Average Error: 0.1 → 0.1
Time: 18.8s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(1 + \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - \cos v \cdot e\right)\right) \cdot \frac{e \cdot \sin v}{\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) \cdot \left(\cos v \cdot e\right) + 1}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(1 + \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - \cos v \cdot e\right)\right) \cdot \frac{e \cdot \sin v}{\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) \cdot \left(\cos v \cdot e\right) + 1}
double f(double e, double v) {
        double r965010 = e;
        double r965011 = v;
        double r965012 = sin(r965011);
        double r965013 = r965010 * r965012;
        double r965014 = 1.0;
        double r965015 = cos(r965011);
        double r965016 = r965010 * r965015;
        double r965017 = r965014 + r965016;
        double r965018 = r965013 / r965017;
        return r965018;
}
↓
double f(double e, double v) {
        double r965019 = 1.0;
        double r965020 = v;
        double r965021 = cos(r965020);
        double r965022 = e;
        double r965023 = r965021 * r965022;
        double r965024 = r965023 * r965023;
        double r965025 = r965024 - r965023;
        double r965026 = r965019 + r965025;
        double r965027 = sin(r965020);
        double r965028 = r965022 * r965027;
        double r965029 = r965024 * r965023;
        double r965030 = r965029 + r965019;
        double r965031 = r965028 / r965030;
        double r965032 = r965026 * r965031;
        return r965032;
}

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. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))