Average Error: 29.4 → 7.3
Time: 20.7s
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 r2999554 = a;
        double r2999555 = r2999554 * r2999554;
        double r2999556 = b;
        double r2999557 = r2999556 * r2999556;
        double r2999558 = r2999555 - r2999557;
        double r2999559 = r2999558 / r2999555;
        double r2999560 = fabs(r2999559);
        double r2999561 = sqrt(r2999560);
        return r2999561;
}
↓
double f(double a, double b) {
        double r2999562 = 1.0;
        double r2999563 = b;
        double r2999564 = a;
        double r2999565 = r2999563 / r2999564;
        double r2999566 = r2999565 * r2999565;
        double r2999567 = r2999562 - r2999566;
        double r2999568 = fabs(r2999567);
        double r2999569 = sqrt(r2999568);
        return r2999569;
}

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 29.4

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

  2. Simplified7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))