Average Error: 29.1 → 7.5
Time: 10.7s
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 r1788712 = a;
        double r1788713 = r1788712 * r1788712;
        double r1788714 = b;
        double r1788715 = r1788714 * r1788714;
        double r1788716 = r1788713 - r1788715;
        double r1788717 = r1788716 / r1788713;
        double r1788718 = fabs(r1788717);
        double r1788719 = sqrt(r1788718);
        return r1788719;
}
↓
double f(double a, double b) {
        double r1788720 = 1.0;
        double r1788721 = b;
        double r1788722 = a;
        double r1788723 = r1788721 / r1788722;
        double r1788724 = r1788723 * r1788723;
        double r1788725 = r1788720 - r1788724;
        double r1788726 = fabs(r1788725);
        double r1788727 = sqrt(r1788726);
        return r1788727;
}

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 29.1

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

  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 2019129 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))