Average Error: 14.2 → 0.0
Time: 3.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)\right)\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)\right)
double f(double a, double b) {
        double r68887 = a;
        double r68888 = r68887 * r68887;
        double r68889 = b;
        double r68890 = r68889 * r68889;
        double r68891 = r68888 - r68890;
        double r68892 = r68891 / r68888;
        double r68893 = fabs(r68892);
        double r68894 = sqrt(r68893);
        return r68894;
}
↓
double f(double a, double b) {
        double r68895 = a;
        double r68896 = b;
        double r68897 = r68895 + r68896;
        double r68898 = r68897 / r68895;
        double r68899 = r68895 - r68896;
        double r68900 = r68899 / r68895;
        double r68901 = r68898 * r68900;
        double r68902 = fabs(r68901);
        double r68903 = sqrt(r68902);
        double r68904 = log1p(r68903);
        double r68905 = expm1(r68904);
        return r68905;
}

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. Using strategy rm

  6. Applied expm1-log1p-u0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)\right)}\]

  7. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)\right)\]

Reproduce

herbie shell --seed 2020039 +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)))))