2022-03-13 23:23:20 -04:00
|
|
|
( fix16.tal )
|
|
|
|
|
( )
|
|
|
|
|
( use a signed 16-bit short as a fixed point number. )
|
|
|
|
|
( )
|
|
|
|
|
( numbers are interpreted as fractions with an implicit )
|
|
|
|
|
( 256 denominator. the upper byte is signed and )
|
|
|
|
|
( represents the "whole" part of the number, and the )
|
|
|
|
|
( lower byte is unsigned and represents the )
|
|
|
|
|
( "fractional" part of the number. )
|
|
|
|
|
( )
|
|
|
|
|
( 16-bit fixed point can represent fractional values )
|
|
|
|
|
( in the range -128 <= x < 128. the smallest fraction it )
|
|
|
|
|
( can represent is 1/256, which is about 0.004. )
|
|
|
|
|
( )
|
|
|
|
|
( SHORT FRACTION DECIMAL )
|
|
|
|
|
( #0000 0/256 0.000 )
|
|
|
|
|
( #0001 1/256 0.004 )
|
|
|
|
|
( #0002 2/256 0.008 )
|
|
|
|
|
( #0040 64/256 0.250 )
|
|
|
|
|
( #0080 128/256 0.500 )
|
|
|
|
|
( #0100 256/256 1.000 )
|
|
|
|
|
( #0700 1792/256 7.000 )
|
|
|
|
|
( #7f00 32512/256 127.000 )
|
|
|
|
|
( #7fff 32767/256 127.996 )
|
|
|
|
|
( #8000 -32768/256 -128.000 )
|
|
|
|
|
( #8001 -32767/256 -127.996 )
|
|
|
|
|
( #8100 -32767/256 -127.000 )
|
|
|
|
|
( #ff00 -256/256 -1.000 )
|
|
|
|
|
( #ffff -1/256 -0.004 )
|
|
|
|
|
( )
|
2022-03-13 23:28:13 -04:00
|
|
|
( many 8.8 operations are equivalent to uin16: )
|
2022-03-13 23:23:20 -04:00
|
|
|
( * addition/subtraction )
|
|
|
|
|
( * division )
|
2022-03-13 23:28:13 -04:00
|
|
|
( or signed int16: )
|
|
|
|
|
( * comparisons/equality )
|
2022-03-13 23:23:20 -04:00
|
|
|
( )
|
|
|
|
|
( but due to 16-bit truncation multiplication differs... )
|
|
|
|
|
( )
|
|
|
|
|
( x*y = x0*y0 + x0*y1/256 + x1*y0/256 + x1*y1/65536 )
|
|
|
|
|
( )
|
|
|
|
|
( since we only have 16-bits: )
|
|
|
|
|
( 1. we need to drop the 8 high bits from x0*y0 )
|
|
|
|
|
( 2. we need to drop the 8 low bits from x1*y1 )
|
|
|
|
|
( 3. we need to use all the bits from x0*y1 and x1*y0 )
|
|
|
|
|
( )
|
|
|
|
|
( that said, if either x or y is whole (i.e. ends in 00) )
|
|
|
|
|
( then we can just shift that argument right by 8 and use )
|
2022-03-13 23:28:13 -04:00
|
|
|
( MUL2. )
|
2022-03-13 23:23:20 -04:00
|
|
|
|
2022-03-17 23:04:40 -04:00
|
|
|
%xyz { ;x16-emit JSR2 #0a #18 DEO }
|
2022-03-13 23:23:20 -04:00
|
|
|
|
|
|
|
|
( useful constants )
|
|
|
|
|
( )
|
|
|
|
|
( to generate your own: )
|
|
|
|
|
( 1. take true value, e.g. 3.14159... )
|
|
|
|
|
( 2. multiply by 256 )
|
|
|
|
|
( 3. round to nearest whole number )
|
|
|
|
|
( 4. emit hex output )
|
|
|
|
|
( )
|
|
|
|
|
( in python: hex(round(x * 256)) )
|
|
|
|
|
%x16-zero { #0000 } ( 0.0 )
|
|
|
|
|
%x16-one { #0100 } ( 1.0 )
|
|
|
|
|
%x16-two { #0200 } ( 2.0 )
|
|
|
|
|
%x16-ten { #0a00 } ( 10.0 )
|
|
|
|
|
%x16-hundred { #6400 } ( 100.0 )
|
|
|
|
|
%x16-minus-one { #7f00 } ( -1.0 )
|
|
|
|
|
%x16-minus-two { #7e00 } ( -2.0 )
|
|
|
|
|
%x16-pi/2 { #0192 } ( 1.57079... )
|
|
|
|
|
%x16-pi { #0324 } ( 3.14159... )
|
|
|
|
|
%x16-pi*2 { #0648 } ( 6.28318... )
|
|
|
|
|
%x16-e { #02b8 } ( 2.71828... )
|
|
|
|
|
%x16-phi { #019e } ( 1.61803... )
|
|
|
|
|
%x16-sqrt-2 { #016a } ( 1.41421... )
|
|
|
|
|
%x16-sqrt-3 { #01bb } ( 1.73205... )
|
|
|
|
|
%x16-epsilon { #0001 } ( 0.00390625 )
|
|
|
|
|
%x16-minimum { #8000 } ( -128.0 )
|
|
|
|
|
%x16-maximum { #7fff } ( 127.99609375 )
|
|
|
|
|
|
|
|
|
|
( useful macros )
|
|
|
|
|
%x16-is-non-neg { x16-minimum LTH2 }
|
|
|
|
|
%x16-is-neg { x16-maximum GTH2 }
|
|
|
|
|
|
2022-03-17 23:04:40 -04:00
|
|
|
%x16-emit-dec { #30 ADD #18 DEO }
|
|
|
|
|
|
|
|
|
|
( |0100
|
|
|
|
|
x16-zero xyz
|
|
|
|
|
x16-one xyz
|
|
|
|
|
x16-two xyz
|
|
|
|
|
x16-ten xyz
|
|
|
|
|
x16-hundred xyz
|
|
|
|
|
x16-pi/2 xyz
|
|
|
|
|
x16-pi xyz
|
|
|
|
|
x16-pi*2 xyz
|
|
|
|
|
x16-e xyz
|
|
|
|
|
x16-phi xyz
|
|
|
|
|
x16-sqrt-2 xyz
|
|
|
|
|
x16-sqrt-3 xyz
|
|
|
|
|
x16-epsilon xyz
|
|
|
|
|
#1234 xyz
|
|
|
|
|
BRK )
|
|
|
|
|
|
|
|
|
|
@x16-emit ( x* -> )
|
|
|
|
|
SWP DUP #64 LTH ,&<100 JCN
|
|
|
|
|
#64 DIVk DUP x16-emit-dec MUL SUB ,&>=10 JMP
|
|
|
|
|
&<100 DUP #0a LTH ,&<10 JCN
|
|
|
|
|
&>=10 #0a DIVk DUP x16-emit-dec MUL SUB
|
|
|
|
|
&<10 x16-emit-dec
|
|
|
|
|
LIT '. #18 DEO
|
|
|
|
|
( emit fractional part )
|
|
|
|
|
#00 SWP ( lo* )
|
|
|
|
|
#000a MUL2 #0100 DIV2k DUP2 NIP x16-emit-dec MUL2 SUB2
|
|
|
|
|
#000a MUL2 #0100 DIV2k DUP2 NIP x16-emit-dec MUL2 SUB2
|
|
|
|
|
#000a MUL2 #0100 DIV2k STH2k MUL2 SUB2 #0080 LTH2 ,&no-round JCN INC2r
|
|
|
|
|
&no-round STH2r NIP x16-emit-dec JMP2r
|
|
|
|
|
|
2022-03-13 23:23:20 -04:00
|
|
|
( comparison between x and y. )
|
|
|
|
|
( - ff: x < y )
|
|
|
|
|
( - 00: x = y )
|
|
|
|
|
( - 01: x > y )
|
|
|
|
|
@x16-cmp ( x* y* -> c^ )
|
|
|
|
|
STH2k x16-is-neg ,&yn JMP
|
|
|
|
|
x16-is-non-neg ,&ypxp ( y>=0 )
|
|
|
|
|
POP2 POP2r #ff JMP2r ( x<0 y>=0 )
|
|
|
|
|
&ypxp ;x16-ucmp JMP2 ( x>=0 y>=0 )
|
|
|
|
|
&yn x16-is-neg ,&ynxn ( y<0 )
|
|
|
|
|
POP2 POP2r #01 JMP2r ( x>=0 y<0 )
|
|
|
|
|
&ynxn SWP2 ;x16-ucmp JMP2 ( x<0 y<0 )
|
|
|
|
|
|
|
|
|
|
( unsigned comparison between x and y. )
|
|
|
|
|
( - ff: x < y )
|
|
|
|
|
( - 00: x = y )
|
|
|
|
|
( - 01: x > y )
|
|
|
|
|
@x16-ucmp ( x* y* -> c^ )
|
|
|
|
|
LTH2k ,< JCN GTH2 JMP2r
|
|
|
|
|
< POP2 POP2 #ff JMP2r
|
|
|
|
|
|
2022-03-17 23:04:40 -04:00
|
|
|
@x16-eq ( x* y* -> x=y^ ) EQU2 JMP2r
|
|
|
|
|
@x16-ne ( x* y* -> x!=0^ ) NEQ2 JMP2r
|
|
|
|
|
@x16-lt ( x* y* -> x<y^ ) ;x16-cmp JSR2 #ff EQU JMP2r
|
|
|
|
|
@x16-lteq ( x* y* -> x<y^ ) ;x16-cmp JSR2 #01 NEQ JMP2r
|
|
|
|
|
@x16-gt ( x* y* -> x<y^ ) ;x16-cmp JSR2 #01 EQU JMP2r
|
|
|
|
|
@x16-gteq ( x* y* -> x<y^ ) ;x16-cmp JSR2 #ff NEQ JMP2r
|
2022-03-13 23:23:20 -04:00
|
|
|
|
|
|
|
|
@x16-is-whole ( x* -> bool^ )
|
|
|
|
|
NIP #00 EQU JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-add ( x* y* -> x+y* )
|
|
|
|
|
ADD2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-sub ( x* y* -> x-y* )
|
|
|
|
|
SUB2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-negate ( x* -> -x* )
|
|
|
|
|
#0000 SWP2 SUB2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-mul ( x* y* -> xy* )
|
|
|
|
|
DUP #00 EQU ,&rhs-whole JCN
|
|
|
|
|
SWP2 DUP #00 EQU ,&rhs-whole JCN
|
|
|
|
|
,&y3 STR ,&y1 STR ,&x3 STR ,&x1 STR
|
|
|
|
|
LIT2 &x2 00 &x3 00 LIT2 &y2 00 &y3 00 MUL2 #08 SFT2
|
|
|
|
|
LIT2 &x0 00 &x1 00 ,&y2 LDR2 MUL2 ADD2
|
|
|
|
|
,&x2 LDR2 LIT2 &y0 00 &y1 00 MUL2 ADD2
|
|
|
|
|
,&x0 LDR2 ,&x0 LDR2 MUL2 #80 SFT2 ADD2 JMP2r
|
|
|
|
|
&rhs-whole #08 SFT2 MUL2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-div ( x* y* -> x/y* )
|
|
|
|
|
SWP2 DUP2 x16-is-non-neg ,&non-negative ( y x )
|
|
|
|
|
;x16-negate JSR2 SWP2 DIV2 ;x16-negate JMP2
|
|
|
|
|
&non-negative
|
|
|
|
|
SWP2 DIV2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-mod ( x* y* -> x%y* )
|
|
|
|
|
;x16-div JSR2 ;x16-mul JSR2 SUB2 JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-mod-div ( x* y* -> x%y* x/y* )
|
|
|
|
|
;x16-div JSR2 STH2k ;x16-mul JSR2 SUB2 STH2r JMP2r
|
|
|
|
|
|
|
|
|
|
@x16-div-mod ( x* y* -> x/y* x%y* )
|
|
|
|
|
;x16-mod-div JSR2 SWP2 JMP2r
|
2022-03-14 00:00:01 -04:00
|
|
|
|
|
|
|
|
( trigonometry )
|
|
|
|
|
( )
|
|
|
|
|
( this uses there different angle representations: )
|
|
|
|
|
( )
|
|
|
|
|
( 1. angle representation (#0080 = pi, #0100 = 2pi) )
|
|
|
|
|
( 2. fix16 representation (#0324 = pi, #0648 = 2pi) )
|
|
|
|
|
( 3. degrees (#00b4 = pi, #0168 = 2pi) )
|
|
|
|
|
( )
|
|
|
|
|
( angles are the most precise, but may require some )
|
|
|
|
|
( conversion. )
|
|
|
|
|
|
|
|
|
|
( calculate the sin of the given angle. )
|
|
|
|
|
( )
|
|
|
|
|
( the input should be an angle and the )
|
|
|
|
|
( result will be 16-bit fixed point. )
|
|
|
|
|
( )
|
|
|
|
|
( cos(x) = sin(x + pi/2) )
|
|
|
|
|
@cos-angle ( x* -> cos(x)* )
|
|
|
|
|
#0080 ADD2 ;sin-angle JMP2
|
|
|
|
|
|
|
|
|
|
( calculate the sin of the given angle. )
|
|
|
|
|
( )
|
|
|
|
|
( the input should be an angle and the )
|
|
|
|
|
( result will be 16-bit fixed point. )
|
|
|
|
|
( )
|
|
|
|
|
( sin table offset math: )
|
|
|
|
|
( 0 <= x < 64 -> table[x] )
|
|
|
|
|
( 64 <= x < 128 -> table[127 - x] )
|
|
|
|
|
( 128 <= x < 192 -> -table[x - 128] )
|
|
|
|
|
( 192 <= x < 256 -> -table[255 - x] )
|
|
|
|
|
@sin-angle ( x* -> sin(x)* )
|
|
|
|
|
NIP DUP #7f GTH ,&d180+ JCN
|
|
|
|
|
#0001 STH2
|
|
|
|
|
DUP #3f GTH ,&d90 ,&load JMP
|
|
|
|
|
&d90 #7f SWP SUB ,&load JMP
|
|
|
|
|
&d180+
|
|
|
|
|
#ffff STH2
|
|
|
|
|
DUP #bf GTH ,&d270 #80 SUB ,&load JMP
|
|
|
|
|
&d270 #ff SWP SUB ,&load JMP
|
|
|
|
|
&load ,sin-table ADD LDR2 STH2r MUL2 JMP2r
|
|
|
|
|
|
|
|
|
|
( sin table with 64 entries for 1/4 of a unit circle. )
|
|
|
|
|
( )
|
|
|
|
|
( there is no need for interpolation when using )
|
|
|
|
|
( angle values, since 256 distinct values can be )
|
|
|
|
|
( produced using reflection. )
|
|
|
|
|
( )
|
|
|
|
|
( these table values go from 0 until pi/2 (i.e. from )
|
|
|
|
|
( angles #00 until #80). )
|
|
|
|
|
@sin-table ( 0 - pi/2: 64 steps )
|
|
|
|
|
0000 0006 000d 0013 0019 001f 0026 002c
|
|
|
|
|
0032 0038 003e 0044 004a 0050 0056 005c
|
|
|
|
|
0062 0068 006d 0073 0079 007e 0084 0089
|
|
|
|
|
008e 0093 0098 009d 00a2 00a7 00ac 00b1
|
|
|
|
|
00b5 00b9 00be 00c2 00c6 00ca 00ce 00d1
|
|
|
|
|
00d5 00d8 00dc 00df 00e2 00e5 00e7 00ea
|
|
|
|
|
00ed 00ef 00f1 00f3 00f5 00f7 00f8 00fa
|
|
|
|
|
00fb 00fc 00fd 00fe 00ff 00ff 0100 0100
|
