Average Error: 30.0 → 7.6
Time: 18.0s
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 r2815455 = a;
        double r2815456 = r2815455 * r2815455;
        double r2815457 = b;
        double r2815458 = r2815457 * r2815457;
        double r2815459 = r2815456 - r2815458;
        double r2815460 = r2815459 / r2815456;
        double r2815461 = fabs(r2815460);
        double r2815462 = sqrt(r2815461);
        return r2815462;
}
↓
double f(double a, double b) {
        double r2815463 = 1.0;
        double r2815464 = b;
        double r2815465 = a;
        double r2815466 = r2815464 / r2815465;
        double r2815467 = r2815466 * r2815466;
        double r2815468 = r2815463 - r2815467;
        double r2815469 = fabs(r2815468);
        double r2815470 = sqrt(r2815469);
        return r2815470;
}

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 30.0

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

  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 2019130 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))