Average Error: 30.0 → 7.9
Time: 11.3s
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 r1636141 = a;
        double r1636142 = r1636141 * r1636141;
        double r1636143 = b;
        double r1636144 = r1636143 * r1636143;
        double r1636145 = r1636142 - r1636144;
        double r1636146 = r1636145 / r1636142;
        double r1636147 = fabs(r1636146);
        double r1636148 = sqrt(r1636147);
        return r1636148;
}
↓
double f(double a, double b) {
        double r1636149 = 1.0;
        double r1636150 = b;
        double r1636151 = a;
        double r1636152 = r1636150 / r1636151;
        double r1636153 = r1636152 * r1636152;
        double r1636154 = r1636149 - r1636153;
        double r1636155 = fabs(r1636154);
        double r1636156 = sqrt(r1636155);
        return r1636156;
}

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 30.0

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

  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 2019132 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))