Average Error: 0.1 → 0.1
Time: 15.2s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v - 1\right) + 1 \cdot 1\right) \cdot \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v - 1\right) + 1 \cdot 1\right) \cdot \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}
double f(double e, double v) {
        double r19514 = e;
        double r19515 = v;
        double r19516 = sin(r19515);
        double r19517 = r19514 * r19516;
        double r19518 = 1.0;
        double r19519 = cos(r19515);
        double r19520 = r19514 * r19519;
        double r19521 = r19518 + r19520;
        double r19522 = r19517 / r19521;
        return r19522;
}
↓
double f(double e, double v) {
        double r19523 = e;
        double r19524 = v;
        double r19525 = cos(r19524);
        double r19526 = r19523 * r19525;
        double r19527 = 1.0;
        double r19528 = r19526 - r19527;
        double r19529 = r19526 * r19528;
        double r19530 = r19527 * r19527;
        double r19531 = r19529 + r19530;
        double r19532 = sin(r19524);
        double r19533 = r19523 * r19532;
        double r19534 = 3.0;
        double r19535 = pow(r19527, r19534);
        double r19536 = pow(r19526, r19534);
        double r19537 = r19535 + r19536;
        double r19538 = r19533 / r19537;
        double r19539 = r19531 * r19538;
        return r19539;
}

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

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

Reproduce

herbie shell --seed 2019298 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))