Average Error: 29.8 → 7.5
Time: 26.6s
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 r12410835 = a;
        double r12410836 = r12410835 * r12410835;
        double r12410837 = b;
        double r12410838 = r12410837 * r12410837;
        double r12410839 = r12410836 - r12410838;
        double r12410840 = r12410839 / r12410836;
        double r12410841 = fabs(r12410840);
        double r12410842 = sqrt(r12410841);
        return r12410842;
}
↓
double f(double a, double b) {
        double r12410843 = 1.0;
        double r12410844 = b;
        double r12410845 = a;
        double r12410846 = r12410844 / r12410845;
        double r12410847 = r12410846 * r12410846;
        double r12410848 = r12410843 - r12410847;
        double r12410849 = fabs(r12410848);
        double r12410850 = sqrt(r12410849);
        return r12410850;
}

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 29.8

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

  2. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019124 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))