Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r11805 = e;
        double r11806 = v;
        double r11807 = sin(r11806);
        double r11808 = r11805 * r11807;
        double r11809 = 1.0;
        double r11810 = cos(r11806);
        double r11811 = r11805 * r11810;
        double r11812 = r11809 + r11811;
        double r11813 = r11808 / r11812;
        return r11813;
}
↓
double f(double e, double v) {
        double r11814 = e;
        double r11815 = v;
        double r11816 = sin(r11815);
        double r11817 = r11814 * r11816;
        double r11818 = 1.0;
        double r11819 = r11818 * r11818;
        double r11820 = cos(r11815);
        double r11821 = r11814 * r11820;
        double r11822 = r11821 * r11821;
        double r11823 = r11819 - r11822;
        double r11824 = r11817 / r11823;
        double r11825 = r11818 - r11821;
        double r11826 = r11824 * r11825;
        return r11826;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))