Average Error: 30.4 → 8.1
Time: 21.9s
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 r4071806 = a;
        double r4071807 = r4071806 * r4071806;
        double r4071808 = b;
        double r4071809 = r4071808 * r4071808;
        double r4071810 = r4071807 - r4071809;
        double r4071811 = r4071810 / r4071807;
        double r4071812 = fabs(r4071811);
        double r4071813 = sqrt(r4071812);
        return r4071813;
}
↓
double f(double a, double b) {
        double r4071814 = 1.0;
        double r4071815 = b;
        double r4071816 = a;
        double r4071817 = r4071815 / r4071816;
        double r4071818 = r4071817 * r4071817;
        double r4071819 = r4071814 - r4071818;
        double r4071820 = fabs(r4071819);
        double r4071821 = sqrt(r4071820);
        return r4071821;
}

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 30.4

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

  2. Simplified8.1

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

  3. Final simplification8.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))