Average Error: 29.7 → 7.5
Time: 18.0s
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 r9787672 = a;
        double r9787673 = r9787672 * r9787672;
        double r9787674 = b;
        double r9787675 = r9787674 * r9787674;
        double r9787676 = r9787673 - r9787675;
        double r9787677 = r9787676 / r9787673;
        double r9787678 = fabs(r9787677);
        double r9787679 = sqrt(r9787678);
        return r9787679;
}
↓
double f(double a, double b) {
        double r9787680 = 1.0;
        double r9787681 = b;
        double r9787682 = a;
        double r9787683 = r9787681 / r9787682;
        double r9787684 = r9787683 * r9787683;
        double r9787685 = r9787680 - r9787684;
        double r9787686 = fabs(r9787685);
        double r9787687 = sqrt(r9787686);
        return r9787687;
}


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