Version 21 (modified by jstolarek, 4 years ago) (diff)

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# Implementing new primitive comparisons to allow branchless algorithms

## 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.

## Solution

This problem was solved by modifying comparison primops to return unboxed unlifted `Int#` instead of `Bool` (which is lifted and thus is returned as a thunk that needs to be evaluated). 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 will have `\$` added before `#`, others will have `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#
```
• a new module `GHC.PrimWrappers` was added to ghc-prim library. This module contains wrappers for 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)

```

Thanks to these wrappers the change is almost backwards compatible. The only thing primop users need to change in their existing code to make it work again is adding import of GHC.PrimWrappers module.

• 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`.

• Other libraries that were modified to work with the new primops are: base, ghc-prim and primitive. The only required modifications were imports of the GHC.PrimWrappers module in modules that use the primops.

## 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 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: