Average Error: 29.7 → 7.4
Time: 10.4s
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 r15163911 = a;
        double r15163912 = r15163911 * r15163911;
        double r15163913 = b;
        double r15163914 = r15163913 * r15163913;
        double r15163915 = r15163912 - r15163914;
        double r15163916 = r15163915 / r15163912;
        double r15163917 = fabs(r15163916);
        double r15163918 = sqrt(r15163917);
        return r15163918;
}
↓
double f(double a, double b) {
        double r15163919 = 1.0;
        double r15163920 = b;
        double r15163921 = a;
        double r15163922 = r15163920 / r15163921;
        double r15163923 = r15163922 * r15163922;
        double r15163924 = r15163919 - r15163923;
        double r15163925 = fabs(r15163924);
        double r15163926 = sqrt(r15163925);
        return r15163926;
}

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 29.7

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

  2. Simplified7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))