Average Error: 0.1 → 0.1
Time: 19.3s
Precision: 64
\[0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 - \left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)} \cdot \left(1 - \cos v \cdot e\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 - \left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)} \cdot \left(1 - \cos v \cdot e\right)
double f(double e, double v) {
        double r798356 = e;
        double r798357 = v;
        double r798358 = sin(r798357);
        double r798359 = r798356 * r798358;
        double r798360 = 1.0;
        double r798361 = cos(r798357);
        double r798362 = r798356 * r798361;
        double r798363 = r798360 + r798362;
        double r798364 = r798359 / r798363;
        return r798364;
}
↓
double f(double e, double v) {
        double r798365 = e;
        double r798366 = v;
        double r798367 = sin(r798366);
        double r798368 = r798365 * r798367;
        double r798369 = 1.0;
        double r798370 = cos(r798366);
        double r798371 = r798370 * r798365;
        double r798372 = r798371 * r798371;
        double r798373 = r798369 - r798372;
        double r798374 = r798368 / r798373;
        double r798375 = r798369 - r798371;
        double r798376 = r798374 * r798375;
        return r798376;
}

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 flip-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))