Average Error: 0.1 → 0.1
Time: 18.6s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{{1}^{3} + {\left(\cos v \cdot e\right)}^{3}} \cdot \left(\left(\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right) - 1 \cdot \left(\cos v \cdot e\right)\right) + 1 \cdot 1\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{{1}^{3} + {\left(\cos v \cdot e\right)}^{3}} \cdot \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)
double f(double e, double v) {
        double r22332 = e;
        double r22333 = v;
        double r22334 = sin(r22333);
        double r22335 = r22332 * r22334;
        double r22336 = 1.0;
        double r22337 = cos(r22333);
        double r22338 = r22332 * r22337;
        double r22339 = r22336 + r22338;
        double r22340 = r22335 / r22339;
        return r22340;
}
↓
double f(double e, double v) {
        double r22341 = e;
        double r22342 = v;
        double r22343 = sin(r22342);
        double r22344 = r22341 * r22343;
        double r22345 = 1.0;
        double r22346 = 3.0;
        double r22347 = pow(r22345, r22346);
        double r22348 = cos(r22342);
        double r22349 = r22348 * r22341;
        double r22350 = pow(r22349, r22346);
        double r22351 = r22347 + r22350;
        double r22352 = r22344 / r22351;
        double r22353 = r22349 * r22349;
        double r22354 = r22345 * r22349;
        double r22355 = r22353 - r22354;
        double r22356 = r22345 * r22345;
        double r22357 = r22355 + r22356;
        double r22358 = r22352 * r22357;
        return r22358;
}

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

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))