Average Error: 14.2 → 0.0
Time: 17.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r71332 = a;
        double r71333 = r71332 * r71332;
        double r71334 = b;
        double r71335 = r71334 * r71334;
        double r71336 = r71333 - r71335;
        double r71337 = r71336 / r71333;
        double r71338 = fabs(r71337);
        double r71339 = sqrt(r71338);
        return r71339;
}
↓
double f(double a, double b) {
        double r71340 = 1.0;
        double r71341 = b;
        double r71342 = a;
        double r71343 = r71341 / r71342;
        double r71344 = r71340 + r71343;
        double r71345 = r71342 - r71341;
        double r71346 = r71345 / r71342;
        double r71347 = r71344 * r71346;
        double r71348 = fabs(r71347);
        double r71349 = sqrt(r71348);
        return r71349;
}

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 14.2

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

  3. Applied difference-of-squares14.2

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

  4. Applied times-frac0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))