Average Error: 14.8 → 0.0
Time: 11.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r56909 = a;
        double r56910 = r56909 * r56909;
        double r56911 = b;
        double r56912 = r56911 * r56911;
        double r56913 = r56910 - r56912;
        double r56914 = r56913 / r56910;
        double r56915 = fabs(r56914);
        double r56916 = sqrt(r56915);
        return r56916;
}
↓
double f(double a, double b) {
        double r56917 = 1.0;
        double r56918 = a;
        double r56919 = b;
        double r56920 = r56918 + r56919;
        double r56921 = r56918 - r56919;
        double r56922 = r56921 / r56918;
        double r56923 = r56920 * r56922;
        double r56924 = r56918 / r56923;
        double r56925 = r56917 / r56924;
        double r56926 = fabs(r56925);
        double r56927 = sqrt(r56926);
        return r56927;
}

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. Using strategy rm

  6. Applied associate-*r/0.0

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

  7. Simplified0.0

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

  9. Applied clear-num0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019298 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))