Average Error: 0.2 → 0.0
Time: 4.3s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 8.62654869172318706 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{1 + \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{x + 1} + 1}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 8.62654869172318706 \cdot 10^{-11}:\\
\;\;\;\;\frac{x}{1 + \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{3}}}\\

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

\end{array}
double f(double x) {
        double r89753 = x;
        double r89754 = 1.0;
        double r89755 = r89753 + r89754;
        double r89756 = sqrt(r89755);
        double r89757 = r89754 + r89756;
        double r89758 = r89753 / r89757;
        return r89758;
}
↓
double f(double x) {
        double r89759 = x;
        double r89760 = 8.626548691723187e-11;
        bool r89761 = r89759 <= r89760;
        double r89762 = 1.0;
        double r89763 = r89759 + r89762;
        double r89764 = sqrt(r89763);
        double r89765 = 3.0;
        double r89766 = pow(r89764, r89765);
        double r89767 = cbrt(r89766);
        double r89768 = r89762 + r89767;
        double r89769 = r89759 / r89768;
        double r89770 = sqrt(r89759);
        double r89771 = r89764 + r89762;
        double r89772 = r89770 / r89771;
        double r89773 = r89770 * r89772;
        double r89774 = r89761 ? r89769 : r89773;
        return r89774;
}

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 < 8.626548691723187e-11

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Simplified0.0

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

    if 8.626548691723187e-11 < x

    1. Initial program 0.5

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

    3. Applied add-cbrt-cube20.9

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

    4. Simplified20.9

      \[\leadsto \frac{x}{1 + \sqrt[3]{\color{blue}{{\left(\sqrt{x + 1}\right)}^{3}}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity20.9

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

    7. Applied add-sqr-sqrt20.8

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

    8. Applied times-frac20.8

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

    9. Simplified20.8

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

    10. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 8.62654869172318706 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{1 + \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{\sqrt{x + 1} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))