Average Error: 0.2 → 0.0
Time: 12.9s
Precision: 64
\[\frac{x}{1 + \sqrt{x + 1}}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 44800399799935003606741155840:\\ \;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 + x} \cdot 1\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{1 + x}}{\sqrt{x}}}\\ \end{array}\]
\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \le 44800399799935003606741155840:\\
\;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 + x} \cdot 1\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\

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

\end{array}
double f(double x) {
        double r4320475 = x;
        double r4320476 = 1.0;
        double r4320477 = r4320475 + r4320476;
        double r4320478 = sqrt(r4320477);
        double r4320479 = r4320476 + r4320478;
        double r4320480 = r4320475 / r4320479;
        return r4320480;
}
↓
double f(double x) {
        double r4320481 = x;
        double r4320482 = 4.4800399799935e+28;
        bool r4320483 = r4320481 <= r4320482;
        double r4320484 = 1.0;
        double r4320485 = r4320484 * r4320484;
        double r4320486 = r4320484 + r4320481;
        double r4320487 = sqrt(r4320486);
        double r4320488 = r4320487 * r4320487;
        double r4320489 = r4320487 * r4320484;
        double r4320490 = r4320488 - r4320489;
        double r4320491 = r4320485 + r4320490;
        double r4320492 = r4320484 * r4320485;
        double r4320493 = r4320487 * r4320486;
        double r4320494 = r4320492 + r4320493;
        double r4320495 = r4320481 / r4320494;
        double r4320496 = r4320491 * r4320495;
        double r4320497 = sqrt(r4320481);
        double r4320498 = r4320484 + r4320487;
        double r4320499 = r4320498 / r4320497;
        double r4320500 = r4320497 / r4320499;
        double r4320501 = r4320483 ? r4320496 : r4320500;
        return r4320501;
}

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 < 4.4800399799935e+28

    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

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

    5. Simplified0.0

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

    if 4.4800399799935e+28 < x

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 44800399799935003606741155840:\\ \;\;\;\;\left(1 \cdot 1 + \left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{1 + x} \cdot 1\right)\right) \cdot \frac{x}{1 \cdot \left(1 \cdot 1\right) + \sqrt{1 + x} \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{1 + \sqrt{1 + x}}{\sqrt{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))