Average Error: 29.2 → 7.5
Time: 19.4s
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 r2973666 = a;
        double r2973667 = r2973666 * r2973666;
        double r2973668 = b;
        double r2973669 = r2973668 * r2973668;
        double r2973670 = r2973667 - r2973669;
        double r2973671 = r2973670 / r2973667;
        double r2973672 = fabs(r2973671);
        double r2973673 = sqrt(r2973672);
        return r2973673;
}
↓
double f(double a, double b) {
        double r2973674 = 1.0;
        double r2973675 = b;
        double r2973676 = a;
        double r2973677 = r2973675 / r2973676;
        double r2973678 = r2973677 * r2973677;
        double r2973679 = r2973674 - r2973678;
        double r2973680 = fabs(r2973679);
        double r2973681 = sqrt(r2973680);
        return r2973681;
}

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 29.2

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

  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 2019158 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))