diff --git a/fix32.tal b/fix32.tal index b677d77..6429b5d 100644 --- a/fix32.tal +++ b/fix32.tal @@ -5,64 +5,125 @@ %DENOM16 { #03e8 } %DENOM32 { #0000 #03e8 } -@x32-eq ( x/** y/** -> bool^ ) !eq32 -@x32-ne ( x/** y/** -> bool^ ) !ne32 +|0100 + ( test cases -- compare the two 32-bit values on wst ) -@x32-is-zero ( x/** -> bool^ ) !is-zero32 -@x32-non-zero ( x/** -> bool^ ) !non-zero32 + #0000 #03e8 ( a=1 ) + #0000 #07d0 ( b=2 ) + x32-add ( a+b ) + #0000 #0bb8 ( c=3 ) + #010e DEO POP4 POP4 #0a18 DEO + + #0000 #07d0 ( a=2 ) + #0000 #0bb8 ( b=3 ) + x32-mul ( a*b ) + #0000 #1770 ( c=6 ) + #010e DEO POP4 POP4 #0a18 DEO + + #0000 #4a38 ( a=19 ) + #0000 #6978 ( b=27 ) + x32-mul ( a*b ) + #0007 #d3d8 ( c=513 ) + #010e DEO POP4 POP4 #0a18 DEO + + #0000 #1d4c ( a=7.5 ) + #0000 #05dc ( b=1.5 ) + x32-div ( a/b ) + #0000 #1388 ( c=5.0 ) + #010e DEO POP4 POP4 #0a18 DEO + + #010e DEO #800f DEO BRK ( ensure stack is empty ) + +@x32-eq ( x/** y/** -> bool^ ) !u32-eq +@x32-ne ( x/** y/** -> bool^ ) !u32-ne + +@x32-is-zero ( x/** -> bool^ ) !u32-is-zero +@x32-non-zero ( x/** -> bool^ ) !u32-non-zero @x32-is-positive ( x/** -> bool^ ) POP2 #8000 LTH2 JMP2r @x32-is-negative ( x/** -> bool^ ) POP2 #7fff GTH2 JMP2r @x32-from-u8 ( x^ -> x/** ) - #0000 ROT OVR SWP DENOM32 !mul32 + #0000 ROT OVR SWP DENOM32 !u32-mul @x32-from-u16 ( x* -> x/** ) - #0000 SWP2 DENOM32 !mul32 + #0000 SWP2 DENOM32 !u32-mul @x32-from-u32 ( x** -> x/** ) - DENOM32 !mul32 + DENOM32 !u32-mul @x32-prepare-cmp ( x/** y/** -> x/** y/** xp^ yp^ ) OVR2 #8000 LTH2 ,&yp STR STH2 STH2 OVR2 #8000 LTH2 ,&xp STR STH2r STH2r LIT2 [ &xp $1 &yp $1 ] JMP2r +( TODO: test these implementations ) +@x32-lt-old ( x** y** -> x x ylo? ) + GTH2 JMP2r ( ; no, is xhi > yhi? ) + } LTH2 #00 EQU JMP2r ( ; yes, is xhi >= yhi? ) + @x32-lt ( x/** y/** -> bool^ ) - x32-prepare-cmp NEQk ?{ POP2 !lt32 } LTH STH POP8 STHr JMP2r + x32-prepare-cmp NEQk ?{ POP2 !u32-lt } LTH STH POP8 STHr JMP2r @x32-gt ( x/** y/** -> bool^ ) - x32-prepare-cmp NEQk ?{ POP2 !gt32 } GTH STH POP8 STHr JMP2r + x32-prepare-cmp NEQk ?{ POP2 !u32-gt } GTH STH POP8 STHr JMP2r @x32-lteq ( x/** y/** -> bool^ ) - x32-prepare-cmp NEQk ?{ POP2 !lteq32 } LTH STH POP8 STHr JMP2r + x32-prepare-cmp NEQk ?{ POP2 !u32-lteq } LTH STH POP8 STHr JMP2r @x32-gteq ( x/** y/** -> bool^ ) - x32-prepare-cmp NEQk ?{ POP2 !gteq32 } GTH STH POP8 STHr JMP2r + x32-prepare-cmp NEQk ?{ POP2 !u32-gteq } GTH STH POP8 STHr JMP2r @x32-add ( x/** y/** -> z/** ) - !add32 + !u32-add @x32-sub ( x/** y/** -> z/** ) - !sub32 + !u32-sub @x32-negate ( x/** y/** -> z/** ) - !negate32 + !u32-negate ( multiply a fixed point number by an unsigned integer ) @x32-scaled-mul32 ( x/** y** -> z/** ) - !mul32 + !u32-mul ( multiply a fixed point number by an unsigned integer ) @x32-scaled-mul16 ( x/** y* -> z/** ) - !mul16 + !u32-mul16 @x32-scaled-div32 ( x/** y** -> z/** ) - !div32 + !u32-div -( TODO ) +( [x * y]/1000 = floor[x/1000] + [[x%1000]*y]/1000 ) @x32-mul ( x/** y/** -> z/** ) + STH2 STH2 DENOM32 ( x/** 1000** [ylo* yhi*] ) + u32-divmod ( q** r** [ylo* yhi*] ) + STH2kr OVR2r STH2r u32-mul ( q** ry** [ylo* yhi*] ) + DENOM32 u32-div ( q** ry1000** [ylo* yhi*] ) + ROT2 STH2 ROT2 STH2r ( ry1000** q** [ylo* yhi*] ) + STH2r STH2r u32-mul ( ry1000** qy** ) + u32-add ( qy+ry/1000** ) + JMP2r ( z/** ) -( TODO ) +( [x * 1000]/y = floor[x/y]*1000 + [[x%y]*1000]/y ) @x32-div ( x/** y/** -> z/** ) + STH2k OVR2 STH2 ( x/** y/** [ylo* yhi*] ) + u32-divmod + DENOM32 u32-mul ( q** r1000** [ylo* yhi*] ) + STH2r STH2r u32-div ( q** r1000/y** [ylo* yhi*] ) + ROT2 STH2 ROT2 STH2r ( r1000/y** q** [ylo* yhi*] ) + DENOM32 u32-mul ( r1000/y** q1000** ) + u32-add ( q+r1000/y** ) + JMP2r ( z/** ) ~math32.tal diff --git a/math32.tal b/math32.tal index 0afb5df..2196173 100644 --- a/math32.tal +++ b/math32.tal @@ -20,176 +20,176 @@ ( Operations supported: ) ( ) ( NAME STACK EFFECT DEFINITION ) -( 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** -> q** x / y ) -( mod32 x** y** -> r** x % y ) -( divmod32 x** y** -> q** r** x / y, x % y ) -( gcd32 x** y** -> z** gcd[x, y] ) -( negate32 x** -> z** -x ) -( lshift32 x** n^ -> z** x< 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 ) -( bitcount8 x^ -> bool^ floor[log2[x]]+1 ) -( bitcount16 x* -> bool^ floor[log2[x]]+1 ) -( bitcount32 x** -> bool^ floor[log2[x]]+1 ) +( u32-add x** y** -> z** x + y ) +( u32-sub x** y** -> z** x - y ) +( u32-mul x** y** -> z** x * y ) +( u32-mul16 x* y* -> z** x * y ) +( u32-div x** y** -> q** x / y ) +( u32-mod x** y** -> r** x % y ) +( u32-divmod x** y** -> q** r** x / y, x % y ) +( u32-gcd x** y** -> z** gcd[x, y] ) +( u32-negate x** -> z** -x ) +( u32-lshift x** n^ -> z** x< z** x>>n ) +( u32-and x** y** -> z** x & y ) +( u32-or x** y** -> z** x | y ) +( u32-xor x** y** -> z** x ^ y ) +( u32-complement x** -> z** ~x ) +( u32-eq x** y** -> bool^ x == y ) +( u32-ne x** y** -> bool^ x != y ) +( u32-is-zero x** -> bool^ x == 0 ) +( u32-non-zero x** -> bool^ x != 0 ) +( u32-lt x** y** -> bool^ x < y ) +( u32-gt x** y** -> bool^ x > y ) +( u32-lteq x** y** -> bool^ x <= y ) +( u32-gteq x** y** -> bool^ x >= y ) +( u8-bitcount x^ -> bool^ floor[log2[x]]+1 ) +( u16-bitcount x* -> bool^ floor[log2[x]]+1 ) +( u32-bitcount x** -> bool^ floor[log2[x]]+1 ) ( ) ( bitcount: number of bits needed to represent the number. ) ( this is equivalent to floor[log2[x]] + 1 ) -@bitcount8 ( x^ -> n^ ) +@u8-bitcount ( x^ -> n^ ) LITr 00 &loop DUP ?{ POP STHr JMP2r } #01 SFT INCr !&loop -@bitcount16 ( x* -> n^ ) +@u16-bitcount ( x* -> n^ ) LITr 00 &loop ORAk ?{ POP2 STHr JMP2r } #01 SFT2 INCr !&loop -@bitcount32 ( x** -> n^ ) - SWP2 bitcount16 DUP ?{ POP !bitcount16 } #10 NIP2 ADD JMP2r +@u32-bitcount ( x** -> n^ ) + SWP2 u16-bitcount DUP ?{ POP !u16-bitcount } #10 NIP2 ADD JMP2r ( -- equality ) ( x == y ) -@eq32 ( xhi* xlo* yhi* ylo* -> bool^ ) +@u32-eq ( xhi* xlo* yhi* ylo* -> bool^ ) ROT2 EQU2 STH EQU2 STHr AND JMP2r ( x != y ) -@ne32 ( xhi* xlo* yhi* ylo* -> bool^ ) +@u32-ne ( xhi* xlo* yhi* ylo* -> bool^ ) ROT2 NEQ2 STH NEQ2 STHr ORA JMP2r ( x == 0 ) -@is-zero32 ( x** -> bool^ ) +@u32-is-zero ( x** -> bool^ ) ORA2 #0000 EQU2 JMP2r ( x != 0 ) -@non-zero32 ( x** -> bool^ ) +@u32-non-zero ( x** -> bool^ ) ORA2 ORA JMP2r ( -- comparisons ) ( x < y ) -@lt32 ( x** y** -> bool^ ) +@u32-lt ( x** y** -> bool^ ) ROT2 SWP2 LTH2 ?{ LTH2 JMP2r } GTH2 #00 EQU JMP2r ( x <= y ) -@lteq32 ( x** y** -> bool^ ) +@u32-lteq ( x** y** -> bool^ ) ROT2 SWP2 GTH2 ?{ GTH2 #00 EQU JMP2r } LTH2 JMP2r ( x > y ) -@gt32 ( x** y** -> bool^ ) +@u32-gt ( x** y** -> bool^ ) ROT2 SWP2 GTH2 ?{ GTH2 JMP2r } LTH2 #00 EQU JMP2r ( x > y ) -@gteq32 ( x** y** -> bool^ ) +@u32-gteq ( x** y** -> bool^ ) ROT2 SWP2 LTH2 ?{ LTH2 #00 EQU JMP2r } GTH2 JMP2r ( -- bitwise operations ) ( x & y ) -@and32 ( xhi* xlo* yhi* ylo* -> xhi&yhi* xlo&ylo* ) +@u32-and ( xhi* xlo* yhi* ylo* -> xhi&yhi* xlo&ylo* ) ROT2 AND2 STH2 AND2 STH2r JMP2r ( x | y ) -@or32 ( xhi* xlo* yhi* ylo* -> xhi|yhi* xlo|ylo* ) +@u32-or ( xhi* xlo* yhi* ylo* -> xhi|yhi* xlo|ylo* ) ROT2 ORA2 STH2 ORA2 STH2r JMP2r ( x ^ y ) -@xor32 ( xhi* xlo* yhi* ylo* -> xhi^yhi* xlo^ylo* ) +@u32-xor ( xhi* xlo* yhi* ylo* -> xhi^yhi* xlo^ylo* ) ROT2 EOR2 STH2 EOR2 STH2r JMP2r ( ~x ) -@complement32 ( x** -> ~xhi* ~xlo* ) +@u32-complement ( x** -> ~xhi* ~xlo* ) SWP2 #ffff EOR2 SWP2 #ffff EOR2 JMP2r ( -- bit-shifting ) ( x >> n ) -@rshift32 ( x** n^ -> x>>n ) - DUP #08 LTH ?shift32-0 ( x n ) - DUP #10 LTH ?rshift32-1 ( x n ) - DUP #18 LTH ?rshift32-2 ( x n ) - !rshift32-3 ( x n ) +@u32-rshift ( x** n^ -> x>>n ) + DUP #08 LTH ?u32-shift-0 ( x n ) + DUP #10 LTH ?u32-rshift-1 ( x n ) + DUP #18 LTH ?u32-rshift-2 ( x n ) + !u32-rshift-3 ( x n ) ( shift by 0-7 bits; used by both lshift and rshift ) -@shift32-0 ( x** n^ -> x>>n ) +@u32-shift-0 ( x** n^ -> x>>n ) STH DUP2 STHkr SFT2 ,&z2 STR2 POP DUP2 STHkr SFT2 ,&z2 LDR ORA ,&z2 STR ,&z1 STR POP STHr SFT2 ,&z1 LDR ORA ,&z1 STR LIT [ &z1 $1 ] LIT2 [ &z2 $2 ] JMP2r ( shift right by 8-15 bits ) -@rshift32-1 ( x** n^ -> x>>n ) +@u32-rshift-1 ( x** n^ -> x>>n ) #08 SUB STH ( stash [n>>8] ) POP DUP2 STHkr SFT2 ,&z2 STR2 POP STHr SFT2 ,&z2 LDR ORA ,&z2 STR #00 SWP LIT2 [ &z2 $2 ] JMP2r ( shift right by 16-23 bits ) -@rshift32-2 ( x** n^ -> x>>n ) +@u32-rshift-2 ( x** n^ -> x>>n ) #10 SUB STH ( stash [n>>16] ) POP2 STHr SFT2 #0000 SWP2 JMP2r ( shift right by 16-23 bits ) -@rshift32-3 ( x** n^ -> x>>n ) +@u32-rshift-3 ( x** n^ -> x>>n ) #18 SUB STH ( stash [n>>24] ) POP2 POP STH SWPr SFTr #00 #0000 STHr JMP2r ( x << n ) -@lshift32 ( x** n^ -> x< x< x< x< x< x< x< x< x< x< zhi* zlo* ) +@u32-add ( xhi* xlo* yhi* ylo* -> zhi* zlo* ) ROT2 STH2k ADD2 STH2k ROT2 ROT2 GTH2r #00 STHr ADD2 ADD2 SWP2 JMP2r ( -x ) -@negate32 ( x** -> -x** ) - complement32 INC2 ORAk ?{ SWP2 INC2 SWP2 } JMP2r +@u32-negate ( x** -> -x** ) + u32-complement INC2 ORAk ?{ SWP2 INC2 SWP2 } JMP2r ( x - y ) -@sub32 ( x** y** -> z** ) +@u32-sub ( x** y** -> z** ) ROT2 STH2k SWP2 SUB2 STH2k ROT2 ROT2 LTH2r #00 STHr ADD2 SUB2 SWP2 JMP2r ( 16-bit multiplication ) -@mul16 ( x* y* -> z** ) +@u32-mul16 ( x* y* -> z** ) ,&y1 STR ,&y0 STR ( save ylo, yhi ) ,&x1 STR ,&x0 STR ( save xlo, xhi ) #0000 ,&z1 STR ,&w0 STR ( reset z1 and w0 ) @@ -209,12 +209,12 @@ ( add z and a<<8 ) #00 LIT2 [ &z1 $1 &z2 $1 ] LIT [ &z3 $1 ] LIT2 [ &w0 $1 &w1 $1 ] LIT [ &w2 $1 ] #00 - !add32 + !u32-add ( x * y ) -@mul32 ( x** y** -> z** ) +@u32-mul ( x** y** -> z** ) ROT2k ( x0* x1* y0* y1* y0* y1* x1* ) - mul16 ,&z1 STR2 ,&z0 STR2 POP2 ( x0* x1* y0* y1* ; sum = [x1*y1] ) + u32-mul16 ,&z1 STR2 ,&z0 STR2 POP2 ( x0* x1* y0* y1* ; sum = [x1*y1] ) STH2 ROT2 STH2 ( x1* y0* [y1* x0*] ) MUL2r MUL2 STH2r ADD2 ( x1*y0+y1*x0* ) ( [x0*y0]<<32 will completely overflow ) @@ -222,62 +222,62 @@ LIT2 [ &z1 $2 ] JMP2r ( x / y ) -@div32 ( x** y** -> q** ) - z_divmod32 ;z_divmod32/quo0 LDA2 ;z_divmod32/quo1 LDA2 JMP2r +@u32-div ( x** y** -> q** ) + z_u32-divmod ;z_u32-divmod/quo0 LDA2 ;z_u32-divmod/quo1 LDA2 JMP2r ( x % y ) -@mod32 ( x** y** -> r** ) - z_divmod32 ;z_divmod32/rem0 LDA2 ;z_divmod32/rem1 LDA2 JMP2r +@u32-mod ( x** y** -> r** ) + z_u32-divmod ;z_u32-divmod/rem0 LDA2 ;z_u32-divmod/rem1 LDA2 JMP2r ( x / y, x % y ) -@divmod32 ( x** y** -> q** r** ) - z_divmod32 - ;z_divmod32/quo0 LDA2 ;z_divmod32/quo1 LDA2 - ;z_divmod32/rem0 LDA2 ;z_divmod32/rem1 LDA2 +@u32-divmod ( x** y** -> q** r** ) + z_u32-divmod + ;z_u32-divmod/quo0 LDA2 ;z_u32-divmod/quo1 LDA2 + ;z_u32-divmod/rem0 LDA2 ;z_u32-divmod/rem1 LDA2 JMP2r ( private: calculate and store x / y and x % y ) -@z_divmod32 ( x** y** -> ) +@z_u32-divmod ( x** y** -> ) ( ; store y and x for repeated use ) #0000 DUP2 ,&quo0 STR2 ,&quo1 STR2 ( x** y** ; quo<-0 ) STH2k ,&div1 STR2 STH2k ,&div0 STR2 ( x** [ylo* yhi*] ; div<-y ) OVR2 OVR2 ,&rem1 STR2 ,&rem0 STR2 ( x** [ylo* yhi*] ; rem<-x ) OVR2 OVR2 STH2r STH2r ( x** x** y** ) OVR2 OVR2 STH2 STH2 ( x** x** y** [ylo* yhi*] ) - gteq32 ?{ POP2 POP2 POP2r POP2r JMP2r } ( x** [ylo* yhi*] ; return if x < y ) + u32-gteq ?{ POP2 POP2 POP2r POP2r JMP2r } ( x** [ylo* yhi*] ; return if x < y ) ( ; bitcount[x] - bitcount[y] determines largest multiple of y to try ) - bitcount32 STH2r STH2r bitcount32 SUB ( shift=rbits-dits^ ) + u32-bitcount STH2r STH2r u32-bitcount SUB ( shift=rbits-dits^ ) #00 DUP2 ( shift^ 0^ shift^ 0^ ) #0000 INC2k ROT2 POP ( shift^ 0^ 0* 1* shift^ ) - lshift32 ,&cur1 STR2 ,&cur0 STR2 ( shift^ 0^ ; cur<-1<= cur [current divisor], we can subtract it and add to quotient ) ( ; otherwise, skip that iteration and reduce cur. ) LIT2 [ &rem0 $2 ] LIT2 [ &rem1 $2 ] ,&div0 LDR2 ,&div1 LDR2 - lt32 ?{ + u32-lt ?{ ( ; since rem >= div, we have found a multiple of y that divides x ) ,&rem0 LDR2 ,&rem1 LDR2 ( rem** ) LIT2 [ &div0 $2 ] LIT2 [ &div1 $2 ] ( rem** div** ) - sub32 ,&rem1 STR2 ,&rem0 STR2 ( ; rem<-rem-div** ) + u32-sub ,&rem1 STR2 ,&rem0 STR2 ( ; rem<-rem-div** ) LIT2 [ &quo0 $2 ] LIT2 [ &quo1 $2 ] ( quo** ) LIT2 [ &cur0 $2 ] LIT2 [ &cur1 $2 ] ( quo** cur** ) - add32 ,&quo1 STR2 ,&quo0 STR2 ( ; quo<-quo+cur** ) + u32-add ,&quo1 STR2 ,&quo0 STR2 ( ; quo<-quo+cur** ) } - ,&div0 LDR2 ,&div1 LDR2 #01 rshift32 ( div>>1** ) + ,&div0 LDR2 ,&div1 LDR2 #01 u32-rshift ( div>>1** ) ,&div1 STR2 ,&div0 STR2 ( ; div<-div>>1 ) - ,&cur0 LDR2 ,&cur1 LDR2 #01 rshift32 ( cur>>1** ) + ,&cur0 LDR2 ,&cur1 LDR2 #01 u32-rshift ( cur>>1** ) OVR2 OVR2 ,&cur1 STR2 ,&cur0 STR2 ( cur>>1** ; cur<-cur>>1 ) - non-zero32 ?&loop JMP2r ( ; loop if cur>0, else we're done ) + u32-non-zero ?&loop JMP2r ( ; loop if cur>0, else we're done ) ( greatest common divisor - euclidean algorithm ) -@gcd32 ( x** y** -> z** ) - &loop OVR2 OVR2 is-zero32 ?{ ( x** y** ) +@u32-gcd ( x** y** -> z** ) + &loop OVR2 OVR2 u32-is-zero ?{ ( x** y** ) OVR2 OVR2 STH2 STH2 ( x** y** [y**] ) - mod32 ( r=x%y** [y**] ) + u32-mod ( r=x%y** [y**] ) STH2r ROT2 ROT2 ( yhi* rhi* rlo* [ylo*] ) STH2r ROT2 ROT2 !&loop ( y** r** ) } POP2 POP2 JMP2r ( z** )