Average Error: 14.3 → 0.0
Time: 11.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}
double f(double a, double b) {
        double r84978 = a;
        double r84979 = r84978 * r84978;
        double r84980 = b;
        double r84981 = r84980 * r84980;
        double r84982 = r84979 - r84981;
        double r84983 = r84982 / r84979;
        double r84984 = fabs(r84983);
        double r84985 = sqrt(r84984);
        return r84985;
}
↓
double f(double a, double b) {
        double r84986 = a;
        double r84987 = b;
        double r84988 = r84986 + r84987;
        double r84989 = r84988 / r84986;
        double r84990 = 1.0;
        double r84991 = r84986 - r84987;
        double r84992 = r84986 / r84991;
        double r84993 = r84990 / r84992;
        double r84994 = r84989 * r84993;
        double r84995 = fabs(r84994);
        double r84996 = sqrt(r84995);
        return r84996;
}

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. Using strategy rm

  3. Applied difference-of-squares14.3

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

  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

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))