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<html><head><link rel="stylesheet" href="style.css"></head><body><div class="page">
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<h1>20161018 - Fixed Point Rounding</h1>
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<br>
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<center><i>On the list of things to relearn for FPGA based machines.</i></center>
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<center><i>Posting a few more notes while in the background I continue to work towards the next design try.</i></center>
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<br>
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<pre>
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=====================
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DSP FUNCTIONALITY
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=====================
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Some of the Xilinx 7 series DSP functionality and base pipeline (where p as input is from clk-1),
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clk-2 clk-1 clk
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===== ======== ===========
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c= p{op}=p
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c= p{op}=(p>>17)
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c= p{op}=c
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c= p=c
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a=,b= m=a*b p=m
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a=,b= m=a*b p+=m
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a=,b= m=a*b p-=m
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a=,b= m=a*b p=(p>>17)+m
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a=,b= m=a*b p=(p>>17)-m
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a=,b= m=a*b,c= p=c+m
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a=,b= m=a*b,c= p=c-m
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a=,b= t=a:b p{op}=t
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a=,b= t=a:b p=(p>>17){op}t
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a=,b= t=a:b,c= p=c{op}t
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For an 18-bit word accumulator based machine, it is possible to fast min/max
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Uses the internal DSP result forwarding paths
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// signed min and max in 3 cycles
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p=min(p,c); -> p-=c; p= (p>>17)&p; p+=c;
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p=max(p,c); -> p-=c; p=~(p>>17)&p; p+=c;
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Forwarding is going to be useful for going to one thread and mitigating pipelining
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// branch free, where x and y are immediates in 2 cycles
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p=(p< 0)?x:y; -> p= (p>>17)&(x-y); p+=y;
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p=(p>=0)?x:y; -> p=~(p>>17)&(x-y); p+=y;
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====================
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DSP AND ROUNDING
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====================
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Didn't see how to efficiently implement "round half to even"
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However "round half away from zero" seems easy to implement
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Logic for rounding (hopefully I got that right),
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if(x>=0) x++; // cin in second example
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x+=(1<<(n-1))-1; // c input in second example
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x>>=n;
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DSP supports CIN (carry in) support
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Which can take either the inverted sign of P
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(dsp output feed back in the next cycle),
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or the inverted sign of a*b
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DSP can do the following {multiply, round, shift}
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at a throughput of 0.5 clocks,
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p=a*b+cin+c; // throughput of one clock
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p=c+(p>>17); // gets dedicated forwarding path
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Common usage case,
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// divide by a constant via multiply by reciprocal,
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// or multiply by {0 to 131071} representing {0.0 to nearly 1.0}
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a=number; // up to 25-bit signed number on 7 series DSP
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b=fraction; // second argument is 18-bits signed
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p=a*b+cin+65535;
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p=p>>17;
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For reference
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Full table for 3-bit signed numbers a*b multiply
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with round away from zero before shift right by 2,
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a b float int binary a*b+cin +round output
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------------------------ ------------------------------------------
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-4 * -1.00 = 4.00 -> 4 100 * 100 = 010001 -> 0010010 -> 0100 = 4 (overflows)
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-4 * -0.75 = 3.00 -> 3 100 * 101 = 001101 -> 0001110 -> 0011 = 3
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-4 * -0.50 = 2.00 -> 2 100 * 110 = 001001 -> 0001010 -> 0010 = 2
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-4 * -0.25 = 1.00 -> 1 100 * 111 = 000101 -> 0000110 -> 0001 = 1
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-4 * 0.00 = -0.00 -> 0 100 * 000 = 000001 -> 0000010 -> 0000 = 0
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-4 * 0.25 = -1.00 -> -1 100 * 001 = 111100 -> 1111101 -> 1111 = -1
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-4 * 0.50 = -2.00 -> -2 100 * 010 = 111000 -> 1111001 -> 1110 = -2
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-4 * 0.75 = -3.00 -> -3 100 * 011 = 110100 -> 1110101 -> 1101 = -3
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------------------------ ------------------------------------------
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-3 * -1.00 = 3.00 -> 3 101 * 100 = 001101 -> 0001110 -> 0011 = 3
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-3 * -0.75 = 2.25 -> 2 101 * 101 = 001010 -> 0001011 -> 0010 = 2
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-3 * -0.50 = 1.50 -> 2 101 * 110 = 000111 -> 0001000 -> 0010 = 2
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-3 * -0.25 = 0.75 -> 1 101 * 111 = 000100 -> 0000101 -> 0001 = 1
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-3 * 0.00 = -0.00 -> 0 101 * 000 = 000001 -> 0000010 -> 0000 = 0
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-3 * 0.25 = -0.75 -> -1 101 * 001 = 111101 -> 1111110 -> 1111 = -1
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-3 * 0.50 = -1.50 -> -2 101 * 010 = 111010 -> 1111011 -> 1110 = -2
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-3 * 0.75 = -2.25 -> -2 101 * 011 = 110111 -> 1111000 -> 1110 = -2
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------------------------ ------------------------------------------
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-2 * -1.00 = 2.00 -> 2 110 * 100 = 001001 -> 0001010 -> 0010 = 2
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-2 * -0.75 = 1.50 -> 2 110 * 101 = 000111 -> 0001000 -> 0010 = 2
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-2 * -0.50 = 1.00 -> 1 110 * 110 = 000101 -> 0000110 -> 0001 = 1
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-2 * -0.25 = 0.50 -> 1 110 * 111 = 000011 -> 0000100 -> 0001 = 1
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-2 * 0.00 = -0.00 -> 0 110 * 000 = 000001 -> 0000010 -> 0000 = 0
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-2 * 0.25 = -0.50 -> -1 110 * 001 = 111110 -> 1111111 -> 1111 = -1
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-2 * 0.50 = -1.00 -> -1 110 * 010 = 111100 -> 1111101 -> 1111 = -1
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-2 * 0.75 = -1.50 -> -2 110 * 011 = 111010 -> 1111011 -> 1110 = -2
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------------------------ ------------------------------------------
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-1 * -1.00 = 1.00 -> 1 111 * 100 = 000101 -> 0000110 -> 0001 = 1
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-1 * -0.75 = 0.75 -> 1 111 * 101 = 000100 -> 0000101 -> 0001 = 1
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-1 * -0.50 = 0.50 -> 1 111 * 110 = 000011 -> 0000100 -> 0001 = 1
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-1 * -0.25 = 0.25 -> 0 111 * 111 = 000010 -> 0000011 -> 0000 = 0
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-1 * 0.00 = -0.00 -> 0 111 * 000 = 000001 -> 0000010 -> 0000 = 0
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-1 * 0.25 = -0.25 -> 0 111 * 001 = 111111 -> 0000000 -> 0000 = 0
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-1 * 0.50 = -0.50 -> -1 111 * 010 = 111110 -> 1111111 -> 1111 = -1
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-1 * 0.75 = -0.75 -> -1 111 * 011 = 111101 -> 1111110 -> 1111 = -1
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------------------------ ------------------------------------------
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0 * -1.00 = -0.00 -> 0 000 * 100 = 000001 -> 0000010 -> 0000 = 0
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0 * -0.75 = -0.00 -> 0 000 * 101 = 000001 -> 0000010 -> 0000 = 0
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0 * -0.50 = -0.00 -> 0 000 * 110 = 000001 -> 0000010 -> 0000 = 0
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0 * -0.25 = -0.00 -> 0 000 * 111 = 000001 -> 0000010 -> 0000 = 0
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0 * 0.00 = 0.00 -> 0 000 * 000 = 000001 -> 0000010 -> 0000 = 0
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0 * 0.25 = 0.00 -> 0 000 * 001 = 000001 -> 0000010 -> 0000 = 0
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0 * 0.50 = 0.00 -> 0 000 * 010 = 000001 -> 0000010 -> 0000 = 0
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0 * 0.75 = 0.00 -> 0 000 * 011 = 000001 -> 0000010 -> 0000 = 0
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------------------------ ------------------------------------------
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1 * -1.00 = -1.00 -> -1 001 * 100 = 111100 -> 1111101 -> 1111 = -1
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1 * -0.75 = -0.75 -> -1 001 * 101 = 111101 -> 1111110 -> 1111 = -1
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1 * -0.50 = -0.50 -> -1 001 * 110 = 111110 -> 1111111 -> 1111 = -1
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1 * -0.25 = -0.25 -> 0 001 * 111 = 111111 -> 0000000 -> 0000 = 0
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1 * 0.00 = 0.00 -> 0 001 * 000 = 000001 -> 0000010 -> 0000 = 0
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1 * 0.25 = 0.25 -> 0 001 * 001 = 000010 -> 0000011 -> 0000 = 0
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1 * 0.50 = 0.50 -> 1 001 * 010 = 000011 -> 0000100 -> 0001 = 1
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1 * 0.75 = 0.75 -> 1 001 * 011 = 000100 -> 0000101 -> 0001 = 1
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------------------------ ------------------------------------------
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2 * -1.00 = -2.00 -> -2 010 * 100 = 111000 -> 1111001 -> 1110 = -2
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2 * -0.75 = -1.50 -> -2 010 * 101 = 111010 -> 1111011 -> 1110 = -2
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2 * -0.50 = -1.00 -> -1 010 * 110 = 111100 -> 1111101 -> 1111 = -1
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2 * -0.25 = -0.50 -> -1 010 * 111 = 111110 -> 1111111 -> 1111 = -1
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2 * 0.00 = 0.00 -> 0 010 * 000 = 000001 -> 0000010 -> 0000 = 0
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2 * 0.25 = 0.50 -> 1 010 * 001 = 000011 -> 0000100 -> 0001 = 1
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2 * 0.50 = 1.00 -> 1 010 * 010 = 000101 -> 0000110 -> 0001 = 1
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2 * 0.75 = 1.50 -> 2 010 * 011 = 000111 -> 0001000 -> 0010 = 2
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------------------------ ------------------------------------------
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3 * -1.00 = -3.00 -> -3 011 * 100 = 110100 -> 1110101 -> 1101 = -3
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3 * -0.75 = -2.25 -> -2 011 * 101 = 110111 -> 1111000 -> 1110 = -2
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3 * -0.50 = -1.50 -> -2 011 * 110 = 111010 -> 1111011 -> 1110 = -2
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3 * -0.25 = -0.75 -> -1 011 * 111 = 111101 -> 1111110 -> 1111 = -1
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3 * 0.00 = 0.00 -> 0 011 * 000 = 000001 -> 0000010 -> 0000 = 0
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3 * 0.25 = 0.75 -> 1 011 * 001 = 000100 -> 0000101 -> 0001 = 1
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3 * 0.50 = 1.50 -> 2 011 * 010 = 000111 -> 0001000 -> 0010 = 2
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3 * 0.75 = 2.25 -> 2 011 * 011 = 001010 -> 0001011 -> 0010 = 2
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</pre>
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<br>
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<center><i>Keeping the prior rough draft as well..</i></center>
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<br>
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<br>
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<b>Round Half Away From Zero</b>
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<br>
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