Average Error: 29.4 → 7.3
Time: 21.2s
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 r3894883 = a;
        double r3894884 = r3894883 * r3894883;
        double r3894885 = b;
        double r3894886 = r3894885 * r3894885;
        double r3894887 = r3894884 - r3894886;
        double r3894888 = r3894887 / r3894884;
        double r3894889 = fabs(r3894888);
        double r3894890 = sqrt(r3894889);
        return r3894890;
}
↓
double f(double a, double b) {
        double r3894891 = 1.0;
        double r3894892 = b;
        double r3894893 = a;
        double r3894894 = r3894892 / r3894893;
        double r3894895 = r3894894 * r3894894;
        double r3894896 = r3894891 - r3894895;
        double r3894897 = fabs(r3894896);
        double r3894898 = sqrt(r3894897);
        return r3894898;
}

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. Simplified7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))