Average Error: 0.2 → 0.0
Time: 8.0s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9228484514852318:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(1 + \sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 9228484514852318:\\
\;\;\;\;\frac{x}{\sqrt[3]{{\left(1 + \sqrt{x + 1}\right)}^{3}}}\\

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

\end{array}
double code(double x) {
	return (x / (1.0 + sqrt((x + 1.0))));
}
↓
double code(double x) {
	double VAR;
	if ((x <= 9228484514852318.0)) {
		VAR = (x / cbrt(pow((1.0 + sqrt((x + 1.0))), 3.0)));
	} else {
		VAR = (sqrt(x) * (sqrt(x) / (1.0 + sqrt((x + 1.0)))));
	}
	return VAR;
}

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 < 9228484514852318.0

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    if 9228484514852318.0 < 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-sqrt0.1

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 9228484514852318:\\ \;\;\;\;\frac{x}{\sqrt[3]{{\left(1 + \sqrt{x + 1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{\sqrt{x}}{1 + \sqrt{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1 (sqrt (+ x 1)))))