Average Error: 30.4 → 8.1
Time: 22.4s
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 r4963059 = a;
        double r4963060 = r4963059 * r4963059;
        double r4963061 = b;
        double r4963062 = r4963061 * r4963061;
        double r4963063 = r4963060 - r4963062;
        double r4963064 = r4963063 / r4963060;
        double r4963065 = fabs(r4963064);
        double r4963066 = sqrt(r4963065);
        return r4963066;
}
↓
double f(double a, double b) {
        double r4963067 = 1.0;
        double r4963068 = b;
        double r4963069 = a;
        double r4963070 = r4963068 / r4963069;
        double r4963071 = r4963070 * r4963070;
        double r4963072 = r4963067 - r4963071;
        double r4963073 = fabs(r4963072);
        double r4963074 = sqrt(r4963073);
        return r4963074;
}

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 30.4

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

  2. Simplified8.1

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

  3. Final simplification8.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))