Average Error: 30.1 → 7.7
Time: 15.9s
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 r1455280 = a;
        double r1455281 = r1455280 * r1455280;
        double r1455282 = b;
        double r1455283 = r1455282 * r1455282;
        double r1455284 = r1455281 - r1455283;
        double r1455285 = r1455284 / r1455281;
        double r1455286 = fabs(r1455285);
        double r1455287 = sqrt(r1455286);
        return r1455287;
}
↓
double f(double a, double b) {
        double r1455288 = 1.0;
        double r1455289 = b;
        double r1455290 = a;
        double r1455291 = r1455289 / r1455290;
        double r1455292 = r1455291 * r1455291;
        double r1455293 = r1455288 - r1455292;
        double r1455294 = fabs(r1455293);
        double r1455295 = sqrt(r1455294);
        return r1455295;
}

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 30.1

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

  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 2019143 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))