Average Error: 29.0 → 7.5
Time: 27.0s
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 r2659321 = a;
        double r2659322 = r2659321 * r2659321;
        double r2659323 = b;
        double r2659324 = r2659323 * r2659323;
        double r2659325 = r2659322 - r2659324;
        double r2659326 = r2659325 / r2659322;
        double r2659327 = fabs(r2659326);
        double r2659328 = sqrt(r2659327);
        return r2659328;
}
↓
double f(double a, double b) {
        double r2659329 = 1.0;
        double r2659330 = b;
        double r2659331 = a;
        double r2659332 = r2659330 / r2659331;
        double r2659333 = r2659332 * r2659332;
        double r2659334 = r2659329 - r2659333;
        double r2659335 = fabs(r2659334);
        double r2659336 = sqrt(r2659335);
        return r2659336;
}

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 29.0

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

  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 2019144 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))