214 lines
6.1 KiB
Forth
214 lines
6.1 KiB
Forth
# Floating Point
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On Unix hosts, the floating point stack provides a set of
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words building on the C `double` data type. In most cases,
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this will be a 64-bit, IEEE 754 double precision floating
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point format.
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~~~
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{{
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'FPU var
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:identify
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@FPU n:zero? 0; drop
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#2 io:scan-for dup n:negative?
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[ drop 'IO_DEVICE_TYPE_0002_NOT_FOUND s:put nl ]
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[ !FPU ] choose ;
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---reveal---
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:float:operation identify @FPU io:invoke ;
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}}
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~~~
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The initial set of words build directly over the core
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floating point device functionality, which on a Unix host
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maps closely to C and `libm`.
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~~~
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:n:to-float (n-_f:-n) #0 float:operation ;
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:s:to-float (s-_f:-n) #1 float:operation ;
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:f:to-number (f:a-__-n) #2 float:operation ;
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:f:to-string (f:n-__-s) s:empty dup #3 float:operation ;
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:f:+ (f:ab-c) #4 float:operation ;
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:f:- (f:ab-c) #5 float:operation ;
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:f:* (f:ab-c) #6 float:operation ;
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:f:/ (f:ab-c) #7 float:operation ;
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:f:floor (f:ab-c) #8 float:operation ;
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:f:ceiling (f:f-f) #9 float:operation ;
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:f:sqrt (f:f-f) #10 float:operation ;
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:f:eq? (f:ab-c) #11 float:operation ;
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:f:-eq? (f:ab-c) #12 float:operation ;
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:f:lt? (f:ab-c) #13 float:operation ;
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:f:gt? (f:ab-c) #14 float:operation ;
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:f:depth (-n) #15 float:operation ;
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:f:dup (f:a-aa) #16 float:operation ;
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:f:drop (f:a-) #17 float:operation ;
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:f:swap (f:ab-ba) #18 float:operation ;
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:f:log (f:ab-c) #19 float:operation ;
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:f:power (f:ab-c) #20 float:operation ;
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:f:sin (f:f-f) #21 float:operation ;
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:f:cos (f:f-f) #22 float:operation ;
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:f:tan (f:f-f) #23 float:operation ;
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:f:asin (f:f-f) #24 float:operation ;
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:f:acos (f:f-f) #25 float:operation ;
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:f:atan (f:f-f) #26 float:operation ;
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:f:push (f:f-) #27 float:operation ;
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:f:pop (f:-f) #28 float:operation ;
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:f:adepth (-n) #29 float:operation ;
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~~~
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Above this, additional functions are defined. First are words
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to aid in structuring the floating point stack.
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~~~
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:f:over (f:ab-aba) f:push f:dup f:pop f:swap ;
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:f:tuck (f:ab-bab) f:dup f:push f:swap f:pop ;
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:f:nip (f:ab-b) f:swap f:drop ;
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:f:drop-pair (f:ab-) f:drop f:drop ;
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:f:dup-pair (f:ab-abab) f:over f:over ;
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:f:rot (f:abc-bca) f:push f:swap f:pop f:swap ;
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~~~
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Then a word to allow creation of floating point values via a
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`.` prefix.
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~~~
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:prefix:. (s-__f:-a)
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compiling? &s:keep &s:temp choose &s:to-float class:word ; immediate
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~~~
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~~~
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:f:square (f:n-m) f:dup f:* ;
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:f:positive? (-f__f:a-) #0 n:to-float f:gt? ;
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:f:negative? (-f__f:a-) #0 n:to-float f:lt? ;
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:f:negate (f:a-b) #-1 n:to-float f:* ;
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:f:abs (f:a-b) f:dup f:negative? &f:negate if ;
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:f:put (f:a-) f:to-string s:put ;
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:f:PI (f:-F) .3.141592 ;
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:f:E (f:-F) .2.718281 ;
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:f:NAN (f:-n) .0 .0 f:/ ;
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:f:INF (f:-n) .1.0 .0 f:/ ;
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:f:-INF (f:-n) .-1.0 .0 f:/ ;
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:f:nan? (f:n-,-f) f:dup f:-eq? ;
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:f:inf? (f:n-,-f) f:INF f:eq? ;
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:f:-inf? (f:n-,-f) f:-INF f:eq? ;
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:f:round (-|f:a-b)
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f:dup f:negative?
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[ .0.5 f:- f:ceiling ]
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[ .0.5 f:+ f:floor ] choose ;
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:f:min (f:nn-n) f:dup-pair f:lt? &f:drop &f:nip choose ;
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:f:max (f:nn-n) f:dup-pair f:gt? &f:drop &f:nip choose ;
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:f:limit (f:nlu-n) f:swap f:push f:min f:pop f:max ;
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:f:between? (f:nlu-n) f:rot f:dup f:push f:rot f:rot f:limit f:pop f:eq? ;
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:f:inc (f:n-n) .1 f:+ ;
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:f:dec (f:n-n) .1 f:- ;
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:f:case (f:ff-,q-)
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f:over f:eq? [ f:drop call #-1 ] [ drop #0 ] choose 0; pop drop-pair ;
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:f:sign (-n|f:a-)
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f:dup .0 f:eq? [ #0 f:drop ] if;
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.0 f:gt? [ #1 ] [ #-1 ] choose ;
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~~~
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---------------------------------------------------------------
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# Floating Point Encoding
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This implements a means of encoding floating point values
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into signed integer cells. The technique is described in
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the paper titled "Encoding floating point numbers to shorter
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integers" by Kiyoshi Yoneda and Charles Childers.
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This will extend the `f:` vocabulary and adds a new `e:`
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vocabulary.
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## Code & Commentary
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Define some constants. The range is slightly reduced from
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the standard integer range as the smallest value is used
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for NaN.
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~~~
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n:MAX n:dec 'e:MAX const
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n:MAX n:dec n:negate 'e:MIN const
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n:MIN 'e:NAN const
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n:MAX 'e:INF const
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n:MAX n:negate 'e:-INF const
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~~~
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~~~
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:e:n? (u-f) e:MIN n:inc e:MAX n:dec n:between? ;
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:e:max? (u-f) e:MAX eq? ;
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:e:min? (u-f) e:MIN eq? ;
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:e:zero? (u-f) n:zero? ;
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:e:nan? (u-f) e:NAN eq? ;
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:e:inf? (u-f) e:INF eq? ;
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:e:-inf? (u-f) e:-INF eq? ;
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:e:clip (u-u) e:MIN e:MAX n:limit ;
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~~~
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Since 32-bit cells take about 9 decimal digits, if you set
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`[ .1e5 ] &f:E1 set-hook`
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you will have 5 decimal digits left for the integer part of
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the encoded number, which corresponds to 8 decimal digits
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decoded.
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Encode/decode words to secure dynamic range. This portion
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is the essence of the method.
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~~~
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:f:E1 (-|f:-n)_e-unit_in_float hook .1.e5 ; (decimal_digits_to_shift_left
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:f:signed-sqrt (|f:n-n) f:dup f:sign f:abs f:sqrt n:to-float f:* ;
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:f:signed-square (|f:n-n) f:dup f:sign f:dup f:* n:to-float f:* ;
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~~~
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Deal with special cases.
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~~~
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{{
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:f:-shift (|f:n-n)_shift_left f:E1 f:* ;
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:f:+shift (|f:n-n)_shift_right f:E1 f:/ ;
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:f:+encode (|f:n-n) f:signed-sqrt f:-shift ;
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:f:-encode (|f:n-n) f:dup f:sign f:+shift f:dup f:* n:to-float f:* ;
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---reveal---
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:f:to-e (-e|f:n-)
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f:dup f:nan? [ f:drop e:NAN ] if;
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f:dup f:inf? [ f:drop e:INF ] if;
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f:dup f:-inf? [ f:drop e:-INF ] if;
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f:+encode f:round f:to-number e:clip (e
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e:MIN &f:drop case
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e:MAX &f:drop case ;
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:e:to-f (e-|f:-n)
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e:NAN &f:NAN case
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e:INF &f:INF case
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e:-INF &f:-INF case
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n:to-float f:-encode ;
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}}
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~~~
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~~~
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:f:store (a-|f:n-) &f:to-e dip store ;
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:f:fetch (a-|f:-n) fetch e:to-f ;
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~~~
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~~~
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:f:dump-stack (-)
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f:depth dup &f:push times
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[ f:pop f:dup f:put sp ] times ;
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:f:dump-astack (-)
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f:adepth dup &f:pop times
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[ f:dup f:put sp f:push ] times ;
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~~~
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~~~
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:e:put (e-)
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e:MAX [ 'e:MAX s:put ] case
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e:MIN [ 'e:MIN s:put ] case
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#0 [ 'e:0 s:put ] case
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e:NAN [ 'e:NAN s:put ] case
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e:INF [ 'e:INF s:put ] case
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e:-INF [ 'e:-INF s:put ] case
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e:to-f f:put ;
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~~~
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