2022-03-13 23:23:20 -04:00
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( fix16.tal )
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( )
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( use a signed 16-bit short as a fixed point number. )
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( )
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( numbers are interpreted as fractions with an implicit )
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( 256 denominator. the upper byte is signed and )
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( represents the "whole" part of the number, and the )
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( lower byte is unsigned and represents the )
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( "fractional" part of the number. )
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( )
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( 16-bit fixed point can represent fractional values )
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( in the range -128 <= x < 128. the smallest fraction it )
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2022-11-06 21:49:02 -05:00
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( can represent is 1/256, which is about 0.0039. )
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2022-03-13 23:23:20 -04:00
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( )
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( SHORT FRACTION DECIMAL )
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2022-11-06 21:49:02 -05:00
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( #0000 0/256 0.0000 )
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( #0001 1/256 0.0039 )
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( #0002 2/256 0.0078 )
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( #0040 64/256 0.2500 )
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( #0080 128/256 0.5000 )
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( #0100 256/256 1.0000 )
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( #0700 1792/256 7.0000 )
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( #7f00 32512/256 127.0000 )
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( #7fff 32767/256 127.9961 )
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( #8000 -32768/256 -128.0000 )
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( #8001 -32767/256 -127.9961 )
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( #8100 -32767/256 -127.0000 )
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( #ff00 -256/256 -1.0000 )
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( #ffff -1/256 -0.0039 )
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2022-03-13 23:23:20 -04:00
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( )
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2022-10-23 15:07:48 -04:00
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( many 8.8 operations are equivalent to unsigned int16: )
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( * addition )
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( * subtraction )
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2022-03-13 23:28:13 -04:00
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( or signed int16: )
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( * comparisons/equality )
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2022-03-13 23:23:20 -04:00
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( )
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( but due to 16-bit truncation multiplication differs... )
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( )
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( x*y = x0*y0 + x0*y1/256 + x1*y0/256 + x1*y1/65536 )
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( )
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( since we only have 16-bits: )
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( 1. we need to drop the 8 high bits from x0*y0 )
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( 2. we need to drop the 8 low bits from x1*y1 )
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( 3. we need to use all the bits from x0*y1 and x1*y0 )
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( )
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( that said, if either x or y is whole (i.e. ends in 00) )
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( then we can just shift that argument right by 8 and use )
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2022-03-13 23:28:13 -04:00
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( MUL2. )
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2022-10-23 15:07:48 -04:00
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( )
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( similarly with division we have: )
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2022-11-06 22:01:45 -05:00
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( )
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( x = x'/256 )
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( y = y'/256 )
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( x/y = z = z'/256 )
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( (x'/256)/(y'/256) = (x'*256 / y)/256 )
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( z' = (x' * 256 / y)/256 )
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2022-03-13 23:23:20 -04:00
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( useful constants )
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( )
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( to generate your own: )
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2022-11-06 21:49:02 -05:00
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( )
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2022-03-13 23:23:20 -04:00
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( 1. take true value, e.g. 3.14159... )
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( 2. multiply by 256 )
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( 3. round to nearest whole number )
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( 4. emit hex output )
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( )
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( in python: hex(round(x * 256)) )
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%x16-zero { #0000 } ( 0.0 )
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%x16-one { #0100 } ( 1.0 )
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%x16-two { #0200 } ( 2.0 )
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%x16-ten { #0a00 } ( 10.0 )
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%x16-hundred { #6400 } ( 100.0 )
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%x16-minus-one { #7f00 } ( -1.0 )
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%x16-minus-two { #7e00 } ( -2.0 )
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%x16-pi/2 { #0192 } ( 1.57079... )
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%x16-pi { #0324 } ( 3.14159... )
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%x16-pi*2 { #0648 } ( 6.28318... )
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%x16-e { #02b8 } ( 2.71828... )
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%x16-phi { #019e } ( 1.61803... )
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%x16-sqrt-2 { #016a } ( 1.41421... )
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%x16-sqrt-3 { #01bb } ( 1.73205... )
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2022-11-06 21:49:02 -05:00
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%x16-epsilon { #0001 } ( 0.00390... )
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2022-03-13 23:23:20 -04:00
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%x16-minimum { #8000 } ( -128.0 )
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2022-11-06 21:49:02 -05:00
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%x16-maximum { #7fff } ( 127.99609... )
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%x16-max-whole { #7f00 } ( 127.0 )
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2022-03-13 23:23:20 -04:00
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( useful macros )
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%x16-is-non-neg { x16-minimum LTH2 }
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%x16-is-neg { x16-maximum GTH2 }
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2022-03-17 23:04:40 -04:00
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%x16-emit-dec { #30 ADD #18 DEO }
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@x16-emit ( x* -> )
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2022-10-23 15:07:48 -04:00
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DUP2 #8000 EQU2 ,&is-min JCN
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DUP2 #8000 GTH2 ,&is-neg JCN
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2022-03-17 23:04:40 -04:00
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SWP DUP #64 LTH ,&<100 JCN
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#64 DIVk DUP x16-emit-dec MUL SUB ,&>=10 JMP
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2022-10-23 15:07:48 -04:00
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&is-min POP2
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LIT "- #18 DEO LIT "1 #18 DEO LIT "2 #18 DEO LIT "8 #18 DEO
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LIT ". #18 DEO LIT "0 #18 DEO LIT "0 #18 DEO LIT "0 #18 DEO
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JMP2r
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&is-neg
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LIT "- #18 DEO #ffff EOR2 INC2 ,x16-emit JMP
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2022-03-17 23:04:40 -04:00
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&<100 DUP #0a LTH ,&<10 JCN
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&>=10 #0a DIVk DUP x16-emit-dec MUL SUB
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&<10 x16-emit-dec
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LIT '. #18 DEO
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( emit fractional part )
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#00 SWP ( lo* )
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#000a MUL2 #0100 DIV2k DUP2 NIP x16-emit-dec MUL2 SUB2
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#000a MUL2 #0100 DIV2k DUP2 NIP x16-emit-dec MUL2 SUB2
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2022-11-06 21:49:02 -05:00
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#000a MUL2 #0100 DIV2k DUP2 NIP x16-emit-dec MUL2 SUB2
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2022-03-17 23:04:40 -04:00
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#000a MUL2 #0100 DIV2k STH2k MUL2 SUB2 #0080 LTH2 ,&no-round JCN INC2r
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&no-round STH2r NIP x16-emit-dec JMP2r
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2022-03-13 23:23:20 -04:00
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( comparison between x and y. )
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( - ff: x < y )
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( - 00: x = y )
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( - 01: x > y )
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@x16-cmp ( x* y* -> c^ )
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2022-11-06 21:49:02 -05:00
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STH2k x16-is-neg ,&yn JCN ( x* [y*] ; ? )
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DUP2 x16-is-non-neg ,&same JCN ( x* [y*] ; y>=0 )
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POP2 POP2r #ff JMP2r ( -1 ; x<0 y>=0 )
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&yn DUP2 x16-is-neg ,&same JCN ( x* [y*] ; y<0 )
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POP2 POP2r #01 JMP2r ( 1 ; x>=0 y<0 )
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&same STH2r ;x16-ucmp JMP2 ( res ; x<0 y<0 b )
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2022-03-13 23:23:20 -04:00
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( unsigned comparison between x and y. )
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( - ff: x < y )
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( - 00: x = y )
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( - 01: x > y )
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@x16-ucmp ( x* y* -> c^ )
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LTH2k ,< JCN GTH2 JMP2r
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< POP2 POP2 #ff JMP2r
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2022-03-17 23:04:40 -04:00
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@x16-eq ( x* y* -> x=y^ ) EQU2 JMP2r
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@x16-ne ( x* y* -> x!=0^ ) NEQ2 JMP2r
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@x16-lt ( x* y* -> x<y^ ) ;x16-cmp JSR2 #ff EQU JMP2r
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@x16-lteq ( x* y* -> x<y^ ) ;x16-cmp JSR2 #01 NEQ JMP2r
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@x16-gt ( x* y* -> x<y^ ) ;x16-cmp JSR2 #01 EQU JMP2r
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@x16-gteq ( x* y* -> x<y^ ) ;x16-cmp JSR2 #ff NEQ JMP2r
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2022-03-13 23:23:20 -04:00
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@x16-is-whole ( x* -> bool^ )
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NIP #00 EQU JMP2r
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@x16-add ( x* y* -> x+y* )
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ADD2 JMP2r
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@x16-sub ( x* y* -> x-y* )
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SUB2 JMP2r
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@x16-negate ( x* -> -x* )
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#0000 SWP2 SUB2 JMP2r
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@x16-mul ( x* y* -> xy* )
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DUP #00 EQU ,&rhs-whole JCN
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SWP2 DUP #00 EQU ,&rhs-whole JCN
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,&y3 STR ,&y1 STR ,&x3 STR ,&x1 STR
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LIT2 &x2 00 &x3 00 LIT2 &y2 00 &y3 00 MUL2 #08 SFT2
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LIT2 &x0 00 &x1 00 ,&y2 LDR2 MUL2 ADD2
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,&x2 LDR2 LIT2 &y0 00 &y1 00 MUL2 ADD2
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2022-11-06 21:49:02 -05:00
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,&x0 LDR2 ,&y0 LDR2 MUL2 #80 SFT2 ADD2 JMP2r
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2022-03-13 23:23:20 -04:00
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&rhs-whole #08 SFT2 MUL2 JMP2r
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@x16-div ( x* y* -> x/y* )
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2022-11-06 21:49:02 -05:00
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DIV2k STH2k ( x y x/y {x/y} )
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LITr 80 SFT2r ( x y x/y {div=(x/y)<<8 )
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OVR2 STH2 ( x y x/y {y div} )
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MUL2 SUB2 ( x%y {y div} )
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STH2r LIT2r 0100 ( x%y y {0100 div} )
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( we know x%y < y, so start right-shifting y )
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&loop
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DUP2 #0000 EQU2 ,&done JCN
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#01 SFT2 LITr 01 SFT2r ( rem yi {shifti div} )
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LTH2k ,&loop JCN ( rem yi {shifti div} )
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SWP2 OVR2 SUB2 SWP2 ( rem-yi yi {shifti div} )
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DUP2r ROT2r ADD2r SWP2r ( rem-yi yi {shifti div+shifti} )
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,&loop JMP ( rem-yi yi {shifti div+shifti} )
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&done
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POP2 POP2 ( {shiftk div} )
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POP2r STH2r JMP2r ( div )
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@x16-quotient ( x* y* -> x//y* )
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DIV2 #80 SFT2 JMP2r
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@x16-remainder ( x* y* -> x%y* )
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DIV2k MUL2 SUB2 JMP2r
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