Average Error: 14.3 → 0.0
Time: 11.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a - \frac{b}{\sqrt{a}} \cdot \frac{b}{\sqrt{a}}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a - \frac{b}{\sqrt{a}} \cdot \frac{b}{\sqrt{a}}}{a}\right|}
double f(double a, double b) {
        double r63321 = a;
        double r63322 = r63321 * r63321;
        double r63323 = b;
        double r63324 = r63323 * r63323;
        double r63325 = r63322 - r63324;
        double r63326 = r63325 / r63322;
        double r63327 = fabs(r63326);
        double r63328 = sqrt(r63327);
        return r63328;
}
↓
double f(double a, double b) {
        double r63329 = a;
        double r63330 = b;
        double r63331 = sqrt(r63329);
        double r63332 = r63330 / r63331;
        double r63333 = r63332 * r63332;
        double r63334 = r63329 - r63333;
        double r63335 = r63334 / r63329;
        double r63336 = fabs(r63335);
        double r63337 = sqrt(r63336);
        return r63337;
}

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 14.3

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

  2. Simplified0.7

    \[\leadsto \color{blue}{\sqrt{\left|\frac{a - \frac{b \cdot b}{a}}{a}\right|}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.7

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

  5. Applied times-frac0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))