Average Error: 14.4 → 0.0
Time: 3.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)\right)\right|}
double f(double a, double b) {
        double r57998 = a;
        double r57999 = r57998 * r57998;
        double r58000 = b;
        double r58001 = r58000 * r58000;
        double r58002 = r57999 - r58001;
        double r58003 = r58002 / r57999;
        double r58004 = fabs(r58003);
        double r58005 = sqrt(r58004);
        return r58005;
}
↓
double f(double a, double b) {
        double r58006 = a;
        double r58007 = b;
        double r58008 = r58006 + r58007;
        double r58009 = r58008 / r58006;
        double r58010 = r58006 - r58007;
        double r58011 = r58010 / r58006;
        double r58012 = r58009 * r58011;
        double r58013 = log1p(r58012);
        double r58014 = expm1(r58013);
        double r58015 = fabs(r58014);
        double r58016 = sqrt(r58015);
        return r58016;
}

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 14.4

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

  3. Applied difference-of-squares14.4

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

  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

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +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)))))