Average Error: 14.7 → 0.0
Time: 3.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{a - b}}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{a - b}}\right)\right)\right|}
double f(double a, double b) {
        double r67315 = a;
        double r67316 = r67315 * r67315;
        double r67317 = b;
        double r67318 = r67317 * r67317;
        double r67319 = r67316 - r67318;
        double r67320 = r67319 / r67316;
        double r67321 = fabs(r67320);
        double r67322 = sqrt(r67321);
        return r67322;
}
↓
double f(double a, double b) {
        double r67323 = a;
        double r67324 = b;
        double r67325 = r67323 + r67324;
        double r67326 = r67325 / r67323;
        double r67327 = 1.0;
        double r67328 = r67323 - r67324;
        double r67329 = r67323 / r67328;
        double r67330 = r67327 / r67329;
        double r67331 = log1p(r67330);
        double r67332 = expm1(r67331);
        double r67333 = r67326 * r67332;
        double r67334 = fabs(r67333);
        double r67335 = sqrt(r67334);
        return r67335;
}

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 14.7

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

  3. Applied difference-of-squares14.7

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

  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. Using strategy rm

  8. Applied expm1-log1p-u0.0

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{a - b}}\right)\right)}\right|}\]

  9. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{a - b}}\right)\right)\right|}\]

Reproduce

herbie shell --seed 2020056 +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)))))