Average Error: 30.9 → 7.9
Time: 15.9s
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 r3054708 = a;
        double r3054709 = r3054708 * r3054708;
        double r3054710 = b;
        double r3054711 = r3054710 * r3054710;
        double r3054712 = r3054709 - r3054711;
        double r3054713 = r3054712 / r3054709;
        double r3054714 = fabs(r3054713);
        double r3054715 = sqrt(r3054714);
        return r3054715;
}
↓
double f(double a, double b) {
        double r3054716 = 1.0;
        double r3054717 = b;
        double r3054718 = a;
        double r3054719 = r3054717 / r3054718;
        double r3054720 = r3054719 * r3054719;
        double r3054721 = r3054716 - r3054720;
        double r3054722 = fabs(r3054721);
        double r3054723 = sqrt(r3054722);
        return r3054723;
}

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 30.9

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

  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 2019171 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))