Average Error: 29.4 → 7.6
Time: 18.0s
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 r4328380 = a;
        double r4328381 = r4328380 * r4328380;
        double r4328382 = b;
        double r4328383 = r4328382 * r4328382;
        double r4328384 = r4328381 - r4328383;
        double r4328385 = r4328384 / r4328381;
        double r4328386 = fabs(r4328385);
        double r4328387 = sqrt(r4328386);
        return r4328387;
}
↓
double f(double a, double b) {
        double r4328388 = b;
        double r4328389 = a;
        double r4328390 = r4328388 / r4328389;
        double r4328391 = 1.0;
        double r4328392 = r4328390 + r4328391;
        double r4328393 = r4328391 - r4328390;
        double r4328394 = r4328392 * r4328393;
        double r4328395 = fabs(r4328394);
        double r4328396 = sqrt(r4328395);
        return r4328396;
}

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)))))