Tricking Java into Adding Up Arrays Faster

There is currently what I like to think of as a “crop circle” in Java 9: you can make hash codes faster by manually unrolling some multiplications, which is tracked by an OpenJDK ticket. This one is even weirder. Imagine you have an int[] and want to compute the sum of its elements. You could do exactly that, or, supposing your values are small enough not to overflow, you can make your loop much faster by multiplying each element by 2 inside the loop, and dividing the result by 2 at the end. This is because autovectorisation, with strength reductions to shifts and additions to boot, kicks in for loops that look like a dot product, whereas summations of arrays don’t seem to be optimised at all – see this post for an analysis. Don’t believe me? Run the code at github.


    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int NaiveSum(IntData state) {
        int value = 0;
        int[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += data[i];
        }
        return value;
    }

    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int CropCircle(IntData state) {
        int value = 0;
        int[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += 2 * data[i];
        }
        return value / 2;
    }

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
CropCircle thrpt 1 10 29.922687 0.383028 ops/ms 100000
CropCircle thrpt 1 10 2.065812 0.120089 ops/ms 1000000
NaiveSum thrpt 1 10 26.241689 0.660850 ops/ms 100000
NaiveSum thrpt 1 10 1.868644 0.244081 ops/ms 1000000

How much Algebra does C2 Know? Part 2: Distributivity

In part one of this series of posts, I looked at how important associativity and independence are for fast loops. C2 seems to utilise these properties to generate unrolled and pipelined machine code for loops, achieving higher throughput even in cases where the kernel of the loop is 3x slower according to vendor advertised instruction throughputs. C2 has a weird and wonderful relationship with distributivity, and hints from the programmer can both and help hinder the generation of good quality machine code.

Viability and Correctness

Distributivity is the simple notion of factoring out brackets. Is this, in general, a viable loop rewrite strategy? This can be utilised to transform the method Scale into FactoredScale, both of which perform floating point arithmetic:


    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public double Scale(DoubleData state) {
        double value = 0D;
        double[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += 3.14159 * data[i];
        }
        return value;
    }

    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public double FactoredScale(DoubleData state) {
        double value = 0D;
        double[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += data[i];
        }
        return 3.14159 * value;
    }

Running the project at github with the argument --include .*scale.*, there may be a performance gain to be had from this rewrite, but it isn’t clear cut:

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
FactoredScale thrpt 1 10 7.011606 0.274742 ops/ms 100000
FactoredScale thrpt 1 10 0.621515 0.026853 ops/ms 1000000
Scale thrpt 1 10 6.962434 0.240180 ops/ms 100000
Scale thrpt 1 10 0.671042 0.011686 ops/ms 1000000

With the real numbers it would be completely valid, but floating point arithmetic is not associative. Joseph Darcy explains why in this deep dive on floating point semantics. Broken associativity of addition entails broken distributivity of any operation over it, so the two loops are not equivalent, and they give different outputs (e.g. 15662.513298516365 vs 15662.51329851632 for one sample input). The rewrite isn’t correct even for floating point data, so it isn’t an optimisation that could be applied in good faith, except in a very small number of cases. You have to rewrite the loop yourself and figure out if the small but inevitable differences are acceptable.

Counterintuitive Performance

Integer multiplication is distributive over addition, and we can check if C2 does this rewrite by running the same code with 32 bit integer values, for now fixing a scale factor of 10 (which seems like an innocuous value, no?)


    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int Scale_Int(IntData state) {
        int value = 0;
        int[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += 10 * data[i];
        }
        return value;
    }

    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int FactoredScale_Int(IntData state) {
        int value = 0;
        int[] data = state.data1;
        for (int i = 0; i < data.length; ++i) {
            value += data[i];
        }
        return 10 * value;
    }

The results are fascinating:

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
FactoredScale_Int thrpt 1 10 28.339699 0.608075 ops/ms 100000
FactoredScale_Int thrpt 1 10 2.392579 0.506413 ops/ms 1000000
Scale_Int thrpt 1 10 33.335721 0.295334 ops/ms 100000
Scale_Int thrpt 1 10 2.838242 0.448213 ops/ms 1000000

The code is doing thousands more multiplications in less time when the multiplication is not factored out of the loop. So what the devil is going on? Inspecting the assembly for the faster loop is revealing

  0x000001c89e499400: vmovdqu ymm8,ymmword ptr [rbp+r13*4+10h]
  0x000001c89e499407: movsxd  r10,r13d       
  0x000001c89e49940a: vmovdqu ymm9,ymmword ptr [rbp+r10*4+30h]
  0x000001c89e499411: vmovdqu ymm13,ymmword ptr [rbp+r10*4+0f0h]
  0x000001c89e49941b: vmovdqu ymm12,ymmword ptr [rbp+r10*4+50h]
  0x000001c89e499422: vmovdqu ymm4,ymmword ptr [rbp+r10*4+70h]
  0x000001c89e499429: vmovdqu ymm3,ymmword ptr [rbp+r10*4+90h]
  0x000001c89e499433: vmovdqu ymm2,ymmword ptr [rbp+r10*4+0b0h]
  0x000001c89e49943d: vmovdqu ymm0,ymmword ptr [rbp+r10*4+0d0h]
  0x000001c89e499447: vpslld  ymm11,ymm8,1h  
  0x000001c89e49944d: vpslld  ymm1,ymm0,1h   
  0x000001c89e499452: vpslld  ymm0,ymm0,3h   
  0x000001c89e499457: vpaddd  ymm5,ymm0,ymm1 
  0x000001c89e49945b: vpslld  ymm0,ymm2,3h   
  0x000001c89e499460: vpslld  ymm7,ymm3,3h   
  0x000001c89e499465: vpslld  ymm10,ymm4,3h 
  0x000001c89e49946a: vpslld  ymm15,ymm12,3h
  0x000001c89e499470: vpslld  ymm14,ymm13,3h
  0x000001c89e499476: vpslld  ymm1,ymm9,3h  
  0x000001c89e49947c: vpslld  ymm2,ymm2,1h  
  0x000001c89e499481: vpaddd  ymm6,ymm0,ymm2   
  0x000001c89e499485: vpslld  ymm0,ymm3,1h     
  0x000001c89e49948a: vpaddd  ymm7,ymm7,ymm0   
  0x000001c89e49948e: vpslld  ymm0,ymm4,1h     
  0x000001c89e499493: vpaddd  ymm10,ymm10,ymm0
  0x000001c89e499497: vpslld  ymm0,ymm12,1h   
  0x000001c89e49949d: vpaddd  ymm12,ymm15,ymm0
  0x000001c89e4994a1: vpslld  ymm0,ymm13,1h   
  0x000001c89e4994a7: vpaddd  ymm4,ymm14,ymm0 
  0x000001c89e4994ab: vpslld  ymm0,ymm9,1h    
  0x000001c89e4994b1: vpaddd  ymm2,ymm1,ymm0  
  0x000001c89e4994b5: vpslld  ymm0,ymm8,3h    
  0x000001c89e4994bb: vpaddd  ymm8,ymm0,ymm11 
  0x000001c89e4994c0: vphaddd ymm0,ymm8,ymm8  
  0x000001c89e4994c5: vphaddd ymm0,ymm0,ymm3  
  0x000001c89e4994ca: vextracti128 xmm3,ymm0,1h
  0x000001c89e4994d0: vpaddd  xmm0,xmm0,xmm3   
  0x000001c89e4994d4: vmovd   xmm3,ebx         
  0x000001c89e4994d8: vpaddd  xmm3,xmm3,xmm0   
  0x000001c89e4994dc: vmovd   r10d,xmm3        
  0x000001c89e4994e1: vphaddd ymm0,ymm2,ymm2   
  0x000001c89e4994e6: vphaddd ymm0,ymm0,ymm3   
  0x000001c89e4994eb: vextracti128 xmm3,ymm0,1h
  0x000001c89e4994f1: vpaddd  xmm0,xmm0,xmm3   
  0x000001c89e4994f5: vmovd   xmm3,r10d        
  0x000001c89e4994fa: vpaddd  xmm3,xmm3,xmm0   
  0x000001c89e4994fe: vmovd   r11d,xmm3        
  0x000001c89e499503: vphaddd ymm2,ymm12,ymm12  
  0x000001c89e499508: vphaddd ymm2,ymm2,ymm0    
  0x000001c89e49950d: vextracti128 xmm0,ymm2,1h 
  0x000001c89e499513: vpaddd  xmm2,xmm2,xmm0    
  0x000001c89e499517: vmovd   xmm0,r11d         
  0x000001c89e49951c: vpaddd  xmm0,xmm0,xmm2    
  0x000001c89e499520: vmovd   r10d,xmm0         
  0x000001c89e499525: vphaddd ymm0,ymm10,ymm10  
  0x000001c89e49952a: vphaddd ymm0,ymm0,ymm3   
  0x000001c89e49952f: vextracti128 xmm3,ymm0,1h
  0x000001c89e499535: vpaddd  xmm0,xmm0,xmm3
  0x000001c89e499539: vmovd   xmm3,r10d   
  0x000001c89e49953e: vpaddd  xmm3,xmm3,xmm0   
  0x000001c89e499542: vmovd   r11d,xmm3        
  0x000001c89e499547: vphaddd ymm2,ymm7,ymm7   
  0x000001c89e49954c: vphaddd ymm2,ymm2,ymm0   
  0x000001c89e499551: vextracti128 xmm0,ymm2,1h
  0x000001c89e499557: vpaddd  xmm2,xmm2,xmm0 
  0x000001c89e49955b: vmovd   xmm0,r11d      
  0x000001c89e499560: vpaddd  xmm0,xmm0,xmm2 
  0x000001c89e499564: vmovd   r10d,xmm0      
  0x000001c89e499569: vphaddd ymm0,ymm6,ymm6   
  0x000001c89e49956e: vphaddd ymm0,ymm0,ymm3   
  0x000001c89e499573: vextracti128 xmm3,ymm0,1h
  0x000001c89e499579: vpaddd  xmm0,xmm0,xmm3   
  0x000001c89e49957d: vmovd   xmm3,r10d        
  0x000001c89e499582: vpaddd  xmm3,xmm3,xmm0   
  0x000001c89e499586: vmovd   r11d,xmm3        
  0x000001c89e49958b: vphaddd ymm2,ymm5,ymm5   
  0x000001c89e499590: vphaddd ymm2,ymm2,ymm0   
  0x000001c89e499595: vextracti128 xmm0,ymm2,1h
  0x000001c89e49959b: vpaddd  xmm2,xmm2,xmm0
  0x000001c89e49959f: vmovd   xmm0,r11d     
  0x000001c89e4995a4: vpaddd  xmm0,xmm0,xmm2
  0x000001c89e4995a8: vmovd   r10d,xmm0
  0x000001c89e4995ad: vphaddd ymm2,ymm4,ymm4 
  0x000001c89e4995b2: vphaddd ymm2,ymm2,ymm1
  0x000001c89e4995b7: vextracti128 xmm1,ymm2,1h
  0x000001c89e4995bd: vpaddd  xmm2,xmm2,xmm1
  0x000001c89e4995c1: vmovd   xmm1,r10d  
  0x000001c89e4995c6: vpaddd  xmm1,xmm1,xmm2    
  0x000001c89e4995ca: vmovd   ebx,xmm1          

The loop is aggressively unrolled, pipelined, and vectorised. Moreover, the multiplication by ten results not in a multiplication but two left shifts (see VPSLLD) and an addition. Note that x << 1 + x << 3 = x * 10 and C2 seems to know it; this rewrite can be applied because it can be proven statically that the factor is always 10. The “optimised” loop doesn’t vectorise at all (and I have no idea why not – isn’t this a bug? Yes it is.)

  0x000002bbebeda3c8: add     ebx,dword ptr [rbp+r8*4+14h]
  0x000002bbebeda3cd: add     ebx,dword ptr [rbp+r8*4+18h]
  0x000002bbebeda3d2: add     ebx,dword ptr [rbp+r8*4+1ch]
  0x000002bbebeda3d7: add     ebx,dword ptr [rbp+r8*4+20h]
  0x000002bbebeda3dc: add     ebx,dword ptr [rbp+r8*4+24h]
  0x000002bbebeda3e1: add     ebx,dword ptr [rbp+r8*4+28h]
  0x000002bbebeda3e6: add     ebx,dword ptr [rbp+r8*4+2ch]
  0x000002bbebeda3eb: add     r13d,8h           
  0x000002bbebeda3ef: cmp     r13d,r11d         
  0x000002bbebeda3f2: jl      2bbebeda3c0h      
  

This is a special case: data is usually dynamic and variable, so the loop cannot always be proven to be equivalent to a linear combination of bit shifts. The routine is compiled for all possible parameters, not just statically contrived cases like the one above, so you may never see this assembly in the wild. However, even with random factors, the slow looking loop is aggressively optimised in a way the hand “optimised” code is not:


    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int Scale_Int_Dynamic(ScaleState state) {
        int value = 0;
        int[] data = state.data;
        int factor = state.randomFactor();
        for (int i = 0; i < data.length; ++i) {
            value += factor * data[i];
        }
        return value;
    }

    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    @Benchmark
    public int FactoredScale_Int_Dynamic(ScaleState state) {
        int value = 0;
        int[] data = state.data;
        int factor = state.randomFactor();
        for (int i = 0; i < data.length; ++i) {
            value += data[i];
        }
        return factor * value;
    }

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
FactoredScale_Int_Dynamic thrpt 1 10 26.100439 0.340069 ops/ms 100000
FactoredScale_Int_Dynamic thrpt 1 10 1.918011 0.297925 ops/ms 1000000
Scale_Int_Dynamic thrpt 1 10 30.219809 2.977389 ops/ms 100000
Scale_Int_Dynamic thrpt 1 10 2.314159 0.378442 ops/ms 1000000

Far from seeking to exploit distributivity to reduce the number of multiplication instructions, it seems to almost embrace the extraneous operations as metadata to drive optimisations. The assembly for Scale_Int_Dynamic confirms this (it shows vectorised multiplication, not shifts, within the loop):


  0x000001f5ca2fa200: vmovdqu ymm0,ymmword ptr [r13+r14*4+10h]
  0x000001f5ca2fa207: vpmulld ymm11,ymm0,ymm2   
  0x000001f5ca2fa20c: movsxd  r10,r14d          
  0x000001f5ca2fa20f: vmovdqu ymm0,ymmword ptr [r13+r10*4+30h]
  0x000001f5ca2fa216: vmovdqu ymm1,ymmword ptr [r13+r10*4+0f0h]
  0x000001f5ca2fa220: vmovdqu ymm3,ymmword ptr [r13+r10*4+50h]
  0x000001f5ca2fa227: vmovdqu ymm7,ymmword ptr [r13+r10*4+70h]
  0x000001f5ca2fa22e: vmovdqu ymm6,ymmword ptr [r13+r10*4+90h]
  0x000001f5ca2fa238: vmovdqu ymm5,ymmword ptr [r13+r10*4+0b0h]
  0x000001f5ca2fa242: vmovdqu ymm4,ymmword ptr [r13+r10*4+0d0h]
  0x000001f5ca2fa24c: vpmulld ymm9,ymm0,ymm2    
  0x000001f5ca2fa251: vpmulld ymm4,ymm4,ymm2    
  0x000001f5ca2fa256: vpmulld ymm5,ymm5,ymm2    
  0x000001f5ca2fa25b: vpmulld ymm6,ymm6,ymm2    
  0x000001f5ca2fa260: vpmulld ymm8,ymm7,ymm2    
  0x000001f5ca2fa265: vpmulld ymm10,ymm3,ymm2   
  0x000001f5ca2fa26a: vpmulld ymm3,ymm1,ymm2    
  0x000001f5ca2fa26f: vphaddd ymm1,ymm11,ymm11  
  0x000001f5ca2fa274: vphaddd ymm1,ymm1,ymm0    
  0x000001f5ca2fa279: vextracti128 xmm0,ymm1,1h 
  0x000001f5ca2fa27f: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa283: vmovd   xmm0,ebx          
  0x000001f5ca2fa287: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa28b: vmovd   r10d,xmm0         
  0x000001f5ca2fa290: vphaddd ymm1,ymm9,ymm9    
  0x000001f5ca2fa295: vphaddd ymm1,ymm1,ymm0    
  0x000001f5ca2fa29a: vextracti128 xmm0,ymm1,1h 
  0x000001f5ca2fa2a0: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa2a4: vmovd   xmm0,r10d         
  0x000001f5ca2fa2a9: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa2ad: vmovd   r11d,xmm0         
  0x000001f5ca2fa2b2: vphaddd ymm0,ymm10,ymm10  
  0x000001f5ca2fa2b7: vphaddd ymm0,ymm0,ymm1    
  0x000001f5ca2fa2bc: vextracti128 xmm1,ymm0,1h 
  0x000001f5ca2fa2c2: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa2c6: vmovd   xmm1,r11d         
  0x000001f5ca2fa2cb: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa2cf: vmovd   r10d,xmm1         
  0x000001f5ca2fa2d4: vphaddd ymm1,ymm8,ymm8    
  0x000001f5ca2fa2d9: vphaddd ymm1,ymm1,ymm0    
  0x000001f5ca2fa2de: vextracti128 xmm0,ymm1,1h 
  0x000001f5ca2fa2e4: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa2e8: vmovd   xmm0,r10d         
  0x000001f5ca2fa2ed: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa2f1: vmovd   r11d,xmm0         
  0x000001f5ca2fa2f6: vphaddd ymm0,ymm6,ymm6    
  0x000001f5ca2fa2fb: vphaddd ymm0,ymm0,ymm1    
  0x000001f5ca2fa300: vextracti128 xmm1,ymm0,1h 
  0x000001f5ca2fa306: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa30a: vmovd   xmm1,r11d         
  0x000001f5ca2fa30f: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa313: vmovd   r10d,xmm1         
  0x000001f5ca2fa318: vphaddd ymm1,ymm5,ymm5    
  0x000001f5ca2fa31d: vphaddd ymm1,ymm1,ymm0    
  0x000001f5ca2fa322: vextracti128 xmm0,ymm1,1h 
  0x000001f5ca2fa328: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa32c: vmovd   xmm0,r10d         
  0x000001f5ca2fa331: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa335: vmovd   r11d,xmm0         
  0x000001f5ca2fa33a: vphaddd ymm0,ymm4,ymm4    
  0x000001f5ca2fa33f: vphaddd ymm0,ymm0,ymm1    
  0x000001f5ca2fa344: vextracti128 xmm1,ymm0,1h 
  0x000001f5ca2fa34a: vpaddd  xmm0,xmm0,xmm1    
  0x000001f5ca2fa34e: vmovd   xmm1,r11d         
  0x000001f5ca2fa353: vpaddd  xmm1,xmm1,xmm0    
  0x000001f5ca2fa357: vmovd   r10d,xmm1         
  0x000001f5ca2fa35c: vphaddd ymm1,ymm3,ymm3    
  0x000001f5ca2fa361: vphaddd ymm1,ymm1,ymm7    
  0x000001f5ca2fa366: vextracti128 xmm7,ymm1,1h 
  0x000001f5ca2fa36c: vpaddd  xmm1,xmm1,xmm7   
  0x000001f5ca2fa370: vmovd   xmm7,r10d        
  0x000001f5ca2fa375: vpaddd  xmm7,xmm7,xmm1   
  0x000001f5ca2fa379: vmovd   ebx,xmm7         

There are two lessons to be learnt here. The first is that what you see is not what you get. The second is about the correctness of asymptotic analysis. If hierarchical cache renders asymptotic analysis bullshit (linear time but cache friendly algorithms can, and do, outperform logarithmic algorithms with cache misses), optimising compilers render the field practically irrelevant.

Confusing Sets and Lists

I have often seen the roles of lists and sets confused. An application can be brought to its knees – that is, cease to deliver commercial value – if List.contains is called frequently enough on big enough lists. And then there is the workaround… When I moved over to Java from C++ several years ago, it seemed utterly crazy that there was even a Collection interface – exactly what Scott Meier’s Effective STL said not to do. I still think it’s crazy. Sets and lists cannot be substituted, and when you add up the marginal costs, as well as the costs of any compensatory workarounds, confusing them is responsible for a lot of performance bugs. As an application developer, it is part of your job to choose. Here are a few simple examples of when to use each collection type.

Contains

Is an element in the collection?

Never ever do this with a List. This operation is often referred to as being O(n). Which means in your worst case will touch every element in the list (technically, at least once). You have a choice between HashSet and a TreeSet, and both have costs and benefits.

If you choose a HashSet, your best case is O(1): you evaluate a hash code, take its modulus with respect to the size of an array, and look up a bucket containing only one element. The worst case occurs with a degenerate hash code which maps all elements to the same bucket. This is again O(n): you probe a linked list testing each element for equality. On average you get something between these two cases and it depends on the uniformity of your hash code implementation.

If you choose a TreeSet you get a worst case O(log n): this is effectively just a binary search through the nodes in a red black tree. Performance is limited by the cost of the comparator, and suffers systematically from cache misses for large sets (like any kind of pointer chasing, branch prediction and prefetching is difficult if not impossible).

If you’re working with numbers, and small to medium collections, a sorted primitive array may be the best approach, so long as it fits in cache. If you’re working with integers, you can do this in constant time in the worst case by using a BitSet.

Select

What is the value of the element at a given index with respect to a sort order?

This is an obvious use case for a List: it’s O(1) – this is just a lookup at an array offset.

You couldn’t even write the code with a HashSet without copying the data into an intermediate ordered structure, at which point you would probably think again. You see this sort of thing done in code written by inexpensive programmers at large outsourcing consultancies, who were perhaps just under unreasonable pressure to deliver to arbitrary deadlines.

SortedSet, and anything conceptually similar, is the wrong data structure for this operation. The only way to compute this is O(n): you iterate through the set incrementing a counter until you reach the index, and then return the element you’ve iterated to. If you reach the end of the set, you throw. If you do this a lot, you’ll notice.

Rank

How many predecessors does an element have with respect to a sort order?

Another classic operation for List, so long as you keep it sorted. Use Collections.binarySearch to find the index of the element in the collection with complexity O(log n). This is its rank. If you can get away with it, primitive arrays will be much better here, especially if they are small enough to fit in cache.

Once again, there are creativity points available for the solution involving a HashSet, and they do exist in the wild, but a clean looking solution is at least possible with a SortedSet. However, it involves an iteration with another check against an incrementing counter. It’s O(n) and if you do it a lot, you’ll blow your performance profile, so use a sorted list instead.

What if you had the source code?

Is this fundamental or just a limitation of the Collections framework? Maybe if you had the source code you could just make these data structures optimal for every use case, without having to choose the right one? Not without creating a Frankenstein, and not without a lot of memory. Optimisation isn’t free.

How much Algebra does C2 Know? Part 1: Associativity

Making loops execute faster is firmly rooted in algebra, but how much does C2 know or care about? When building a highly optimised query engine, a critical concern is the quality of assembly code generated for loops. There is a lot more to JIT compilation than loop optimisation; inlining, class hierarchy analysis, escape analysis to name but a few. Moreover, everything it does has to be fast since it shares resources with the application itself; it can’t spend time unless it brings a net benefit. Being such a generalist, does C2, the JIT compiler used in server applications, know high school algebra?

Specific knowledge of maths is not always worthwhile to program execution, even when it leads to high performance gains. As a motivating example, there is no way to refer directly to the natural numbers in any programming language I have ever used. For instance, the sum of the first n natural numbers is ((n+1) * n)/2, and most high school students know it. This expression is intuitively much faster to evaluate than the equivalent algorithm:


int sum(int n) {
    int total = 0;
    for (int i = 0; i <= n; ++i) {
        total += i;
    }
    return total;
}

But would this loop rewrite be a worthwhile optimisation? The expression takes about 3.5ns to compute the sum of the first million natural numbers, whereas the loop takes 350┬Ás, so we can conclude that C2 does not know this formula and prefers brute force. I would be aghast if time had been spent on optimisations like this: unless your application spends a lot of time adding up contiguous ranges of natural numbers, the marginal benefit is negligible. If this is what your application does most, you should do it yourself. The possibility of an optimisation doesn’t imply its viability: there needs to be a benefit when considering engineering effort, speed improvement, reliability and ubiquity. While this optimisation fails miserably on the grounds of ubiquity, there’s useful schoolboy maths that C2 does seem to know.

Associativity and Dependencies

Each x86 instruction has a throughput – the number of cycles it takes to complete – and a latency – the number of cycles it takes before the result is available to the next instruction in a chain. These numbers are produced by processor vendors, but there are independent numbers like these from Agner Fog, which also includes more detailed definitions of terms like latency. At first, the latency number feels a bit like a scam: what use is an advertised throughput if we can’t use the result immediately? This is where pipelining comes in: independent instructions can be interleaved. If a loop operation is associative and there are no dependencies between iterations, then it can be unrolled, which enables pipelining. If a loop operation is also commutative, then out of order execution is permitted. Evidence of an unrolled loop suggests that the compiler has realised that an operation is at least associative.

To see this in action it’s necessary to find an associative loop reduction that the compiler can’t vectorise. I took an example from the RoaringBitmap library – computing the cardinality of a bitmap container – which is a perfect example to capture this behaviour, because bit counts cannot be vectorised in Java.


  /**
   * Recomputes the cardinality of the bitmap.
   */
  protected void computeCardinality() {
    this.cardinality = 0;
    for (int k = 0; k < this.bitmap.length; k++) {
      this.cardinality += Long.bitCount(this.bitmap[k]);
    }
  }

we can see evidence of loop unrolling and out of order execution when looking at the assembly code emitted. The popcnt instructions are executed on the array out of order, and do not wait for the addition to the accumulator.

popcnt  r9,qword ptr [rbx+r13*8+10h]

movsxd  r8,r13d

popcnt  r10,qword ptr [rbx+r8*8+28h]

popcnt  r11,qword ptr [rbx+r8*8+18h]

popcnt  rdx,qword ptr [rbx+r8*8+20h]
 
movsxd  r8,r9d

add     r8,rbp

movsxd  r9,edx

To generate this assembly code you can run the project at github with the arguments

--include .*popcnt.* 
--print-assembly

The compiler does a very good job in this case: you can try unrolling the loop yourself, but you can only match performance if you guess the loop stride correctly. It’s impossible to prove a negative proposition, but it’s likely you’ll only make it worse if you try. C2 graduates with flying colours here: it definitely understands associativity and dependence.

The catch with pipelining is that an instruction must always wait for its operands. While the operation is associative, there is no way to reorder the code below.


    private int[] prefixSum(int[] data) {
        int[] result = new int[data.length];
        for (int i = 1; i < result.length; ++i) {
            result[i] = result[i - 1] + data[i];
        }
        return result;
    }

What happens with a prefix sum? There’s no unrolling: you can see the loop control statements have not been removed (look for commands like cmp ebx, inc ebx). The loop is also executed in order because it is sequentially dependent.


  0x000001c21215bbc8: mov     r9d,dword ptr [r8+0ch]  
  0x000001c21215bbcc: mov     ebp,dword ptr [r13+rbx*4+0ch]
  0x000001c21215bbd1: cmp     ebx,r9d           
  0x000001c21215bbd4: jnb     1c21215bce1h      
  0x000001c21215bbda: add     ebp,dword ptr [r8+rbx*4+10h]
  0x000001c21215bbdf: cmp     ebx,edi          

Does this harm performance? add takes 0.33 cycles, whereas popcnt takes 1 cycle per instruction. Shouldn’t a prefix sum be faster to calculate than a population count, on the same length of array and same width of integer? They can be compared head to head (implementing prefix sum for long[] to keep word width constant)

--include .*prefix.PrefixSum.PrefixSumLong|.*popcnt.PopCount.PopCount$

Far from having 3x throughput, the prefix sum is much worse. This is entirely because there is no loop unrolling and no pipelining. When possible, C2 applies aggressive unrolling optimisations unavailable to the programmer. For vectorisable operations (requiring linear independence and countability), loop unrolling further marks the loop as a candidate for auto-vectorisation.

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit Param: size
PopCount thrpt 1 10 9.174499 0.394487 ops/ms 100000
PopCount thrpt 1 10 1.217521 0.513734 ops/ms 1000000
PrefixSumLong thrpt 1 10 6.807279 0.925282 ops/ms 100000
PrefixSumLong thrpt 1 10 0.443974 0.053544 ops/ms 1000000

If the dependencies need to fetch data from RAM the latency can be much higher than loading from registers or from prefetched cache. Even when fetching from RAM, the worst case scenario, during this delay independent instructions can complete, unless they have a false dependency.

Zeroing Negative Values in Arrays Efficiently

Replacing negatives with zeroes in large arrays of values is a primitive function of several complex financial risk measures, including potential future exposure (PFE) and the liquidity coverage ratio (LCR). While this is not an interesting operation by any stretch of the imagination, it is useful and there is significant benefit in its performance. This is an operation that can be computed very efficiently using the instruction VMAXPD. For Intel Xeon processors, this instruction requires half a cycle to calculate and has a latency (how long before another instruction can use its result) of four cycles. There is currently no way to trick Java into using this instruction for this simple operation, though there is a placeholder implementation on the current DoubleVector prototype in Project Panama which may do so.

C++ Intel Intrinsics

It’s possible to target instructions from different processor vendors, in my case Intel, by using intrinsic functions which expose instructions as high level functions. The code looks incredibly ugly but it works. Here is a C++ function for 256 bit ymm registers:


void zero_negatives(const double* source, double* target, const size_t length) {
  for (size_t i = 0; i + 3 < length; i += 4) {
    __m256d vector = _mm256_load_pd(source + i);
    __m256d zeroed = _mm256_max_pd(vector, _mm256_setzero_pd());
    _mm256_storeu_pd(target + i, zeroed);
  }
}

The function loads doubles into 256 bit vectors, within each vector replaces the negative values with zero, and writes them back into an array. It generates the following assembly code (which, incidentally, is less of a shit show to access than in Java):


void zero_negatives(const double* source, double* target, const size_t length) {
00007FF746EE5110  mov         qword ptr [rsp+18h],r8  
00007FF746EE5115  mov         qword ptr [rsp+10h],rdx  
00007FF746EE511A  mov         qword ptr [rsp+8],rcx  
00007FF746EE511F  push        r13  
00007FF746EE5121  push        rbp  
00007FF746EE5122  push        rdi  
00007FF746EE5123  sub         rsp,250h  
00007FF746EE512A  mov         r13,rsp  
00007FF746EE512D  lea         rbp,[rsp+20h]  
00007FF746EE5132  and         rbp,0FFFFFFFFFFFFFFE0h  
00007FF746EE5136  mov         rdi,rsp  
00007FF746EE5139  mov         ecx,94h  
00007FF746EE513E  mov         eax,0CCCCCCCCh  
00007FF746EE5143  rep stos    dword ptr [rdi]  
00007FF746EE5145  mov         rcx,qword ptr [rsp+278h]  
  for (size_t i = 0; i + 3 < length; i += 4) {
00007FF746EE514D  mov         qword ptr [rbp+8],0  
00007FF746EE5155  jmp         zero_negatives+53h (07FF746EE5163h)  
00007FF746EE5157  mov         rax,qword ptr [rbp+8]  
00007FF746EE515B  add         rax,4  
00007FF746EE515F  mov         qword ptr [rbp+8],rax  
00007FF746EE5163  mov         rax,qword ptr [rbp+8]  
00007FF746EE5167  add         rax,3  
00007FF746EE516B  cmp         rax,qword ptr [length]  
00007FF746EE5172  jae         zero_negatives+0DDh (07FF746EE51EDh)  
    __m256d vector = _mm256_load_pd(source + i);
00007FF746EE5174  mov         rax,qword ptr [source]  
00007FF746EE517B  mov         rcx,qword ptr [rbp+8]  
00007FF746EE517F  lea         rax,[rax+rcx*8]  
00007FF746EE5183  vmovupd     ymm0,ymmword ptr [rax]  
00007FF746EE5187  vmovupd     ymmword ptr [rbp+180h],ymm0  
00007FF746EE518F  vmovupd     ymm0,ymmword ptr [rbp+180h]  
00007FF746EE5197  vmovupd     ymmword ptr [rbp+40h],ymm0  
    __m256d zeroed = _mm256_max_pd(vector, _mm256_setzero_pd());
00007FF746EE519C  vxorpd      xmm0,xmm0,xmm0  
00007FF746EE51A0  vmovupd     ymmword ptr [rbp+200h],ymm0  
00007FF746EE51A8  vmovupd     ymm0,ymmword ptr [rbp+40h]  
00007FF746EE51AD  vmaxpd      ymm0,ymm0,ymmword ptr [rbp+200h]  
00007FF746EE51B5  vmovupd     ymmword ptr [rbp+1C0h],ymm0  
00007FF746EE51BD  vmovupd     ymm0,ymmword ptr [rbp+1C0h]  
00007FF746EE51C5  vmovupd     ymmword ptr [rbp+80h],ymm0  
    _mm256_storeu_pd(target + i, zeroed);
00007FF746EE51CD  mov         rax,qword ptr [target]  
00007FF746EE51D4  mov         rcx,qword ptr [rbp+8]  
00007FF746EE51D8  lea         rax,[rax+rcx*8]  
00007FF746EE51DC  vmovupd     ymm0,ymmword ptr [rbp+80h]  
00007FF746EE51E4  vmovupd     ymmword ptr [rax],ymm0  
  }
00007FF746EE51E8  jmp         zero_negatives+47h (07FF746EE5157h)  
}
00007FF746EE51ED  lea         rsp,[r13+250h]  
00007FF746EE51F4  pop         rdi  
00007FF746EE51F5  pop         rbp  
00007FF746EE51F6  pop         r13  
00007FF746EE51F8  ret    

This code is noticeably fast. I measured the throughput averaged over 1000 iterations, with an array of 10 million doubles (800MB) uniformly distributed between +/- 1E7, to quantify the throughput in GB/s and iterations/s. This code does between 4.5 and 5 iterations per second, which translates to processing approximately 4GB/s. This seems high, and since I am unaware of best practices in C++, if the measurement is flawed, I would gratefully be educated in the comments.


void benchmark() {
  const size_t length = 1E8;
  double* values = new double[length];
  fill_array(values, length);
  double* zeroed = new double[length];
  auto start = std::chrono::high_resolution_clock::now();
  int iterations = 1000;
  for (int i = 0; i < iterations; ++i) {
    zero_negatives(values, zeroed, length);
  }
  auto end = std::chrono::high_resolution_clock::now();
  auto nanos = std::chrono::duration_cast<std::chrono::nanoseconds>(end - start).count();
  double thrpt_s = (iterations * 1E9) / nanos;
  double thrpt_gbps = (thrpt_s * sizeof(double) * length) / 1E9;
  std::cout << thrpt_s << "/s" << std::endl;
  std::cout << thrpt_gbps << "GB/s" << std::endl;
  delete[] values;
  delete[] zeroed;
}

While I am sure there are various ways an expert could tweak this for performance, this code can’t get much faster unless there are 512 bit zmm registers available, in which case it would be wasteful. While the code looks virtually the same for AVX512 (just replace “256” with “512”), portability and efficiency are at odds. Handling the mess of detecting the best instruction set for the deployed architecture is the main reason for using Java in performance sensitive (but not critical) applications. But this is not the code the JVM generates.

Java Auto-Vectorisation (Play Your Cards Right)

There is currently no abstraction modelling vectorisation in Java. The only access available is if the compiler engineers implement an intrinsic, or auto-vectorisation, which will try, and sometimes succeed admirably, to translate your code to a good vector implementation. There is currently a prototype project for explicit vectorisation in Project Panama. There are a few ways to skin this cat, and it’s worth looking at the code they generate and the throughput available from each approach.

There is a choice between copying the array and zeroing out the negatives, and allocating a new array and only writing the non-negative values. There is another choice between an if statement and branchless code using Math.max. This results in the following four implementations which I measure on comparable data to the C++ benchmark (10 million doubles, normally distributed with mean zero). To be fair to the Java code, as in the C++ benchmarks, the cost of allocation is isolated by writing into an array pre-allocated once per benchmark. This penalises the approaches where the array is copied first and then zeroed wherever the value is negative. The code is online at github.


    @Benchmark
    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    public double[] BranchyCopyAndMask(ArrayWithNegatives state) {
        double[] data = state.data;
        double[] result = state.target;
        System.arraycopy(data, 0, result, 0, data.length);
        for (int i = 0; i < result.length; ++i) {
            if (result[i] < 0D) {
                result[i] = 0D;
            }
        }
        return result;
    }

    @Benchmark
    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    public double[] BranchyNewArray(ArrayWithNegatives state) {
        double[] data = state.data;
        double[] result = state.target;
        for (int i = 0; i < result.length; ++i) {
            result[i] = data[i] < 0D ? 0D : data[i];
        }
        return result;
    }

    @Benchmark
    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    public double[] NewArray(ArrayWithNegatives state) {
        double[] data = state.data;
        double[] result = state.target;
        for (int i = 0; i < result.length; ++i) {
            result[i] = Math.max(data[i], 0D);
        }
        return result;
    }

    @Benchmark
    @CompilerControl(CompilerControl.Mode.DONT_INLINE)
    public double[] CopyAndMask(ArrayWithNegatives state) {
        double[] data = state.data;
        double[] result = state.target;
        System.arraycopy(data, 0, result, 0, data.length);
        for (int i = 0; i < result.length; ++i) {
            result[i] = Math.max(result[i], 0D);
        }
        return result;
    }

None of these implementations comes close to the native code above. The best implementation performs 1.8 iterations per second which equates to processing approximately 1.4GB/s, vastly inferior to the 4GB/s achieved with Intel intrinsics. The results are below:

Benchmark Mode Threads Samples Score Score Error (99.9%) Unit
BranchyCopyAndMask thrpt 1 10 1.314845 0.061662 ops/s
BranchyNewArray thrpt 1 10 1.802673 0.061835 ops/s
CopyAndMask thrpt 1 10 1.146630 0.018903 ops/s
NewArray thrpt 1 10 1.357020 0.116481 ops/s

As an aside, there is a very interesting observation to make, worthy of its own post: if the array consists only of positive values, the “branchy” implementations run very well, at speeds comparable to the zero_negatives (when it ran with 50% negatives). The ratio of branch hits to misses is an orthogonal explanatory variable, and the input data, while I often don’t think about it enough, is very important.

I only looked at the assembly emitted for the fastest version (BranchyNewArray) and it doesn’t look anything like zero_negatives, though it does use some vectorisation – as pointed out by Daniel Lemire in the comments, this code has probably not been vectorised and is probably using SSE2 (indeed only quad words are loaded into 128 bit registers):

  0x000002ae309c3d5c: vmovsd  xmm0,qword ptr [rdx+rax*8+10h]
  0x000002ae309c3d62: vxorpd  xmm1,xmm1,xmm1    
  0x000002ae309c3d66: vucomisd xmm0,xmm1        

I don’t really understand, and haven’t thought about, the intent of the emitted code, but it makes extensive use of the instruction VUCOMISD for comparisons with zero, which has a lower latency but lower throughput than VMAXPD. It would certainly be interesting to see how Project Panama does this. Perhaps this should just be made available as a fail-safe intrinsic like Arrays.mismatch?