Average Error: 29.3 → 7.0
Time: 27.9s
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 r10426195 = a;
        double r10426196 = r10426195 * r10426195;
        double r10426197 = b;
        double r10426198 = r10426197 * r10426197;
        double r10426199 = r10426196 - r10426198;
        double r10426200 = r10426199 / r10426196;
        double r10426201 = fabs(r10426200);
        double r10426202 = sqrt(r10426201);
        return r10426202;
}
↓
double f(double a, double b) {
        double r10426203 = 1.0;
        double r10426204 = b;
        double r10426205 = a;
        double r10426206 = r10426204 / r10426205;
        double r10426207 = r10426206 * r10426206;
        double r10426208 = r10426203 - r10426207;
        double r10426209 = fabs(r10426208);
        double r10426210 = sqrt(r10426209);
        return r10426210;
}

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