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# Implementing new primitive comparisons to allow branchless algorithms

This page gathers the notes about implementing new primitive logical operations and thus resolving ticket #6135.

## The problem

Consider following fragment of code:

case (x <# 0#) || (x >=# width) || (y <# 0#) || (y >=# height) of True -> E1 False -> E2

This kind of code is common in image processing (and array programming in general) where one needs to check whether the `(x,y)`

coordinates are within the image. Primitive comparison operators `<#`

and `>=#`

have type `Int# -> Int# -> Bool`

. Logical OR operator `(||)`

is defined as:

(||) :: Bool -> Bool -> Bool True || _ = True False || x = x

in GHC.Classes (ghc-prim library) which is equivalent of:

(||) x y = case x of True -> True False -> y

During the compilation process (assuming the optimizations are turned on) the definition of `(||)`

gets inlined and then case-of-case transform is performed successively. This results in following Core (cleaned up for clarity):

case <# x 0 of _ { False -> case >=# x width of _ { False -> case <# y 0 of _ { False -> case >=# y height of _ { False -> E2 True -> E1 }; True -> E1 }; True -> E1 }; True -> E1 };

and in following assembler code:

.Lc1rf: testq %r14,%r14 jl .Lc1rk cmpq %rdi,%r14 jge .Lc1rp testq %rsi,%rsi jl .Lc1ru cmpq %r8,%rsi jge .Lc1rz movl $Main_g2_closure+1,%ebx jmp *0(%rbp) .Lc1rk: movl $Main_g1_closure+1,%ebx jmp *0(%rbp) .Lc1rp: movl $Main_g1_closure+1,%ebx jmp *0(%rbp) .Lc1ru: movl $Main_g1_closure+1,%ebx jmp *0(%rbp) .Lc1rz: movl $Main_g1_closure+1,%ebx jmp *0(%rbp)

There are five possible branches to take, although four of them have the same result. This is caused by code duplication introduced by case-of-case transform (see this blog post for a step by step derivation). According to Ben Lippmeier, who submitted the original bug report, mis-predicted branches are bad in object code because they stall the pipeline.

Note: this example was produced with GHC 7.6.3. At the moment of merging new primops into HEAD, there was no code duplication at the assembly level when using old primops. However, avoiding code duplication is not the main problem new primops are meant to solve. That problem are conditional branches.

## Solution

This problem was solved by modifying comparison primops to return unboxed unlifted `Int#`

instead of `Bool`

. Having `Int#`

returned as a result of logical comparison will allow to use branchless bitwise logical operators instead of branching logical operators defined by Haskell.

## Implementation details

Below is a summary of implementation details and decisions:

- the new comparison primops return a value of type
`Int#`

:`1#`

represents`True`

and`0#`

represents`False`

. The`Int#`

type was chosen because on Haskell it is more common to use signed Int type insetad of unsigned Word. By using`Int#`

the users can easily convert unboxed result into a boxed value, without need to use`word2Int#`

and`int2word#`

primops. - as a small side-task, four new logical bitwise primops have been implemented:
`andI#`

,`orI#`

,`xorI#`

and`negI#`

(#7689). These operate on values of type`Int#`

. Earlier we had only bitwise logical primops operating on values of type`Word#`

. - names of the existing comparison primops were changed. Operators had
`$`

added before`#`

, others had`I`

added before the`#`

(this is a mnemonic denoting that this primop returns and`Int#`

). Examples:

>=$# :: Int# -> Int# -> Int# /=$## :: Double# -> Double# -> Int# gtCharI# :: Char# -> Char# -> Int# eqWordI# :: Word# -> Word# -> Int# ltFloatI# :: Float# -> Float# -> Int# leAddrI# :: Addr# -> Addr# -> Int# sameMutableArrayI# :: MutableArray# s a -> MutableArray# s a -> Int#

- built in
`GHC.Prim`

modules was renamed to`GHC.Prim.BuiltIn`

. In`ghc-prim`

we added a module`GHC.Prim`

which re-exports all definitions from`GHC.Prim.BuiltIn`

but also adds wrappers for new comparison primops. These wrappers have names identical to removed primops and return a`Bool`

. Examples:

gtChar# :: Char# -> Char# -> Bool gtChar# a b = tagToEnum# (a `gtCharI#` b) (>=#) :: Int# -> Int# -> Bool (>=#) a b = tagToEnum# (a >=$# b) eqWord# :: Word# -> Word# -> Bool eqWord# a b = tagToEnum# (a `eqWordI#` b) (/=##) :: Double# -> Double# -> Bool (/=##) a b = tagToEnum# (a /=$## b) ltFloat# :: Float# -> Float# -> Bool ltFloat# a b = tagToEnum# (a `ltFloatI#` b) leAddr# :: Addr# -> Addr# -> Bool leAddr# a b = tagToEnum# (a `leAddrI#` b) sameMutableArray# :: MutableArray# s a -> MutableArray# s a -> Int# sameMutableArray# a b = tagToEnum# (a `sameMutableArrayI#` b)

Thanks to renaming of previously existing `GHC.Prim`

module and adding wrappers in new `GHC.Prim`

module **the whole change of primops is backwards compatible**.

- functions for comparing
`Integer`

type, implemented in`integer-gmp`

and`integer-simple`

libraries, received a similar treatment. Technically they are not primops, because they are implemented in Haskell (in case of`integer-gmp`

also with FFI), but they pretend to be ones. There are six primops for comparing`Integer`

values:eqInteger# :: Integer -> Integer -> Int# neqInteger# :: Integer -> Integer -> Int# leInteger# :: Integer -> Integer -> Int# ltInteger# :: Integer -> Integer -> Int# gtInteger# :: Integer -> Integer -> Int# geInteger# :: Integer -> Integer -> Int#

Each of these functions has a wrapper that calls `tagToEnum#`

and returns a `Bool`

. These wrappers are: `eqInteger`

, `neqInteger`

, `leInteger`

, `ltInteger`

, `gtInteger`

and `geInteger`

.

- This change also required some small adjustments in
`base`

package.

## Eliminating branches using new primops

With the new primops we can rewrite the original expression that motivated the problem:

case (x <# 0#) || (x >=# width) || (y <# 0#) || (y >=# height) of True -> E1 False -> E2

as

case (x <$# 0#) `orI#` (x >=$# width) `orI#` (y <$# 0#) `orI#` (y >=$# height) of True -> E1 False -> E2

Let's analyze how that code gets compiled by the LLVM backend. For purposes of this analysis I will convert the above case expression to the following function:

f :: Int# -> Int# -> Int# -> Int# -> String f x y width height = case (x <$# 0#) `orI#` (x >=$# width) `orI#` (y <$# 0#) `orI#` (y >=$# height) of 1# -> "one" 0# -> "zero"

By dumping intermediate Cmm representation we can determine how variables are mapped to CPU registers. Arguments are passed to the function using a stack:

Main.f_slow() // [R1] { info_tbl: [] stack_info: arg_space: 0 updfr_space: Nothing } {offset cWY: R5 = I64[Sp + 24]; R4 = I64[Sp + 16]; R3 = I64[Sp + 8]; R2 = I64[Sp]; R1 = R1; Sp = Sp + 32; call Main.f_info(R5, R4, R3, R2, R1) args: 8, res: 0, upd: 8; } }

`R2`

contains the first argument `x`

, `R3`

contains `y`

, `R4`

contains `width`

and `R5`

contains `height`

. We can verify that by looking at the body of `Main.f_info`

:

_sV9::I64 = R3; _sV3::I64 = R2; _sWx::I64 = %MO_S_Lt_W64(_sV3::I64, 0) | %MO_S_Ge_W64(_sV3::I64, R4) | %MO_S_Lt_W64(_sV9::I64, 0) | %MO_S_Ge_W64(_sV9::I64, R5);

Mappings between Cmm's `R(2/3/4/5)`

registers and machine registers are defined in `includes/stg/MachRegs.h`

:

#define REG_R2 r14 #define REG_R3 rsi #define REG_R4 rdi #define REG_R5 r8

Knowing that we can dump the assembly generated by LLVM backend:

Main_f_info: # BB#0: movq %rsi, %rax orq %r14, %rax shrq $63, %rax cmpq %rdi, %r14 setge %cl movzbl %cl, %ecx orq %rax, %rcx cmpq %r8, %rsi setge %al movzbl %al, %eax orq %rcx, %rax jne .LBB4_1 # BB#3: movq Main_f2_closure(%rip), %rax movl $Main_f2_closure, %ebx jmpq *%rax # TAILCALL .LBB4_1: cmpq $1, %rax jne .LBB4_2 # BB#4: movq Main_f1_closure(%rip), %rax movl $Main_f1_closure, %ebx jmpq *%rax # TAILCALL .LBB4_2: movq Main_f3_closure(%rip), %rax movl $Main_f3_closure, %ebx jmpq *%rax # TAILCALL

Let's analyze line by line the part responsible for evaluating the scrutinee:

movq %rsi, %rax # load y (stored in %rsi) into %rax register orq %r14, %rax # perform bitwise OR between y (now in %rax) and x (in %r14) # and store result in %rax shrq $63, %rax # shift %rax 63 bits to the right. If both x and y were # positive numbers, then their sign bits (MSB) were set to # 0 and so %rax is now 0. If at least one of them was # negative then its sign bit must have been 1 and so %rax # is now 1 cmpq %rdi, %r14 # compare width (in %rdi) with x (in %r14) and set the flags # in the flag register according to the result. setge %cl # if, based on flags set by cmpq, x was greater or equal to # width then we set %cl to 1. Otherwise it is set to 0. movzbl %cl, %ecx # zero the bits of %ecx register except the lowest 8 bits # containg the result of previous operation orq %rax, %rcx # perform logical OR on results of the previous test and # store the result in %rcx. At this point if %rcx is 1 # then either x was negative, or y was negative or # x was greater or equal to width cmpq %r8, %rsi # now we are checking whether y is greter or equal to # height. This is the same as previosly for x and width. setge %al # This time we set LSB of %al to 1 if y >= height movzbl %al, %eax # as previously, clear bits of %eax except lowest 8 bits orq %rcx, %rax # perform logical OR which combines result of previous # three comparisons with the last one jne .LBB2_1 # if ZF is not set it this means that either %rcx or %rax # was not zero, which means that at least one condition # in the scrutinee was true

The assembly does not contain comparisons and branches in the scrutinee of the case expression, but still uses jumps to select an appropriate branch of the case expression.

### Benchmarks

Below is a benchmark for the proof-of-concept branchless filter function that demonstrates performance gains possible with the new primops:

{-# LANGUAGE BangPatterns, MagicHash #-} module Main ( main ) where import Control.Monad.ST (runST) import Criterion.Config (Config, cfgPerformGC, defaultConfig, ljust) import Criterion.Main import Data.Vector.Unboxed.Mutable (unsafeNew, unsafeSlice, unsafeWrite) import Data.Vector.Unboxed as U (Vector, filter, foldM', fromList, length, unsafeFreeze) import GHC.Exts (Int (I#), (>=$#)) import System.Random (RandomGen, mkStdGen, randoms) import Prelude hiding (filter, length) filterN :: U.Vector Int -> U.Vector Int filterN vec = runST $ do let !size = length vec fVec <- unsafeNew size let put i x = do let !(I# v) = x inc = I# (v >=$# 0#) unsafeWrite fVec i x return $ i + inc fSize <- foldM' put 0 vec unsafeFreeze $ unsafeSlice 0 fSize fVec main :: IO () main = return (mkStdGen 1232134332) >>= defaultMainWith benchConfig (return ()) . benchmarks benchmarks :: RandomGen g => g -> [Benchmark] benchmarks gen = let dataSize = 10 ^ (7 :: Int) inputList = take dataSize . randoms $ gen :: [Int] inputVec = fromList inputList isPositive = (> 0) in [ bgroup "Filter" [ bench "New" $ whnf (filterN) inputVec , bench "Vector" $ whnf (filter isPositive) inputVec ] ] benchConfig :: Config benchConfig = defaultConfig { cfgPerformGC = ljust True }

Compile and run with:

ghc -O2 -fllvm -optlo-O3 Main.hs ./Main -o report.html

Benchmarking shows that `filterN`

function is about 55-65% faster than the `filter`

function based on stream fusion (tested for unboxed vectors containing 10 thousand and 10 million elements). Below is an example benchmarking report from criterion:

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