Average Error: 0.1 → 0.1
Time: 5.1s
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 r12371 = e;
        double r12372 = v;
        double r12373 = sin(r12372);
        double r12374 = r12371 * r12373;
        double r12375 = 1.0;
        double r12376 = cos(r12372);
        double r12377 = r12371 * r12376;
        double r12378 = r12375 + r12377;
        double r12379 = r12374 / r12378;
        return r12379;
}
↓
double f(double e, double v) {
        double r12380 = e;
        double r12381 = v;
        double r12382 = sin(r12381);
        double r12383 = r12380 * r12382;
        double r12384 = 1.0;
        double r12385 = r12384 * r12384;
        double r12386 = cos(r12381);
        double r12387 = r12380 * r12386;
        double r12388 = r12387 * r12387;
        double r12389 = r12385 - r12388;
        double r12390 = r12383 / r12389;
        double r12391 = r12384 - r12387;
        double r12392 = r12390 * r12391;
        return r12392;
}

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 2019352 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))