Average Error: 0.2 → 0.0
Time: 4.1s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.7359094076016717 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.7359094076016717 \cdot 10^{-13}:\\
\;\;\;\;\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)\\

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

\end{array}
double f(double x) {
        double r117694 = x;
        double r117695 = 1.0;
        double r117696 = r117694 + r117695;
        double r117697 = sqrt(r117696);
        double r117698 = r117695 + r117697;
        double r117699 = r117694 / r117698;
        return r117699;
}
↓
double f(double x) {
        double r117700 = x;
        double r117701 = 2.735909407601672e-13;
        bool r117702 = r117700 <= r117701;
        double r117703 = 1.0;
        double r117704 = 3.0;
        double r117705 = pow(r117703, r117704);
        double r117706 = r117700 + r117703;
        double r117707 = sqrt(r117706);
        double r117708 = pow(r117707, r117704);
        double r117709 = r117705 + r117708;
        double r117710 = r117700 / r117709;
        double r117711 = r117703 * r117703;
        double r117712 = r117707 * r117707;
        double r117713 = r117703 * r117707;
        double r117714 = r117712 - r117713;
        double r117715 = r117711 + r117714;
        double r117716 = r117710 * r117715;
        double r117717 = sqrt(r117700);
        double r117718 = r117703 + r117707;
        double r117719 = r117717 / r117718;
        double r117720 = r117717 * r117719;
        double r117721 = r117702 ? r117716 : r117720;
        return r117721;
}

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 < 2.735909407601672e-13

    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

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

    if 2.735909407601672e-13 < 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-sqrt0.2

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

    5. Applied times-frac0.1

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

    6. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.7359094076016717 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))