Average Error: 14.7 → 0.0
Time: 3.7s
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 r90800 = a;
        double r90801 = r90800 * r90800;
        double r90802 = b;
        double r90803 = r90802 * r90802;
        double r90804 = r90801 - r90803;
        double r90805 = r90804 / r90801;
        double r90806 = fabs(r90805);
        double r90807 = sqrt(r90806);
        return r90807;
}
↓
double f(double a, double b) {
        double r90808 = a;
        double r90809 = b;
        double r90810 = r90808 + r90809;
        double r90811 = r90808 - r90809;
        double r90812 = r90811 / r90808;
        double r90813 = r90810 * r90812;
        double r90814 = r90813 / r90808;
        double r90815 = fabs(r90814);
        double r90816 = sqrt(r90815);
        return r90816;
}

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 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 2020056 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))