nxu/math32.tal

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( math32.tal )
( )
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( This library supports arithmetic on 32-bit unsigned integers, )
( also known as long values. )
( )
( 32-bit long values are represented by two 16-bit short values: )
( )
( decimal hexadecimal uxn literals )
( 0 0x00000000 #0000 #0000 )
( 1 0x00000001 #0000 #0001 )
( 4660 0x00001234 #0000 #1234 )
( 65535 0x0000ffff #0000 #ffff )
( 65536 0x00010000 #0001 #0000 )
( 16777215 0x00ffffff #00ff #ffff )
( 4294967295 0xffffffff #ffff #ffff )
( )
( The most significant 16-bit, the "high bits", are stored first. )
( We document long values as x** -- equivalent to xhi* xlo*. )
( )
( Operations supported: )
( )
( - add32 x** y** -> z** [x + y] )
( - sub32 x** y** -> z** [x - y] )
( - mul16 x* y* -> z** [x * y] )
( - mul32 x** y** -> z** [x * y] )
( - div32 x** y** -> z** [x / y] )
( - negate32 x** -> z** [-x] )
( - left-shift x** n^ -> z** [x<<n] )
( - right-shift x** n^ -> z** [x>>n] )
( - and32 x** y** -> z** [x & y] )
( - or32 x** y** -> z** [x | y] )
( - xor32 x** y** -> z** [x ^ y] )
( - complement32 x** -> z** [~x] )
( - eq32 x** y** -> bool^ [x == y] )
( - ne32 x** y** -> bool^ [x != y] )
( - is-zero32 x** -> bool^ [x == 0] )
( - non-zero32 x** -> bool^ [x != 0] )
( - lt32 x** y** -> bool^ [x < y] )
( - gt32 x** y** -> bool^ [x > y] )
( - lteq32 x** y** -> bool^ [x <= y] )
( - gteq32 x** y** -> bool^ [x >= y] )
( - bitcount x** -> bool^ log2+1 )
( )
( In addition to the code this file uses 44 bytes of registers )
( to store temporary state: )
( )
( - shared memory, 16 bytes )
( - mul32 memory, 12 bytes )
( - div32 memory, 16 bytes )
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%DEBUG { #ff #0e DEO }
%RTN { JMP2r }
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%TOR { ROT ROT } ( a b c -> c a b )
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%TOR2 { ROT2 ROT2 }
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%COMPLEMENT32 { SWP2 #ffff EOR2 SWP2 #ffff EOR2 }
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( bitcount: number of bits needed to represent number )
( equivalent to floor[log2[x]] + 1 )
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@bitcount8 ( x^ -> n^ )
#00 SWP ( n x )
&loop
DUP #00 EQU ( n x x=0 )
,&done JCN ( n x )
#01 SFT ( n x>>1 )
SWP INC SWP ( n+1 x>>1 )
,&loop JMP
&done
POP ( n )
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RTN
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@bitcount16 ( x* -> n^ )
SWP ( xlo xhi )
;bitcount8 JSR2 ( xlo nhi )
DUP #00 NEQ ( xlo nhi nhi!=0 )
,&hi-set JCN ( xlo nhi )
SWP ;bitcount8 JSR2 ADD ( nhi+nlo )
RTN
&hi-set
SWP POP #08 ADD ( nhi+8 )
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RTN
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@bitcount32 ( x** -> n^ )
SWP2 ( xlo* xhi* )
;bitcount16 JSR2 ( xlo* nhi )
DUP #00 NEQ ( xlo* nhi nhi!=0 )
,&hi-set JCN ( xlo* nhi )
TOR ;bitcount16 JSR2 ADD RTN ( nhi+nlo )
&hi-set
TOR POP2 #10 ADD ( nhi+16 )
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RTN
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( equality )
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( x == y )
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@eq32 ( xhi* xlo* yhi* ylo* -> bool^ )
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ROT2 EQU2 STH
EQU2 STHr AND RTN
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( x != y )
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@ne32 ( xhi* xlo* yhi* ylo* -> bool^ )
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ROT2 NEQ2 STH
NEQ2 STHr ORA RTN
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( x == 0 )
@is-zero32 ( x** -> bool^ )
ORA2 #0000 EQU2 RTN
( x != 0 )
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@non-zero32 ( x** -> bool^ )
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ORA2 #0000 NEQ2 RTN
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( comparisons )
( x < y )
@lt32 ( x** y** -> bool^ )
ROT2 SWP2 ( xhi yhi xlo ylo )
LTH2 ,&lt-lo JCN ( xhi yhi )
LTH2 RTN
&lt-lo
GTH2 #00 EQU RTN
( x <= y )
@lteq32 ( x** y** -> bool^ )
ROT2 SWP2 ( xhi yhi xlo ylo )
GTH2 ,&gt-lo JCN ( xhi yhi )
GTH2 #00 EQU RTN
&gt-lo
LTH2 RTN
( x > y )
@gt32 ( x** y** -> bool^ )
ROT2 SWP2 ( xhi yhi xlo ylo )
GTH2 ,&gt-lo JCN ( xhi yhi )
GTH2 RTN
&gt-lo
LTH2 #00 EQU RTN
( x > y )
@gteq32 ( x** y** -> bool^ )
ROT2 SWP2 ( xhi yhi xlo ylo )
LTH2 ,&lt-lo JCN ( xhi yhi )
LTH2 #00 EQU RTN
&lt-lo
GTH2 RTN
( bitwise operations )
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( x & y )
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@and32 ( xhi* xlo* yhi* ylo* -> xhi|yhi* xlo|ylo* )
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ROT2 AND2 STH2 AND2 STH2r RTN
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( x | y )
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@or32 ( xhi* xlo* yhi* ylo* -> xhi|yhi* xlo|ylo* )
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ROT2 ORA2 STH2 ORA2 STH2r RTN
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( x ^ y )
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@xor32 ( xhi* xlo* yhi* ylo* -> xhi|yhi* xlo|ylo* )
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ROT2 EOR2 STH2 EOR2 STH2r RTN
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( ~x )
@complement32 ( x** -> ~x** )
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COMPLEMENT32 RTN
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( temporary registers )
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( used by most operations, except mul32 and div32 )
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@sh [ &x0 $1 &x1 $1 &x2 $1 &x3 $1
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&y0 $1 &y1 $1 &y2 $1 &y3 $1
&z0 $1 &z1 $1 &z2 $1 &z3 $1
&a0 $1 &a1 $1 &a2 $2 ]
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( bit shifting )
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( x >> n )
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@right-shift ( x** n^ -> x<<n )
DUP #08 LTH ;right-shift0 JCN2 ( x n )
DUP #10 LTH ;right-shift1 JCN2 ( x n )
DUP #18 LTH ;right-shift2 JCN2 ( x n )
;right-shift3 JMP2 ( x n )
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RTN
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( shift right by 0-7 bits )
@right-shift0 ( x** n^ -> x<<n )
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STHk SFT ;sh/z3 STA ( write z3 )
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#00 STHkr SFT2 #00 ;sh/z3 LDA ORA2 ;sh/z2 STA2 ( write z2,z3 )
#00 STHkr SFT2 #00 ;sh/z2 LDA ORA2 ;sh/z1 STA2 ( write z1,z2 )
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#00 STHr SFT2 #00 ;sh/z1 LDA ORA2 ( compute z0,z1 )
;sh/z2 LDA2
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RTN
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( shift right by 8-15 bits )
@right-shift1 ( x** n^ -> x<<n )
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#08 SUB STH POP
STHkr SFT ;sh/z3 STA ( write z3 )
#00 STHkr SFT2 #00 ;sh/z3 LDA ORA2 ;sh/z2 STA2 ( write z2,z3 )
#00 STHr SFT2 #00 ;sh/z2 LDA ORA2 ( compute z1,z2 )
#00 TOR ;sh/z3 LDA
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RTN
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( shift right by 16-23 bits )
@right-shift2 ( x** n^ -> x<<n )
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#10 SUB STH POP2
STHkr SFT ;sh/z3 STA ( write z3 )
#00 STHr SFT2 #00 ;sh/z3 LDA ORA2 ( compute z2,z3 )
#0000 SWP2
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RTN
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( shift right by 16-23 bits )
@right-shift3 ( x** n^ -> x<<n )
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#18 SUB STH POP2 POP ( x0 )
#00 SWP #0000 SWP2 ( 00 00 00 x0 )
STHr SFT
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RTN
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( x << n )
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@left-shift ( x** n^ -> x<<n )
DUP #08 LTH ;left-shift0 JCN2 ( x n )
DUP #10 LTH ;left-shift1 JCN2 ( x n )
DUP #18 LTH ;left-shift2 JCN2 ( x n )
;left-shift3 JMP2 ( x n )
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RTN
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( shift left by 0-7 bits )
@left-shift0 ( x** n^ -> x<<n )
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#40 SFT STH ( stash n<<4 )
#00 SWP STHkr SFT2 ;sh/z2 STA2 ( store z2,z3 )
#00 SWP STHkr SFT2 #00 ;sh/z2 LDA ORA2 ;sh/z1 STA2 ( store z1,z2 )
#00 SWP STHkr SFT2 #00 ;sh/z1 LDA ORA2 ;sh/z0 STA2 ( store z0,z1 )
STHr SFT ;sh/z0 LDA ORA ( calculate z0 )
;sh/z1 LDA ;sh/z2 LDA2
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RTN
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( shift left by 8-15 bits )
@left-shift1 ( x** n^ -> x<<n )
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#08 SUB #40 SFT STH ( stash [n-8]<<4 )
#00 SWP STHkr SFT2 ;sh/z1 STA2 ( store z1,z2 )
#00 SWP STHkr SFT2 #00 ;sh/z1 LDA ORA2 ;sh/z0 STA2 ( store z0,z1 )
STHr SFT ;sh/z0 LDA ORA ( calculate z0 )
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SWP POP ( x0 unused )
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;sh/z1 LDA2 #00
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RTN
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( shift left by 16-23 bits )
@left-shift2 ( x** n^ -> x<<n )
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#10 SUB #40 SFT STH ( stash [n-16]<<4 )
#00 SWP STHkr SFT2 ;sh/z0 STA2 ( store z0,z1 )
STHr SFT ;sh/z0 LDA ORA ( calculate z0 )
STH POP2 STHr
;sh/z1 LDA #0000
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RTN
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( shift left by 24-31 bits )
@left-shift3 ( x** n^ -> x<<n )
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#18 SUB #40 SFT ( x0 x1 x2 x3 r=[n-24]<<4 )
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SFT ( x0 x1 x2 x3<<r )
SWP2 POP2 SWP POP #0000 #00
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RTN
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( arithmetic )
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( x + y )
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@add32 ( xhi* xlo* yhi* ylo* -> zhi* zlo* )
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;sh/y2 STA2 ;sh/y0 STA2 ( save ylo, yhi )
;sh/x2 STA2 ;sh/x0 STA2 ( save xlo, xhi )
#0000 #0000 ;sh/z0 STA2 ;sh/z2 STA2 ( reset zhi, zlo )
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( x3 + y3 => z2z3 )
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#00 ;sh/x3 LDA #00 ;sh/y3 LDA ADD2 ;sh/z2 STA2
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( x2 + y2 + z2 => z1z2 )
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#00 ;sh/x2 LDA ;sh/z1 LDA2 ADD2 ;sh/z1 STA2
#00 ;sh/y2 LDA ;sh/z1 LDA2 ADD2 ;sh/z1 STA2
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( x1 + y1 + z1 => z0z1 )
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#00 ;sh/x1 LDA ;sh/z0 LDA2 ADD2 ;sh/z0 STA2
#00 ;sh/y1 LDA ;sh/z0 LDA2 ADD2 ;sh/z0 STA2
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( x0 + y0 + z0 => z0 )
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;sh/x0 LDA ;sh/z0 LDA ADD ;sh/z0 STA
;sh/y0 LDA ;sh/z0 LDA ADD ;sh/z0 STA
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( load zhi,zlo )
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;sh/z0 LDA2 ;sh/z2 LDA2
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RTN
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( -x )
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@negate32 ( x** -> -x** )
COMPLEMENT32
INC2 ( ~xhi -xlo )
DUP2 #0000 NEQ2 ( ~xhi -xlo non-zero? )
,&done JCN ( xlo non-zero => don't inc hi )
SWP2 INC2 SWP2 ( -xhi -xlo )
&done
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RTN
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( x - y )
@sub32 ( x** y** -> z** )
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;negate32 JSR2 ;add32 JSR2 RTN
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( 16-bit multiplication )
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@mul16 ( x* y* -> z** )
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;sh/y1 STA ;sh/y0 STA ( save ylo, yhi )
;sh/x1 STA ;sh/x0 STA ( save xlo, xhi )
#0000 #00 ;sh/z1 STA2 ;sh/z3 STA ( reset z1,z2,z3 )
#0000 #00 ;sh/a0 STA2 ;sh/a2 STA ( reset a0,a1,a2 )
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( x1 * y1 => z1z2 )
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#00 ;sh/x1 LDA #00 ;sh/y1 LDA MUL2 ;sh/z2 STA2
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( x0 * y1 => z0z1 )
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#00 ;sh/x0 LDA #00 ;sh/y1 LDA MUL2 ;sh/z1 LDA2 ADD2 ;sh/z1 STA2
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( x1 * y0 => a1a2 )
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#00 ;sh/x1 LDA #00 ;sh/y0 LDA MUL2 ;sh/a1 STA2
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( x0 * y0 => a0a1 )
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#00 ;sh/x0 LDA #00 ;sh/y0 LDA MUL2 ;sh/a0 LDA2 ADD2 ;sh/a0 STA2
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( add z and a<<8 )
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#00 ;sh/z1 LDA2 ;sh/z3 LDA
;sh/a0 LDA2 ;sh/a2 LDA #00
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;add32 JSR2
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RTN
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( x * y )
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@mul32 ( x** y** -> z** )
,&y1 STR2 ,&y0 STR2 ( save ylo, yhi )
,&x1 STR2 ,&x0 STR2 ( save xlo, xhi )
,&y1 LDR2 ,&x1 LDR2 ;mul16 JSR2 ( [x1*y1] )
,&z1 STR2 ,&z0 STR2 ( sum = x1*y1, save zlo, zhi )
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,&y1 LDR2 ,&x0 LDR2 MUL2 ( [x0*y1]<<16 )
,&y0 LDR2 ,&x1 LDR2 MUL2 ( [x1*y0]<<16 )
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( [x0*y0]<<32 will completely overflow )
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ADD2 ,&z0 LDR2 ADD2 ( sum += x0*y1<<16 + x1*y0<<16 )
,&z1 LDR2
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RTN
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[ &x0 $2 &x1 $2
&y0 $2 &y1 $2
&z0 $2 &z1 $2 ]
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( x / y )
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@div32 ( x** y** -> q** )
( store y and x for repeated use )
;div32/div1 STA2 ;div32/div0 STA2 ( y -> div )
;div32/rem1 STA2 ;div32/rem0 STA2 ( x -> rem )
( if x < y then the answer is 0 )
;div32/rem0 LDA2 ;div32/rem1 LDA2
;div32/div0 LDA2 ;div32/div1 LDA2
;lt32 JSR2 ,&is-zero JCN ,&not-zero JMP
&is-zero
#0000 #0000 RTN
( x >= y so the answer is >= 1 )
&not-zero
#0000 ;div32/quo0 STA2 #0000 ;div32/quo1 STA2 ( 0 -> quo )
( bitcount[x] - bitcount[y] determines the largest multiple of y to try )
;div32/rem0 LDA2 ;div32/rem1 LDA2 ;bitcount32 JSR2 ( rbits^ )
;div32/div0 LDA2 ;div32/div1 LDA2 ;bitcount32 JSR2 ( rbits^ dbits^ )
SUB ( shift=rbits-dits )
#00 DUP2 ( shift 0 shift 0 )
( 1<<shift -> cur )
#0000 #0001 ROT2 POP
;left-shift JSR2 ;div32/cur1 STA2 ;div32/cur0 STA2
( div<<shift -> div )
;div32/div0 LDA2 ;div32/div1 LDA2 ROT2 POP
;left-shift JSR2 ;div32/div1 STA2 ;div32/div0 STA2
&loop
( if rem >= the current divisor, we can subtract it and add to quotient )
,&rem0 LDR2 ,&rem1 LDR2 ,&div0 LDR2 ,&div1 LDR2 ;lt32 JSR2 ( rem<div? )
,&rem-lt JCN ( if rem < div skip this iteration )
( if rem >= div, then we have found a multiple of y that divides x )
,&rem0 LDR2 ,&rem1 LDR2 ,&div0 LDR2 ,&div1 LDR2 ;sub32 JSR2 ,&rem1 STR2 ,&rem0 STR2 ( rem -= div )
,&quo0 LDR2 ,&quo1 LDR2 ,&cur0 LDR2 ,&cur1 LDR2 ;add32 JSR2 ,&quo1 STR2 ,&quo0 STR2 ( quo += cur )
&rem-lt
,&div0 LDR2 ,&div1 LDR2 #01 ;right-shift JSR2 ,&div1 STR2 ,&div0 STR2 ( div >>= 1 )
,&cur0 LDR2 ,&cur1 LDR2 #01 ;right-shift JSR2 ,&cur1 STR2 ,&cur0 STR2 ( cur >>= 1 )
,&cur0 LDR2 ,&cur1 LDR2 ;non-zero32 JSR2 ,&loop JCN ( if cur>0, loop. else we're done )
,&quo0 LDR2 ,&quo1 LDR2 ( TODO: consider making this divmod32 )
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RTN
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[ &div0 $2 &div1 $2
&rem0 $2 &rem1 $2
&quo0 $2 &quo1 $2
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&cur0 $2 &cur1 $2 ]