Average Error: 0.1 → 0.1
Time: 21.5s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)
double f(double e, double v) {
        double r19595 = e;
        double r19596 = v;
        double r19597 = sin(r19596);
        double r19598 = r19595 * r19597;
        double r19599 = 1.0;
        double r19600 = cos(r19596);
        double r19601 = r19595 * r19600;
        double r19602 = r19599 + r19601;
        double r19603 = r19598 / r19602;
        return r19603;
}
↓
double f(double e, double v) {
        double r19604 = e;
        double r19605 = v;
        double r19606 = sin(r19605);
        double r19607 = r19604 * r19606;
        double r19608 = 1.0;
        double r19609 = r19608 * r19608;
        double r19610 = 2.0;
        double r19611 = pow(r19604, r19610);
        double r19612 = cos(r19605);
        double r19613 = pow(r19612, r19610);
        double r19614 = r19611 * r19613;
        double r19615 = r19609 - r19614;
        double r19616 = r19607 / r19615;
        double r19617 = r19604 * r19612;
        double r19618 = r19608 - r19617;
        double r19619 = r19616 * r19618;
        return r19619;
}

Error

Bits error versus e

Bits error versus v

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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

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}}} \cdot \left(1 - e \cdot \cos v\right)\]

  6. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{1 \cdot 1 - {e}^{2} \cdot {\left(\cos v\right)}^{2}} \cdot \left(1 - e \cdot \cos v\right)\]

Reproduce

herbie shell --seed 2019326 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))