Average Error: 0.1 → 0.1
Time: 18.3s
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 r19465 = e;
        double r19466 = v;
        double r19467 = sin(r19466);
        double r19468 = r19465 * r19467;
        double r19469 = 1.0;
        double r19470 = cos(r19466);
        double r19471 = r19465 * r19470;
        double r19472 = r19469 + r19471;
        double r19473 = r19468 / r19472;
        return r19473;
}
↓
double f(double e, double v) {
        double r19474 = e;
        double r19475 = v;
        double r19476 = sin(r19475);
        double r19477 = r19474 * r19476;
        double r19478 = 1.0;
        double r19479 = 3.0;
        double r19480 = pow(r19478, r19479);
        double r19481 = cos(r19475);
        double r19482 = r19481 * r19474;
        double r19483 = pow(r19482, r19479);
        double r19484 = r19480 + r19483;
        double r19485 = r19477 / r19484;
        double r19486 = r19482 * r19482;
        double r19487 = r19478 * r19482;
        double r19488 = r19486 - r19487;
        double r19489 = r19478 * r19478;
        double r19490 = r19488 + r19489;
        double r19491 = r19485 * r19490;
        return r19491;
}

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)))))