Average Error: 29.5 → 7.8
Time: 58.2s
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 r9999364 = a;
        double r9999365 = r9999364 * r9999364;
        double r9999366 = b;
        double r9999367 = r9999366 * r9999366;
        double r9999368 = r9999365 - r9999367;
        double r9999369 = r9999368 / r9999365;
        double r9999370 = fabs(r9999369);
        double r9999371 = sqrt(r9999370);
        return r9999371;
}
↓
double f(double a, double b) {
        double r9999372 = 1.0;
        double r9999373 = b;
        double r9999374 = a;
        double r9999375 = r9999373 / r9999374;
        double r9999376 = r9999375 * r9999375;
        double r9999377 = r9999372 - r9999376;
        double r9999378 = fabs(r9999377);
        double r9999379 = sqrt(r9999378);
        return r9999379;
}

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 29.5

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

  2. Simplified7.8

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

  3. Final simplification7.8

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))