Average Error: 0.2 → 0.0
Time: 9.6s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 44800399799935003606741155840:\\ \;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - 1 \cdot \sqrt{1 + x}\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{1 + x}}{\sqrt{x}}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 44800399799935003606741155840:\\
\;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - 1 \cdot \sqrt{1 + x}\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{1 + x}}{\sqrt{x}}}\\

\end{array}
double f(double x) {
        double r5531802 = x;
        double r5531803 = 1.0;
        double r5531804 = r5531802 + r5531803;
        double r5531805 = sqrt(r5531804);
        double r5531806 = r5531803 + r5531805;
        double r5531807 = r5531802 / r5531806;
        return r5531807;
}
↓
double f(double x) {
        double r5531808 = x;
        double r5531809 = 4.4800399799935e+28;
        bool r5531810 = r5531808 <= r5531809;
        double r5531811 = 1.0;
        double r5531812 = r5531811 * r5531811;
        double r5531813 = r5531811 + r5531808;
        double r5531814 = sqrt(r5531813);
        double r5531815 = r5531814 * r5531814;
        double r5531816 = r5531811 * r5531814;
        double r5531817 = r5531815 - r5531816;
        double r5531818 = r5531812 + r5531817;
        double r5531819 = r5531811 * r5531812;
        double r5531820 = r5531814 * r5531813;
        double r5531821 = r5531819 + r5531820;
        double r5531822 = r5531808 / r5531821;
        double r5531823 = r5531818 * r5531822;
        double r5531824 = sqrt(r5531808);
        double r5531825 = r5531811 + r5531814;
        double r5531826 = r5531825 / r5531824;
        double r5531827 = r5531824 / r5531826;
        double r5531828 = r5531810 ? r5531823 : r5531827;
        return r5531828;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 4.4800399799935e+28

    1. Initial program 0.0

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

    3. Applied flip3-+0.0

      \[\leadsto \frac{x}{\color{blue}{\frac{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)}}}\]

    4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\left(1 \cdot 1\right) \cdot 1 + \left(1 + x\right) \cdot \sqrt{1 + x}}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]

    if 4.4800399799935e+28 < x

    1. Initial program 0.5

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1 + \sqrt{x + 1}}{\sqrt{x}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 44800399799935003606741155840:\\ \;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - 1 \cdot \sqrt{1 + x}\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{1 + x}}{\sqrt{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))