Average Error: 29.4 → 7.6
Time: 17.5s
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 r5609826 = a;
        double r5609827 = r5609826 * r5609826;
        double r5609828 = b;
        double r5609829 = r5609828 * r5609828;
        double r5609830 = r5609827 - r5609829;
        double r5609831 = r5609830 / r5609827;
        double r5609832 = fabs(r5609831);
        double r5609833 = sqrt(r5609832);
        return r5609833;
}
↓
double f(double a, double b) {
        double r5609834 = 1.0;
        double r5609835 = b;
        double r5609836 = a;
        double r5609837 = r5609835 / r5609836;
        double r5609838 = r5609837 * r5609837;
        double r5609839 = r5609834 - r5609838;
        double r5609840 = fabs(r5609839);
        double r5609841 = sqrt(r5609840);
        return r5609841;
}

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