Average Error: 15.0 → 0.0
Time: 18.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}
double f(double a, double b) {
        double r92832 = a;
        double r92833 = r92832 * r92832;
        double r92834 = b;
        double r92835 = r92834 * r92834;
        double r92836 = r92833 - r92835;
        double r92837 = r92836 / r92833;
        double r92838 = fabs(r92837);
        double r92839 = sqrt(r92838);
        return r92839;
}
↓
double f(double a, double b) {
        double r92840 = 1.0;
        double r92841 = b;
        double r92842 = a;
        double r92843 = r92841 / r92842;
        double r92844 = r92841 * r92843;
        double r92845 = r92844 / r92842;
        double r92846 = r92840 - r92845;
        double r92847 = fabs(r92846);
        double r92848 = sqrt(r92847);
        return r92848;
}

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 15.0

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

  2. Simplified15.0

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b \cdot b}{a \cdot a}\right|}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.6

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))