Average Error: 29.3 → 7.2
Time: 20.7s
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 r3542905 = a;
        double r3542906 = r3542905 * r3542905;
        double r3542907 = b;
        double r3542908 = r3542907 * r3542907;
        double r3542909 = r3542906 - r3542908;
        double r3542910 = r3542909 / r3542906;
        double r3542911 = fabs(r3542910);
        double r3542912 = sqrt(r3542911);
        return r3542912;
}
↓
double f(double a, double b) {
        double r3542913 = 1.0;
        double r3542914 = b;
        double r3542915 = a;
        double r3542916 = r3542914 / r3542915;
        double r3542917 = r3542916 * r3542916;
        double r3542918 = r3542913 - r3542917;
        double r3542919 = fabs(r3542918);
        double r3542920 = sqrt(r3542919);
        return r3542920;
}

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. Simplified7.2

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

  3. Final simplification7.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))