Average Error: 28.2 → 7.3
Time: 14.2s
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 r1245148 = a;
        double r1245149 = r1245148 * r1245148;
        double r1245150 = b;
        double r1245151 = r1245150 * r1245150;
        double r1245152 = r1245149 - r1245151;
        double r1245153 = r1245152 / r1245149;
        double r1245154 = fabs(r1245153);
        double r1245155 = sqrt(r1245154);
        return r1245155;
}
↓
double f(double a, double b) {
        double r1245156 = 1.0;
        double r1245157 = b;
        double r1245158 = a;
        double r1245159 = r1245157 / r1245158;
        double r1245160 = r1245159 * r1245159;
        double r1245161 = r1245156 - r1245160;
        double r1245162 = fabs(r1245161);
        double r1245163 = sqrt(r1245162);
        return r1245163;
}

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 28.2

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

  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 2019155 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))