Average Error: 0.1 → 0.1
Time: 5.1s
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 r11753 = e;
        double r11754 = v;
        double r11755 = sin(r11754);
        double r11756 = r11753 * r11755;
        double r11757 = 1.0;
        double r11758 = cos(r11754);
        double r11759 = r11753 * r11758;
        double r11760 = r11757 + r11759;
        double r11761 = r11756 / r11760;
        return r11761;
}
↓
double f(double e, double v) {
        double r11762 = e;
        double r11763 = v;
        double r11764 = sin(r11763);
        double r11765 = r11762 * r11764;
        double r11766 = 1.0;
        double r11767 = r11766 * r11766;
        double r11768 = cos(r11763);
        double r11769 = r11762 * r11768;
        double r11770 = r11769 * r11769;
        double r11771 = r11767 - r11770;
        double r11772 = r11765 / r11771;
        double r11773 = r11766 - r11769;
        double r11774 = r11772 * r11773;
        return r11774;
}

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 2020035 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))