Average Error: 29.7 → 7.5
Time: 27.8s
Precision: 64
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
double f(double a, double b) {
        double r6096169 = a;
        double r6096170 = r6096169 * r6096169;
        double r6096171 = b;
        double r6096172 = r6096171 * r6096171;
        double r6096173 = r6096170 - r6096172;
        double r6096174 = r6096173 / r6096170;
        double r6096175 = fabs(r6096174);
        double r6096176 = sqrt(r6096175);
        return r6096176;
}
↓
double f(double a, double b) {
        double r6096177 = 1.0;
        double r6096178 = b;
        double r6096179 = a;
        double r6096180 = r6096178 / r6096179;
        double r6096181 = r6096180 * r6096180;
        double r6096182 = r6096177 - r6096181;
        double r6096183 = fabs(r6096182);
        double r6096184 = sqrt(r6096183);
        return r6096184;
}


\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 29.7

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

  2. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019101 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))