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 r19728 = e;
        double r19729 = v;
        double r19730 = sin(r19729);
        double r19731 = r19728 * r19730;
        double r19732 = 1.0;
        double r19733 = cos(r19729);
        double r19734 = r19728 * r19733;
        double r19735 = r19732 + r19734;
        double r19736 = r19731 / r19735;
        return r19736;
}
↓
double f(double e, double v) {
        double r19737 = e;
        double r19738 = v;
        double r19739 = sin(r19738);
        double r19740 = r19737 * r19739;
        double r19741 = 1.0;
        double r19742 = r19741 * r19741;
        double r19743 = 2.0;
        double r19744 = pow(r19737, r19743);
        double r19745 = cos(r19738);
        double r19746 = pow(r19745, r19743);
        double r19747 = r19744 * r19746;
        double r19748 = r19742 - r19747;
        double r19749 = r19740 / r19748;
        double r19750 = r19737 * r19745;
        double r19751 = r19741 - r19750;
        double r19752 = r19749 * r19751;
        return r19752;
}

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

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

  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)))))