Average Error: 29.9 → 7.5
Time: 17.6s
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 r6147515 = a;
        double r6147516 = r6147515 * r6147515;
        double r6147517 = b;
        double r6147518 = r6147517 * r6147517;
        double r6147519 = r6147516 - r6147518;
        double r6147520 = r6147519 / r6147516;
        double r6147521 = fabs(r6147520);
        double r6147522 = sqrt(r6147521);
        return r6147522;
}
↓
double f(double a, double b) {
        double r6147523 = 1.0;
        double r6147524 = b;
        double r6147525 = a;
        double r6147526 = r6147524 / r6147525;
        double r6147527 = r6147526 * r6147526;
        double r6147528 = r6147523 - r6147527;
        double r6147529 = fabs(r6147528);
        double r6147530 = sqrt(r6147529);
        return r6147530;
}

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 29.9

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

  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 2019121 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))