Average Error: 29.3 → 7.0
Time: 37.7s
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 r7478757 = a;
        double r7478758 = r7478757 * r7478757;
        double r7478759 = b;
        double r7478760 = r7478759 * r7478759;
        double r7478761 = r7478758 - r7478760;
        double r7478762 = r7478761 / r7478758;
        double r7478763 = fabs(r7478762);
        double r7478764 = sqrt(r7478763);
        return r7478764;
}
↓
double f(double a, double b) {
        double r7478765 = 1.0;
        double r7478766 = b;
        double r7478767 = a;
        double r7478768 = r7478766 / r7478767;
        double r7478769 = r7478768 * r7478768;
        double r7478770 = r7478765 - r7478769;
        double r7478771 = fabs(r7478770);
        double r7478772 = sqrt(r7478771);
        return r7478772;
}

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. Simplified7.0

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

  3. Final simplification7.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))