Average Error: 29.2 → 7.5
Time: 15.2s
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 r3606723 = a;
        double r3606724 = r3606723 * r3606723;
        double r3606725 = b;
        double r3606726 = r3606725 * r3606725;
        double r3606727 = r3606724 - r3606726;
        double r3606728 = r3606727 / r3606724;
        double r3606729 = fabs(r3606728);
        double r3606730 = sqrt(r3606729);
        return r3606730;
}
↓
double f(double a, double b) {
        double r3606731 = 1.0;
        double r3606732 = b;
        double r3606733 = a;
        double r3606734 = r3606732 / r3606733;
        double r3606735 = r3606734 * r3606734;
        double r3606736 = r3606731 - r3606735;
        double r3606737 = fabs(r3606736);
        double r3606738 = sqrt(r3606737);
        return r3606738;
}

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 29.2

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

  2. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019158 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))