Average Error: 14.4 → 0.0
Time: 2.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}
double f(double a, double b) {
        double r65769 = a;
        double r65770 = r65769 * r65769;
        double r65771 = b;
        double r65772 = r65771 * r65771;
        double r65773 = r65770 - r65772;
        double r65774 = r65773 / r65770;
        double r65775 = fabs(r65774);
        double r65776 = sqrt(r65775);
        return r65776;
}
↓
double f(double a, double b) {
        double r65777 = a;
        double r65778 = b;
        double r65779 = r65777 + r65778;
        double r65780 = r65777 - r65778;
        double r65781 = r65780 / r65777;
        double r65782 = r65779 * r65781;
        double r65783 = r65782 / r65777;
        double r65784 = fabs(r65783);
        double r65785 = sqrt(r65784);
        return r65785;
}

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 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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019356 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))