Average Error: 14.5 → 0.0
Time: 2.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}
double f(double a, double b) {
        double r74762 = a;
        double r74763 = r74762 * r74762;
        double r74764 = b;
        double r74765 = r74764 * r74764;
        double r74766 = r74763 - r74765;
        double r74767 = r74766 / r74763;
        double r74768 = fabs(r74767);
        double r74769 = sqrt(r74768);
        return r74769;
}
↓
double f(double a, double b) {
        double r74770 = a;
        double r74771 = b;
        double r74772 = r74770 + r74771;
        double r74773 = r74772 / r74770;
        double r74774 = 1.0;
        double r74775 = r74770 - r74771;
        double r74776 = r74770 / r74775;
        double r74777 = r74774 / r74776;
        double r74778 = r74773 * r74777;
        double r74779 = fabs(r74778);
        double r74780 = sqrt(r74779);
        return r74780;
}

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 clear-num0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))