Average Error: 29.2 → 7.3
Time: 12.0s
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 r2656359 = a;
        double r2656360 = r2656359 * r2656359;
        double r2656361 = b;
        double r2656362 = r2656361 * r2656361;
        double r2656363 = r2656360 - r2656362;
        double r2656364 = r2656363 / r2656360;
        double r2656365 = fabs(r2656364);
        double r2656366 = sqrt(r2656365);
        return r2656366;
}
↓
double f(double a, double b) {
        double r2656367 = 1.0;
        double r2656368 = b;
        double r2656369 = a;
        double r2656370 = r2656368 / r2656369;
        double r2656371 = r2656370 * r2656370;
        double r2656372 = r2656367 - r2656371;
        double r2656373 = fabs(r2656372);
        double r2656374 = sqrt(r2656373);
        return r2656374;
}

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. Simplified7.3

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

  3. Final simplification7.3

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))