Average Error: 29.9 → 7.5
Time: 12.0s
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 r1547849 = a;
        double r1547850 = r1547849 * r1547849;
        double r1547851 = b;
        double r1547852 = r1547851 * r1547851;
        double r1547853 = r1547850 - r1547852;
        double r1547854 = r1547853 / r1547850;
        double r1547855 = fabs(r1547854);
        double r1547856 = sqrt(r1547855);
        return r1547856;
}
↓
double f(double a, double b) {
        double r1547857 = 1.0;
        double r1547858 = b;
        double r1547859 = a;
        double r1547860 = r1547858 / r1547859;
        double r1547861 = r1547860 * r1547860;
        double r1547862 = r1547857 - r1547861;
        double r1547863 = fabs(r1547862);
        double r1547864 = sqrt(r1547863);
        return r1547864;
}

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 29.9

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

  2. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))