Now that the ICFP deadline is past, I've returned to working on adding SIMD
support to GHC and associated libraries. The short-term goal is to be able to
leverage SIMD instructions from the `vector`

and—hopefully
transparently—from
Data Parallel Haskell. Back in November I added a fairly complete set of
primops to GHC
that generate SSE instructions when using the LLVM back-end. If you're
interested in the original design for SIMD support in GHC and the state of
implementation, you can read about it
here. It's a bit of a hack
and requires the LLVM back-end, but it does work.

Primops are necessary, but what we'd really like a higher-level interface to these low-level instructions. This post is a very short introduction to my recent efforts in that direction. All the code I describe is public—you can find directions for getting up and running with the GHC simd branch on github.

Because this is Haskell, we'll start by
introducing a data type for a SIMD vector that is indexed by the type of the
scalar values it contains. The term "vector" is already overloaded, so, at Simon
Peyton Jones' suggestion, we call a SIMD vector containing scalars of type `a`

a
`Multi a`

. Because we want to choose a different primitive representation for
each different `a`

, `Multi`

is a
type family (actually
an *associated* type family). Along with `Multi`

, we define a type class
`MultiPrim`

that allows us to treat primitive operations on `Multi`

's in a
uniform way, just as the `Prim`

type class defined by the
`primitive`

library allows for
scalars. Here's the first part of the definition of the `MultiPrim`

type class
and the `Multi`

associated family. You can see that it defines functions for
replicating scalars across `Multi`

's, folding a function over the scalar
elements of a `Multi`

, reading a `Multi`

out of a `ByteArray#`

, etc. Right now
there are instance definitions for `Multi Float`

, `Multi Double`

, `Multi Int32`

,
`Multi Int64`

, and `Multi Int`

. This type class and the rest of the code I'll be
showing are
actually part of the `simd`

branch of the `vector`

library that I've but on
github. You can go look there for
further details, like the `Num`

instances defined for the `Multi`

's.

class (Prim a, Prim (Multi a)) => MultiPrim a where data Multi a -- | The number of elements of type @a@ in a @Multi a@. multiplicity :: Multi a -> Int -- | Replicate a scalar across a @Multi a@. multireplicate :: a -> Multi a -- | Map a function over the elements of a @Multi a@. multimap :: (a -> a) -> Multi a -> Multi a -- | Fold a function over the elements of a @Multi a@. multifold :: (b -> a -> b) -> b -> Multi a -> b -- | Read a multi-value from the array. The offset is in elements of type -- @a@ rather than in elements of type @Multi a@. indexByteArrayAsMulti# :: ByteArray# -> Int# -> Multi a

Now that we have the `Multi`

type, we would like to use it operate over
`Vector`

's—that is, vector types from the
`vector`

library. A `Vector`

has
scalar elements, so for us to be able to use SIMD operations on these scalars we
need to know something about the representation the `Vector`

uses, namely that
it lays out scalars *contiguously* in memory. The `PackedVector`

type class lets
us express this constraint in Haskell's type system, and I won't say anything
more about it here, but instances are defined for the appropriate vector types
in the `Data.Vector.Unboxed`

and `Data.Vector.Storable`

modules.

Of course the next step is to define appropriate versions of our old friends, map, zip, and fold, that will let us exploit SIMD operations. Here they are.

mmap :: (PackedVector v a, PackedVector v b) => (a -> b) -> (Multi a -> Multi b) -> v a -> v b

mzipWith :: (PackedVector v a, PackedVector v b, PackedVector v c) => (a -> b -> c) -> (Multi a -> Multi b -> Multi c) -> v a -> v b -> v c

mfoldl' :: PackedVector v b => (a -> b -> a) -> (a -> Multi b -> a) -> a -> v b -> a

If you're familiar with the `vector`

library, you may know it uses
stream fusion to generate very efficient code—many operations are typically
compiled
to tight loops similar to what one would get from a C compiler. Stream fusion
works by re-expressing high-level operations like map, zip, and fold in terms of
"step" functions. Each step function takes some state and an element and
produces either some new state and a new element, just some new state, or a
value that says it is done processing elements. To support computing over
vectors
using SIMD operations, I have added a new "stream" variant so that step
functions can receive not just scalar elements, but `Multi`

elements. That is,
at every step, the stream consumer could be handed either a scalar or a `Multi`

and must be prepared for either case. `mmap`

, `mzipWith`

, and `mfoldl`

are
almost exactly like their scalar-only counterparts, but they each take an extra
function argument for handling `Multi`

's.

Let's see if this stuff actually works by starting off with something easy---
summing up all the elements in a vector. The following code uses the new
`vector`

library primitives `multifold`

and `U.mfoldl`

to exploit SIMD
instructions.

import qualified Data.Vector.Unboxed as U import Data.Primitive.Multi multisum :: U.Vector Float -> Float multisum v = multifold (+) s ms where s :: Float ms :: Multi Float (s, ms) = U.mfoldl' plus1 plusm (0, 0) v plusm (x, mx) my = (x, mx + my) plus1 (x, mx) y = (x + y, mx)

We'll compare it with five other versions. "Scalar" and "Scalar (C)" are plain
old scalar versions written in Haskell and C, respectively. "Manual" and "Manual
(C)" are hand-written Haskell and C versions, respectively. The Haskell version
explicitly iterates over the vector instead of using a fold. The `vector`

version is the code we just saw, and the `multivector`

version is based on a
library I wrote to test out fusion when I first added SSE support to GHC. It
implements a small subset of the `vector`

library API. Here we go ^{1}

Not bad. The following table gives the timings for vectors with 2^{24} elements.
In this case, Haskell is as fast as C. This isn't too surprising, as we've
seen before that Haskell can be as fast as C.

Variant | Time (ms) |
---|---|

Scalar | 19.7 ± 0.2 |

Scalar (C) | 19.7 ± 0.4 |

Manual | 4.62 ± 0.03 |

Manual (C) | 4.58 ± 0.02 |

vector | 4.62 ± 0.02 |

multivector | 4.62 ± 0.02 |

Of course, summing up the elements in a vector isn't so hard. The great thing
about the `vector`

library is that you can write high-level Haskell code and,
through the magic of fusion, you end up with a tight inner loop that looks like
what you might have gotten out of a C compiler had you chosen to write in C.
Let's try s slightly more difficult computation that will require fusion—dot
product.

Computing the dot product efficiently requires fusing two loops to perform a combined addition and multiplication. Here is the scalar version in Haskell

import qualified Data.Vector.Unboxed as U dotp :: U.Vector Float -> U.Vector Float -> Float dotp v w = U.sum $ U.zipWith (*) v w

And here is our first cut at a SIMD version.

import qualified Data.Vector.Unboxed as U import Data.Primitive.Multi multidotp :: U.Vector Float -> U.Vector Float -> Float multidotp v w = multifold (+) s ms where s :: Float ms :: Multi Float (s, ms) = U.mfoldl' plus1 plusm (0, 0) $ U.mzipWith (*) (*) v w plusm (x, mx) my = (x, mx + my) plus1 (x, mx) y = (x + y, mx)

Let's look at performance once more. Again, "Manual" is a Haskell version that manually iterates over the vector once and fuses the addition and multiplication, the idea being that this is what we would hope to get out of GHC after fusion, inlining, constructor specialization, etc.

For reference, here are the timings for the case with n = 2^{24} again.

Variant | Time (ms) |
---|---|

Scalar | 16.98 ± 0.08 |

Scalar (C) | 16.63 ± 0.09 |

Manual | 8.87 ± 0.03 |

Manual (C) | 8.64 ± 0.02 |

vector | 13.03 ± 0.07 |

multivector | 9.5 ± 0.1 |

Not so hot. Although our hand-written Haskell implementation ("Manual" in the
plot and table) is competitive with C, the `vector`

version is not.
Interestingly, the "multivector" version *is* competitive. What could be going
on?

The first things that jumps to mind is that fusion might not be kicking in: I
could've screwed up the implementation of the SIMD-enabled combinators! To check
this hypothesis, let's look at the
GHC core generated for the main loop in `multidotp`

(this is the loop that
iterates over elements SIMD-vector-wise):

1: letrec { 2: $s$wmfoldlM_loopm_s4ri [Occ=LoopBreaker] 3: :: GHC.Prim.Int# 4: -> GHC.Prim.Int# 5: -> GHC.Prim.~# 6: * 7: Data.Primitive.Multi.FloatX4.FloatX4 8: (Data.Primitive.Multi.Multi GHC.Types.Float) 9: -> GHC.Prim.FloatX4# 10: -> GHC.Prim.Float# 11: -> (# GHC.Types.Float, 12: Data.Primitive.Multi.Multi GHC.Types.Float #) 13: [LclId, Arity=5, Str=DmdType LLLLL] 14: $s$wmfoldlM_loopm_s4ri = 15: \ (sc_s4nR :: GHC.Prim.Int#) 16: (sc1_s4nS :: GHC.Prim.Int#) 17: (sg_s4nT 18: :: GHC.Prim.~# 19: * 20: Data.Primitive.Multi.FloatX4.FloatX4 21: (Data.Primitive.Multi.Multi GHC.Types.Float)) 22: (sc2_s4nU :: GHC.Prim.FloatX4#) 23: (sc3_s4nV :: GHC.Prim.Float#) -> 24: case GHC.Prim.>=# sc1_s4nS ipv7_aHm of _ { 25: GHC.Types.False -> 26: case GHC.Prim.indexFloatArrayAsFloatX4# 27: ipv2_s4kn (GHC.Prim.+# ipv_s4kl sc1_s4nS) 28: of wild_a4j3 { __DEFAULT -> 29: case GHC.Prim.>=# sc_s4nR ipv6_XI3 of _ { 30: GHC.Types.False -> 31: case GHC.Prim.indexFloatArrayAsFloatX4# 32: ipv5_s4l7 (GHC.Prim.+# ipv3_s4l5 sc_s4nR) 33: of wild3_X4jF { __DEFAULT -> 34: $s$wmfoldlM_loopm_s4ri 35: (GHC.Prim.+# sc_s4nR 4) 36: (GHC.Prim.+# sc1_s4nS 4) 37: @~ (Sym (Data.Primitive.Multi.NTCo:R:MultiFloat) ; Sym 38: (Data.Primitive.Multi.TFCo:R:MultiFloat)) 39: (GHC.Prim.plusFloatX4# 40: sc2_s4nU (GHC.Prim.timesFloatX4# wild_a4j3 wild3_X4jF)) 41: sc3_s4nV 42: }; 43: GHC.Types.True -> ... 44: } 45: }; 46: GHC.Types.True -> ... 47: };

We can see that the two loops have been fused. I won't show the core for the
other Haskell implementations, but I'll note that it looks pretty much the same
except for one thing: `multidotp`

is carrying around *two* pieces of state
during the fold it performs, a scalar `Float`

and a `Multi Float`

. That
shouldn't make a difference though—these guys should just live in two separate
registers. There's only one reasonable thing left to do: look at some assembly.

Just so we have an idea of what we *want* to see, let's examine the inner loop
of the C version first:

.L3: movaps (%rdi,%rax), %xmm0 mulps (%rdx,%rax), %xmm0 addq $16, %rax cmpq %r8, %rax addps %xmm0, %xmm1 jne .L3

Cool. Our array pointers live in `rdi`

and `rdx`

, our index in `rax`

, and the
array bounds in `r8`

. Now on to the "manual" Haskell version.

.LBB5_3: # %n5oi # =>This Inner Loop Header: Depth=1 movups (%rcx), %xmm2 movups (%rdx), %xmm1 mulps %xmm2, %xmm1 addps %xmm1, %xmm0 addq $16, %rcx addq $16, %rdx addq $4, %r14 cmpq %r14, %rax jg .LBB5_3

Still pretty good. This time our array pointers live in `rcx`

and `rdx`

, our
index in `r14`

, and our bounds in `rax`

. Note that the index is now measured in
`float`

's instead of bytes. How about the "multivector" version?

1: .LBB1_2: # %n3JW.i 2: # in Loop: Header=BB1_1 Depth=1 3: cmpq %rax, %r8 4: jle .LBB1_5 5: # BB#3: # %n3K9.i 6: # in Loop: Header=BB1_1 Depth=1 7: movq 8(%rcx), %rdx 8: addq %rax, %rdx 9: movq 16(%rcx), %rdi 10: movups 16(%rdi,%rdx,4), %xmm2 11: movups (%rbx), %xmm1 12: mulps %xmm2, %xmm1 13: addps %xmm1, %xmm0 14: movups %xmm0, -56(%rcx) # 15: addq $16, %rbx 16: addq $4, %rax 17: .LBB1_1: # %tailrecurse.i 18: # =>This Inner Loop Header: Depth=1 19: cmpq %rax, %r9 20: jg .LBB1_2

There is definitely more junk here. Still, not horrible except for line
14 where we spill the result to the stack. Apparently the
spill doesn't cost us much ^{2}. Now the "vector" version that had performance issues.

1: .LBB4_2: # %n4H3 2: # in Loop: Header=BB4_1 Depth=1 3: cmpq %r14, 43(%rbx) 4: jle .LBB4_5 5: # BB#3: # %n4Hw 6: # in Loop: Header=BB4_1 Depth=1 7: movq 35(%rbx), %rdx 8: addq %r14, %rdx 9: movq 3(%rbx), %rcx 10: movq 11(%rbx), %rdi 11: movups 16(%rdi,%rdx,4), %xmm0 12: movq 27(%rbx), %rdx 13: addq %rsi, %rdx 14: movups 16(%rcx,%rdx,4), %xmm1 15: mulps %xmm0, %xmm1 16: movups (%rbp), %xmm0 # 17: addps %xmm1, %xmm0 18: movups %xmm0, (%rbp) # 19: addq $4, %r14 20: addq $4, %rsi 21: .LBB4_1: # %tailrecurse 22: # =>This Inner Loop Header: Depth=1 23: cmpq %rsi, %rax 24: jg .LBB4_2

Ah-hah, there's our likely culprit: our accumulator is loaded from the stack in
line 16 and spilled back in line 18.
Yuck! It looks like carrying around that extra bit of state really cost us. I'm
not sure why LLVM didn't spill the `Float`

portion of the state to the stack
temporarily so that it could use the register for the main loop, but it seems
likely that it is related to the
GHC calling convention used by the LLVM back-end.

I'm disappointed that we weren't able to get C-competitive performance from our high-level Haskell code, especially since it seems so tantalizingly close. At least there is hope that with some prodding we can convince LLVM to keep our accumulating parameter in a register.

## Footnotes:

^{1} All
timings were done on a laptop with a 2.70GHz Intel® Core™ i7-2620M CPU with
frequency scaling disabled. 100 trials were performed at each data point. C code
was compiled by GCC at `-O3`

, and `llc`

and `opt`

were invoked with `-O3`

.

^{2} On an AMD machine I have access to this spill
*does* incur a penalty.