Average Error: 29.4 → 7.1
Time: 18.4s
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 r2921650 = a;
        double r2921651 = r2921650 * r2921650;
        double r2921652 = b;
        double r2921653 = r2921652 * r2921652;
        double r2921654 = r2921651 - r2921653;
        double r2921655 = r2921654 / r2921651;
        double r2921656 = fabs(r2921655);
        double r2921657 = sqrt(r2921656);
        return r2921657;
}
↓
double f(double a, double b) {
        double r2921658 = 1.0;
        double r2921659 = b;
        double r2921660 = a;
        double r2921661 = r2921659 / r2921660;
        double r2921662 = r2921661 * r2921661;
        double r2921663 = r2921658 - r2921662;
        double r2921664 = fabs(r2921663);
        double r2921665 = sqrt(r2921664);
        return r2921665;
}

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 29.4

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

  2. Simplified7.1

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

  3. Final simplification7.1

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))