Average Error: 0.2 → 0.0
Time: 18.6s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 16902064928340788:\\ \;\;\;\;\frac{\left(\left(\left(1 + x\right) - 1 \cdot \sqrt{1 + x}\right) + 1 \cdot 1\right) \cdot x}{\sqrt{1 + x} \cdot \left(1 + x\right) + 1 \cdot \left(1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{1 + x}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 16902064928340788:\\
\;\;\;\;\frac{\left(\left(\left(1 + x\right) - 1 \cdot \sqrt{1 + x}\right) + 1 \cdot 1\right) \cdot x}{\sqrt{1 + x} \cdot \left(1 + x\right) + 1 \cdot \left(1 \cdot 1\right)}\\

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

\end{array}
double f(double x) {
        double r6075150 = x;
        double r6075151 = 1.0;
        double r6075152 = r6075150 + r6075151;
        double r6075153 = sqrt(r6075152);
        double r6075154 = r6075151 + r6075153;
        double r6075155 = r6075150 / r6075154;
        return r6075155;
}
↓
double f(double x) {
        double r6075156 = x;
        double r6075157 = 16902064928340788.0;
        bool r6075158 = r6075156 <= r6075157;
        double r6075159 = 1.0;
        double r6075160 = r6075159 + r6075156;
        double r6075161 = sqrt(r6075160);
        double r6075162 = r6075159 * r6075161;
        double r6075163 = r6075160 - r6075162;
        double r6075164 = r6075159 * r6075159;
        double r6075165 = r6075163 + r6075164;
        double r6075166 = r6075165 * r6075156;
        double r6075167 = r6075161 * r6075160;
        double r6075168 = r6075159 * r6075164;
        double r6075169 = r6075167 + r6075168;
        double r6075170 = r6075166 / r6075169;
        double r6075171 = sqrt(r6075156);
        double r6075172 = r6075159 + r6075161;
        double r6075173 = r6075171 / r6075172;
        double r6075174 = r6075171 * r6075173;
        double r6075175 = r6075158 ? r6075170 : r6075174;
        return r6075175;
}

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 < 16902064928340788.0

    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

      \[\leadsto \color{blue}{\frac{x}{1 \cdot \left(1 \cdot 1\right) + \left(1 + x\right) \cdot \sqrt{1 + x}}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\]
    6. Using strategy rm

    7. Applied associate-*l/0.0

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

    8. Simplified0.0

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

    if 16902064928340788.0 < x

    1. Initial program 0.5

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

    3. Applied *-un-lft-identity0.5

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 16902064928340788:\\ \;\;\;\;\frac{\left(\left(\left(1 + x\right) - 1 \cdot \sqrt{1 + x}\right) + 1 \cdot 1\right) \cdot x}{\sqrt{1 + x} \cdot \left(1 + x\right) + 1 \cdot \left(1 \cdot 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{1 + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))