Average Error: 28.8 → 7.4
Time: 26.7s
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 r2973800 = a;
        double r2973801 = r2973800 * r2973800;
        double r2973802 = b;
        double r2973803 = r2973802 * r2973802;
        double r2973804 = r2973801 - r2973803;
        double r2973805 = r2973804 / r2973801;
        double r2973806 = fabs(r2973805);
        double r2973807 = sqrt(r2973806);
        return r2973807;
}
↓
double f(double a, double b) {
        double r2973808 = 1.0;
        double r2973809 = b;
        double r2973810 = a;
        double r2973811 = r2973809 / r2973810;
        double r2973812 = r2973811 * r2973811;
        double r2973813 = r2973808 - r2973812;
        double r2973814 = fabs(r2973813);
        double r2973815 = sqrt(r2973814);
        return r2973815;
}

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 28.8

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

  2. Simplified7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2019139 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))