Average Error: 15.0 → 0.0
Time: 17.2s
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 r93722 = a;
        double r93723 = r93722 * r93722;
        double r93724 = b;
        double r93725 = r93724 * r93724;
        double r93726 = r93723 - r93725;
        double r93727 = r93726 / r93723;
        double r93728 = fabs(r93727);
        double r93729 = sqrt(r93728);
        return r93729;
}
↓
double f(double a, double b) {
        double r93730 = a;
        double r93731 = b;
        double r93732 = r93730 + r93731;
        double r93733 = r93732 / r93730;
        double r93734 = r93730 - r93731;
        double r93735 = r93734 / r93730;
        double r93736 = r93733 * r93735;
        double r93737 = fabs(r93736);
        double r93738 = sqrt(r93737);
        double r93739 = log1p(r93738);
        double r93740 = expm1(r93739);
        return r93740;
}

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 15.0

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

  3. Applied difference-of-squares15.0

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

  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 2020047 +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)))))