Average Error: 30.9 → 7.9
Time: 15.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r3050706 = a;
        double r3050707 = r3050706 * r3050706;
        double r3050708 = b;
        double r3050709 = r3050708 * r3050708;
        double r3050710 = r3050707 - r3050709;
        double r3050711 = r3050710 / r3050707;
        double r3050712 = fabs(r3050711);
        double r3050713 = sqrt(r3050712);
        return r3050713;
}
↓
double f(double a, double b) {
        double r3050714 = 1.0;
        double r3050715 = b;
        double r3050716 = a;
        double r3050717 = r3050715 / r3050716;
        double r3050718 = r3050717 * r3050717;
        double r3050719 = r3050714 - r3050718;
        double r3050720 = fabs(r3050719);
        double r3050721 = sqrt(r3050720);
        return r3050721;
}

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 30.9

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

  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 2019171 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))