Average Error: 0.2 → 0.0
Time: 37.4s
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 r5524466 = x;
        double r5524467 = 1.0;
        double r5524468 = r5524466 + r5524467;
        double r5524469 = sqrt(r5524468);
        double r5524470 = r5524467 + r5524469;
        double r5524471 = r5524466 / r5524470;
        return r5524471;
}
↓
double f(double x) {
        double r5524472 = x;
        double r5524473 = 1614.0612239232446;
        bool r5524474 = r5524472 <= r5524473;
        double r5524475 = 1.0;
        double r5524476 = r5524475 + r5524472;
        double r5524477 = sqrt(r5524476);
        double r5524478 = r5524477 * r5524476;
        double r5524479 = cbrt(r5524478);
        double r5524480 = r5524475 + r5524479;
        double r5524481 = r5524472 / r5524480;
        double r5524482 = sqrt(r5524472);
        double r5524483 = r5524475 + r5524477;
        double r5524484 = r5524482 / r5524483;
        double r5524485 = r5524484 * r5524482;
        double r5524486 = r5524474 ? r5524481 : r5524485;
        return r5524486;
}

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 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))