Average Error: 29.7 → 7.4
Time: 3.6s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
double f(double a, double b) {
        double r1484683 = a;
        double r1484684 = r1484683 * r1484683;
        double r1484685 = b;
        double r1484686 = r1484685 * r1484685;
        double r1484687 = r1484684 - r1484686;
        double r1484688 = r1484687 / r1484684;
        double r1484689 = fabs(r1484688);
        double r1484690 = sqrt(r1484689);
        return r1484690;
}
↓
double f(double a, double b) {
        double r1484691 = 1.0;
        double r1484692 = b;
        double r1484693 = a;
        double r1484694 = r1484692 / r1484693;
        double r1484695 = r1484694 * r1484694;
        double r1484696 = r1484691 - r1484695;
        double r1484697 = fabs(r1484696);
        double r1484698 = sqrt(r1484697);
        return r1484698;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.7

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified7.4

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

  3. Final simplification7.4

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))