Average Error: 29.7 → 7.4
Time: 9.7s
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 r12369981 = a;
        double r12369982 = r12369981 * r12369981;
        double r12369983 = b;
        double r12369984 = r12369983 * r12369983;
        double r12369985 = r12369982 - r12369984;
        double r12369986 = r12369985 / r12369982;
        double r12369987 = fabs(r12369986);
        double r12369988 = sqrt(r12369987);
        return r12369988;
}
↓
double f(double a, double b) {
        double r12369989 = 1.0;
        double r12369990 = b;
        double r12369991 = a;
        double r12369992 = r12369990 / r12369991;
        double r12369993 = r12369992 * r12369992;
        double r12369994 = r12369989 - r12369993;
        double r12369995 = fabs(r12369994);
        double r12369996 = sqrt(r12369995);
        return r12369996;
}

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 29.7

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

  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 2019128 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))