Average Error: 14.4 → 0.0
Time: 4.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r94909 = a;
        double r94910 = r94909 * r94909;
        double r94911 = b;
        double r94912 = r94911 * r94911;
        double r94913 = r94910 - r94912;
        double r94914 = r94913 / r94910;
        double r94915 = fabs(r94914);
        double r94916 = sqrt(r94915);
        return r94916;
}
↓
double f(double a, double b) {
        double r94917 = 1.0;
        double r94918 = a;
        double r94919 = b;
        double r94920 = r94918 + r94919;
        double r94921 = r94918 / r94920;
        double r94922 = r94917 / r94921;
        double r94923 = r94918 - r94919;
        double r94924 = r94923 / r94918;
        double r94925 = r94922 * r94924;
        double r94926 = fabs(r94925);
        double r94927 = sqrt(r94926);
        return r94927;
}

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 14.4

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

  3. Applied difference-of-squares14.4

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

  4. Applied times-frac0.0

    \[\leadsto \sqrt{\left|\color{blue}{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right|}\]
  5. Using strategy rm

  6. Applied clear-num0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))