 = r880914 * r880915;
        double r880917 = r880916 * r880916;
        double r880918 = 1.0;
        double r880919 = r880918 - r880916;
        double r880920 = r880917 + r880919;
        double r880921 = sqrt(r880920);
        double r880922 = r880913 * r880921;
        double r880923 = r880916 + r880918;
        double r880924 = sqrt(r880923);
        double r880925 = r880922 / r880924;
        double r880926 = 3.0;
        double r880927 = pow(r880916, r880926);
        double r880928 = r880918 + r880927;
        double r880929 = sqrt(r880928);
        double r880930 = r880915 / r880929;
        double r880931 = r880925 * r880930;
        return r880931;
}

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-frac0.2

    \[\leadsto \color{blue}{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}}\]
  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. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019139 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))