Average Error: 0.1 → 0.1
Time: 18.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\left(\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - 1 \cdot \left(\cos v \cdot e\right)\right) + 1 \cdot 1\right) \cdot \frac{\sin v}{\left(\cos v \cdot e\right) \cdot \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) + 1 \cdot \left(1 \cdot 1\right)}\right) \cdot e\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\left(\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - 1 \cdot \left(\cos v \cdot e\right)\right) + 1 \cdot 1\right) \cdot \frac{\sin v}{\left(\cos v \cdot e\right) \cdot \left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right) + 1 \cdot \left(1 \cdot 1\right)}\right) \cdot e
double f(double e, double v) {
        double r1200323 = e;
        double r1200324 = v;
        double r1200325 = sin(r1200324);
        double r1200326 = r1200323 * r1200325;
        double r1200327 = 1.0;
        double r1200328 = cos(r1200324);
        double r1200329 = r1200323 * r1200328;
        double r1200330 = r1200327 + r1200329;
        double r1200331 = r1200326 / r1200330;
        return r1200331;
}
↓
double f(double e, double v) {
        double r1200332 = v;
        double r1200333 = cos(r1200332);
        double r1200334 = e;
        double r1200335 = r1200333 * r1200334;
        double r1200336 = r1200335 * r1200335;
        double r1200337 = 1.0;
        double r1200338 = r1200337 * r1200335;
        double r1200339 = r1200336 - r1200338;
        double r1200340 = r1200337 * r1200337;
        double r1200341 = r1200339 + r1200340;
        double r1200342 = sin(r1200332);
        double r1200343 = r1200335 * r1200336;
        double r1200344 = r1200337 * r1200340;
        double r1200345 = r1200343 + r1200344;
        double r1200346 = r1200342 / r1200345;
        double r1200347 = r1200341 * r1200346;
        double r1200348 = r1200347 * r1200334;
        return r1200348;
}

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

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))