Average Error: 29.7 → 8.1
Time: 24.3s
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 r2491365 = a;
        double r2491366 = r2491365 * r2491365;
        double r2491367 = b;
        double r2491368 = r2491367 * r2491367;
        double r2491369 = r2491366 - r2491368;
        double r2491370 = r2491369 / r2491366;
        double r2491371 = fabs(r2491370);
        double r2491372 = sqrt(r2491371);
        return r2491372;
}
↓
double f(double a, double b) {
        double r2491373 = 1.0;
        double r2491374 = b;
        double r2491375 = a;
        double r2491376 = r2491374 / r2491375;
        double r2491377 = r2491376 * r2491376;
        double r2491378 = r2491373 - r2491377;
        double r2491379 = fabs(r2491378);
        double r2491380 = sqrt(r2491379);
        return r2491380;
}

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 29.7

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

  2. Simplified8.1

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

  3. Final simplification8.1

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

Reproduce

herbie shell --seed 2019138 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))