Average Error: 0.1 → 0.1
Time: 19.4s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\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)
double f(double e, double v) {
        double r21416 = e;
        double r21417 = v;
        double r21418 = sin(r21417);
        double r21419 = r21416 * r21418;
        double r21420 = 1.0;
        double r21421 = cos(r21417);
        double r21422 = r21416 * r21421;
        double r21423 = r21420 + r21422;
        double r21424 = r21419 / r21423;
        return r21424;
}
↓
double f(double e, double v) {
        double r21425 = e;
        double r21426 = v;
        double r21427 = sin(r21426);
        double r21428 = r21425 * r21427;
        double r21429 = 1.0;
        double r21430 = 3.0;
        double r21431 = pow(r21429, r21430);
        double r21432 = cos(r21426);
        double r21433 = r21425 * r21432;
        double r21434 = pow(r21433, r21430);
        double r21435 = r21431 + r21434;
        double r21436 = r21428 / r21435;
        double r21437 = r21429 * r21429;
        double r21438 = r21433 * r21433;
        double r21439 = r21429 * r21433;
        double r21440 = r21438 - r21439;
        double r21441 = r21437 + r21440;
        double r21442 = r21436 * r21441;
        return r21442;
}

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

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))