Average Error: 0.2 → 0.0
Time: 34.8s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1614.061223923244597244774922728538513184:\\ \;\;\;\;\frac{x}{1 + \sqrt[3]{\sqrt{1 + x} \cdot \left(1 + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{1 + \sqrt{1 + x}} \cdot \sqrt{x}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1614.061223923244597244774922728538513184:\\
\;\;\;\;\frac{x}{1 + \sqrt[3]{\sqrt{1 + x} \cdot \left(1 + x\right)}}\\

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

\end{array}
double f(double x) {
        double r3865574 = x;
        double r3865575 = 1.0;
        double r3865576 = r3865574 + r3865575;
        double r3865577 = sqrt(r3865576);
        double r3865578 = r3865575 + r3865577;
        double r3865579 = r3865574 / r3865578;
        return r3865579;
}
↓
double f(double x) {
        double r3865580 = x;
        double r3865581 = 1614.0612239232446;
        bool r3865582 = r3865580 <= r3865581;
        double r3865583 = 1.0;
        double r3865584 = r3865583 + r3865580;
        double r3865585 = sqrt(r3865584);
        double r3865586 = r3865585 * r3865584;
        double r3865587 = cbrt(r3865586);
        double r3865588 = r3865583 + r3865587;
        double r3865589 = r3865580 / r3865588;
        double r3865590 = sqrt(r3865580);
        double r3865591 = r3865583 + r3865585;
        double r3865592 = r3865590 / r3865591;
        double r3865593 = r3865592 * r3865590;
        double r3865594 = r3865582 ? r3865589 : r3865593;
        return r3865594;
}

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 < 1614.0612239232446

    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

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

    if 1614.0612239232446 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1614.061223923244597244774922728538513184:\\ \;\;\;\;\frac{x}{1 + \sqrt[3]{\sqrt{1 + x} \cdot \left(1 + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{1 + \sqrt{1 + x}} \cdot \sqrt{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))