Average Error: 31.2 → 8.3
Time: 13.5s
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 r1998706 = a;
        double r1998707 = r1998706 * r1998706;
        double r1998708 = b;
        double r1998709 = r1998708 * r1998708;
        double r1998710 = r1998707 - r1998709;
        double r1998711 = r1998710 / r1998707;
        double r1998712 = fabs(r1998711);
        double r1998713 = sqrt(r1998712);
        return r1998713;
}
↓
double f(double a, double b) {
        double r1998714 = 1.0;
        double r1998715 = b;
        double r1998716 = a;
        double r1998717 = r1998715 / r1998716;
        double r1998718 = r1998717 * r1998717;
        double r1998719 = r1998714 - r1998718;
        double r1998720 = fabs(r1998719);
        double r1998721 = sqrt(r1998720);
        return r1998721;
}

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 31.2

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

  2. Simplified8.3

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

  3. Final simplification8.3

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

Reproduce

herbie shell --seed 2019172 +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)))))