Average Error: 29.4 → 7.6
Time: 17.9s
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 r5922353 = a;
        double r5922354 = r5922353 * r5922353;
        double r5922355 = b;
        double r5922356 = r5922355 * r5922355;
        double r5922357 = r5922354 - r5922356;
        double r5922358 = r5922357 / r5922354;
        double r5922359 = fabs(r5922358);
        double r5922360 = sqrt(r5922359);
        return r5922360;
}
↓
double f(double a, double b) {
        double r5922361 = 1.0;
        double r5922362 = b;
        double r5922363 = a;
        double r5922364 = r5922362 / r5922363;
        double r5922365 = r5922364 * r5922364;
        double r5922366 = r5922361 - r5922365;
        double r5922367 = fabs(r5922366);
        double r5922368 = sqrt(r5922367);
        return r5922368;
}

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 29.4

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

  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 2019107 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))