Portability | portable |
---|---|
Stability | experimental |
Maintainer | erkokl@gmail.com |
Computing population-counts (number of set bits) and autimatically generating C code.
- popCount_Slow :: SWord64 -> SWord8
- popCount :: SWord64 -> SWord8
- pop8 :: [SWord8]
- fastPopCountIsCorrect :: SWord64 -> SBool
- genPopCountInC :: IO ()
Reference: Slow but obviously correct
popCount_Slow :: SWord64 -> SWord8Source
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:
>>>
popCount_Slow 0x0123456789ABCDEF
32 :: SWord8
Faster: Using a look-up table
popCount :: SWord64 -> SWord8Source
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.
Look-up table, containing population counts for all possible 8-bit value, from 0 to 255.
Verification
While we would like to use popCount
instead of popCount_Slow
, how can
we be absolutely sure that we got the constants correctly computed and
stored in pop8
? It is entirely possible that one of the entries might
be incorrect, causing a hard-to-detect bug. However, using SBV we can equivalence
check that the faster version computes precisely the same answers as the slower
reference version.
fastPopCountIsCorrect :: SWord64 -> SBoolSource
States the correctness of faster population-count algorithm, with respect to the reference slow version. We have:
>>>
prove fastPopCountIsCorrect
Q.E.D.
Code generation
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 popCount
:
/* Header file for popCount. Automatically generated by SBV. Do not edit! */ #ifndef __popCount__HEADER_INCLUDED__ #define __popCount__HEADER_INCLUDED__ #include <inttypes.h> #include <stdint.h> /* Unsigned bit-vectors */ typedef uint8_t SBool ; typedef uint8_t SWord8 ; typedef uint16_t SWord16; typedef uint32_t SWord32; typedef uint64_t SWord64; /* Signed bit-vectors */ typedef int8_t SInt8 ; typedef int16_t SInt16; typedef int32_t SInt32; typedef int64_t SInt64; /* Entry point prototype: */ SWord8 popCount(const SWord64 x); #endif /* __popCount__HEADER_INCLUDED__ */
The generated C function:
/* File: "popCount.c". Automatically generated by SBV. Do not edit! */ #include <inttypes.h> #include <stdint.h> #include "popCount.h" 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; return s40; }
SBV will also generate a driver program for test purposes:
/* Example driver program for popCount. */ /* Automatically generated by SBV. Edit as you see fit! */ #include <inttypes.h> #include <stdint.h> #include <stdio.h> #include "popCount.h" int main(void) { const SWord8 pc = popCount(0x0123456789abcdefULL); printf("popCount(0x0123456789abcdefULL) = %"PRIu8"\n", pc); return 0; }
And a Makefile
to simplify compilation:
# Makefile for popCount. Automatically generated by SBV. Do not edit! CC=gcc CCFLAGS=-Wall -O3 -DNDEBUG -fomit-frame-pointer all: popCount_driver popCount.o: popCount.c popCount.h ${CC} ${CCFLAGS} -c popCount.c -o popCount.o popCount_driver.o: popCount_driver.c ${CC} ${CCFLAGS} -c popCount_driver.c -o popCount_driver.o popCount_driver: popCount.o popCount_driver.o ${CC} ${CCFLAGS} popCount.o popCount_driver.o -o popCount_driver clean: rm -f popCount_driver.o popCount.o veryclean: clean rm -f popCount_driver