Average Error: 30.6 → 7.7
Time: 45.8s
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 r3436455 = a;
        double r3436456 = r3436455 * r3436455;
        double r3436457 = b;
        double r3436458 = r3436457 * r3436457;
        double r3436459 = r3436456 - r3436458;
        double r3436460 = r3436459 / r3436456;
        double r3436461 = fabs(r3436460);
        double r3436462 = sqrt(r3436461);
        return r3436462;
}
↓
double f(double a, double b) {
        double r3436463 = 1.0;
        double r3436464 = b;
        double r3436465 = a;
        double r3436466 = r3436464 / r3436465;
        double r3436467 = r3436466 * r3436466;
        double r3436468 = r3436463 - r3436467;
        double r3436469 = fabs(r3436468);
        double r3436470 = sqrt(r3436469);
        return r3436470;
}

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 30.6

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

  2. Simplified7.7

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

  3. Final simplification7.7

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

Reproduce

herbie shell --seed 2019168 +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)))))