Average Error: 14.5 → 0.0
Time: 3.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r69428 = a;
        double r69429 = r69428 * r69428;
        double r69430 = b;
        double r69431 = r69430 * r69430;
        double r69432 = r69429 - r69431;
        double r69433 = r69432 / r69429;
        double r69434 = fabs(r69433);
        double r69435 = sqrt(r69434);
        return r69435;
}
↓
double f(double a, double b) {
        double r69436 = 1.0;
        double r69437 = a;
        double r69438 = b;
        double r69439 = r69437 + r69438;
        double r69440 = r69437 - r69438;
        double r69441 = r69440 / r69437;
        double r69442 = r69439 * r69441;
        double r69443 = r69437 / r69442;
        double r69444 = r69436 / r69443;
        double r69445 = fabs(r69444);
        double r69446 = sqrt(r69445);
        return r69446;
}

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 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. Using strategy rm

  9. Applied clear-num0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 +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)))))