Average Error: 30.0 → 7.6
Time: 16.3s
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 r2694379 = a;
        double r2694380 = r2694379 * r2694379;
        double r2694381 = b;
        double r2694382 = r2694381 * r2694381;
        double r2694383 = r2694380 - r2694382;
        double r2694384 = r2694383 / r2694380;
        double r2694385 = fabs(r2694384);
        double r2694386 = sqrt(r2694385);
        return r2694386;
}
↓
double f(double a, double b) {
        double r2694387 = 1.0;
        double r2694388 = b;
        double r2694389 = a;
        double r2694390 = r2694388 / r2694389;
        double r2694391 = r2694390 * r2694390;
        double r2694392 = r2694387 - r2694391;
        double r2694393 = fabs(r2694392);
        double r2694394 = sqrt(r2694393);
        return r2694394;
}

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 30.0

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

  2. Simplified7.6

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

  3. Final simplification7.6

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

Reproduce

herbie shell --seed 2019130 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))