Average Error: 0.1 → 0.1
Time: 12.3s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r11732 = e;
        double r11733 = v;
        double r11734 = sin(r11733);
        double r11735 = r11732 * r11734;
        double r11736 = 1.0;
        double r11737 = cos(r11733);
        double r11738 = r11732 * r11737;
        double r11739 = r11736 + r11738;
        double r11740 = r11735 / r11739;
        return r11740;
}
↓
double f(double e, double v) {
        double r11741 = e;
        double r11742 = v;
        double r11743 = sin(r11742);
        double r11744 = r11741 * r11743;
        double r11745 = 1.0;
        double r11746 = r11745 * r11745;
        double r11747 = 2.0;
        double r11748 = pow(r11741, r11747);
        double r11749 = cos(r11742);
        double r11750 = pow(r11749, r11747);
        double r11751 = r11748 * r11750;
        double r11752 = r11746 - r11751;
        double r11753 = r11744 / r11752;
        double r11754 = r11741 * r11749;
        double r11755 = r11745 - r11754;
        double r11756 = r11753 * r11755;
        return r11756;
}

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. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019351 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))