Average Error: 29.3 → 7.5
Time: 12.6s
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 r2402614 = a;
        double r2402615 = r2402614 * r2402614;
        double r2402616 = b;
        double r2402617 = r2402616 * r2402616;
        double r2402618 = r2402615 - r2402617;
        double r2402619 = r2402618 / r2402615;
        double r2402620 = fabs(r2402619);
        double r2402621 = sqrt(r2402620);
        return r2402621;
}
↓
double f(double a, double b) {
        double r2402622 = 1.0;
        double r2402623 = b;
        double r2402624 = a;
        double r2402625 = r2402623 / r2402624;
        double r2402626 = r2402625 * r2402625;
        double r2402627 = r2402622 - r2402626;
        double r2402628 = fabs(r2402627);
        double r2402629 = sqrt(r2402628);
        return r2402629;
}

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. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))