Support java compilation sharding for target side.

Add "LOCAL_JAVAC_SHARD_SIZE" to represent the number of java
source path entries in each shard.

Sharding is not allowed when "LOCAL_JAVAC_SHARD_SIZE" and
"LOCAL_JAR_PROCESSORS" are both enabled.

Limitation:
1. 0 <= LOCAL_JAVAC_SHARD_SIZE <= 8192
1. 0 < NUM(java_sources) <= 8192
2. 0 <= NUM(shards) <= 100

Performance Compare:
<Unsharded Build------------------------------------------------>
1. Build from clean state
rm -r -f out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/
&& time m
out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar

real	1m2.720s user	5m26.604s sys	0m39.552s

2. Incremental build
m out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar &&
touch frameworks/base/core/java/android/net/http/HttpResponseCache.java && time m
out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar

real	0m37.586s user	5m47.804s sys	0m50.388s

<Sharded Build-------------------------------------------------->
1. Build from clean state
rm -r -f out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/
&& time m
out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar

Javac Shard Size: 50----real	1m10.163s user	25m59.008s sys	1m58.460s
Javac Shard Size: 100---real	1m2.115s user	21m3.600s sys	1m15.964s
Javac Shard Size: 150---real	0m59.520s user	18m10.544s sys	1m12.628s
Javac Shard Size: 200---real	0m56.894s user	15m39.244s sys	1m11.608s
Javac Shard Size: 250---real	0m55.991s user	14m38.716s sys	1m2.292s
Javac Shard Size: 300---real	0m55.114s user	13m6.568s sys	1m8.200s
Javac Shard Size: 350---real	0m53.144s user	12m7.740s sys	1m3.836s
Javac Shard Size: 400---real	0m54.929s user	12m9.324s sys	1m4.340s
Javac Shard Size: 450---real	1m30.194s user	25m31.468s sys	1m52.416s
Javac Shard Size: 500---real	0m53.976s user	10m35.500s sys	0m55.160s

2. Incremental build
m out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar &&
touch frameworks/base/core/java/android/net/http/HttpResponseCache.java && time m
out/target/common/obj/JAVA_LIBRARIES/framework_intermediates/classes-full-debug.jar

Javac Shard Size: 50-----real	0m16.322s user	1m8.648s sys	0m31.700s
Javac Shard Size: 100----real	0m16.163s user	1m22.932s sys	0m29.440s
Javac Shard Size: 150----real	0m16.611s user	1m37.828s sys	0m21.168s
Javac Shard Size: 200----real	0m16.936s user	1m49.248s sys	0m28.636s
Javac Shard Size: 250----real	0m17.509s user	1m54.944s sys	0m32.768s
Javac Shard Size: 300----real	0m18.868s user	1m54.088s sys	0m28.824s
Javac Shard Size: 350----real	0m17.629s user	1m54.108s sys	0m31.056s
Javac Shard Size: 400----real	0m18.658s user	2m7.712s sys	0m30.636s
Javac Shard Size: 450----real	0m18.874s user	2m8.808s sys	0m33.540s
Javac Shard Size: 500----real	0m19.432s user	2m24.400s sys	0m30.368s

time m nothing:
real 0m5.799s user 0m7.236s sys 0m3.068s

Test: m clean && m -j checkbuild
Bug: b/67424047
Change-Id: Id0766d2b7de7c4546d29bbc7f8a0dd0e4b9ad45b
diff --git a/core/math.mk b/core/math.mk
index 047d046..57cb681 100644
--- a/core/math.mk
+++ b/core/math.mk
@@ -42,6 +42,11 @@
 $(if $(call math_is_number,$(1)),,$(error Only positive integers <= 100 are supported (not $(1))))
 endef
 
+# return a list containing integers ranging from [$(1),$(2)]
+define int_range_list
+$(call _math_check_valid,$(1))$(call _math_check_valid,$(2))$(wordlist $(1),$(2),$(__MATH_NUMBERS))
+endef
+
 #$(call _math_check_valid,0)
 #$(call _math_check_valid,1)
 #$(call _math_check_valid,100)
@@ -75,3 +80,60 @@
 define inc_and_print
 $(strip $(eval $(1) := $($(1)) .)$(words $($(1))))
 endef
+
+_INT_LIMIT_WORDS := $(foreach a,x x,$(foreach b,x x x x x x x x x x x x x x x x,\
+  $(foreach c,x x x x x x x x x x x x x x x x,x x x x x x x x x x x x x x x x)))
+
+define _int_encode
+$(if $(filter $(words x $(_INT_LIMIT_WORDS)),$(words $(wordlist 1,$(1),x $(_INT_LIMIT_WORDS)))),\
+  $(call pretty-error,integer greater than $(words $(_INT_LIMIT_WORDS)) is not supported!),\
+    $(wordlist 1,$(1),$(_INT_LIMIT_WORDS)))
+endef
+
+# _int_max returns the maximum of the two arguments
+# input: two (x) lists; output: one (x) list
+# integer cannot be passed in directly. It has to be converted using _int_encode.
+define _int_max
+$(subst xx,x,$(join $(1),$(2)))
+endef
+
+# first argument is greater than second argument
+# output: non-empty if true
+# integer cannot be passed in directly. It has to be converted using _int_encode.
+define _int_greater-than
+$(filter-out $(words $(2)),$(words $(call _int_max,$(1),$(2))))
+endef
+
+# first argument equals to second argument
+# output: non-empty if true
+# integer cannot be passed in directly. It has to be converted using _int_encode.
+define _int_equal
+$(filter $(words $(1)),$(words $(2)))
+endef
+
+# first argument is greater than or equal to second argument
+# output: non-empty if true
+# integer cannot be passed in directly. It has to be converted using _int_encode.
+define _int_greater-or-equal
+$(call _int_greater-than,$(1),$(2))$(call _int_equal,$(1),$(2))
+endef
+
+define int_plus
+$(words $(call _int_encode,$(1)) $(call _int_encode,$(2)))
+endef
+
+define int_subtract
+$(if $(call _int_greater-or-equal,$(call _int_encode,$(1)),$(call _int_encode,$(2))),\
+  $(words $(filter-out xx,$(join $(call _int_encode,$(1)),$(call _int_encode,$(2))))),\
+    $(call pretty-error,$(1) subtract underflow $(2)))
+endef
+
+define int_multiply
+$(words $(foreach a,$(call _int_encode,$(1)),$(call _int_encode,$(2))))
+endef
+
+define int_divide
+$(if $(filter 0,$(2)),$(call pretty-error,division by zero is not allowed!),$(strip \
+  $(if $(call _int_greater-or-equal,$(call _int_encode,$(1)),$(call _int_encode,$(2))), \
+    $(call int_plus,$(call int_divide,$(call int_subtract,$(1),$(2)),$(2)),1),0)))
+endef