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