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I built Lua 5.3 rc3 under Cygwin by adding the following build option to the Makefile in the src folder:

cygwin:
	$(MAKE) "LUA_A=cyglua-5.3.dll.a" "LUA_T=lua.exe" \
	"AR=$(CC) -shared -o" "RANLIB=strip --strip-unneeded" \
	"MYCFLAGS=-DLUA_USE_LINUX -DLUA_BUILD_AS_DLL" \
	"MYLIBS=-lreadline -lhistory" "MYLDFLAGS=-s" lua.exe

Someone had posted this online a while ago when they were trying to build Lua 5.1 and 5.2 under Cygwin.  I just changed the version number to build a Lua 5.3 library .  I also had to add "cygwin" to the platform list (PLATS) in this Makefile as well as the top level Makefile.  I built the Lua 5.3 rc3 library by issuing the command "make cygwin" at the top level and I didn't change any settings in any of the .h files or anything else in the Makefile (besides adding the Cygwin support). 
 
I modified my Lua 5.1 application and the associated C libraries so that they would run under Lua 5.3 and everything worked fine when I ran them.

I then downloaded the Lua 5.3 rc3 test suite, however, when I ran the basic tests (lua -e"_U=true" all.lua), the following error message appeared:

***** FILE 'math.lua'*****
.testing numbers and math lib
64-bit integers, 53-bit (mantissa) floats
lua53: math.lua:155: assertion failed!
stack traceback:
        [C]: in function 'assert'
        math.lua:155: in main chunk
        (...tail calls...)
        all.lua:177: in main chunk
        [C]: in ?
.>>> closing state <<<

Looking at the math.lua source code, here is the section where the assert occurs:

-- negative exponents
do
  assert(2^-3 == 1 / 2^3)
  assert(eq((-3)^-3, 1 / (-3)^3))
  for i = -3, 3 do    -- variables avoid constant folding
    for j = -3, 3 do
      assert(eq(i^j, 1 / i^(-j)))   <-- THIS IS THE ASSERT AT LINE 155
     end
  end
end

-----Original Message-----
From: lua-l-bounces@lists.lua.org [mailto:lua-l-bounces@lists.lua.org] On Behalf Of Luiz Henrique de Figueiredo
Sent: Friday, January 02, 2015 8:37 AM
To: lua-l@lists.lua.org
Subject: [ANN] Lua 5.3.0 (rc3) now available

Lua 5.3.0 (rc3) is now available for testing at
	http://www.lua.org/work/lua-5.3.0-rc3.tar.gz

MD5	5dd17d29fbb70139d315e2cea751a128  -
SHA1	83e26c3c21abe7c309c5e9df08b6a7a7db85ef2a  -

This is a release candidate for the final release of Lua 5.3.0.

A few things have changed since beta that we'd like to test in the wild.
In particular, we made some changes in luaconf.h and in the Makefile that we'd like to test. Please try compiling the current code in as many platforms as possible. We expect the compilation will go smoothly as usual but please report any warnings or other glitches.

We'd also like feedback on the documentation:
	http://www.lua.org/work/doc/readme.html
	http://www.lua.org/work/doc/contents.html
	http://www.lua.org/work/doc/manual.html

Finally, for those of you into this, please test luac -l -l on your scripts, just to make sure we haven't missed anything with the new VM instructions etc.

The main change in Lua 5.3.0 is the introduction of integers. See also
	http://www.lua.org/work/doc/#changes

The complete diffs are available at
	http://www.lua.org/work/diffs-lua-5.3.0-rc2-rc3.txt

An updated test suite is available at
	http://www.lua.org/work/lua-5.3.0-rc3-tests.tar.gz

All feedback welcome. Thanks.
--lhf