Average Error: 29.4 → 7.6
Time: 16.8s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}
double f(double a, double b) {
        double r4476744 = a;
        double r4476745 = r4476744 * r4476744;
        double r4476746 = b;
        double r4476747 = r4476746 * r4476746;
        double r4476748 = r4476745 - r4476747;
        double r4476749 = r4476748 / r4476745;
        double r4476750 = fabs(r4476749);
        double r4476751 = sqrt(r4476750);
        return r4476751;
}
↓
double f(double a, double b) {
        double r4476752 = b;
        double r4476753 = a;
        double r4476754 = r4476752 / r4476753;
        double r4476755 = 1.0;
        double r4476756 = r4476754 + r4476755;
        double r4476757 = r4476755 - r4476754;
        double r4476758 = r4476756 * r4476757;
        double r4476759 = fabs(r4476758);
        double r4476760 = sqrt(r4476759);
        return r4476760;
}

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 29.4

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

  2. Simplified7.5

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity7.5

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

  5. Applied difference-of-squares7.6

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

  6. Final simplification7.6

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

Reproduce

herbie shell --seed 2019163 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))