Average Error: 14.1 → 0.0
Time: 12.9s
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 r73402 = a;
        double r73403 = r73402 * r73402;
        double r73404 = b;
        double r73405 = r73404 * r73404;
        double r73406 = r73403 - r73405;
        double r73407 = r73406 / r73403;
        double r73408 = fabs(r73407);
        double r73409 = sqrt(r73408);
        return r73409;
}
↓
double f(double a, double b) {
        double r73410 = a;
        double r73411 = b;
        double r73412 = r73410 + r73411;
        double r73413 = r73412 / r73410;
        double r73414 = 1.0;
        double r73415 = r73410 - r73411;
        double r73416 = r73410 / r73415;
        double r73417 = r73414 / r73416;
        double r73418 = r73413 * r73417;
        double r73419 = fabs(r73418);
        double r73420 = sqrt(r73419);
        return r73420;
}

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 2019297 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))