Average Error: 29.8 → 7.7
Time: 35.2s
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 r2876550 = a;
        double r2876551 = r2876550 * r2876550;
        double r2876552 = b;
        double r2876553 = r2876552 * r2876552;
        double r2876554 = r2876551 - r2876553;
        double r2876555 = r2876554 / r2876551;
        double r2876556 = fabs(r2876555);
        double r2876557 = sqrt(r2876556);
        return r2876557;
}
↓
double f(double a, double b) {
        double r2876558 = 1.0;
        double r2876559 = b;
        double r2876560 = a;
        double r2876561 = r2876559 / r2876560;
        double r2876562 = r2876561 * r2876561;
        double r2876563 = r2876558 - r2876562;
        double r2876564 = fabs(r2876563);
        double r2876565 = sqrt(r2876564);
        return r2876565;
}

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 29.8

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

  2. Simplified7.7

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

  3. Final simplification7.7

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))