Average Error: 0.2 → 0.0
Time: 14.0s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \le 1012629.646108792512677609920501708984375:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \le 1012629.646108792512677609920501708984375:\\
\;\;\;\;\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 r112645 = x;
        double r112646 = 1.0;
        double r112647 = r112645 + r112646;
        double r112648 = sqrt(r112647);
        double r112649 = r112646 + r112648;
        double r112650 = r112645 / r112649;
        return r112650;
}
↓
double f(double x) {
        double r112651 = x;
        double r112652 = 1.0;
        double r112653 = r112651 + r112652;
        double r112654 = sqrt(r112653);
        double r112655 = r112652 + r112654;
        double r112656 = r112651 / r112655;
        double r112657 = 1012629.6461087925;
        bool r112658 = r112656 <= r112657;
        double r112659 = 3.0;
        double r112660 = pow(r112652, r112659);
        double r112661 = pow(r112654, r112659);
        double r112662 = r112660 + r112661;
        double r112663 = r112651 / r112662;
        double r112664 = r112652 * r112652;
        double r112665 = r112654 * r112654;
        double r112666 = r112652 * r112654;
        double r112667 = r112665 - r112666;
        double r112668 = r112664 + r112667;
        double r112669 = r112663 * r112668;
        double r112670 = sqrt(r112651);
        double r112671 = r112670 / r112655;
        double r112672 = r112670 * r112671;
        double r112673 = r112658 ? r112669 : r112672;
        return r112673;
}

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 (+ 1.0 (sqrt (+ x 1.0)))) < 1012629.6461087925

    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 1012629.6461087925 < (/ x (+ 1.0 (sqrt (+ x 1.0))))

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \le 1012629.646108792512677609920501708984375:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))