Average Error: 30.4 → 8.1
Time: 15.3s
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 r3907778 = a;
        double r3907779 = r3907778 * r3907778;
        double r3907780 = b;
        double r3907781 = r3907780 * r3907780;
        double r3907782 = r3907779 - r3907781;
        double r3907783 = r3907782 / r3907779;
        double r3907784 = fabs(r3907783);
        double r3907785 = sqrt(r3907784);
        return r3907785;
}
↓
double f(double a, double b) {
        double r3907786 = 1.0;
        double r3907787 = b;
        double r3907788 = a;
        double r3907789 = r3907787 / r3907788;
        double r3907790 = r3907789 * r3907789;
        double r3907791 = r3907786 - r3907790;
        double r3907792 = fabs(r3907791);
        double r3907793 = sqrt(r3907792);
        return r3907793;
}

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