Average Error: 29.2 → 7.5
Time: 17.5s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r3717541 = a;
        double r3717542 = r3717541 * r3717541;
        double r3717543 = b;
        double r3717544 = r3717543 * r3717543;
        double r3717545 = r3717542 - r3717544;
        double r3717546 = r3717545 / r3717542;
        double r3717547 = fabs(r3717546);
        double r3717548 = sqrt(r3717547);
        return r3717548;
}
↓
double f(double a, double b) {
        double r3717549 = 1.0;
        double r3717550 = b;
        double r3717551 = a;
        double r3717552 = r3717550 / r3717551;
        double r3717553 = r3717552 * r3717552;
        double r3717554 = r3717549 - r3717553;
        double r3717555 = fabs(r3717554);
        double r3717556 = sqrt(r3717555);
        return r3717556;
}

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 29.2

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

  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 2019158 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))