Average Error: 29.3 → 7.9
Time: 21.3s
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 r2799764 = a;
        double r2799765 = r2799764 * r2799764;
        double r2799766 = b;
        double r2799767 = r2799766 * r2799766;
        double r2799768 = r2799765 - r2799767;
        double r2799769 = r2799768 / r2799765;
        double r2799770 = fabs(r2799769);
        double r2799771 = sqrt(r2799770);
        return r2799771;
}
↓
double f(double a, double b) {
        double r2799772 = 1.0;
        double r2799773 = b;
        double r2799774 = a;
        double r2799775 = r2799773 / r2799774;
        double r2799776 = r2799775 * r2799775;
        double r2799777 = r2799772 - r2799776;
        double r2799778 = fabs(r2799777);
        double r2799779 = sqrt(r2799778);
        return r2799779;
}

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