Average Error: 14.4 → 0.0
Time: 24.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right|}
double f(double a, double b) {
        double r72403 = a;
        double r72404 = r72403 * r72403;
        double r72405 = b;
        double r72406 = r72405 * r72405;
        double r72407 = r72404 - r72406;
        double r72408 = r72407 / r72404;
        double r72409 = fabs(r72408);
        double r72410 = sqrt(r72409);
        return r72410;
}
↓
double f(double a, double b) {
        double r72411 = 1.0;
        double r72412 = a;
        double r72413 = b;
        double r72414 = r72413 / r72412;
        double r72415 = r72413 * r72414;
        double r72416 = r72412 - r72415;
        double r72417 = r72412 / r72416;
        double r72418 = r72411 / r72417;
        double r72419 = fabs(r72418);
        double r72420 = sqrt(r72419);
        return r72420;
}

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 14.4

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

  3. Applied clear-num14.4

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 +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)))))