Average Error: 29.7 → 8.1
Time: 24.7s
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 r2594695 = a;
        double r2594696 = r2594695 * r2594695;
        double r2594697 = b;
        double r2594698 = r2594697 * r2594697;
        double r2594699 = r2594696 - r2594698;
        double r2594700 = r2594699 / r2594696;
        double r2594701 = fabs(r2594700);
        double r2594702 = sqrt(r2594701);
        return r2594702;
}
↓
double f(double a, double b) {
        double r2594703 = 1.0;
        double r2594704 = b;
        double r2594705 = a;
        double r2594706 = r2594704 / r2594705;
        double r2594707 = r2594706 * r2594706;
        double r2594708 = r2594703 - r2594707;
        double r2594709 = fabs(r2594708);
        double r2594710 = sqrt(r2594709);
        return r2594710;
}

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. Simplified8.1

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

  3. Final simplification8.1

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))