wip
This commit is contained in:
parent
f3c520c176
commit
e3905178a4
|
|
@ -0,0 +1,305 @@
|
||||||
|
( bfloat16.tal )
|
||||||
|
( )
|
||||||
|
( This file implements the bfloat16 format. )
|
||||||
|
( )
|
||||||
|
( This differs from IEEE float-16 by providing more exponent bits )
|
||||||
|
( in exchange for fewer mantissa bits. In other words it trades )
|
||||||
|
( coarser precision for larger numerical range. )
|
||||||
|
( )
|
||||||
|
( The bfloat16 value uses 16-bits divided as follows: )
|
||||||
|
( - sign (1 bit, 0-1) )
|
||||||
|
( - exponent (8 bits, 0-255) )
|
||||||
|
( - mantissa (7 bits, 0-127) )
|
||||||
|
( )
|
||||||
|
( Kinds of values: )
|
||||||
|
( - zeros (exponent==0 mantissa==0) )
|
||||||
|
( - subnormal (exponent==0 mantissa!=0) )
|
||||||
|
( - infinities (exponent==255 mantissa==0) )
|
||||||
|
( - nans (exponent==255 mantissa!=0) )
|
||||||
|
( - normal (everything else) )
|
||||||
|
( )
|
||||||
|
( Equations: )
|
||||||
|
( - normal = -1^sign * 2^(exponent - 127) * (1 + mantissa/128) )
|
||||||
|
( - subnormal = -1^sign * 2^-126 * mantissa/128 )
|
||||||
|
( (exponent ranges from 1 to 254 since 0 and 255 are special) )
|
||||||
|
( )
|
||||||
|
( VALUE SIGN EXPONENT MANTISSA NOTES )
|
||||||
|
( 0 0 00000000 0000000 )
|
||||||
|
( -0 1 00000000 0000000 mostly equivalent to zero )
|
||||||
|
( 1 0 01111111 0000000 )
|
||||||
|
( 2 0 10000000 0000000 )
|
||||||
|
( 3 0 10000000 1000000 )
|
||||||
|
( -1 1 01111111 0000000 )
|
||||||
|
( 17 0 10000011 0001000 )
|
||||||
|
( ~9.184e-41 0 00000000 0000001 smallest positive value )
|
||||||
|
( ~1.689e38 0 11111110 1111111 largest finite value )
|
||||||
|
( +inf 0 11111111 0000000 positive infinity )
|
||||||
|
( -inf 1 11111111 0000000 negative infinity )
|
||||||
|
( nan * 11111111 ******* lots of nans; * is wild )
|
||||||
|
( )
|
||||||
|
( This code doesn't distinguish between quiet and signaling NaNs. )
|
||||||
|
( )
|
||||||
|
( Bfloat16 values are emitted in a hexadecimal format: )
|
||||||
|
( )
|
||||||
|
( HEXADECIMAL SIGN EXPONENT MANTISSA DECIMAL )
|
||||||
|
( 0x1.00p+00 1 10000000 0000000 1.0 )
|
||||||
|
( 0x0.01p-7f 0 00000000 0000001 ~9.184e-41 )
|
||||||
|
( -0x1.80p+02 1 10000010 1000000 -6.0 )
|
||||||
|
( 0x1.c0p+02 0 10000010 1100000 7.0 )
|
||||||
|
( )
|
||||||
|
( Eventually I'd like to display integral part of the number )
|
||||||
|
( in a more natural way but the 1.xx format is OK for now. )
|
||||||
|
( )
|
||||||
|
( For consistency zeros are emitted as 0x00p+00 and -0x00p+00. )
|
||||||
|
( Infinities are "inf" and "-inf" and NaN is "nan". )
|
||||||
|
|
||||||
|
%EMIT { #18 DEO }
|
||||||
|
%SPACE { #20 EMIT }
|
||||||
|
%NEWLINE { #0a EMIT }
|
||||||
|
%DEBUG { #ff #0e DEO }
|
||||||
|
|
||||||
|
|0100
|
||||||
|
( #01 ;byte-to-bf16 JSR2 ;test JSR2
|
||||||
|
#02 ;byte-to-bf16 JSR2 ;test JSR2 )
|
||||||
|
( #437d -> 0 01010110 1111101 )
|
||||||
|
( #437c -> 0 01010110 1111100 )
|
||||||
|
( #00 #86 #7f ;bf16-join JSR2 ;emit-bf16 JSR2 NEWLINE )
|
||||||
|
( #ff ;byte-to-bf16 JSR2 ;test JSR2 )
|
||||||
|
( #ff ;byte-to-bf16 JSR2 #01 ;round-shift JSR2 ;test JSR2
|
||||||
|
#03 ;byte-to-bf16 JSR2 ;test JSR2
|
||||||
|
#7f80 ;test JSR2
|
||||||
|
#ff80 ;test JSR2
|
||||||
|
#ff81 ;test JSR2
|
||||||
|
#0000 ;test JSR2
|
||||||
|
#8000 ;test JSR2
|
||||||
|
#0001 ;test JSR2
|
||||||
|
#8001 ;test JSR2
|
||||||
|
#3f80 ;test JSR2
|
||||||
|
#bf80 ;test JSR2
|
||||||
|
#4000 ;test JSR2
|
||||||
|
#4080 ;test JSR2 )
|
||||||
|
#00 #00 DIV ( exit )
|
||||||
|
BRK
|
||||||
|
|
||||||
|
@test ( x* -> )
|
||||||
|
DUP2 ;emit-u16 JSR2 SPACE LIT '- EMIT LIT '> EMIT SPACE ;emit-bf16 JSR2 NEWLINE JMP2r
|
||||||
|
|
||||||
|
@emit-digit ( d^ -> )
|
||||||
|
DUP #0a LTH
|
||||||
|
,<-10 JCN #27 ADD
|
||||||
|
<-10 #30 ADD EMIT
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
@emit-u8 ( n^ -> )
|
||||||
|
DUP #04 SFT ;emit-digit JSR2
|
||||||
|
#0f AND ;emit-digit JSR2
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
@emit-u16 ( x* -> )
|
||||||
|
SWP ;emit-u8 JSR2
|
||||||
|
;emit-u8 JSR2
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
@emit-s8 ( x^ -> )
|
||||||
|
DUP #07 SFT ,&is-negative JCN LIT '+ EMIT ;emit-u8 JSR2 JMP2r
|
||||||
|
&is-negative LIT '- EMIT #7f AND #80 SWP SUB ;emit-u8 JSR2 JMP2r
|
||||||
|
|
||||||
|
@emit-s16 ( x* -> )
|
||||||
|
DUP2 #0f SFT2 SWP POP ,&is-negative JCN LIT '+ EMIT ;emit-u16 JSR2 JMP2r
|
||||||
|
&is-negative LIT '- EMIT #7fff AND2 #8000 SWP2 SUB2 ;emit-u16 JSR2 JMP2r
|
||||||
|
|
||||||
|
@emit-bf16 ( x* -> )
|
||||||
|
;bf16-split JSR2 ( sgn exp mnt )
|
||||||
|
|
||||||
|
( sentinel or value )
|
||||||
|
OVR #ff NEQ ,&non-sentinal JCN
|
||||||
|
,&is-nan JCN POP #00 EQU ,&pos-inf JCN LIT '- EMIT
|
||||||
|
&pos-inf LIT 'i EMIT LIT 'n EMIT LIT 'f EMIT JMP2r
|
||||||
|
&is-nan LIT 'n EMIT LIT 'a EMIT LIT 'n EMIT JMP2r
|
||||||
|
|
||||||
|
( zero or non-zero )
|
||||||
|
&non-sentinal DUP2 ORA ,&non-zero JCN
|
||||||
|
POP2 ,&is-negative-zero JCN ,&zero-suffix JMP
|
||||||
|
&is-negative-zero LIT '- EMIT
|
||||||
|
&zero-suffix LIT '0 EMIT LIT 'x EMIT LIT '0 EMIT LIT '. EMIT
|
||||||
|
#00 ;emit-u8 JSR2 LIT 'p EMIT #00 ;emit-s8 JSR2 JMP2r
|
||||||
|
|
||||||
|
( normal or subnormal )
|
||||||
|
&non-zero ROT ,&is-negative JCN ,&post-sgn JMP
|
||||||
|
&is-negative LIT '- EMIT
|
||||||
|
&post-sgn LIT '0 EMIT LIT 'x EMIT
|
||||||
|
OVR ,&is-normal JCN LIT '0 ,&suffix JMP &is-normal LIT '1
|
||||||
|
&suffix EMIT LIT '. EMIT ;emit-u8 JSR2
|
||||||
|
LIT 'p EMIT #7f SUB ;emit-s8 JSR2
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
@bf16-join ( sgn^ exp^ mta^ -> x* )
|
||||||
|
STH #00 #01 SFT2 ( sgn^ exp* [mta^] )
|
||||||
|
ROT #00 #01 SFT2 NIP #00 ORA2 ( sgn|exp* [mta^] )
|
||||||
|
#00 STHr ORA2 ( sgn|exp|mta* )
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
( sgn: 0-1, exp: 0-255, mta: 0-127 )
|
||||||
|
@bf16-split ( x* -> sgn^ exp^ mta^ )
|
||||||
|
OVR #07 SFT STH ( xhi xlo [sgn] )
|
||||||
|
#10 SFT2 SWP STHr ( mnt<1 exp sgn )
|
||||||
|
SWP ( mnt<1 exp sgn )
|
||||||
|
ROT ( exp sgn mnt<1 )
|
||||||
|
#01 SFT ( sgn exp mnt )
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
%SIGN { POP #07 SFT }
|
||||||
|
%EXPONENT { #10 SFT2 POP }
|
||||||
|
%MANTISSA { NIP #7f AND }
|
||||||
|
|
||||||
|
( returns sign: #00 or #01 )
|
||||||
|
@sign ( x* -> sgn^ )
|
||||||
|
SIGN JMP2r
|
||||||
|
|
||||||
|
( returns exp: #00 to #ff )
|
||||||
|
@exponent ( x* -> exp^ )
|
||||||
|
EXPONENT JMP2r
|
||||||
|
|
||||||
|
( returns mta: #00 to #7f )
|
||||||
|
@mantissa ( x* -> mta^ )
|
||||||
|
MANTISSA JMP2r
|
||||||
|
|
||||||
|
( returns full mta: #00 to #ff )
|
||||||
|
( normal numbers will be >= #80 )
|
||||||
|
( subnormal numbers will be < #80 )
|
||||||
|
@full-mantissa ( x* -> fmta^ )
|
||||||
|
DUP2 MANTISSA STH
|
||||||
|
EXPONENT ,&is-normal JCN STHr JMP2r
|
||||||
|
&is-normal #80 STHr ORA JMP2r
|
||||||
|
|
||||||
|
@negate-bf16 ( x* -> z* )
|
||||||
|
#8000 EOR2 JMP2r
|
||||||
|
|
||||||
|
@is-zero ( x* -> bool^ )
|
||||||
|
#7fff AND2 #0000 EQU2
|
||||||
|
|
||||||
|
@non-zero ( x* -> bool^ )
|
||||||
|
#7fff AND2 #0000 NEQ2
|
||||||
|
|
||||||
|
( Shift mantissa m right by n bits, with rounding )
|
||||||
|
( )
|
||||||
|
( We round differently depending on the value to be lost: )
|
||||||
|
( )
|
||||||
|
( 1. If the bits to be removed are > 0.5 we round up )
|
||||||
|
( 2. If the bits to be removed are < 0.5 we round down )
|
||||||
|
( 3. If the bits to be removed are = 0.5 we: )
|
||||||
|
( a. Round up if doing so produces an even mantissa )
|
||||||
|
( b. Round down if diong so produces an even mantissa )
|
||||||
|
( )
|
||||||
|
( This method is useful when adding two values that have )
|
||||||
|
( different exponents. We will want to truncate the value )
|
||||||
|
( with the smaller exponent to try to shift the mantissa )
|
||||||
|
( into the range of the larger value. )
|
||||||
|
( )
|
||||||
|
( It's important to remember to include the mantissa's )
|
||||||
|
( leading one value (if any) before calling this method )
|
||||||
|
@round-shift ( mta^ n^ -> z* )
|
||||||
|
STH2k ( mta n [mta n] )
|
||||||
|
#08 SWP SUB ( mta 8-n [mta n] )
|
||||||
|
STHk ( mta 8-n [8-n mta n] )
|
||||||
|
#7f SWP SFT ( mta mask=7f>>(8-n) )
|
||||||
|
AND ( mta&mask )
|
||||||
|
STHr #01 SWP #10 MUL SFT ( mta&mask lim=1<<(8-n) [mta n] )
|
||||||
|
DUP2 LTH ,&rnd-down JCN ( masked limit [mta n] )
|
||||||
|
GTH ,&rnd-up JCN ( [mta n] )
|
||||||
|
( round-to-even )
|
||||||
|
STH2r #01 SUB SFT ( mta>>(n-1) )
|
||||||
|
INC #01 SFT ( (mta>>(n-1)+1)>>1 )
|
||||||
|
JMP2r
|
||||||
|
&rnd-down ( masked limit [mta n] )
|
||||||
|
POP2 STH2r SFT JMP2r
|
||||||
|
&rnd-up ( [mta n] )
|
||||||
|
STH2r SFT INC JMP2r
|
||||||
|
|
||||||
|
@is-nan ( x* -> bool^ )
|
||||||
|
SWP ( xlo xhi )
|
||||||
|
#7f ANDk ( xlo xhi 7f xhi&7f )
|
||||||
|
EQU ( xlo xhi xhi&7f=7f )
|
||||||
|
NIP SWP #00 NEQ ( xhi&7f=7f xlo!=0 )
|
||||||
|
AND JMP2r
|
||||||
|
|
||||||
|
@non-nan ( x* -> bool^ )
|
||||||
|
#00 EQU STH ( xhi [xlo=0] )
|
||||||
|
#7f ANDk NEQ ( xhi 7f!=xhi&7f [xlo=0] )
|
||||||
|
NIP STHr ORA ( 7f!=xhi&7f|xlo=0 )
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
@add-bf16 ( x* y* -> z* )
|
||||||
|
DUP2 ;non-nan
|
||||||
|
|
||||||
|
@byte-to-bf16 ( n^ -> x* )
|
||||||
|
#86 SWP ( exp n )
|
||||||
|
&loop
|
||||||
|
DUP #7f GTH ,&ready JCN
|
||||||
|
#10 SFT SWP #01 SUB SWP
|
||||||
|
,&loop JMP
|
||||||
|
&ready
|
||||||
|
#7f AND STH ( exp [n&7f] )
|
||||||
|
#00 #01 SFT2 #00 STHr ORA2
|
||||||
|
JMP2r
|
||||||
|
|
||||||
|
( rules: )
|
||||||
|
( 1. nan = nan is false )
|
||||||
|
( 2. x = x is true )
|
||||||
|
( 3. (x = y) is (y = x) )
|
||||||
|
( 4. -0 = +0 is true )
|
||||||
|
@eq-bf16 ( x* y* -> bool^ )
|
||||||
|
DUP2 ;non-nan JSR2 STH SWP2 ( is y not nan? )
|
||||||
|
DUP2 ;non-nan JSR2 STH SWP2 ( is x not nan? )
|
||||||
|
STH2r ORA ,¬-nan JCN ( is either x or y not nan? )
|
||||||
|
POP2 POP2 #00 JMP2r ( else return false )
|
||||||
|
¬-nan
|
||||||
|
DUP2 ;non-zero JSR2 ,¬-zero JCN ( is y non-zero? )
|
||||||
|
POP2 ;is-zero JSR2 JMP2r ( if y is zero, return x-is-zero )
|
||||||
|
¬-zero
|
||||||
|
EQU2 JMP2r ( if not nan or zero, standard comparison )
|
||||||
|
|
||||||
|
@ne-bf16 ( x* y* -> bool^ )
|
||||||
|
;eq-bf16 JSR2 #00 EQU JMP2r
|
||||||
|
|
||||||
|
( rules for sentinels (in order): )
|
||||||
|
( 1. x < x is false )
|
||||||
|
( 2a. nan < x is false )
|
||||||
|
( 2b. x < nan is false )
|
||||||
|
( 3a. x < +inf is true )
|
||||||
|
( 3b. +inf < x is false )
|
||||||
|
( 4a. -inf < x is true )
|
||||||
|
( 4b. x < -inf is false )
|
||||||
|
( 5. -0 < +0 is false )
|
||||||
|
( 6. -x < +x (or 0) )
|
||||||
|
( 7. 0 < +x )
|
||||||
|
( 8. x*2^p < y*2^q if p < q )
|
||||||
|
( 9. x*2^p < y*2^p if x < y )
|
||||||
|
@lt-bf16 ( x* y* -> bool^ )
|
||||||
|
,&y STR2 ,&x STR2
|
||||||
|
DUP2 ;non-nan JSR2 STH SWP2 ( is y not nan? )
|
||||||
|
DUP2 ;non-nan JSR2 STH SWP2 ( is x not nan? )
|
||||||
|
STH2r ORA ,¬-nan JCN ( is either x or y not nan? )
|
||||||
|
POP2 POP2 #00 JMP2r ( else return false )
|
||||||
|
¬-nan
|
||||||
|
DUP2 ;non-zero JSR2 STH SWP2 ( is y non-zero? )
|
||||||
|
DUP2 ;non-zero JSR2 STH SWP2 ( is x non-zero? )
|
||||||
|
STH2r ORA ,¬-zero JCN ( both x and y non-zero? )
|
||||||
|
POP2 POP2 #00 JMP2r
|
||||||
|
¬-zero
|
||||||
|
DUP2 ;sign JSR2 STH SWP2
|
||||||
|
DUP2 ;sign JSR2 STH SWP2
|
||||||
|
STH2r EQUk ,&same-sign JCN
|
||||||
|
GTH STH POP2 POP2 STHr JMP2r
|
||||||
|
[ &x $2 &y $2 ]
|
||||||
|
&same-sign
|
||||||
|
DUP2 ;exponent JSR2 STH SWP2
|
||||||
|
DUP2 ;exponent JSR2 STH SWP2
|
||||||
|
STH2r EQUk ,&exp-eq JCN
|
||||||
|
LTH STH POP2 POP2 STHr JMP2r
|
||||||
|
&exp-eq
|
||||||
|
POP2
|
||||||
|
;mantissa JSR2 STH
|
||||||
|
;exponent JSR2 STHr
|
||||||
|
LTH JMP2r
|
||||||
|
|
||||||
Loading…
Reference in New Issue