Average Error: 14.0 → 0.0
Time: 10.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a - \frac{b}{a} \cdot b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a - \frac{b}{a} \cdot b}}\right|}
double f(double a, double b) {
        double r86298 = a;
        double r86299 = r86298 * r86298;
        double r86300 = b;
        double r86301 = r86300 * r86300;
        double r86302 = r86299 - r86301;
        double r86303 = r86302 / r86299;
        double r86304 = fabs(r86303);
        double r86305 = sqrt(r86304);
        return r86305;
}
↓
double f(double a, double b) {
        double r86306 = 1.0;
        double r86307 = a;
        double r86308 = b;
        double r86309 = r86308 / r86307;
        double r86310 = r86309 * r86308;
        double r86311 = r86307 - r86310;
        double r86312 = r86307 / r86311;
        double r86313 = r86306 / r86312;
        double r86314 = fabs(r86313);
        double r86315 = sqrt(r86314);
        return r86315;
}

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 14.0

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied clear-num14.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))