Scale modified Bessel functions to then unscale later So I have some variables $\,x_{1},\, x_{2},\, \nu\, =\, 12.654,\, 13.487,\, 0\,$ and the following function:
$\dfrac{(x_{1}\cdot(-BesselK(\nu,x_{1}\cdot125))\cdot BesselI(\nu,x_{2}\cdot125))-(x_{2}\cdot BesselK(\nu+1,x_{1}\cdot 125)\cdot BesselI(\nu+1,x_{2}\cdot 125))}{\dfrac{-1}{125\cdot x_{1}}}$
What's my problem? Well even though the final output is a reasonable number, the output of the Bessel functions are either super-huge or super-small and the calculation requires quad-precision (which is very sluggish). I need to scale down the value going into the Bessel function but I need to be able to get the end result to be the same as if I used quad-precision.
Thank you.

If $125 x_1$ and $125 x_2$ are fairly large compared to $\nu^2$, you might want to use the asymptotic forms of these Bessel functions:
$$ \eqalign{K_\nu(x) &\sim \sqrt{\frac{\pi}{2x}} e^{-x} \left(1+{\frac {4\,{\nu}^{2}-1^2}{1!\; 8x}}+{\frac {(4 \nu^2-1^2)(4 \nu^2-3^2)}{2!\; (8x)^{2}}}\right.\cr &\left.+{\frac {(4 \nu^2 - 1^2)(4 \nu^2 - 3^2)(4 \nu^2 - 5^2)}{3! \;(8x)^{3}}}+{\frac {(4 \nu^2 - 1^2)(4 \nu^2 - 3^2)(4 \nu^2 - 5^2)(4 \nu^2 - 7^2)}{4! \;(8x)^{4}}}+\ldots 
\right)\cr
I_\nu(x) &\sim  \frac{e^x}{\sqrt{2\pi x}}\left(1-{\frac {4\,{\nu}^{2}-1^2}{1!\; 8x}}+{\frac {(4 \nu^2-1^2)(4 \nu^2-3^2)}{2!\; (8x)^{2}}}\right.\cr &\left.-{\frac {(4 \nu^2 - 1^2)(4 \nu^2 - 3^2)(4 \nu^2 - 5^2)}{3! \;(8x)^{3}}}+{\frac {(4 \nu^2 - 1^2)(4 \nu^2 - 3^2)(4 \nu^2 - 5^2)(4 \nu^2 - 7^2)}{4! \;(8x)^{4}}}-\ldots 
\right)\cr}$$
