Average Error: 14.7 → 0.0
Time: 13.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}
double f(double a, double b) {
        double r76922 = a;
        double r76923 = r76922 * r76922;
        double r76924 = b;
        double r76925 = r76924 * r76924;
        double r76926 = r76923 - r76925;
        double r76927 = r76926 / r76923;
        double r76928 = fabs(r76927);
        double r76929 = sqrt(r76928);
        return r76929;
}
↓
double f(double a, double b) {
        double r76930 = a;
        double r76931 = b;
        double r76932 = r76930 + r76931;
        double r76933 = r76932 / r76930;
        double r76934 = 1.0;
        double r76935 = r76930 - r76931;
        double r76936 = r76930 / r76935;
        double r76937 = r76934 / r76936;
        double r76938 = r76933 * r76937;
        double r76939 = fabs(r76938);
        double r76940 = sqrt(r76939);
        return r76940;
}

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 14.7

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

  3. Applied difference-of-squares14.7

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

  4. Applied times-frac0.0

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

  6. Applied clear-num0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019291 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))