Average Error: 29.5 → 7.5
Time: 37.8s
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 r3313566 = a;
        double r3313567 = r3313566 * r3313566;
        double r3313568 = b;
        double r3313569 = r3313568 * r3313568;
        double r3313570 = r3313567 - r3313569;
        double r3313571 = r3313570 / r3313567;
        double r3313572 = fabs(r3313571);
        double r3313573 = sqrt(r3313572);
        return r3313573;
}
↓
double f(double a, double b) {
        double r3313574 = 1.0;
        double r3313575 = b;
        double r3313576 = a;
        double r3313577 = r3313575 / r3313576;
        double r3313578 = r3313577 * r3313577;
        double r3313579 = r3313574 - r3313578;
        double r3313580 = fabs(r3313579);
        double r3313581 = sqrt(r3313580);
        return r3313581;
}

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 29.5

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

  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 2019149 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))