Experiments in program optimisation

Optimising division-based hash functions

Story: Conway's Game of Life

Tags: Java optimisation hashmap life

17 Sep 2015

In “Game of Life, hash tables and hash codes” we implemented Conway’s game of Life using Java’s HashMap and discovered that some hash functions cause higher execution speed than the others. We introduced several hash functions and measured their distribution of hash codes. In the next article (“Hash functions: disassembling and measuring performance”) we learned how to achieve stable performance of micro-benchmarks (those that measure execution speed of individual hash functions), and measured the speed of all those functions. We found several abnormalities, one of which was a big difference in performance of the division-based hash function between Java 7 and Java 8. Let’s look at this case.

The function in question

Let’s repeat the source code of LongPoint6.hashCode():

public int hashCode ()
    return (int) (v % 946840871);

The constant 946840871 was a completely arbitrary choice of a divisor. It’s a prime number and it is relatively big (close to one billion), bigger than any realistic hash table size. There is nothing special about this number, some other number may work better. It is also possible that some other number will allow faster calculations. We don’t use any specific properties of this number here, all following arguments are equally applicable to any number.

The test involving this method ran for 578 ms on Java 7 and 1661 ms on Java 8. The only way to find out why is by looking at the generated code (see the previous article for the instructions how to do it).

Let’s start with the Java 7:

0x00007f88e10684a0: sub    rsp,0x18
0x00007f88e10684a7: mov    QWORD PTR [rsp+0x10],rbp  ;*synchronization entry
                                              ; - LongPoint6::hashCode@-1 (line 30)
0x00007f88e10684ac: mov    r10,QWORD PTR [rsi+0x10]  ;*getfield v
                                              ; - LongPoint6::hashCode@1 (line 30)
0x00007f88e10684b0: mov    r11,r10
0x00007f88e10684b3: sar    r11,0x3f
0x00007f88e10684b7: mov    rax,0x2449f0232c624b0b
0x00007f88e10684c1: imul   r10
0x00007f88e10684c4: sar    rdx,0x1b
0x00007f88e10684c8: sub    rdx,r11
0x00007f88e10684cb: imul   r11,rdx,0x386fa527
0x00007f88e10684d2: sub    r10,r11
0x00007f88e10684d5: mov    eax,r10d           ;*l2i  ; - LongPoint6::hashCode@8 (line 30)
0x00007f88e10684d8: add    rsp,0x10
0x00007f88e10684dc: pop    rbp
0x00007f88e10684dd: test   DWORD PTR [rip+0xb74fb1d],eax        # 0x00007f88ec7b8000
                                              ;   {poll_return}
0x00007f88e10684e3: ret    

There aren’t any division instructions in the code. The pseudo-code looks like this:

public int hashCode ()
    long sign = v >> 63;
    long div = ((v ** 2614885092524444427L) >> 91) - sign;
    return (int) (v - div * 946840871);

I call this pseudo-code rather than just code because it contains an operation that is not accessible in Java: a 128-bit multiplication, which takes two 64-bit numbers and produces a 128-bit result (I wrote it as **). There is such operation in the instruction set – both MUL and IMUL have 128-bit versions. The result is placed into rdx and rax, we are using rdx, that’s why the total shift is 0x1B + 64 = 91.

What we see is the result of a well-known optimisation: replacing of division by a constant divisor with a multiplication and some shifts. It is well described in Agner Fog. Optimizing subroutines in assembly language. An optimization guide for x86 platforms. The HotSpot uses slightly different algorithm (they use two bits less, and compensate for signed values). If the code was written for unsigned numbers, it could save three instructions here.

The Java 8 code looks like this:

0x0000000002989a00: mov    DWORD PTR [rsp-0x6000],eax
0x0000000002989a07: push   rbp
0x0000000002989a08: sub    rsp,0x50           ;*aload_0
                                              ; - LongPoint6::hashCode@0 (line 30)

0x0000000002989a0c: mov    rcx,QWORD PTR [rdx+0x10]  ;*getfield v
                                              ; - LongPoint6::hashCode@1 (line 30)

0x0000000002989a10: mov    rdx,rcx
0x0000000002989a13: movabs rcx,0x386fa527
0x0000000002989a1d: mov    rsi,rcx
0x0000000002989a20: mov    rcx,rsi
0x0000000002989a23: cmp    rsi,0x0
0x0000000002989a27: je     0x0000000002989a40
0x0000000002989a2d: call   0x000000006fe379a0  ;*lrem
                                              ; - LongPoint6::hashCode@7 (line 30)
                                              ;   {runtime_call}
0x0000000002989a32: mov    eax,eax
0x0000000002989a34: add    rsp,0x50
0x0000000002989a38: pop    rbp
0x0000000002989a39: test   DWORD PTR [rip+0xfffffffffd7a66c1],eax        # 0x0000000000130100
                                              ;   {poll_return}
0x0000000002989a3f: ret
0x0000000002989a40: call   0x0000000002959c80  ; OopMap{off=101}
                                              ; - LongPoint6::hashCode@7 (line 30)
                                              ;   {runtime_call}

This code does not employ any optimisations. It does not even compile the remainder operation into DIV or IDIV instructions. It calls a runtime routine. It even goes as far as checking that freshly loaded constant isn’t zero. In short, this is really awful code, no wonder it runs slowly. Perhaps, HotSpot developers considered division by a constant a rare and unimportant case, and removed corresponding optimisation to keep their code short.

Optimising division: unsigned case

The code Java 7 generates is perfect. The only way it could be improved is by using unsigned division instead of signed one, but this can’t be done in Java.

Is there any way to improve the Java 8 case? If our numbers were 32-bit, there would be an obvious way – to write code similar to the pseudo-code above; instead of 128-bit multiplication, we would need a 64-bit one, and this is a normal long operation. Unfortunately, we can’t do it that easily in the 64-bit case, because, as I said, there is no ** operation in Java.

This doesn’t mean we must stop here: after all, we have a general-purpose computer, and there is nothing mystical about 128-bit multiplication. It can be programmed in Java using available arithmetic.

Let’s first simplify the problem by considering our numbers unsigned.

Let x and y be unsigned 64-bit values, and A, B, C and D be unsigned 32-bit values, where

x = 232A + B
y = 232C + D


xy = (232A + B) (232C + D) = 264AC + 232(AD + BC) + BD

or, graphically,

high 64 bits of xy low 64 bits of xy
hi (A*C) lo (A*C)
hi (A*D) lo (A*D)
hi (B*C) lo (B*C)
hi (B*D) lo (B*D)

This is a 128-bit unsigned number, which we will represent as two longs, one for high and one for low part of the result. Both will be interpreted as unsigned 64-bit values. Each of the terms AC, AD, BC, BD is a 64-bit number. The low part of the result can be calculated by the obvious code:

long mult_unsigned_lopart (long x, long y)
    long A = uhi (x);
    long B = ulo (x);
    long C = uhi (y);
    long D = ulo (y);
    return B*D + ((A*D + B*C) << 32);

long uhi (long x)
    return hi (x) & 0xFFFFFFFFL;

long ulo (long x)
    return lo (x) & 0xFFFFFFFFL;

Note the uhi and ulo functions (“unsigned hi” and “unsigned lo”) and their difference from hi and lo that we used to extract integer parts of a value stored as long:

public static int hi (long w)
    return (int) (w >>> 32);

public static int lo (long w)
    return (int) w;

These functions return int, which, if directly converted to long, will be sign-extended, which will corrupt the results. Here is a simple example. Let x be 0x0000000000000001 and y be 0xFFFFFFFFFFFFFFFF. Obviously, the 128-bit result should be 0x0000000000000000FFFFFFFFFFFFFFFF, and mult_unsigned_lopart() must return 0xFFFFFFFFFFFFFFFF (which will be printed as -1). However, with the sign-extended versions of hi and lo we would have

A = 0
B = 1
C = −1
D = −1
BD + ((AD + BC) << 32)
    = −1 + ((0 + −1) << 32)

The code using uhi, ulo works correctly:

A = 0
B = 1
BD + ((AD + BC) << 32)
    = 0xFFFFFFFF + ((0 + 0xFFFFFFFF) << 32)
    = 0xFFFFFFFF + 0xFFFFFFFF00000000

The mult_unsigned_lopart was shown here for completeness; we don’t need it for our remainder problem. The pseudo-code of hashCode() above (the one using the ** operation) shifts the multiplication result by 91, which totally ignores the low part of the product.

It may seem that the following code will calculate the high part:

static long mult_unsigned_hipart (long x, long y)
    long A = uhi (x);
    long B = ulo (x);
    long C = uhi (y);
    long D = ulo (y);
    return A*C + ((A*D + B*C) >>> 32);

This, however, does not work – for two reasons. One is that (A*D + B*C) can occupy 65 bits (a carry bit may be produced during addition), and another one is that, although B*D term is not included directly into the result, it can produce carry when its high part is added to the low part of A*D + B*C. We have to perform addition in 32-bit blocks and add all the carry bits manually, according to the picture above:

static long mult_unsigned_hipart (long x, long y)
    long A = uhi (x);
    long B = ulo (x);
    long C = uhi (y);
    long D = ulo (y);

    long AC = A * C;
    long AD = A * D;
    long BC = B * C;
    long BD = B * D;

    long ADl_BCl_BDh = ulo (AD) + ulo (BC) + uhi (BD);
    return AC + uhi (AD) + uhi (BC) + uhi (ADl_BCl_BDh);

Now it is very easy to program the hash function:

public int hashCode ()
    long div = mult_unsigned_hipart (v, 2614885092524444427L) >> 27;
    return (int) (v - div * 946840871);

It seems very unlikely that this monster will work faster than the original code. It contains five multiplications, one shift and six additions. This makes it a very interesting test: how fast can it possibly run?

Signed or unsigned?

We have just calculated an unsigned version of the remainder-based hash function. However, the code of LongPoint6.hashCode() uses regular Java’s arithmetic, which is signed. How important is that? What impact does it have on the quality of hashing? In Java dividing a negative number by a positive one produces negative remainder:

25 % 3 = 1
−25 % 3 = −1

Strictly speaking, this isn’t what is usually called “a remainder” in mathematics. Proper mathematical remainder isn’t ever negative:

0 ≤ x % y < y

and should be 2 for −25 % 3.

To check the quality of unsigned hashing, we must emulate unsigned calculations in Java. A positive signed 64-bit number keeps its value when interpreted as unsigned. A negative number becomes a big positive, 264 added to its value, so we must add (264 mod 946840871) = 518863773 to the remainder. Since the remainder was zero or negative before, it won’t exceed the divisor, but it may stay negative, in which case we need to add a divisor again:

public int hashCode ()
    int r = (int) (v % 946840871);
    if (v < 0) r += 518863773;
    if (r < 0) r += 946840871;
    return r;

Here are the slot counts we got with signed division (see here):

Field size: 1034; slots used:  982; avg= 1.05
Count size: 3938; slots used: 3236; avg= 1.22

And here are the results for unsigned division:

Field size: 1034; slots used:  968; avg= 1.07
Count size: 3938; slots used: 3133; avg= 1.26

The results haven’t changed much for field but have changed for count: they became worse. It does not mean they became very bad. Previously, the result was equal to E + 5.28σ, where E is the expected value 3126.69, and σ is the standard deviation 20.68. Now the result is E + 0.3σ, which still falls well within good statistical limits and performs better than some other hash functions. It just does not perform as well as the signed division, which I remarked on back then as demonstrating unusually good behaviour.

This means that if we manage to write an incredibly fast version of the hash function based on the unsigned division, we have chances to improve the overall speed, otherwise the signed version is a better choice.

Another approach is to change our OFFSET, which is now 0x80000000, to some positive value. This will leave less bits for x and y parts, but the long values will always be positive, and unsigned version will produce the same result as the signed one. Some of these values perform better than the others: for instance, 0x40000000 gives 3118 (E − 0.42σ) as count size, while 0x8000000 gives 3234 (E + 5.18σ). We won’t do it to keep the story short.

Optimising division: signed case

We’ve seen that the signed version has some potential advantages. Can we modify the code to work in signed case?

The formulae we used above:

x = 232A + B
y = 232C + D
xy = (232A + B) (232C + D)
    = 264AC + 232(AD + BC) + BD

are still applicable if x and y are signed 64-bit numbers. In this case A and C are signed 32-bit values, while B and D are unsigned. The picture stays the same, except for important change: AD and BC can now be negative, so the values must be sign-extended before addition:

high 64 bits of xy low 64 bits of xy
hi (A*C) lo (A*C)
sign (A*D) hi (A*D) lo (A*D)
sign (B*C) hi (B*C) lo (B*C)
hi (B*D) lo (B*D)

Here sign(v) is a 32-bit sign extension of v. Note that BD must not be sign-extended, since it is a positive value. These sign manipulations can be implemented very easily by using hi and lo instead of uhi and ulo where appropriate:

static long mult_signed_hipart (long x, long y)
    long A = hi (x);
    long B = ulo (x);
    long C = hi (y);
    long D = ulo (y);
    long AC = A * C;
    long AD = A * D;
    long BC = B * C;
    long BD = B * D;
    long ADl_BCl_BDh = ulo (AD) + ulo (BC) + uhi (BD);
    return AC + hi (AD) + hi (BC) + uhi (ADl_BCl_BDh);

Finally, the signed version of the hash function:

public int hashCode ()
    long sign = v >> 63;
    long div = (mult_signed_hipart (v, 2614885092524444427L) >> 27) - sign;
    return (int) (v - div * 946840871);

Performance measurements

Let’s add new unsigned and signed versions to our hash function test suit (they’ll be called LongPoint60 and LongPoint61 to indicate the fact that they are modified versions of LongPoint6).

The full new test suite is here. The class to run is HashTime1.

Here are the results (in milliseconds for 100M iterations):

Class name Comment Time, Java 7Time, Java 8
Point Multiply by 3, 5 548 485
Long Long default 547 486
LongPoint Multiply by 3, 5 547 486
LongPoint3 Multiply by 11, 17 547 485
LongPoint4 Multiply by two big primes 547 486
LongPoint5 Multiply by one big prime 547 486
LongPoint6 Modulo big prime 578 1655
LongPoint60 Modulo optimised, unsigned 640 610
LongPoint61 Modulo optimised, signed 729 727
LongPoint7 java.util.zip.CRC32 10109 1882
NullPoint Null function (does nothing) 532 455

The results are surprising. Our optimisation did work! The code above, being rather long and complex, still runs faster than the modulo operation as implemented by Java 8. And quite a bit faster: if we subtract the calling overhead (measured as the time of NullPoint), we get:

Class name Comment Time, Java 7Time, Java 8
LongPoint6 Modulo big prime 46 1200
LongPoint60 Modulo optimised, unsigned 108 155
LongPoint61 Modulo optimised, signed 197 272

Some observations:

Class name Comment Cycles, Java 7Cycles, Java 8
LongPoint6 Modulo big prime 1.2 31.2
LongPoint60 Modulo optimised, unsigned 2.8 4.0
LongPoint61 Modulo optimised, signed 5.1 7.0

Even though Java 8 figures are bigger than those of Java 7, they are still surprisingly small. The processor must be very advanced to calculate our unsigned version in 4 cycles.

The disassembly

Just out of curiosity, let’s look at the code generated for our functions. Each of them is compiled twice by Java 8, we’ll look at the second output. This is the code for the unsigned version (LongPoint60):

0x00000000027be8e0: sub    rsp,0x18
0x00000000027be8e7: mov    QWORD PTR [rsp+0x10],rbp  ;*synchronization entry
                                                ; - LongPoint60::hashCode@-1 (line 46)

0x00000000027be8ec: mov    r10,QWORD PTR [rdx+0x10]  ;*getfield v
                                              ; - LongPoint60::hashCode@1 (line 46)

0x00000000027be8f0: mov    r11,r10
0x00000000027be8f3: shr    r11,0x20           ;*lushr
                                              ; - LongUtil::uhi@3 (line 25)
                                              ; - LongPoint60::mult_unsigned_hipart@1 (line 29)
                                              ; - LongPoint60::hashCode@7 (line 46)

0x00000000027be8f7: mov    r8d,r10d           ;*land
                                              ; - LongUtil::ulo@4 (line 30)
                                              ; - LongPoint60::mult_unsigned_hipart@7 (line 30)
                                              ; - LongPoint60::hashCode@7 (line 46)

0x00000000027be8fa: imul   r9,r11,0x2449f023
0x00000000027be901: imul   rcx,r8,0x2c624b0b
0x00000000027be908: imul   r8,r8,0x2449f023   ;*lmul
                                              ; - LongPoint60::mult_unsigned_hipart@42 (line 36)
                                              ; - LongPoint60::hashCode@7 (line 46)

0x00000000027be90f: shr    rcx,0x20
0x00000000027be913: mov    rbx,r8
0x00000000027be916: shr    rbx,0x20
0x00000000027be91a: mov    r8d,r8d
0x00000000027be91d: imul   r11,r11,0x2c624b0b  ;*lmul
                                              ; - LongPoint60::mult_unsigned_hipart@35 (line 35)
                                              ; - LongPoint60::hashCode@7 (line 46)

0x00000000027be924: mov    edi,r11d
0x00000000027be927: add    rdi,r8
0x00000000027be92a: add    rdi,rcx
0x00000000027be92d: shr    r11,0x20
0x00000000027be931: add    r9,r11
0x00000000027be934: add    r9,rbx
0x00000000027be937: shr    rdi,0x20
0x00000000027be93b: add    r9,rdi
0x00000000027be93e: sar    r9,0x1b
0x00000000027be942: imul   r11,r9,0x386fa527
0x00000000027be949: sub    r10,r11
0x00000000027be94c: mov    eax,r10d           ;*l2i  ; - LongPoint60::hashCode@24 (line 47)

0x00000000027be94f: add    rsp,0x10
0x00000000027be953: pop    rbp
0x00000000027be954: test   DWORD PTR [rip+0xfffffffffd9816a6],eax        # 0x0000000000140000
                                              ;   {poll_return}
0x00000000027be95a: ret    

This is the code for the signed version (LongPoint61):

0x00000000025949e0: sub    rsp,0x18
0x00000000025949e7: mov    QWORD PTR [rsp+0x10],rbp  ;*synchronization entry
                                              ; - LongPoint61::hashCode@-1 (line 46)

0x00000000025949ec: mov    rbx,QWORD PTR [rdx+0x10]  ;*getfield v
                                              ; - LongPoint61::hashCode@1 (line 46)

0x00000000025949f0: mov    r10,rbx
0x00000000025949f3: shr    r10,0x20
0x00000000025949f7: mov    rdi,rbx
0x00000000025949fa: sar    rdi,0x3f
0x00000000025949fe: mov    r11d,r10d
0x0000000002594a01: mov    r10d,ebx           ;*land
                                              ; - LongUtil::ulo@4 (line 30)
                                              ; - LongPoint61::mult_signed_hipart@8 (line 30)
                                              ; - LongPoint61::hashCode@15 (line 47)

0x0000000002594a04: movsxd r11,r11d           ;*i2l  ; - LongPoint61::mult_signed_hipart@4 (line 29)
                                              ; - LongPoint61::hashCode@15 (line 47)

0x0000000002594a07: imul   r8,r10,0x2c624b0b
0x0000000002594a0e: imul   r9,r11,0x2c624b0b  ;*lmul
                                              ; - LongPoint61::mult_signed_hipart@37 (line 35)
                                              ; - LongPoint61::hashCode@15 (line 47)

0x0000000002594a15: shr    r8,0x20
0x0000000002594a19: mov    rcx,r9
0x0000000002594a1c: shr    rcx,0x20
0x0000000002594a20: mov    edx,r9d
0x0000000002594a23: mov    r9d,ecx
0x0000000002594a26: imul   r11,r11,0x2449f023
0x0000000002594a2d: movsxd r9,r9d
0x0000000002594a30: add    r11,r9
0x0000000002594a33: imul   r10,r10,0x2449f023  ;*lmul
                                              ; - LongPoint61::mult_signed_hipart@44 (line 36)
                                              ; - LongPoint61::hashCode@15 (line 47)

0x0000000002594a3a: mov    r9d,r10d
0x0000000002594a3d: add    rdx,r9
0x0000000002594a40: add    rdx,r8
0x0000000002594a43: shr    r10,0x20
0x0000000002594a47: shr    rdx,0x20
0x0000000002594a4b: mov    r8d,r10d
0x0000000002594a4e: movsxd r10,r8d
0x0000000002594a51: add    r11,r10
0x0000000002594a54: add    r11,rdx
0x0000000002594a57: sar    r11,0x1b
0x0000000002594a5b: sub    r11,rdi
0x0000000002594a5e: imul   r10,r11,0x386fa527
0x0000000002594a65: sub    rbx,r10
0x0000000002594a68: mov    eax,ebx            ;*l2i  ; - LongPoint61::hashCode@34 (line 48)
0x0000000002594a6a: add    rsp,0x10
0x0000000002594a6e: pop    rbp
0x0000000002594a6f: test   DWORD PTR [rip+0xfffffffffdc9b58b],eax        # 0x0000000000230000
                                                ;   {poll_return}
0x0000000002594a75: ret    

Both of these pieces look like a very good code for me. It is a mystery why the second one runs so much slower than the first one. It is still amazing that both run much faster than a runtime call, which performs s DIV instruction.

A full test

How do the optimised versions perform in the full test (a Life application)? Let’s try them. Here are the times (in milliseconds for 10,000 iterations). The times for other versions are taken from the original article:

Class name Comment Time, Java 7Time, Java 8
Point Multiply by 3, 5 2743 4961
Long Long default 4273 6755
LongPoint Multiply by 3, 5 2602 4836
LongPoint3 Multiply by 11, 17 2074 2028
LongPoint4 Multiply by two big primes 2000 1585
LongPoint5 Multiply by one big prime 1979 1553
LongPoint6 Modulo big prime 1890 1550
LongPoint60 Modulo optimised, unsigned 2124 1608
LongPoint61 Modulo optimised, signed 2198 1689
LongPoint7 java.util.zip.CRC32 10115 3206

The new versions do not perform badly, compared to other hash functions, but they are not nearly as fast as the microbenchmarks suggest. In particular, they don’t outperform the original division-based version. Why this happend is a mystery, perhaps Java 8 is capable of some magic we don’t know about yet.


Coming soon

Is there any way to improve division even more? Can we outperform the original division-based version on the full test this way, and if not, why? We’ll see soon.

comments powered by Disqus