Average Error: 14.8 → 0.0
Time: 3.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r49827 = a;
        double r49828 = r49827 * r49827;
        double r49829 = b;
        double r49830 = r49829 * r49829;
        double r49831 = r49828 - r49830;
        double r49832 = r49831 / r49828;
        double r49833 = fabs(r49832);
        double r49834 = sqrt(r49833);
        return r49834;
}
↓
double f(double a, double b) {
        double r49835 = 1.0;
        double r49836 = b;
        double r49837 = a;
        double r49838 = r49836 / r49837;
        double r49839 = r49835 + r49838;
        double r49840 = r49837 - r49836;
        double r49841 = r49840 / r49837;
        double r49842 = r49839 * r49841;
        double r49843 = fabs(r49842);
        double r49844 = sqrt(r49843);
        return r49844;
}

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 14.8

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.8

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

  4. Applied times-frac0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))