Average Error: 30.8 → 7.8
Time: 23.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}
double f(double a, double b) {
        double r2227718 = a;
        double r2227719 = r2227718 * r2227718;
        double r2227720 = b;
        double r2227721 = r2227720 * r2227720;
        double r2227722 = r2227719 - r2227721;
        double r2227723 = r2227722 / r2227719;
        double r2227724 = fabs(r2227723);
        double r2227725 = sqrt(r2227724);
        return r2227725;
}
↓
double f(double a, double b) {
        double r2227726 = 1.0;
        double r2227727 = b;
        double r2227728 = a;
        double r2227729 = r2227727 / r2227728;
        double r2227730 = r2227726 - r2227729;
        double r2227731 = r2227729 + r2227726;
        double r2227732 = r2227730 * r2227731;
        double r2227733 = fabs(r2227732);
        double r2227734 = sqrt(r2227733);
        return r2227734;
}

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 30.8

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

  2. Simplified7.8

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

  4. Applied *-un-lft-identity7.8

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

  5. Applied difference-of-squares7.8

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

  6. Final simplification7.8

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

Reproduce

herbie shell --seed 2019200 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))