Average Error: 30.8 → 8.3
Time: 14.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r3625620 = a;
        double r3625621 = r3625620 * r3625620;
        double r3625622 = b;
        double r3625623 = r3625622 * r3625622;
        double r3625624 = r3625621 - r3625623;
        double r3625625 = r3625624 / r3625621;
        double r3625626 = fabs(r3625625);
        double r3625627 = sqrt(r3625626);
        return r3625627;
}
↓
double f(double a, double b) {
        double r3625628 = 1.0;
        double r3625629 = b;
        double r3625630 = a;
        double r3625631 = r3625629 / r3625630;
        double r3625632 = r3625631 * r3625631;
        double r3625633 = r3625628 - r3625632;
        double r3625634 = fabs(r3625633);
        double r3625635 = sqrt(r3625634);
        return r3625635;
}

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 30.8

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

  2. Simplified8.3

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

  3. Final simplification8.3

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

Reproduce

herbie shell --seed 2019169 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))