Average Error: 14.5 → 0.0
Time: 13.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b \cdot \frac{b}{a}}{a}\right)\right)\right|}
double f(double a, double b) {
        double r94894 = a;
        double r94895 = r94894 * r94894;
        double r94896 = b;
        double r94897 = r94896 * r94896;
        double r94898 = r94895 - r94897;
        double r94899 = r94898 / r94895;
        double r94900 = fabs(r94899);
        double r94901 = sqrt(r94900);
        return r94901;
}
↓
double f(double a, double b) {
        double r94902 = a;
        double r94903 = b;
        double r94904 = r94903 / r94902;
        double r94905 = r94903 * r94904;
        double r94906 = r94902 - r94905;
        double r94907 = r94906 / r94902;
        double r94908 = expm1(r94907);
        double r94909 = log1p(r94908);
        double r94910 = fabs(r94909);
        double r94911 = sqrt(r94910);
        return r94911;
}

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 14.5

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

  3. Applied associate-/r*14.7

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

  4. Simplified0.0

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

  6. Applied log1p-expm1-u0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019351 +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)))))