Average Error: 0.1 → 0.1
Time: 17.9s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \left(\frac{\sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
e \cdot \left(\frac{\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)\right)
double f(double e, double v) {
        double r1555315 = e;
        double r1555316 = v;
        double r1555317 = sin(r1555316);
        double r1555318 = r1555315 * r1555317;
        double r1555319 = 1.0;
        double r1555320 = cos(r1555316);
        double r1555321 = r1555315 * r1555320;
        double r1555322 = r1555319 + r1555321;
        double r1555323 = r1555318 / r1555322;
        return r1555323;
}
↓
double f(double e, double v) {
        double r1555324 = e;
        double r1555325 = v;
        double r1555326 = sin(r1555325);
        double r1555327 = 1.0;
        double r1555328 = r1555327 * r1555327;
        double r1555329 = cos(r1555325);
        double r1555330 = r1555324 * r1555329;
        double r1555331 = r1555330 * r1555330;
        double r1555332 = r1555328 - r1555331;
        double r1555333 = r1555326 / r1555332;
        double r1555334 = r1555327 - r1555330;
        double r1555335 = r1555333 * r1555334;
        double r1555336 = r1555324 * r1555335;
        return r1555336;
}

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 flip-+0.1

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

  8. Applied associate-/r/0.1

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

  9. Final simplification0.1

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

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)))))