Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
\[0.0 \le e \le 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
↓
\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)
double f(double e, double v) {
        double r12570 = e;
        double r12571 = v;
        double r12572 = sin(r12571);
        double r12573 = r12570 * r12572;
        double r12574 = 1.0;
        double r12575 = cos(r12571);
        double r12576 = r12570 * r12575;
        double r12577 = r12574 + r12576;
        double r12578 = r12573 / r12577;
        return r12578;
}
↓
double f(double e, double v) {
        double r12579 = e;
        double r12580 = v;
        double r12581 = sin(r12580);
        double r12582 = r12579 * r12581;
        double r12583 = 1.0;
        double r12584 = 3.0;
        double r12585 = pow(r12583, r12584);
        double r12586 = cos(r12580);
        double r12587 = r12579 * r12586;
        double r12588 = pow(r12587, r12584);
        double r12589 = r12585 + r12588;
        double r12590 = r12582 / r12589;
        double r12591 = r12583 * r12583;
        double r12592 = r12587 * r12587;
        double r12593 = r12583 * r12587;
        double r12594 = r12592 - r12593;
        double r12595 = r12591 + r12594;
        double r12596 = r12590 * r12595;
        return r12596;
}

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 flip3-+0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}}{1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)}}}\]

  4. Applied associate-/r/0.1

    \[\leadsto \color{blue}{\frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)}\]

  5. Final simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{{1}^{3} + {\left(e \cdot \cos v\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right) - 1 \cdot \left(e \cdot \cos v\right)\right)\right)\]

Reproduce

herbie shell --seed 2020059 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (<= 0.0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))