Average Error: 0.2 → 0.0
Time: 10.9s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \le 1012629.646108792512677609920501708984375:\\ \;\;\;\;\frac{x}{{1}^{3} + {\left(\sqrt{x + 1}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - 1 \cdot \sqrt{x + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{x + 1}}{\sqrt{x}}}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{x + 1}}{\sqrt{x}}}\\

\end{array}
double f(double x) {
        double r93139 = x;
        double r93140 = 1.0;
        double r93141 = r93139 + r93140;
        double r93142 = sqrt(r93141);
        double r93143 = r93140 + r93142;
        double r93144 = r93139 / r93143;
        return r93144;
}
↓
double f(double x) {
        double r93145 = x;
        double r93146 = 1.0;
        double r93147 = r93145 + r93146;
        double r93148 = sqrt(r93147);
        double r93149 = r93146 + r93148;
        double r93150 = r93145 / r93149;
        double r93151 = 1012629.6461087925;
        bool r93152 = r93150 <= r93151;
        double r93153 = 3.0;
        double r93154 = pow(r93146, r93153);
        double r93155 = pow(r93148, r93153);
        double r93156 = r93154 + r93155;
        double r93157 = r93145 / r93156;
        double r93158 = r93146 * r93146;
        double r93159 = r93148 * r93148;
        double r93160 = r93146 * r93148;
        double r93161 = r93159 - r93160;
        double r93162 = r93158 + r93161;
        double r93163 = r93157 * r93162;
        double r93164 = sqrt(r93145);
        double r93165 = r93149 / r93164;
        double r93166 = r93164 / r93165;
        double r93167 = r93152 ? r93163 : r93166;
        return r93167;
}

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 add-sqr-sqrt0.1

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

    4. Applied associate-/l*0.0

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

  4. Final simplification0.0

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

Reproduce

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