Average Error: 29.5 → 7.9
Time: 39.7s
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 r20695706 = a;
        double r20695707 = r20695706 * r20695706;
        double r20695708 = b;
        double r20695709 = r20695708 * r20695708;
        double r20695710 = r20695707 - r20695709;
        double r20695711 = r20695710 / r20695707;
        double r20695712 = fabs(r20695711);
        double r20695713 = sqrt(r20695712);
        return r20695713;
}
↓
double f(double a, double b) {
        double r20695714 = 1.0;
        double r20695715 = b;
        double r20695716 = a;
        double r20695717 = r20695715 / r20695716;
        double r20695718 = r20695717 * r20695717;
        double r20695719 = r20695714 - r20695718;
        double r20695720 = fabs(r20695719);
        double r20695721 = sqrt(r20695720);
        return r20695721;
}


\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 29.5

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

  2. Simplified7.9

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

  3. Final simplification7.9

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))