Average Error: 29.3 → 7.9
Time: 11.1s
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 r1401882 = a;
        double r1401883 = r1401882 * r1401882;
        double r1401884 = b;
        double r1401885 = r1401884 * r1401884;
        double r1401886 = r1401883 - r1401885;
        double r1401887 = r1401886 / r1401883;
        double r1401888 = fabs(r1401887);
        double r1401889 = sqrt(r1401888);
        return r1401889;
}
↓
double f(double a, double b) {
        double r1401890 = 1.0;
        double r1401891 = b;
        double r1401892 = a;
        double r1401893 = r1401891 / r1401892;
        double r1401894 = r1401893 * r1401893;
        double r1401895 = r1401890 - r1401894;
        double r1401896 = fabs(r1401895);
        double r1401897 = sqrt(r1401896);
        return r1401897;
}

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 29.3

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

  2. Simplified7.9

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

  3. Final simplification7.9

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

Reproduce

herbie shell --seed 2019152 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))