Average Error: 29.3 → 7.6
Time: 24.2s
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 r5740663 = a;
        double r5740664 = r5740663 * r5740663;
        double r5740665 = b;
        double r5740666 = r5740665 * r5740665;
        double r5740667 = r5740664 - r5740666;
        double r5740668 = r5740667 / r5740664;
        double r5740669 = fabs(r5740668);
        double r5740670 = sqrt(r5740669);
        return r5740670;
}
↓
double f(double a, double b) {
        double r5740671 = 1.0;
        double r5740672 = b;
        double r5740673 = a;
        double r5740674 = r5740672 / r5740673;
        double r5740675 = r5740674 * r5740674;
        double r5740676 = r5740671 - r5740675;
        double r5740677 = fabs(r5740676);
        double r5740678 = sqrt(r5740677);
        return r5740678;
}

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. Simplified7.6

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

  3. Final simplification7.6

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))