Average Error: 29.3 → 7.6
Time: 21.4s
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 r4897821 = a;
        double r4897822 = r4897821 * r4897821;
        double r4897823 = b;
        double r4897824 = r4897823 * r4897823;
        double r4897825 = r4897822 - r4897824;
        double r4897826 = r4897825 / r4897822;
        double r4897827 = fabs(r4897826);
        double r4897828 = sqrt(r4897827);
        return r4897828;
}
↓
double f(double a, double b) {
        double r4897829 = 1.0;
        double r4897830 = b;
        double r4897831 = a;
        double r4897832 = r4897830 / r4897831;
        double r4897833 = r4897832 * r4897832;
        double r4897834 = r4897829 - r4897833;
        double r4897835 = fabs(r4897834);
        double r4897836 = sqrt(r4897835);
        return r4897836;
}

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 29.3

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

  2. Simplified7.6

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

  3. Final simplification7.6

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

Reproduce

herbie shell --seed 2019112 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))