Average Error: 13.7 → 0.0
Time: 4.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r80915 = a;
        double r80916 = r80915 * r80915;
        double r80917 = b;
        double r80918 = r80917 * r80917;
        double r80919 = r80916 - r80918;
        double r80920 = r80919 / r80916;
        double r80921 = fabs(r80920);
        double r80922 = sqrt(r80921);
        return r80922;
}
↓
double f(double a, double b) {
        double r80923 = a;
        double r80924 = b;
        double r80925 = r80923 + r80924;
        double r80926 = r80925 / r80923;
        double r80927 = r80923 - r80924;
        double r80928 = r80927 / r80923;
        double r80929 = r80926 * r80928;
        double r80930 = fabs(r80929);
        double r80931 = sqrt(r80930);
        return r80931;
}

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 13.7

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

  3. Applied difference-of-squares13.7

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))