Average Error: 29.9 → 7.5
Time: 11.8s
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 r1436095 = a;
        double r1436096 = r1436095 * r1436095;
        double r1436097 = b;
        double r1436098 = r1436097 * r1436097;
        double r1436099 = r1436096 - r1436098;
        double r1436100 = r1436099 / r1436096;
        double r1436101 = fabs(r1436100);
        double r1436102 = sqrt(r1436101);
        return r1436102;
}
↓
double f(double a, double b) {
        double r1436103 = 1.0;
        double r1436104 = b;
        double r1436105 = a;
        double r1436106 = r1436104 / r1436105;
        double r1436107 = r1436106 * r1436106;
        double r1436108 = r1436103 - r1436107;
        double r1436109 = fabs(r1436108);
        double r1436110 = sqrt(r1436109);
        return r1436110;
}

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 29.9

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

  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 2019154 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))