Average Error: 0.2 → 0.3
Time: 4.9s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\frac{1}{\frac{1 + \sqrt{x + 1}}{x}}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\frac{1}{\frac{1 + \sqrt{x + 1}}{x}}
double f(double x) {
        double r136605 = x;
        double r136606 = 1.0;
        double r136607 = r136605 + r136606;
        double r136608 = sqrt(r136607);
        double r136609 = r136606 + r136608;
        double r136610 = r136605 / r136609;
        return r136610;
}
↓
double f(double x) {
        double r136611 = 1.0;
        double r136612 = 1.0;
        double r136613 = x;
        double r136614 = r136613 + r136612;
        double r136615 = sqrt(r136614);
        double r136616 = r136612 + r136615;
        double r136617 = r136616 / r136613;
        double r136618 = r136611 / r136617;
        return r136618;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{x}{1 + \sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied clear-num0.3

    \[\leadsto \color{blue}{\frac{1}{\frac{1 + \sqrt{x + 1}}{x}}}\]

  4. Final simplification0.3

    \[\leadsto \frac{1}{\frac{1 + \sqrt{x + 1}}{x}}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))