Data.SBV.Examples.CodeGeneration.PopulationCount

Computing population-counts (number of set bits) and autimatically generating C code.

Reference: Slow but obviously correct

popCountSlow :: SWord64 -> SWord8

Given a 64-bit quantity, the simplest (and obvious) way to count the number of bits that are set in it is to simply walk through all the bits and add 1 to a running count. This is slow, as it requires 64 iterations, but is simple and easy to convince yourself that it is correct. For instance:

>>> popCountSlow 0x0123456789ABCDEF
32 :: SWord8

Faster: Using a look-up table

popCountFast :: SWord64 -> SWord8

Faster version. This is essentially the same algorithm, except we go 8 bits at a time instead of one by one, by using a precomputed table of population-count values for each byte. This algorithm loops only 8 times, and hence is at least 8 times more efficient.

pop8 :: [SWord8]

Look-up table, containing population counts for all possible 8-bit value, from 0 to 255. Note that we do not "hard-code" the values, but merely use the slow version to compute them.

Verification

We prove that popCountFast and popCountSlow are functionally equivalent. This is essential as we will automatically generate C code from popCountFast, and we would like to make sure that the fast version is correct with respect to the slower reference version.

fastPopCountIsCorrect :: SWord64 -> SBool

States the correctness of faster population-count algorithm, with respect to the reference slow version. (We use yices here as it's quite fast for this problem. Z3 seems to take much longer.) We have:

>>> proveWith yices fastPopCountIsCorrect
Q.E.D.

Code generation

genPopCountInC :: IO ()

Not only we can prove that faster version is correct, but we can also automatically generate C code to compute population-counts for us. This action will generate all the C files that you will need, including a driver program for test purposes.

Below is the generated header file for popCountFast:

>>> genPopCountInC
== BEGIN: "Makefile" ================
# Makefile for popCount. Automatically generated by SBV. Do not edit!

# include any user-defined .mk file in the current directory.
-include *.mk

CC=gcc
CCFLAGS?=-Wall -O3 -DNDEBUG -fomit-frame-pointer

all: popCount_driver

popCount.o: popCount.c popCount.h
	${CC} ${CCFLAGS} -c $< -o $@

popCount_driver.o: popCount_driver.c
	${CC} ${CCFLAGS} -c $< -o $@

popCount_driver: popCount.o popCount_driver.o
	${CC} ${CCFLAGS} $^ -o $@

clean:
	rm -f *.o

veryclean: clean
	rm -f popCount_driver
== END: "Makefile" ==================
== BEGIN: "popCount.h" ================

BLANKLINE #ifndef __popCount__HEADER_INCLUDED__ #define __popCount__HEADER_INCLUDED__ BLANKLINE #include inttypes.h #include stdint.h #include stdbool.h BLANKLINE

typedef bool SBool; BLANKLINE

typedef uint8_t SWord8 ; typedef uint16_t SWord16; typedef uint32_t SWord32; typedef uint64_t SWord64; BLANKLINE

typedef int8_t SInt8 ; typedef int16_t SInt16; typedef int32_t SInt32; typedef int64_t SInt64; BLANKLINE

SWord8 popCount(const SWord64 x); BLANKLINE #endif == END: popCount.h ================== == BEGIN: popCount_driver.c ================

BLANKLINE #include inttypes.h #include stdint.h #include stdbool.h #include stdio.h #include popCount.h BLANKLINE int main(void) { const SWord8 __result = popCount(0x1b02e143e4f0e0e5ULL); BLANKLINE printf(popCount(0x1b02e143e4f0e0e5ULL) = %PRIu8n, __result); BLANKLINE return 0; } == END: popCount_driver.c ================== == BEGIN: popCount.c ================

BLANKLINE #include inttypes.h #include stdint.h #include stdbool.h #include popCount.h BLANKLINE SWord8 popCount(const SWord64 x) { const SWord64 s0 = x; static const SWord8 table0[] = { 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8 }; const SWord64 s11 = s0 & 0x00000000000000ffULL; const SWord8 s12 = table0[s11]; const SWord64 s13 = s0 >> 8; const SWord64 s14 = 0x00000000000000ffULL & s13; const SWord8 s15 = table0[s14]; const SWord8 s16 = s12 + s15; const SWord64 s17 = s13 >> 8; const SWord64 s18 = 0x00000000000000ffULL & s17; const SWord8 s19 = table0[s18]; const SWord8 s20 = s16 + s19; const SWord64 s21 = s17 >> 8; const SWord64 s22 = 0x00000000000000ffULL & s21; const SWord8 s23 = table0[s22]; const SWord8 s24 = s20 + s23; const SWord64 s25 = s21 >> 8; const SWord64 s26 = 0x00000000000000ffULL & s25; const SWord8 s27 = table0[s26]; const SWord8 s28 = s24 + s27; const SWord64 s29 = s25 >> 8; const SWord64 s30 = 0x00000000000000ffULL & s29; const SWord8 s31 = table0[s30]; const SWord8 s32 = s28 + s31; const SWord64 s33 = s29 >> 8; const SWord64 s34 = 0x00000000000000ffULL & s33; const SWord8 s35 = table0[s34]; const SWord8 s36 = s32 + s35; const SWord64 s37 = s33 >> 8; const SWord64 s38 = 0x00000000000000ffULL & s37; const SWord8 s39 = table0[s38]; const SWord8 s40 = s36 + s39; BLANKLINE return s40; } == END: popCount.c ==================