[build system]: Fix USE_SYSTEM_LIBM=1 build (#37832)

Make.inc

@@ -610,13 +610,15 @@ endif
 # On Windows, we want shared library files to end up in $(build_bindir), instead of $(build_libdir)
 # We also don't really have a private bindir on windows right now, due to lack of RPATH.
 ifeq ($(OS),WINNT)
-build_shlibdir := $(build_bindir)
 shlibdir := $(bindir)
 private_shlibdir := $(bindir)
+build_shlibdir := $(build_bindir)
+build_private_shlibdir := $(build_bindir)
 else
-build_shlibdir := $(build_libdir)
 shlibdir := $(libdir)
 private_shlibdir := $(private_libdir)
+build_shlibdir := $(build_libdir)
+build_private_shlibdir := $(build_private_libdir)
 endif
 
 # If we're on windows, don't do versioned shared libraries.  If we're on OSX,
@@ -1489,21 +1491,25 @@ LIBGCC_NAME := libgcc_s_sjlj-1.$(SHLIB_EXT)
 else
 LIBGCC_NAME := libgcc_s_seh-1.$(SHLIB_EXT)
 endif
-LIBOPENLIBM_NAME := libopenlibm.$(SHLIB_EXT)
 endif
 ifeq ($(OS),Darwin)
 LIBGCC_NAME := libgcc_s.1.$(SHLIB_EXT)
-LIBOPENLIBM_NAME := libopenlibm.3.$(SHLIB_EXT)
 endif
 ifneq ($(findstring $(OS),Linux FreeBSD),)
 LIBGCC_NAME := libgcc_s.$(SHLIB_EXT).1
-LIBOPENLIBM_NAME := libopenlibm.$(SHLIB_EXT).3
 endif
 
+
 LIBGCC_BUILD_DEPLIB := $(call dep_lib_path,$(build_bindir),$(build_shlibdir)/$(LIBGCC_NAME))
 LIBGCC_INSTALL_DEPLIB := $(call dep_lib_path,$(bindir),$(private_shlibdir)/$(LIBGCC_NAME))
-LIBOPENLIBM_BUILD_DEPLIB := $(call dep_lib_path,$(build_bindir),$(build_shlibdir)/$(LIBOPENLIBM_NAME))
-LIBOPENLIBM_INSTALL_DEPLIB := $(call dep_lib_path,$(bindir),$(private_shlibdir)/$(LIBOPENLIBM_NAME))
+
+# USE_SYSTEM_LIBM causes it to get symlinked into build_private_shlibdir
+ifeq ($(USE_SYSTEM_LIBM),1)
+LIBM_BUILD_DEPLIB := $(call dep_lib_path,$(build_bindir),$(build_private_shlibdir)/$(LIBMNAME).$(SHLIB_EXT))
+else
+LIBM_BUILD_DEPLIB := $(call dep_lib_path,$(build_bindir),$(build_shlibdir)/$(LIBMNAME).$(SHLIB_EXT))
+endif
+LIBM_INSTALL_DEPLIB := $(call dep_lib_path,$(bindir),$(private_shlibdir)/$(LIBMNAME).$(SHLIB_EXT))
 
 # We list:
 #  * libgcc_s, because FreeBSD needs to load ours, not the system one.
@@ -1514,10 +1520,10 @@ LIBOPENLIBM_INSTALL_DEPLIB := $(call dep_lib_path,$(bindir),$(private_shlibdir)/
 #  * debug builds must link against libjuliadebug, not libjulia
 #  * install time relative paths are not equal to build time relative paths (../lib vs. ../lib/julia)
 # That second point will no longer be true for most deps once they are placed within Artifacts directories.
-LOADER_BUILD_DEP_LIBS = $(LIBGCC_BUILD_DEPLIB):$(LIBOPENLIBM_BUILD_DEPLIB):$(LIBJULIA_BUILD_DEPLIB)
-LOADER_DEBUG_BUILD_DEP_LIBS = $(LIBGCC_BUILD_DEPLIB):$(LIBOPENLIBM_BUILD_DEPLIB):$(LIBJULIA_DEBUG_BUILD_DEPLIB)
-LOADER_INSTALL_DEP_LIBS = $(LIBGCC_INSTALL_DEPLIB):$(LIBOPENLIBM_INSTALL_DEPLIB):$(LIBJULIA_INSTALL_DEPLIB)
-LOADER_DEBUG_INSTALL_DEP_LIBS = $(LIBGCC_INSTALL_DEPLIB):$(LIBOPENLIBM_INSTALL_DEPLIB):$(LIBJULIA_DEBUG_INSTALL_DEPLIB)
+LOADER_BUILD_DEP_LIBS = $(LIBGCC_BUILD_DEPLIB):$(LIBM_BUILD_DEPLIB):$(LIBJULIA_BUILD_DEPLIB)
+LOADER_DEBUG_BUILD_DEP_LIBS = $(LIBGCC_BUILD_DEPLIB):$(LIBM_BUILD_DEPLIB):$(LIBJULIA_DEBUG_BUILD_DEPLIB)
+LOADER_INSTALL_DEP_LIBS = $(LIBGCC_INSTALL_DEPLIB):$(LIBM_INSTALL_DEPLIB):$(LIBJULIA_INSTALL_DEPLIB)
+LOADER_DEBUG_INSTALL_DEP_LIBS = $(LIBGCC_INSTALL_DEPLIB):$(LIBM_INSTALL_DEPLIB):$(LIBJULIA_DEBUG_INSTALL_DEPLIB)
 
 # Colors for make
 ifndef VERBOSE