Average Error: 0.1 → 0.1
Time: 20.8s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\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)
double f(double e, double v) {
        double r864339 = e;
        double r864340 = v;
        double r864341 = sin(r864340);
        double r864342 = r864339 * r864341;
        double r864343 = 1.0;
        double r864344 = cos(r864340);
        double r864345 = r864339 * r864344;
        double r864346 = r864343 + r864345;
        double r864347 = r864342 / r864346;
        return r864347;
}
↓
double f(double e, double v) {
        double r864348 = e;
        double r864349 = v;
        double r864350 = sin(r864349);
        double r864351 = r864348 * r864350;
        double r864352 = 1.0;
        double r864353 = r864352 * r864352;
        double r864354 = cos(r864349);
        double r864355 = r864348 * r864354;
        double r864356 = r864355 * r864355;
        double r864357 = r864353 - r864356;
        double r864358 = r864351 / r864357;
        double r864359 = r864352 - r864355;
        double r864360 = r864358 * r864359;
        return r864360;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0.0 e 1.0)
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))