Average Error: 29.2 → 7.3
Time: 10.3s
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 r2104914 = a;
        double r2104915 = r2104914 * r2104914;
        double r2104916 = b;
        double r2104917 = r2104916 * r2104916;
        double r2104918 = r2104915 - r2104917;
        double r2104919 = r2104918 / r2104915;
        double r2104920 = fabs(r2104919);
        double r2104921 = sqrt(r2104920);
        return r2104921;
}
↓
double f(double a, double b) {
        double r2104922 = 1.0;
        double r2104923 = b;
        double r2104924 = a;
        double r2104925 = r2104923 / r2104924;
        double r2104926 = r2104925 * r2104925;
        double r2104927 = r2104922 - r2104926;
        double r2104928 = fabs(r2104927);
        double r2104929 = sqrt(r2104928);
        return r2104929;
}

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 29.2

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

  2. Simplified7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2019135 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))