Average Error: 29.3 → 7.5
Time: 15.9s
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 r3744188 = a;
        double r3744189 = r3744188 * r3744188;
        double r3744190 = b;
        double r3744191 = r3744190 * r3744190;
        double r3744192 = r3744189 - r3744191;
        double r3744193 = r3744192 / r3744189;
        double r3744194 = fabs(r3744193);
        double r3744195 = sqrt(r3744194);
        return r3744195;
}
↓
double f(double a, double b) {
        double r3744196 = 1.0;
        double r3744197 = b;
        double r3744198 = a;
        double r3744199 = r3744197 / r3744198;
        double r3744200 = r3744199 * r3744199;
        double r3744201 = r3744196 - r3744200;
        double r3744202 = fabs(r3744201);
        double r3744203 = sqrt(r3744202);
        return r3744203;
}

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. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019165 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))