Average Error: 14.1 → 0.0
Time: 4.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r71215 = a;
        double r71216 = r71215 * r71215;
        double r71217 = b;
        double r71218 = r71217 * r71217;
        double r71219 = r71216 - r71218;
        double r71220 = r71219 / r71216;
        double r71221 = fabs(r71220);
        double r71222 = sqrt(r71221);
        return r71222;
}
↓
double f(double a, double b) {
        double r71223 = 1.0;
        double r71224 = a;
        double r71225 = b;
        double r71226 = r71224 + r71225;
        double r71227 = r71224 / r71226;
        double r71228 = r71223 / r71227;
        double r71229 = r71224 - r71225;
        double r71230 = r71229 / r71224;
        double r71231 = r71228 * r71230;
        double r71232 = fabs(r71231);
        double r71233 = sqrt(r71232);
        return r71233;
}

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 14.1

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

  3. Applied difference-of-squares14.1

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

  4. Applied times-frac0.0

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

  6. Applied clear-num0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))