( 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                           )
(                                                         )
( many 8.8 operations are equivalent to uin16:            )
(   * addition/subtraction                                )
(   * division                                            )
( or signed int16:                                        )
(   * comparisons/equality                                )
(                                                         )
( 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 )
( MUL2.                                                   )

|1000

( 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 }

( 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 ,&lt JCN GTH2 JMP2r
    &lt POP2 POP2 #ff JMP2r

@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

@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