Average Error: 0.1 → 0.1
Time: 18.5s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \left(\frac{\sin v}{1 + \left(\cos v \cdot e\right) \cdot \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right)} \cdot \left(1 + \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - \cos v \cdot e\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \left(\frac{\sin v}{1 + \left(\cos v \cdot e\right) \cdot \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right)} \cdot \left(1 + \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - \cos v \cdot e\right)\right)\right)
double f(double e, double v) {
        double r565592 = e;
        double r565593 = v;
        double r565594 = sin(r565593);
        double r565595 = r565592 * r565594;
        double r565596 = 1.0;
        double r565597 = cos(r565593);
        double r565598 = r565592 * r565597;
        double r565599 = r565596 + r565598;
        double r565600 = r565595 / r565599;
        return r565600;
}
↓
double f(double e, double v) {
        double r565601 = e;
        double r565602 = v;
        double r565603 = sin(r565602);
        double r565604 = 1.0;
        double r565605 = cos(r565602);
        double r565606 = r565605 * r565601;
        double r565607 = r565606 * r565606;
        double r565608 = r565606 * r565607;
        double r565609 = r565604 + r565608;
        double r565610 = r565603 / r565609;
        double r565611 = r565607 - r565606;
        double r565612 = r565604 + r565611;
        double r565613 = r565610 * r565612;
        double r565614 = r565601 * r565613;
        return r565614;
}

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{e}{1} \cdot \frac{\sin v}{1 + e \cdot \cos v}}\]

  5. Simplified0.1

    \[\leadsto \color{blue}{e} \cdot \frac{\sin v}{1 + e \cdot \cos v}\]
  6. Using strategy rm

  7. Applied flip3-+0.1

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

  8. Applied associate-/r/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019133 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))