Average Error: 31.3 → 8.2
Time: 17.4s
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 r3772802 = a;
        double r3772803 = r3772802 * r3772802;
        double r3772804 = b;
        double r3772805 = r3772804 * r3772804;
        double r3772806 = r3772803 - r3772805;
        double r3772807 = r3772806 / r3772803;
        double r3772808 = fabs(r3772807);
        double r3772809 = sqrt(r3772808);
        return r3772809;
}
↓
double f(double a, double b) {
        double r3772810 = 1.0;
        double r3772811 = b;
        double r3772812 = a;
        double r3772813 = r3772811 / r3772812;
        double r3772814 = r3772813 * r3772813;
        double r3772815 = r3772810 - r3772814;
        double r3772816 = fabs(r3772815);
        double r3772817 = sqrt(r3772816);
        return r3772817;
}

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 31.3

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

  2. Simplified8.2

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

  3. Final simplification8.2

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

Reproduce

herbie shell --seed 2019170 +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)))))