Average Error: 29.0 → 7.3
Time: 11.9s
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 r735723 = a;
        double r735724 = r735723 * r735723;
        double r735725 = b;
        double r735726 = r735725 * r735725;
        double r735727 = r735724 - r735726;
        double r735728 = r735727 / r735724;
        double r735729 = fabs(r735728);
        double r735730 = sqrt(r735729);
        return r735730;
}
↓
double f(double a, double b) {
        double r735731 = b;
        double r735732 = a;
        double r735733 = r735731 / r735732;
        double r735734 = 1.0;
        double r735735 = r735733 + r735734;
        double r735736 = r735734 - r735733;
        double r735737 = r735735 * r735736;
        double r735738 = fabs(r735737);
        double r735739 = sqrt(r735738);
        return r735739;
}

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 29.0

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

  2. Simplified7.3

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

  4. Applied *-un-lft-identity7.3

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

  5. Applied difference-of-squares7.3

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

  6. Final simplification7.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))