Average Error: 29.5 → 7.8
Time: 58.4s
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 r9115703 = a;
        double r9115704 = r9115703 * r9115703;
        double r9115705 = b;
        double r9115706 = r9115705 * r9115705;
        double r9115707 = r9115704 - r9115706;
        double r9115708 = r9115707 / r9115704;
        double r9115709 = fabs(r9115708);
        double r9115710 = sqrt(r9115709);
        return r9115710;
}
↓
double f(double a, double b) {
        double r9115711 = 1.0;
        double r9115712 = b;
        double r9115713 = a;
        double r9115714 = r9115712 / r9115713;
        double r9115715 = r9115714 * r9115714;
        double r9115716 = r9115711 - r9115715;
        double r9115717 = fabs(r9115716);
        double r9115718 = sqrt(r9115717);
        return r9115718;
}

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 29.5

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

  2. Simplified7.8

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

  3. Final simplification7.8

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

Reproduce

herbie shell --seed 2019104 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))