Average Error: 14.1 → 0.0
Time: 3.9s
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 r62800 = a;
        double r62801 = r62800 * r62800;
        double r62802 = b;
        double r62803 = r62802 * r62802;
        double r62804 = r62801 - r62803;
        double r62805 = r62804 / r62801;
        double r62806 = fabs(r62805);
        double r62807 = sqrt(r62806);
        return r62807;
}
↓
double f(double a, double b) {
        double r62808 = a;
        double r62809 = b;
        double r62810 = r62808 + r62809;
        double r62811 = r62810 / r62808;
        double r62812 = 1.0;
        double r62813 = r62808 - r62809;
        double r62814 = r62808 / r62813;
        double r62815 = r62812 / r62814;
        double r62816 = r62811 * r62815;
        double r62817 = fabs(r62816);
        double r62818 = sqrt(r62817);
        return r62818;
}

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 14.1

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

  3. Applied difference-of-squares14.1

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

  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 2019353 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))