Average Error: 0.1 → 0.1
Time: 18.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(-\frac{e \cdot \sin v}{-\left(1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}\right)}\right) \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(-\frac{e \cdot \sin v}{-\left(1 \cdot 1 - \left(e \cdot e\right) \cdot {\left(\cos v\right)}^{2}\right)}\right) \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r20872 = e;
        double r20873 = v;
        double r20874 = sin(r20873);
        double r20875 = r20872 * r20874;
        double r20876 = 1.0;
        double r20877 = cos(r20873);
        double r20878 = r20872 * r20877;
        double r20879 = r20876 + r20878;
        double r20880 = r20875 / r20879;
        return r20880;
}
↓
double f(double e, double v) {
        double r20881 = e;
        double r20882 = v;
        double r20883 = sin(r20882);
        double r20884 = r20881 * r20883;
        double r20885 = 1.0;
        double r20886 = r20885 * r20885;
        double r20887 = r20881 * r20881;
        double r20888 = cos(r20882);
        double r20889 = 2.0;
        double r20890 = pow(r20888, r20889);
        double r20891 = r20887 * r20890;
        double r20892 = r20886 - r20891;
        double r20893 = -r20892;
        double r20894 = r20884 / r20893;
        double r20895 = -r20894;
        double r20896 = r20881 * r20888;
        double r20897 = r20885 - r20896;
        double r20898 = r20895 * r20897;
        return r20898;
}

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 frac-2neg0.1

    \[\leadsto \color{blue}{\frac{-e \cdot \sin v}{-\left(1 + e \cdot \cos v\right)}}\]
  4. Using strategy rm

  5. Applied flip-+0.1

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

  6. Applied distribute-neg-frac0.1

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

  7. Applied associate-/r/0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))