Average Error: 14.3 → 0.0
Time: 4.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r69431 = a;
        double r69432 = r69431 * r69431;
        double r69433 = b;
        double r69434 = r69433 * r69433;
        double r69435 = r69432 - r69434;
        double r69436 = r69435 / r69432;
        double r69437 = fabs(r69436);
        double r69438 = sqrt(r69437);
        return r69438;
}
↓
double f(double a, double b) {
        double r69439 = a;
        double r69440 = b;
        double r69441 = r69439 + r69440;
        double r69442 = r69441 / r69439;
        double r69443 = r69439 - r69440;
        double r69444 = r69443 / r69439;
        double r69445 = r69442 * r69444;
        double r69446 = fabs(r69445);
        double r69447 = sqrt(r69446);
        return r69447;
}

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 14.3

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

  3. Applied difference-of-squares14.3

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))