Average Error: 0.2 → 0.0
Time: 4.8s
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 r117646 = x;
        double r117647 = 1.0;
        double r117648 = r117646 + r117647;
        double r117649 = sqrt(r117648);
        double r117650 = r117647 + r117649;
        double r117651 = r117646 / r117650;
        return r117651;
}
↓
double f(double x) {
        double r117652 = x;
        double r117653 = 8.626548691723187e-11;
        bool r117654 = r117652 <= r117653;
        double r117655 = 1.0;
        double r117656 = r117652 + r117655;
        double r117657 = sqrt(r117656);
        double r117658 = 3.0;
        double r117659 = pow(r117657, r117658);
        double r117660 = cbrt(r117659);
        double r117661 = r117655 + r117660;
        double r117662 = r117652 / r117661;
        double r117663 = sqrt(r117652);
        double r117664 = r117657 + r117655;
        double r117665 = r117663 / r117664;
        double r117666 = r117663 * r117665;
        double r117667 = r117654 ? r117662 : r117666;
        return r117667;
}

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