Average Error: 30.6 → 7.9
Time: 13.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r3027797 = a;
        double r3027798 = r3027797 * r3027797;
        double r3027799 = b;
        double r3027800 = r3027799 * r3027799;
        double r3027801 = r3027798 - r3027800;
        double r3027802 = r3027801 / r3027798;
        double r3027803 = fabs(r3027802);
        double r3027804 = sqrt(r3027803);
        return r3027804;
}
↓
double f(double a, double b) {
        double r3027805 = 1.0;
        double r3027806 = b;
        double r3027807 = a;
        double r3027808 = r3027806 / r3027807;
        double r3027809 = r3027808 * r3027808;
        double r3027810 = r3027805 - r3027809;
        double r3027811 = fabs(r3027810);
        double r3027812 = sqrt(r3027811);
        return r3027812;
}

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 30.6

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

  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 2019192 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))