Average Error: 29.3 → 7.9
Time: 12.3s
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 r803613 = a;
        double r803614 = r803613 * r803613;
        double r803615 = b;
        double r803616 = r803615 * r803615;
        double r803617 = r803614 - r803616;
        double r803618 = r803617 / r803614;
        double r803619 = fabs(r803618);
        double r803620 = sqrt(r803619);
        return r803620;
}
↓
double f(double a, double b) {
        double r803621 = 1.0;
        double r803622 = b;
        double r803623 = a;
        double r803624 = r803622 / r803623;
        double r803625 = r803624 * r803624;
        double r803626 = r803621 - r803625;
        double r803627 = fabs(r803626);
        double r803628 = sqrt(r803627);
        return r803628;
}

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 29.3

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

  2. Simplified7.9

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

  3. Final simplification7.9

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))