Average Error: 29.0 → 7.6
Time: 14.7s
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 r891215 = a;
        double r891216 = r891215 * r891215;
        double r891217 = b;
        double r891218 = r891217 * r891217;
        double r891219 = r891216 - r891218;
        double r891220 = r891219 / r891216;
        double r891221 = fabs(r891220);
        double r891222 = sqrt(r891221);
        return r891222;
}
↓
double f(double a, double b) {
        double r891223 = 1.0;
        double r891224 = b;
        double r891225 = a;
        double r891226 = r891224 / r891225;
        double r891227 = r891226 * r891226;
        double r891228 = r891223 - r891227;
        double r891229 = fabs(r891228);
        double r891230 = sqrt(r891229);
        return r891230;
}

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 29.0

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

  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 2019151 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))