Average Error: 14.0 → 0.0
Time: 17.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a - \frac{b}{a} \cdot b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a - \frac{b}{a} \cdot b}{a}\right|}
double f(double a, double b) {
        double r56688 = a;
        double r56689 = r56688 * r56688;
        double r56690 = b;
        double r56691 = r56690 * r56690;
        double r56692 = r56689 - r56691;
        double r56693 = r56692 / r56689;
        double r56694 = fabs(r56693);
        double r56695 = sqrt(r56694);
        return r56695;
}
↓
double f(double a, double b) {
        double r56696 = a;
        double r56697 = b;
        double r56698 = r56697 / r56696;
        double r56699 = r56698 * r56697;
        double r56700 = r56696 - r56699;
        double r56701 = r56700 / r56696;
        double r56702 = fabs(r56701);
        double r56703 = sqrt(r56702);
        return r56703;
}

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 14.0

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

  3. Applied associate-/r*14.2

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))))