Average Error: 14.5 → 0.0
Time: 4.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r101615 = a;
        double r101616 = r101615 * r101615;
        double r101617 = b;
        double r101618 = r101617 * r101617;
        double r101619 = r101616 - r101618;
        double r101620 = r101619 / r101616;
        double r101621 = fabs(r101620);
        double r101622 = sqrt(r101621);
        return r101622;
}
↓
double f(double a, double b) {
        double r101623 = a;
        double r101624 = b;
        double r101625 = r101623 + r101624;
        double r101626 = r101623 - r101624;
        double r101627 = r101626 / r101623;
        double r101628 = r101623 / r101627;
        double r101629 = r101625 / r101628;
        double r101630 = fabs(r101629);
        double r101631 = sqrt(r101630);
        return r101631;
}

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 14.5

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

  3. Applied difference-of-squares14.5

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

  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 associate-/l*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))