Average Error: 0.1 → 0.1
Time: 4.7s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(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)
double f(double e, double v) {
        double r11809 = e;
        double r11810 = v;
        double r11811 = sin(r11810);
        double r11812 = r11809 * r11811;
        double r11813 = 1.0;
        double r11814 = cos(r11810);
        double r11815 = r11809 * r11814;
        double r11816 = r11813 + r11815;
        double r11817 = r11812 / r11816;
        return r11817;
}
↓
double f(double e, double v) {
        double r11818 = e;
        double r11819 = v;
        double r11820 = sin(r11819);
        double r11821 = r11818 * r11820;
        double r11822 = 1.0;
        double r11823 = 3.0;
        double r11824 = pow(r11822, r11823);
        double r11825 = cos(r11819);
        double r11826 = r11818 * r11825;
        double r11827 = pow(r11826, r11823);
        double r11828 = r11824 + r11827;
        double r11829 = r11821 / r11828;
        double r11830 = r11822 * r11822;
        double r11831 = r11826 * r11826;
        double r11832 = r11822 * r11826;
        double r11833 = r11831 - r11832;
        double r11834 = r11830 + r11833;
        double r11835 = r11829 * r11834;
        return r11835;
}

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 flip3-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020065 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))