⟦c609329b4⟧

TextFile

%!PS-Adobe-2.0
%%Title: PubTeX output 1990.12.12:1351
%%Creator: Pubps,       ArborText, Inc.
%%BoundingBox: (atend)
%%Pages: (atend)
%%DocumentFonts: (atend)
%%EndComments

%! 
%  Dvips.pro - included prolog for DviLaser-generated PostScript files.
% 
%  Copyright (c) 1986-89, ArborText, Inc.
%  Permission to copy is granted so long as the PostScript code
%  is not resold or used in a commercial product.
%
%  $Header: pubps.pro,v 1.14 90/04/10 12:27:08 jsg Exp $

systemdict /setpacking known  % use array packing mode if its available
  {/savepackingmode currentpacking def 
   true setpacking}
  if
 
/$DviLaser 400 dict def
 
% Begin document
/BeginDviLaserDoc {
  vmstatus pop pop 0 eq 
    { $DviLaser begin 
      InitializeState }
    { /DviLaserJob save def
      $DviLaser begin
      InitializeState
      /DviLaserFonts save def }
    ifelse
} bind def
 
% End document
/EndDviLaserDoc {
  vmstatus pop pop 0 eq 
    { end }
    { DviLaserFonts restore
      end
      DviLaserJob restore }
    ifelse
} bind def
 
$DviLaser begin

/tempstr 64 string def
/tempint 0 def
/tempmatrix matrix def
 
%
%  Debugging routines
%
/DebugMode false def

/PrintInt {
  tempstr cvs print
} bind def

/PrintLn {
  (\n) print flush
} bind def
 
/PrintVMStats {
  print
  PrintLn
  (VM status - ) print
  vmstatus
  3 copy
  PrintInt (\(total\), ) print
  PrintInt (\(used\), ) print
  pop
  exch sub 
  PrintInt (\(remaining\), ) print
  PrintInt (\(level\)) print
  PrintLn
} bind def
 
/VMS /PrintVMStats load def 
 
/VMSDebug {
  DebugMode
    {PrintVMStats}
    {pop}
    ifelse
} bind def
 
(beginning of common prolog) VMSDebug 

% Make it easy to bind definitions.
/bdef { bind def } bind def
/xdef { exch def } bdef

% Begin page
/BP {
  /Magnification xdef
  /DviLaserPage save def
  (beginning of page) VMSDebug 
} bdef
 
% End page
/EP {
  DviLaserPage restore
} bdef
 
% Exit page (temporarily) to add fonts/characters.
/XP {
  % Save current point information so it can be reset later. 
  /Xpos where {pop Xpos} {0} ifelse
  /Ypos where {pop Ypos} {0} ifelse
  /currentpoint cvx stopped {0 0 moveto currentpoint} if 
  /DviLaserPage where {pop DviLaserPage restore} if
  moveto
  /Ypos xdef
  /Xpos xdef
} bdef
 
% Resume page
/RP {
  /DviLaserPage save def
} bdef
 
% Purge all fonts to reclaim memory space. 
/PF {
  GlobalMode
  LocalMode
} bdef
 
% Switch to base save/restore level, saving state information. 
/GlobalMode {
  /UserSave where {pop UserSave} if  % invoke "UserSave" if available
  PortraitMode 
  PaperWidth 
  PaperHeight 
  PxlResolution 
  Resolution 
  Magnification
  Ymax
  RasterScaleFactor
  % Save current point information so it can be reset later. 
  /currentpoint cvx stopped {0 0 moveto currentpoint} if 
  /DviLaserPage where {pop DviLaserPage restore} if
  DviLaserFonts restore
  RecoverState
} bdef
 
% Preserve state at the base level.
/RecoverState {
  10 copy
  /Ypos xdef
  /Xpos xdef
  /RasterScaleFactor xdef
  /Ymax xdef
  /Magnification xdef
  /Resolution xdef
  /PxlResolution xdef
  /PaperHeight xdef
  /PaperWidth xdef
  /PortraitMode xdef
  DoInitialScaling
  PortraitMode not {PaperWidth 0 SetupLandscape} if
  Xpos Ypos moveto
} bdef

% Initialize state variables to default values.
/InitializeState {
  /Resolution 3600.0 def
  /PxlResolution 300.0 def
  /RasterScaleFactor PxlResolution Resolution div def
  /PortraitMode true def
  11.0 Resolution mul /PaperHeight xdef
  8.5 Resolution mul /PaperWidth xdef
  /Ymax PaperHeight def
  /Magnification 1000.0 def
  /Xpos 0.0 def
  /Ypos 0.0 def
  /InitialMatrix matrix currentmatrix def
} bdef

% Switch from base save/restore level, restoring state information. 
/LocalMode {
  /Ypos xdef
  /Xpos xdef
  /RasterScaleFactor xdef
  /Ymax xdef
  /Magnification xdef
  /Resolution xdef
  /PxlResolution xdef
  /PaperHeight xdef
  /PaperWidth xdef
  /PortraitMode xdef
  DoInitialScaling
  PortraitMode not {PaperWidth 0 SetupLandscape} if
  Xpos Ypos moveto
  /UserRestore where {pop UserRestore} if  % invoke "UserRestore" if available
  /DviLaserFonts save def
  /DviLaserPage save def
} bdef
 
% Abbreviations 
/S /show load def
/SV /save load def
/RST /restore load def
 
/Yadjust {Ymax exch sub} bdef
 
% (x,y) position absolute, just set Xpos & Ypos, don't move.
/SXY {
  Yadjust 
  /Ypos xdef /Xpos xdef
} bdef
 
% (x,y) position absolute
/XY {
  Yadjust 
  2 copy /Ypos xdef /Xpos xdef
  moveto
} bdef
 
% (x,0) position absolute
/X {
  currentpoint exch pop 
  2 copy /Ypos xdef /Xpos xdef
  moveto
} bdef
 
% (0,y) position absolute
/Y {
  currentpoint pop exch Yadjust 
  2 copy /Ypos xdef /Xpos xdef
  moveto
} bdef
 
% (x,y) position relative
/xy {
  neg rmoveto
  currentpoint /Ypos xdef /Xpos xdef
} bdef
 
% (x,0) position relative
/x {
  0.0 rmoveto
  currentpoint /Ypos xdef /Xpos xdef
} bdef
 
% (0,y) position relative
/y {
  0.0 exch neg rmoveto
  currentpoint /Ypos xdef /Xpos xdef
  } bdef
 
% Print a rule.  In order to get correct pixel size and positioning,
% we usually create a temporary font in which the rule is the only character.
% When the rule is large, however, we fill a rectangle instead.
/R {
  /ht xdef
  /wd xdef
  ht 1950 le wd 1950 le and
    {save
    /tfd 6 dict def
    tfd begin
      /FontType 3 def
      /FontMatrix [1 0 0 1 0 0] def
      /FontBBox [0 0 wd ht] def
      /Encoding 256 array dup 97 /a put def
      /BuildChar {
        pop   % ignore character code
        pop   % ignore font dict, too
        wd 0 0 0 wd ht setcachedevice
        wd ht true
        [1 0 0 -1 0 ht] {<FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF>} imagemask
        } def
      end % tfd
    /tf tfd definefont setfont
    (a) show
    restore
    }
    {gsave
    0 setgray
    currentpoint
    newpath
      moveto
      0.0 ht rlineto
      wd 0.0 rlineto
      0.0 ht neg rlineto
      wd neg 0.0 rlineto
    closepath fill
    grestore
    }
  ifelse
  wd 0.0 rmoveto
  currentpoint /Ypos xdef /Xpos xdef
} bdef

%
%  <PXL-file resolution(pix/inch)> <resolution(pix/inch)> RES
%
/RES {
  /Resolution xdef
  /PxlResolution xdef
  /RasterScaleFactor PxlResolution Resolution div def
  DoInitialScaling
} bdef

%
% Do initial scaling.
%
/DoInitialScaling {
  InitialMatrix setmatrix
  72.0 Resolution div dup scale   
} bdef
 
%
%  <paper-height(pix)> <paper-width(pix)> PM 
%
/PM { 
  XP
  /PaperWidth xdef 
  /PaperHeight xdef
  /Ymax PaperHeight def
  /PortraitMode true def
  DoInitialScaling
  RP
} bdef  
 
%
%  <paper-height(pix)> <paper-width(pix)> LM 
%
/LM {
  XP
  /PaperWidth xdef 
  /PaperHeight xdef
  /Ymax PaperWidth def
  /PortraitMode false def
  DoInitialScaling
  PaperWidth 0 SetupLandscape
  RP
} bdef  
  
% Change magnification setting
/MAG {
  XP
  /Magnification xdef
  RP
} bdef
 
%
%  Switch to landscape mode
%
/SetupLandscape {
  translate
  90.0 rotate
} bdef
 
%
%  <mode> SPB - begin "\special" mode
%
%  This is the PostScript procedure used to transfer from the internal
%  environment used for the DVI translation code emitted by DVIPS to
%  a standard PostScript environment.
%
%  Parameters: 0 - Local
%              1 - Global
%              2 - Inline
%
/SPB {
  /spc_mode xdef
  spc_mode 0 eq spc_mode 2 eq or
    {XP}
    {spc_mode 1 eq {GlobalMode} if} 
    ifelse
  Resolution 72.0 div dup scale        % Restore default scaling...
  Magnification 1000.0 div dup scale   % Adjust for any magnification...
  /Xpos Xpos 72.0 Resolution div mul 1000.0 Magnification div mul def
  /Ypos Ypos 72.0 Resolution div mul 1000.0 Magnification div mul def
} bdef
 
%
%  <mode> SPE - end "\special" mode
%
%  This is the PostScript procedure used to reenter the internal
%  environment used for the DVI translation code emitted by DVIPS from 
%  the standard PostScript environment provided for processing user-supplied
%  PostScript code.
%
%  Parameters: 0 - Local
%              1 - Global
%              2 - Inline
%
/SPE {
  /spc_mode xdef
  1000.0 Magnification div dup scale   % Un-adjust for any magnification...
  72.0 Resolution div dup scale        % Restore default internal scaling...
  spc_mode 0 eq spc_mode 2 eq or
    {RP}
    {spc_mode 1 eq {LocalMode} if} 
    ifelse
} bdef
 
%
%  <num-copies> PP
%
/PP {
  /#copies xdef
  showpage
  /#copies 1 def
} bdef
 
%
%  /font-name <point-size(pix)> DMF
%
/DMF {
  /psz xdef
  /nam xdef
  nam findfont psz scalefont setfont
} bdef
 
%
%  /abcd (xxx) str-concat  ==> /abcdxxx
%
/str-concatstr 64 string def

/str-concat {
  /xxx xdef
  /nam xdef
  /namstr nam str-concatstr cvs def
  /newnam namstr length xxx length add string def
  newnam 0 namstr putinterval
  newnam namstr length xxx putinterval
  newnam cvn 
} bdef
 
%
%  /abcdef 2 str-strip ==> /cdef
%
/str-strip {
  /num xdef
  /nam xdef
  /namstr nam tempstr cvs def
  /newlen namstr length num sub def
  namstr num newlen getinterval
  cvn
} bdef
 
%
%  <old-dict> copydict ==> new-dict on stack
%
/copydict {
  dup length 1 add dict /newdict xdef
    {1 index /FID ne
      {newdict 3 1 roll put}
      {pop pop}
     ifelse
    } forall 
  newdict
} bdef
 
%
%  <font-type> DefineCMEncoding
%
/DefineCMEncoding {
  /EncodeType xdef
 
  /CMEncoding 256 array def
  /Times-Roman findfont /Encoding get aload pop CMEncoding astore pop
 
  EncodeType 11 eq {Do-CM-rm-encoding} if
  EncodeType 12 eq {Do-CM-it-encoding} if
  EncodeType 13 eq {Do-CM-tt-encoding} if
} bdef
 
%
%  Do special mappings for the various CM-font types.  Characters that
%  get "covered up" are repositioned in the range (128,128+32).
%
/Do-standard-CM-encodings {
  CMEncoding
  dup 0 /.notdef put
  dup 1 /.notdef put
  dup 2 /.notdef put
  dup 3 /.notdef put
  dup 4 /.notdef put
  dup 5 /.notdef put
  dup 6 /.notdef put
  dup 7 /.notdef put
 
  dup 8 /.notdef put
  dup 9 /.notdef put
  dup 10 /.notdef put
  dup 11 /.notdef put
  dup 12 /fi put
  dup 13 /fl put
  dup 14 /.notdef put
  dup 15 /.notdef put
 
  dup 16 /dotlessi put
  dup 17 /.notdef put
  dup 18 /grave put
  dup 19 /acute put
  dup 20 /caron put
  dup 21 /breve put
  dup 22 /macron put
  dup 23 /ring put
 
  dup 24 /cedilla put
  dup 25 /germandbls put
  dup 26 /ae put
  dup 27 /oe put
  dup 28 /oslash put
  dup 29 /AE put
  dup 30 /OE put
  dup 31 /Oslash put

  dup 127 /dieresis put

  dup 128 /space put
  dup 129 /quotedbl put
  dup 130 /sterling put
  dup 131 /dollar put
  dup 132 /less put
  dup 133 /greater put
  dup 134 /backslash put
  dup 135 /asciicircum put

  dup 136 /underscore put
  dup 137 /braceleft put
  dup 138 /bar put
  dup 139 /braceright put
  dup 140 /asciitilde put
  pop
} bdef
 
/Do-CM-rm-encoding {
  Do-standard-CM-encodings
  CMEncoding
  dup 32 /.notdef put
  dup 34 /quotedblright put
  dup 60 /exclamdown put
  dup 62 /questiondown put
  dup 92 /quotedblleft put
  dup 94 /circumflex put
  dup 95 /dotaccent put
  dup 123 /endash put
  dup 124 /emdash put
  dup 125 /hungarumlaut put
  dup 126 /tilde put
  pop
} bdef
 
/Do-CM-it-encoding {
  Do-standard-CM-encodings
  CMEncoding
  dup 32 /.notdef put
  dup 34 /quotedblright put
  dup 36 /sterling put
  dup 60 /exclamdown put
  dup 62 /questiondown put
  dup 92 /quotedblleft put
  dup 94 /circumflex put
  dup 95 /dotaccent put
  dup 123 /endash put
  dup 124 /emdash put
  dup 125 /hungarumlaut put
  dup 126 /tilde put
  pop
} bdef
 
/Do-CM-tt-encoding {
  Do-standard-CM-encodings
  CMEncoding
  dup 12 /.notdef put
  dup 13 /quotesingle put
  dup 14 /exclamdown put
  dup 15 /questiondown put
  dup 94 /circumflex put
  dup 126 /tilde put
  pop
} bdef

%
% Routines to handle packing/unpacking numbers.
%
%  <target> <pos> <num> PackHW --> <new target>
%
/PackHW {
  /num xdef
  /pos xdef
  /target xdef
  num 16#0000FFFF and 1 pos sub 16 mul bitshift
    target or
} bdef

%
%  <target> <pos> <num> PackByte --> <new target>
%
/PackByte {
  /num xdef
  /pos xdef
  /target xdef
  num 16#000000FF and 3 pos sub 8 mul bitshift
    target or
} bdef

%
%  <pos> <num> UnpkHW --> <unpacked value>
%
/UnpkHW {
  /num xdef
  /pos xdef
  num 1 pos sub -16 mul bitshift 16#0000FFFF and
  dup 16#00007FFF gt {16#00010000 sub} if
} bdef

%
%  <pos> <num> UnpkByte --> <unpacked value>
%
/UnpkByte {
  /num xdef
  /pos xdef
  num 3 pos sub -8 mul bitshift 16#000000FF and
  dup 16#0000007F gt {16#00000100 sub} if
} bdef

%
%  <int-font-name> <ext-font-name> <pt-sz(pix)> <type> <loaded-fg> DefineCMFont
%
%    type 10: "as-is" PostScript font
%    type 11: CM-mapped PostScript font - roman
%    type 12: CM-mapped PostScript font - text italic 
%    type 13: CM-mapped PostScript font - typewriter type 
%
/int-dict-name {int (-dict) str-concat} bdef
/int-dict {int (-dict) str-concat cvx load} bdef

/DF {
  true  % signal that the font is already loaded
  DefineCMFont
} bdef
 
/DNF {
  false  % signal that the font is not already loaded
  DefineCMFont
} bdef

/DefineCMFont {
  /loaded xdef
  /typ xdef
  /psz xdef
  /ext xdef
  /int xdef
 
  typ 10 ne 
    { % font_type = 11, 12, 13
    loaded not
      { /fnam ext 3 str-strip def
        fnam findfont copydict /newdict xdef 
        typ DefineCMEncoding
        newdict /Encoding CMEncoding put
        ext newdict definefont pop
      } if
    int-dict-name ext findfont psz scalefont def
    currentdict int [int-dict /setfont cvx] cvx put
    }
    { % font_type = 10
    /fnam ext def
    int-dict-name fnam findfont psz scalefont def
    currentdict int [int-dict /setfont cvx] cvx put
    }
  ifelse
} bdef 
 
%
%  <int-font-name> <ext-font-name> <pt-sz(pix)> <PXL mag> <num-chars>
%      [llx lly urx ury] <newfont-fg> DefinePXLFont
%

/PXLF {
  true  % signal that the font is already loaded
  DefinePXLFont
} bdef
 
/PXLNF {
  false  % signal that the font is not already loaded
  DefinePXLFont
} bdef

/PXLBuildCharDict 17 dict def
 
/CMEncodingArray 256 array def
0 1 255 {CMEncodingArray exch dup tempstr cvs cvn put} for

/RasterConvert {RasterScaleFactor div} bdef
 
/TransformBBox {
  aload pop
 
  /BB-ury xdef
  /BB-urx xdef
  /BB-lly xdef
  /BB-llx xdef
 
  [BB-llx RasterConvert BB-lly RasterConvert 
   BB-urx RasterConvert BB-ury RasterConvert]
} bdef

/DefinePXLFont {
  /newfont xdef
  /bb xdef
  /num xdef
  /psz xdef
  /dsz xdef
  /pxlmag xdef
  /ext xdef
  /int xdef
 
  /fnam ext (-) str-concat pxlmag tempstr cvs str-concat def
 
  newfont not {
    int-dict-name 13 dict def
   
    int-dict begin
      /FontType 3 def
      /FontMatrix [1 dsz div 0 0 1 dsz div 0 0] def
      /FontBBox bb TransformBBox def
      /Encoding CMEncodingArray def
      /CharDict 1 dict def
      CharDict begin
        /Char-Info num array def
        end
  
      /BuildChar
        { 
          PXLBuildCharDict begin
            /char xdef
            /fontdict xdef
   
            fontdict /CharDict get /Char-Info get char get aload pop
   
            /rasters xdef
            /PackedWord1 xdef
   
            0 PackedWord1 UnpkHW 16#7FFF ne
              { /PackedWord2 xdef
                /wx 0 PackedWord1 UnpkHW def
                /rows 2 PackedWord1 UnpkByte def
                /cols 3 PackedWord1 UnpkByte def
                /llx 0 PackedWord2 UnpkByte def
                /lly 1 PackedWord2 UnpkByte def
                /urx 2 PackedWord2 UnpkByte def
                /ury 3 PackedWord2 UnpkByte def }
              { /PackedWord2 xdef
                /PackedWord3 xdef
                /PackedWord4 xdef
                /wx 1 PackedWord1 UnpkHW def
                /rows 0 PackedWord2 UnpkHW def
                /cols 1 PackedWord2 UnpkHW def
                /llx 0 PackedWord3 UnpkHW def
                /lly 1 PackedWord3 UnpkHW def
                /urx 0 PackedWord4 UnpkHW def
                /ury 1 PackedWord4 UnpkHW def }
               ifelse

            rows 0 lt
              { /rows rows neg def
                /runlength 1 def }
              { /runlength 0 def }
             ifelse
 
            wx 0 
            llx RasterConvert lly RasterConvert 
            urx RasterConvert ury RasterConvert setcachedevice
            rows 0 ne
              {
              gsave
                cols rows true 
                RasterScaleFactor 0 0 RasterScaleFactor neg llx neg ury 
                  tempmatrix astore
                {GenerateRasters} imagemask
              grestore
              } if
            end
        } def
      end
   
      fnam int-dict definefont pop 
    } if 
 
  int-dict-name fnam findfont psz scalefont def
  currentdict int [int-dict /setfont cvx] cvx put
} bdef 
 
%
%  <int-font-name> <code> <wx> <llx> <lly> <urx> <ury> <rows> <cols> <runlength> <rasters> PXLC
%
/PXLC {
 
  /rasters xdef
  /runlength xdef
  /cols xdef
  /rows xdef
  /ury xdef
  /urx xdef
  /lly xdef
  /llx xdef
  /wx xdef
  /code xdef
  /int xdef
 
  % See if the long or short format is required
  true cols CKSZ rows CKSZ ury CKSZ urx CKSZ lly CKSZ llx CKSZ 
    TackRunLengthToRows
    { int-dict /CharDict get /Char-Info get code 
        [0 0 llx PackByte 1 lly PackByte 2 urx PackByte 3 ury PackByte
         0 0 wx PackHW 2 rows PackByte 3 cols PackByte
         rasters] put}
    { int-dict /CharDict get /Char-Info get code 
        [0 0 urx PackHW 1 ury PackHW
         0 0 llx PackHW 1 lly PackHW
         0 0 rows PackHW 1 cols PackHW
         0 0 16#7FFF PackHW 1 wx PackHW
         rasters] put} 
    ifelse
} bdef
 
/CKSZ {abs 127 le and} bdef
/TackRunLengthToRows {runlength 0 ne {/rows rows neg def} if} bdef
 
%
%  <wx> <dsz> <psz> <llx> <lly> <urx> <ury> <rows> <cols> <runlength> <rasters> PLOTC
%
/PLOTC {
  /rasters xdef
  /runlength xdef
  /cols xdef
  /rows xdef
  /ury xdef
  /urx xdef
  /lly xdef
  /llx xdef
  /psz xdef
  /dsz xdef
  /wx xdef
 
  % "Plot" a character's raster pattern.
  rows 0 ne
    {
    gsave
      currentpoint translate
      psz dsz div dup scale
      cols rows true 
      RasterScaleFactor 0 0 RasterScaleFactor neg llx neg ury 
        tempmatrix astore
      {GenerateRasters} imagemask
    grestore
    } if
  wx x
} bdef
 
% Routine to generate rasters for "imagemask".
/GenerateRasters {
  rasters
  runlength 1 eq {RunLengthToRasters} if
} bdef
 
% Routine to convert from runlength encoding back to rasters.
/RunLengthToRasters {
  % ...not done yet...
} bdef
 
%
%  These procedures handle bitmap processing.
%
%  <bitmap columns> <bitmap rows> <bitmap pix/inch> <magnification> BMbeg
%
/BMbeg {
  /BMmagnification xdef
  /BMresolution xdef
  /BMrows xdef
  /BMcols xdef

  /BMcurrentrow 0 def
  gsave
    0.0 setgray
    Resolution BMresolution div dup scale
    currentpoint translate
    BMmagnification 1000.0 div dup scale
    0.0 BMrows moveto
    BMrows dup scale
    currentpoint translate
    /BMCheckpoint save def
  } bdef

/BMend {
  BMCheckpoint restore
  grestore
  } bdef

%
%  <hex raster bitmap> <rows> BMswath 
%
/BMswath {
  /rows xdef
  /rasters xdef

  BMcols rows true
  [BMrows 0 0 BMrows neg 0 BMcurrentrow neg]
  {rasters}
  imagemask

  /BMcurrentrow BMcurrentrow rows add def
  BMcurrentrow % save this on the stack around a restore...
  BMCheckpoint restore
  /BMcurrentrow xdef
  /BMCheckpoint save def
  } bdef

%
%  Procedures for implementing the "rotate <theta>" special:
%  <theta> ROTB -
%        - ROTE -

/ROTB {
  XP
  gsave
  Xpos Ypos translate
  rotate % using <theta> from the stack
  Xpos neg Ypos neg translate
  RP
  } bdef

/ROTE {XP grestore RP} bdef

%
%  Procedures for implementing the "epsfile <filename> [<mag>]" special:
%  <llx> <lly> <mag> EPSB -
%  - EPSE -

/EPSB {
  0 SPB
  save
  4 1 roll % push the savelevel below the parameters
  /showpage {} def
  Xpos Ypos translate
  1000 div dup scale % using <mag> from the stack
  neg exch neg exch translate % using <llx> <lly> from the stack
  } bdef

/EPSE {restore 0 SPE} bdef

%
%  Procedure for implementing revision bars:
%  <bary1> <bary2> <barx> <barw> REVB -
%  The bar is a line of width barw drawn from (barx,bary1) to (barx,bary2).

/REVB {
  /barw xdef
  /barx xdef
  /bary2 xdef
  /bary1 xdef
  gsave
    barw setlinewidth
    barx bary1 Yadjust moveto
    barx bary2 Yadjust lineto
    stroke
  grestore
  } bdef

%
%  A small dictionary to facilitate The Publisher's implementation
%  of gray table cells.

/grpm 40 dict def

%
%  Procedures for implementing the "paper <source>" option:
%  <name> <eop> SPS          -
%         <eop> paper-manual -
%  etc.  The boolean <eop> is passed so that a paper source procedure
%  knows if it is being called at the beginning (false) or end
%  (true) of a page.

/SPS {
  /eop xdef
  /name xdef
  name where {pop eop name cvx exec} if
  } bdef

/paper-manual {
    {statusdict /manualfeed known
      {statusdict /manualfeed true put}
    if}
  if
  } bdef

/paper-automatic {
    {statusdict /manualfeed known
      {statusdict /manualfeed false put}
    if}
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16 x(is)S 16 x(not)S 15 x(a)S 16 x(safe)S 15 x(number)S 15 x(of)S 
16 x F48(FLOAT_TYPE)S 13 x F6(\(like)S
XP /F21 25 24 2 0 23 18 18 24 0
<0FFFF8 1FFFF8 3FFFF0 608400 408400 808C00 010C00 010C00 030C00
 030C00 020C00 061C00 061C00 0E1C00 0C1C00 1C0E00 1C0E00 180C00>
PXLC RP
1490 491 XY F21(\031)S 1 x F6(,)S 15 x(or)S 15 x F48(CYCLE)S -1 x 
F21(=)S F48(4.0)S 15 x F6(for)S 15 x(certain)S 15 x(other)S 15 x
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(next)S 14 x(beyond)S 14 x(the)S 14 x(range)S 14 x(limit)S 14 x(in)S
14 x(the)S 14 x(direction)S 14 x(away)S 14 x(from)S 14 x(the)S 14 x
(interior)S 14 x(of)S 14 x(the)S 14 x(range;)S 14 x(this)S 14 x(is)S
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17 x(can)S 18 x(be)S 17 x(expected)S 17 x(from)S 17 x(a)S 17 x
(portable)S 17 x(implementation.)S 26 x(Ef)S -1 x(fectively,)S 17 x
(therefore,)S 17 x(range)S 17 x(de\014nitions)S 18 x(have)S 17 x
(the)S 50 y 300 X(added)S 15 x(ef)S -1 x(fect)S 15 x(of)S 15 x
(imposing)S 16 x(accuracy)S 15 x(requirements)S 15 x(on)S 15 x
(implementations)S 15 x(above)S 15 x(and)S 15 x(beyond)S 16 x(those)S
15 x(presented)S 16 x(under)S 15 x(the)S 50 y 300 X(heading)S 19 x 
F7(Accuracy)S 19 x F6(in)S 19 x(the)S 19 x(speci\014cations)S 19 x
(\(cf.)S 14 x(Section)S 19 x(10\).)S 820 Y 383 X(The)S 12 x(phrase)S
13 x(\252mathematically)S 12 x(unbounded\272)S 13 x(in)S 12 x(a)S 
12 x(range)S 13 x(de\014nition)S 12 x(indicates)S 12 x(that)S 13 x
(the)S 13 x(range)S 12 x(of)S 12 x(values)S 13 x(of)S 13 x(the)S 
13 x(function)S 50 y 300 X(is)S 14 x(not)S 14 x(bounded)S 13 x(by)S
14 x(its)S 14 x(mathematical)S 14 x(de\014nition.)S 19 x(It)S 14 x
(also)S 14 x(implies)S 14 x(that)S 14 x(the)S 14 x(function)S 14 x
(is)S 14 x(not)S 14 x(mathematically)S 14 x(multivalued.)S 987 Y 
300 X F2(Note:)S 1080 Y -33 x F6(Unbounded)S 11 x(portions)S 11 x
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11 x F48(SQRT)S F6(,)S 10 x F48(LOG)S F6(,)S 11 x F48(ARCSINH)S F6
(,)S 10 x(and)S 12 x F48(ARCCOSH)S F6(,)S 10 x(which)S 11 x(are)S 
11 x(\252contraction\272)S 49 y 300 X(functions)S 19 x(with)S 19 x
(unbounded)S 20 x(or)S 19 x(semi-unbounded)S 19 x(mathematical)S 
19 x(ranges,)S 20 x(are)S 19 x(unreachable)S 19 x(because)S 20 x
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19 x(given)S 50 y 300 X(in)S 16 x(their)S 16 x(range)S 16 x
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50 y 300 X(their)S 14 x(speci\014cations.)S 19 x(Because)S 14 x(of)S
14 x(permitted)S 14 x(variations)S 14 x(in)S 14 x(implementations,)S
14 x(these)S 14 x(reachable)S 14 x(ranges)S 14 x(can)S 14 x(only)S
14 x(be)S 14 x(stated)S 50 y 300 X(approximately.)S 25 x(Also,)S 
17 x(range)S 17 x(constraints)S 17 x(in)S 17 x(the)S 17 x(user's)S
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(function's)S 19 x(range)S 19 x(unreachable.)S 1557 Y 300 X F50(10.)S
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(computers)S 15 x(with)S 15 x(only)S 15 x(\014nite)S 15 x
(precision,)S 15 x(the)S 15 x(functions)S 16 x(provided)S 15 x(in)S
15 x(this)S 50 y 300 X(generic)S 15 x(package)S 15 x(can,)S 16 x(at)S
15 x(best,)S 15 x(only)S 15 x(approximate)S 15 x(the)S 16 x
(corresponding)S 15 x(mathematically)S 15 x(de\014ned)S 15 x
(functions.)S 1789 Y 383 X(The)S 15 x(accuracy)S 16 x(requirements)S
16 x(contained)S 15 x(in)S 15 x(this)S 15 x(standard)S 15 x
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(implementations)S 15 x(are)S 15 x(allowed)S 16 x(in)S 49 y 300 X
(approximating)S 17 x(the)S 17 x(intended)S 16 x(precise)S 16 x
(mathematical)S 16 x(result)S 17 x(with)S 17 x(\015oating-point)S 
16 x(computations.)S 24 x(Accuracy)S 16 x(requirements)S 50 y 300 X
(of)S 18 x(two)S 18 x(kinds)S 18 x(are)S 18 x(stated)S 18 x(under)S
17 x(the)S 18 x(heading)S 17 x F7(Accuracy)S 18 x F6(in)S 18 x(the)S
18 x(speci\014cations.)S 27 x(Additionally,)S 19 x(range)S 18 x(de)S
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(heading)S 16 x F7(Range)S 16 x F6(impose)S 16 x(requirements)S 15 x
(that)S 16 x(constrain)S 16 x(the)S 16 x(values)S 16 x
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16 x(range)S 50 y 300 X(de\014nitions)S 18 x(are)S 17 x(another)S 
17 x(source)S 18 x(of)S 18 x(accuracy)S 18 x(requirements)S 17 x
(\(the)S 17 x(precise)S 17 x(meaning)S 18 x(of)S 18 x(a)S 18 x
(range)S 18 x(limit)S 18 x(that)S 17 x(is)S 18 x(not)S 17 x(a)S 17 x
(safe)S 50 y 300 X(number)S 17 x(of)S 17 x F48(FLOAT_TYPE)S F6(,)S
16 x(as)S 17 x(an)S 17 x(accuracy)S 18 x(requirement,)S 17 x(is)S 
17 x(discussed)S 17 x(above)S 17 x(in)S 17 x(Section)S 18 x(9\).)S
25 x(Every)S 17 x(result)S 17 x(yielded)S 17 x(by)S 50 y 300 X(a)S
18 x(function)S 17 x(is)S 17 x(subject)S 17 x(to)S 18 x(all)S 17 x
(of)S 17 x(the)S 17 x(function's)S 18 x(applicable)S 17 x(accuracy)S
17 x(requirements,)S 18 x(except)S 18 x(in)S 17 x(the)S 17 x(one)S
17 x(case)S 17 x(described)S 49 y 300 X(in)S 17 x(Section)S 16 x
(12,)S 17 x(below.)S 24 x(In)S 16 x(that)S 16 x(case,)S 17 x(the)S
17 x(result)S 16 x(will)S 17 x(satisfy)S 17 x(a)S 17 x(small)S 16 x
(absolute)S 17 x(error)S 17 x(requirement)S 17 x(in)S 17 x(lieu)S 
17 x(of)S 16 x(the)S 17 x(other)S 50 y 300 X(accuracy)S 20 x
(requirements)S 20 x(de\014ned)S 20 x(for)S 20 x(the)S 19 x
(function.)S 2267 Y 383 X(The)S 16 x(\014rst)S 16 x(kind)S 16 x(of)S
16 x(accuracy)S 16 x(requirement)S 16 x(used)S 15 x(under)S 16 x
(the)S 16 x(heading)S 16 x F7(Accuracy)S 16 x F6(in)S 16 x(the)S 
16 x(speci\014cations)S 16 x(is)S 16 x(a)S 16 x(bound)S 15 x(on)S 
50 y 300 X(the)S 19 x(relative)S 19 x(error)S 19 x(in)S 18 x(the)S
19 x(computed)S 18 x(value)S 19 x(of)S 19 x(the)S 19 x(function,)S
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(provided)S 18 x(by)S 19 x(the)S 19 x(rules)S 19 x(in)S 50 y 300 X
(Sections)S 12 x(11)S 12 x(and)S 12 x(12\))S 12 x(for)S 12 x(all)S
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17 x(is)S 17 x(nonzero.)S 25 x(For)S 17 x(a)S 18 x(given)S 17 x
(function)S
XP /F21 102 20 2 -8 23 29 37 24 0
<0001E0 000630 000E78 000CF0 001C60 001C00 001C00 001C00 003C00
 003800 003800 03FFC0 003800 003800 007000 007000 007000 007000
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 01C000 01C000 018000 038000 038000 630000 F30000 F60000 E40000
 780000>
PXLC RP
998 2417 XY F21(f)S 4 x F6(,)S 19 x(the)S 17 x(relative)S 17 x
(error)S
XP /F21 114 19 2 0 18 18 18 16 0
<383C 4E42 4687 470F 8E1E 8E0C 0E00 0E00 1C00 1C00 1C00 1C00 3800
 3800 3800 3800 7000 3000>
PXLC RP
1358 2417 XY F21(r)S
XP /F21 101 19 2 0 16 18 18 16 0
<01F0 0708 0C08 1804 3808 3008 7030 7FC0 E000 E000 E000 E000 E000
 E004 6008 6010 3060 0F80>
PXLC RP
1378 2417 XY F21(e)S
XP /F20 /cmr10 300 41.5 41.5 128 [-3 -11 41 31] PXLNF RP
XP /F20 40 16 3 -11 13 31 42 16 0
<0040 0080 0100 0200 0600 0C00 0C00 1800 1800 3000 3000 7000 6000
 6000 6000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000 E000
 E000 6000 6000 6000 7000 3000 3000 1800 1800 0C00 0C00 0600 0200
 0100 0080 0040>
PXLC RP
1397 2417 XY F20(\()S
XP /F21 88 34 1 0 34 28 28 40 0
<01FFC0FF80 001E003C00 001E003000 000E002000 000F004000 0007008000
 0007010000 0007820000 0003840000 0003C80000 0001D00000 0001F00000
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 0008380000 00103C0000 00201C0000 00401E0000 00800E0000 01800E0000
 03000F0000 0600070000 1E000F8000 FF803FF000>
PXLC RP
1414 2417 XY F21(X)S
XP /F20 41 16 2 -11 12 31 42 16 0
<8000 4000 2000 1000 1800 0C00 0C00 0600 0600 0300 0300 0380 0180
 0180 0180 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0 01C0
 01C0 0180 0180 0180 0380 0300 0300 0600 0600 0C00 0C00 1800 1000
 2000 4000 8000>
PXLC RP
1451 2417 XY F20(\))S 17 x F6(in)S 17 x(a)S 17 x(computed)S 17 x
(result)S
XP /F21 70 27 2 0 30 28 28 32 0
<01FFFFF0 003C00F0 00380030 00380020 00380020 00380020 00700020
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 01C04000 01C04000 01C04000 01C04000 03808000 03800000 03800000
 03800000 07000000 07000000 07000000 07000000 0F000000 FFF00000>
PXLC RP
1860 2417 XY F21(F)S 6 x F20(\()S -1 x F21(X)S 3 x F20(\))S 18 x F6
(at)S 17 x(the)S 17 x(ar)S -1 x(gument)S 49 y 300 X F21(X)S 25 x F6
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2588 Y 1050 X F21(r)S 1 x(e)S F20(\()S F21(X)S 3 x F20(\))S
XP /F20 61 32 2 4 29 16 12 32 0
<7FFFFFC0 FFFFFFE0 00000000 00000000 00000000 00000000 00000000
 00000000 00000000 00000000 FFFFFFE0 7FFFFFC0>
PXLC RP
1171 2588 XY F20(=)S
XP /F23 /cmex10 300 41.5 41.5 128 [-1 -123 60 32] PXLNF RP
XP /F23 12 14 6 -26 8 1 27 8 0
<C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0
 C0 C0 C0 C0>
PXLC RP
1215 2528 XY F23(\014)S 25 y -14 x(\014)S 25 y -14 x(\014)S 25 y 
-14 x(\014)S -43 y 4 x F21(F)S 6 x F20(\()S F21(X)S 3 x F20(\))S
XP /F22 /cmsy10 300 41.5 41.5 128 [-1 -40 45 32] PXLNF RP
XP /F22 0 32 3 9 28 11 2 32 0
<FFFFFF80 FFFFFF80>
PXLC RP
1345 2560 XY F22(\000)S 9 x F21(f)S 4 x F20(\()S 1 x F21(X)S 3 x F20
(\))S 1233 2579 XY 248 2 R 38 y 1310 X F21(f)S 4 x F20(\()S 1 x F21
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(the)S 15 x(following)S 50 y 300 X(speci\014c)S 23 x(cases:)S 1647 Y 
383 X(\(1\))S 35 x F48(LOG\(X\))S 20 x F6(when)S 21 x F48(X)S 19 x 
F20(=)S 20 x F48(0.0)S -1 x F6(;)S 62 y 383 X(\(2\))S 35 x F48
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19 x F48(0.0)S -1 x F6(;)S 63 y 383 X(\(3\))S 35 x F48(X)S 21 x(**)S
20 x(Y)S 19 x F6(when)S 20 x F48(X)S 18 x F20(=)S 18 x F48(0.0)S 
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XP /F21 60 32 3 -2 28 22 24 32 0
<00000380 00000F00 00003C00 0000F000 0003C000 000F0000 003C0000
 00F00000 03C00000 0F000000 3C000000 F0000000 F0000000 3C000000
 0F000000 03C00000 00F00000 003C0000 000F0000 0003C000 0000F000
 00003C00 00000F00 00000380>
PXLC RP
1019 1772 XY F21(<)S 18 x F48(0.0)S F6(;)S 62 y 383 X(\(4\))S 35 x 
F48(TAN\(X,)S 20 x(CYCLE\))S 16 x F6(when)S 17 x F48(X)S 15 x F20(=)S
16 x(\()S
XP /F20 50 21 2 0 18 28 28 16 0
<07C0 1830 201C 400C 400E F00F F80F F807 F807 7007 000F 000E 000E
 001C 001C 0038 0070 0060 00C0 0180 0300 0601 0C01 1801 1002 3FFE
 7FFE FFFE>
PXLC RP
974 1834 XY F20(2)S
XP /F21 107 22 2 0 20 29 29 24 0
<0FC000 01C000 01C000 038000 038000 038000 038000 070000 070000
 070000 070000 0E0700 0E1880 0E21C0 0E23C0 1C4780 1C8300 1D0000
 1E0000 3F8000 39C000 38E000 38E000 70E100 70E100 70E100 70E200
 E06200 603C00>
PXLC RP
995 1834 XY F21(k)S
XP /F20 43 32 2 -4 29 24 28 32 0
<00060000 00060000 00060000 00060000 00060000 00060000 00060000
 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0
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 00060000 00060000 00060000 00060000 00060000 00060000 00060000>
PXLC RP
1027 1834 XY F20(+)S
XP /F20 49 21 4 0 17 28 28 16 0
<0300 0700 3F00 C700 0700 0700 0700 0700 0700 0700 0700 0700 0700
 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700 0700
 0F80 FFF8>
PXLC RP
1068 1834 XY F20(1\))S
XP /F22 1 12 4 8 8 12 4 8 0
<60 F0 F0 60>
PXLC RP
1116 1834 XY F22(\001)S 10 x F48(CYCLE)S -1 x F21(=)S F48(4.0)S -1 x 
F6(,)S 18 x(for)S 17 x(integer)S 17 x F21(k)S 1 x F6(;)S 62 y 383 X
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12 x F22(\001)S 11 x F48(CYCLE)S -1 x F21(=)S
XP /F48 50 22 2 0 19 25 25 24 0
<07E000 1FF800 7FFE00 783F00 F00F00 F00780 F00380 F00380 000380
 000380 000780 000700 000F00 001E00 003C00 007800 00F000 03E000
 078000 0F0000 1E0380 3C0380 7FFF80 FFFF80 7FFF80>
PXLC RP
1147 1958 XY F48(2.0)S -1 x F6(,)S 19 x(for)S 18 x(integer)S 18 x 
F21(k)S 1 x F6(;)S 63 y 383 X(\(7\))S 35 x F48(COTH\(X\))S 20 x F6
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(\(8\))S 35 x F48(ARCTANH\(X\))S 18 x F6(when)S 20 x F48(X)S 18 x 
F20(=)S
XP /F22 6 32 2 -1 29 27 28 32 0
<00060000 00060000 00060000 00060000 00060000 00060000 00060000
 00060000 00060000 00060000 00060000 00060000 00060000 FFFFFFE0
 FFFFFFE0 00060000 00060000 00060000 00060000 00060000 00060000
 00060000 00060000 00060000 00060000 00060000 FFFFFFE0 FFFFFFE0>
PXLC RP
905 2083 XY F22(\006)S F48(1.0)S -1 x F6(;)S 21 x(and)S 62 y 383 X
(\(9\))S 35 x F48(ARCCOTH\(X\))S 18 x F6(when)S 21 x F48(X)S 18 x 
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2339 Y 300 X F2(Notes:)S 2432 Y -53 x F6(This)S 21 x(rule)S 21 x
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20 x(result.)S 2941 Y 383 X(In)S 11 x(the)S 12 x(Ada)S 11 x
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3128 Y 1265 X(9)S
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%%PageBoundingBox: 72 39 540 718

%%PageBoundingBox: (atend)
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1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY
%%EndPageSetup
300 342 XY F50(12.)S 33 x(Under\015ow)S 449 Y 383 X F6
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21 x(minimum.)S 34 x(For)S 22 x(the)S 21 x(type)S 21 x F48
(FLOAT_TYPE)S F6(,)S 19 x(that)S 22 x(minimum)S 22 x(will)S 21 x(be)S
22 x(at)S 21 x(most)S 49 y 300 X F48(FLOAT_TYPE\015SAFE_SMALL)S F6
(.)S 18 x(For)S 22 x(the)S 22 x(functions)S 22 x(de\014ned)S 22 x
(by)S 22 x(this)S 21 x(standard,)S 22 x(whenever)S 21 x(the)S 22 x
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15 x(permit)S 15 x(a)S 15 x(result)S 16 x(whose)S 15 x(absolute)S 
15 x(value)S 16 x(is)S 16 x(less)S 16 x(than)S 16 x F48(FLOAT_TYPE)S
(\015SAFE_SMALL)S 11 x F6(and)S 15 x(a)S 15 x(prescribed)S 16 x
(result)S 50 y 300 X(is)S 20 x(not)S 20 x(stipulated,)S 21 x(the)S
20 x(implementation)S 20 x(may)S 726 Y 383 X(\(1\))S 35 x(yield)S 
17 x(any)S 16 x(result)S 16 x(permitted)S 16 x(by)S 17 x(the)S 17 x
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383 X(\(2\))S 35 x(yield)S 28 x(any)S 28 x(nonzero)S 28 x(result)S
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300 X F2(Notes:)S 1115 Y -53 x F6(Whenever)S 16 x(part)S 17 x(\(2\))S
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50 y 300 X(small)S 16 x(in)S 16 x(relative)S 16 x(terms,)S 17 x(is)S
17 x(small)S 16 x(in)S 16 x(absolute)S 17 x(terms.)S 23 x(The)S 16 x
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50 y 300 X(to)S 16 x F48(FLOAT_TYPE\015SAFE_SMALL)S -4 x F21(=)S F20
(\()S F48(1.0)S 8 x F22(\000)S
XP /F21 109 36 2 0 34 18 18 32 0
<381F81F0 4E20C618 4640E81C 4680F01C 8F00F01C 8E00E01C 0E00E01C
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 380380E1 380380E2 70070064 30030038>
PXLC RP
959 1265 XY F21(mr)S 1 x(e)S F20(\))S 1 x F6(,)S 16 x(where)S 15 x 
F21(mr)S 1 x(e)S 17 x F6(is)S 16 x(the)S 16 x(maximum)S 15 x
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19 x(result)S 19 x(violating)S 19 x(the)S 19 x(maximum)S 19 x
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15 x(exceeds)S 15 x F48(FLOAT_TYPE\015SAFE_SMALL)S 10 x F6(in)S 15 x
(absolute)S 50 y 300 X(value.)S 25 x(Such)S 17 x(ar)S -1 x(guments)S
18 x(must)S 17 x(necessarily)S 17 x(be)S 17 x(very)S 17 x(close)S 
17 x(to)S 17 x(an)S 17 x(ar)S -1 x(gument)S 17 x(for)S 17 x(which)S
17 x(the)S 17 x(mathematical)S 17 x(result)S 17 x(is)S 17 x(less)S
50 y 300 X(than)S 14 x F48(FLOAT_TYPE\015SAFE_SMALL)S 9 x F6(in)S 
13 x(absolute)S 13 x(value.)S 19 x(In)S 14 x(general,)S 14 x(this)S
14 x(is)S 14 x(the)S 13 x(best)S 14 x(that)S 14 x(can)S 14 x(be)S 
14 x(expected)S 14 x(from)S 14 x(a)S 14 x(portable)S 49 y 300 X
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(amount)S 20 x(of)S 20 x(ef)S -1 x(fort.)S 1679 Y 383 X(The)S 18 x
(rule)S 18 x(is)S 18 x(motivated)S 18 x(by)S 18 x(the)S 18 x
(behavior)S 19 x(prescribed)S 18 x(by)S 18 x(the)S 18 x(Ada)S 18 x
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(operations.)S 50 y 300 X(That)S 13 x(is,)S 13 x(when)S 13 x(the)S
13 x(set)S 13 x(of)S 13 x(possible)S 13 x(results)S 13 x(of)S 13 x
(a)S 13 x(prede\014ned)S 13 x(operation)S 13 x(includes)S 13 x(a)S
13 x(nonzero)S 13 x(number)S 13 x(whose)S 13 x(absolute)S 13 x
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(implementation-de)S
(\014ned)S 19 x(minimum,)S 20 x(the)S 19 x(implementation)S 19 x(is)S
19 x(allowed)S 19 x(to)S 19 x(yield)S 19 x(zero)S 19 x(or)S 18 x
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15 x(correct)S 15 x(sign)S 15 x(and)S 15 x(an)S 15 x(absolute)S 15 x
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15 x(that)S 15 x(minimum.)S 21 x(An)S 15 x(exception)S 15 x(is)S 
15 x(never)S 50 y 300 X(raised)S 23 x(in)S 22 x(this)S 23 x(case.)S
2067 Y 300 X F50(13.)S 33 x(Speci\014cations)S 27 x(of)S 27 x(the)S
27 x(functions)S 2174 Y 383 X F6(Under)S 12 x(the)S 12 x(heading)S
12 x F7(De\014nition)S 12 x F6(in)S 12 x(each)S 12 x(of)S 12 x(the)S
12 x(following)S 12 x(subsections,)S 13 x(the)S 12 x(semantics)S 
13 x(of)S 12 x(an)S 12 x(Ada)S 12 x(call)S 12 x(to)S 12 x(the)S 13 x
(function)S 50 y 300 X(being)S 16 x(de\014ned)S 16 x(is)S 17 x
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16 x(approximation.)S 23 x(The)S 16 x(left-hand)S 16 x(side)S 16 x
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14 x(in)S 15 x(the)S 14 x(\014xed-width)S 14 x(font)S 15 x(used)S 
14 x(throughout)S 15 x(this)S 14 x(document)S 15 x(for)S 15 x
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12 x(an)S 12 x(exact)S 12 x(mathematical)S 12 x(formula;)S 12 x(as)S
12 x(such,)S 13 x(it)S 12 x(and)S 11 x(similar)S 12 x(mathematical)S
12 x(formulas)S 12 x(throughout)S 50 y 300 X(this)S 17 x(document)S
17 x(employ)S 17 x(standard)S 18 x(mathematical)S 17 x(symbols,)S 
18 x(notation,)S 18 x(and)S 17 x(fonts)S 17 x(\(except)S 18 x(for)S
17 x(variable)S 17 x(names)S 17 x(and)S 17 x(some)S 50 y 300 X(real)S
15 x(literals,)S 15 x(which)S 15 x(are)S 15 x(set)S 15 x(in)S 15 x
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15 x(function)S 15 x(call)S 14 x(on)S 50 y 300 X(the)S 15 x
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(under)S 19 x(the)S 19 x(heading)S 19 x F7(Accuracy)S F6(,)S 19 x
(as)S 18 x(discussed)S 19 x(in)S 19 x(Section)S 19 x(10.)S 2683 Y 
300 X F50(13.1.)S 33 x(SQRT)S 28 x(\320)S 28 x(Square)S 28 x(Root)S
2839 Y 383 X F2(Declaration:)S 2994 Y 466 X F48(function)S 20 x
(SQRT)S 21 x(\(X)S 22 x(:)S 21 x(FLOAT_TYPE\))S 20 x(return)S 20 x
(FLOAT_TYPE;)S 3128 Y 1254 X F6(10)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 540 717

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(De\014nition:)S 459 Y 466 X F48(SQRT\(X\))S
XP /F22 25 32 2 1 29 20 19 32 0
<07C00020 1FE00020 3FF80020 783C0060 E01F00E0 C00783C0 8003FF80
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 E01F00E0 C00783C0 8003FF80 8000FF00 80007C00>
PXLC RP
639 459 XY F22(\031)S
XP /F22 112 35 2 -40 36 2 42 40 0
<00000000C0 00000001C0 0000000180 0000000380 0000000300 0000000700
 0000000600 0000000E00 0000000C00 0000001C00 0000001800 0000003800
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PXLC RP
692 425 XY F22(p)S 726 425 XY 22 2 R 34 y -22 x F48(X)S 607 Y 383 X 
F2(Usage:)S
XP /F48 90 22 2 0 19 25 25 24 0
<7FFF80 FFFF80 FFFF80 E00700 E00F00 E01E00 E01C00 003C00 007800
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 1C0380 3C0380 780380 700380 FFFF80 FFFF80 FFFF80>
PXLC RP
466 755 XY F48(Z)S 22 x(:=)S 21 x(SQRT\(X\);)S 902 Y 383 X F2
(Domain:)S 1020 Y 466 X F48(X)S
XP /F22 21 32 3 -8 28 26 34 32 0
<C0000000 F0000000 3C000000 0F000000 03C00000 00F00000 003C0000
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 0F000000 3C000000 F0000000 C0000000 00000000 00000000 00000000
 00000000 00000000 00000000 00000000 FFFFFF80 FFFFFF80>
PXLC RP
509 1020 XY F22(\025)S 21 x F48(0.0)S 1168 Y 383 X F2(Range:)S 1285 Y 
-69 x F48(SQRT\(X\))S 19 x F22(\025)S 21 x F48(0.0)S 1433 Y 383 X 
F2(Accuracy:)S 1550 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x(relative)S
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(FLOAT_TYPE)S
(\015BASE\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x F48(SQRT\(0.0\))S
18 x F20(=)S 20 x F48(0.0)S 1760 Y 383 X F2(Notes:)S 1877 Y -53 x 
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16 x(approximately)S 16 x(given)S 16 x(by)S 1979 Y 925 X F48
(SQRT\(X\))S
XP /F22 20 32 3 -8 28 26 34 32 0
<00000180 00000780 00001E00 00007800 0001E000 00078000 001E0000
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 00000000 00000000 00000000 00000000 FFFFFF80 FFFFFF80>
PXLC RP
1089 1979 XY F22(\024)S -33 y 12 x(p)S 1167 1946 XY 458 2 R 33 y 
1167 X F48(FLOAT_TYPE\015SAFE_LARGE)S 2134 Y 300 X F50(13.2.)S 33 x
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(Declaration:)S 2430 Y 466 X F48(function)S 20 x(LOG)S 21 x(\(X)S 
22 x(:)S 21 x(FLOAT_TYPE\))S 20 x(return)S 20 x(FLOAT_TYPE;)S 2578 Y 
383 X F2(De\014nition:)S 2695 Y 466 X F48(LOG\(X\))S 19 x F22(\031)S
XP /F20 108 12 1 0 10 29 29 16 0
<FC00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00
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PXLC RP
670 2695 XY F20(l)S
XP /F20 111 21 1 0 19 18 18 24 0
<03F000 0E1C00 180600 300300 700380 600180 E001C0 E001C0 E001C0
 E001C0 E001C0 E001C0 600180 700380 300300 180600 0E1C00 03F000>
PXLC RP
681 2695 XY F20(o)S
XP /F20 103 21 1 -9 19 19 28 24 0
<000380 03C4C0 0C38C0 1C3880 181800 381C00 381C00 381C00 381C00
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 07F800>
PXLC RP
702 2695 XY F20(g)S
XP /F55 /cmmi7 300 29.1 29.1 128 [-1 -7 32 22] PXLNF RP
XP /F55 101 16 2 0 13 13 13 16 0
<0780 0C40 1020 3040 6040 7F80 6000 4000 C000 4020 6040 2180 1E00>
PXLC RP
723 2705 XY F55(e)S -10 y 9 x F48(X)S 2843 Y 383 X F2(Usage:)S 2991 Y 
-61 x F48(Z)S 22 x(:=)S 21 x(LOG\(X\);)S 64 x(--)S 21 x(natura)S
XP /F48 108 22 2 0 19 25 25 24 0
<FFC000 FFC000 FFC000 01C000 01C000 01C000 01C000 01C000 01C000
 01C000 01C000 01C000 01C000 01C000 01C000 01C000 01C000 01C000
 01C000 01C000 01C000 01C000 FFFF80 FFFF80 FFFF80>
PXLC RP
989 2991 XY F48(l)S 22 x(logarithm)S 3128 Y 23 x F6(11)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 428 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Domain:)S 455 Y 466 X F48(X)S 21 x F22(\025)S 21 x F48
(0.0)S 599 Y 383 X F2(Range:)S 713 Y -69 x F6(Mathematically)S 20 x
(unbounded)S 856 Y 383 X F2(Accuracy:)S 970 Y 466 X F6(\(a\))S 37 x
(Maximum)S 17 x(relative)S 17 x(error)S 17 x F20(=)S 15 x F48(4.0)S
10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 466 X 
F6(\(b\))S 35 x F48(LOG\(1.0\))S 18 x F20(=)S 20 x F48(0.0)S 1176 Y 
383 X F2(Notes:)S 1289 Y -53 x F6(\(a\))S 37 x(The)S 17 x(reachable)S
18 x(range)S 18 x(of)S 18 x F48(LOG)S 17 x F6(is)S 17 x
(approximately)S 18 x(given)S 17 x(by)S 1390 Y 660 X F20(log)S 10 y 
-1 x F55(e)S -10 y 9 x F48(FLOAT_TYPE\015SAFE_SMALL)S 7 x F22(\024)S
12 x F48(LOG\(X\))S 10 x F22(\024)S 12 x F20(log)S 10 y F55(e)S -10 y 
9 x F48(FLOAT_TYPE\015SAFE_LARGE)S 1503 Y 466 X F6(\(b\))S 35 x
(When)S 20 x F48(X)S 18 x F20(=)S 18 x F48(0.0)S F6(,)S 21 x(see)S
20 x(Section)S 20 x(11.)S 1654 Y 300 X F50(13.3.)S 33 x(LOG)S 25 x
(\320)S 24 x(Logarithm)S 25 x(to)S 25 x(an)S 24 x(Arbitrary)S 25 x
(Base)S 1797 Y 383 X F2(Declaration:)S 1941 Y 466 X F48(function)S
20 x(LOG)S 21 x(\(X,)S 22 x(BASE)S 20 x(:)S 22 x(FLOAT_TYPE\))S 20 x
(return)S 20 x(FLOAT_TYPE;)S 2084 Y 383 X F2(De\014nition:)S 2198 Y 
466 X F48(LOG\(X,BASE\))S 17 x F22(\031)S 20 x F20(log)S
XP /F65 /cmtt10 300 41.5 29.1 128 [0 -10 22 29] PXLF RP
XP /F65 66 21 1 0 19 25 25 24 0
<7FF800 FFFE00 7FFF00 1C0F00 1C0780 1C0380 1C0380 1C0380 1C0780
 1C0700 1FFF00 1FFE00 1FFE00 1C1F00 1C0380 1C03C0 1C01C0 1C01C0
 1C01C0 1C01C0 1C03C0 1C0780 7FFF80 FFFF00 7FFC00>
PXLC RP
830 2208 XY F65(B)S
XP /F65 65 21 1 0 20 25 25 24 0
<00E000 01F000 01F000 01B000 01B000 03B800 03B800 03B800 031800
 071C00 071C00 071C00 071C00 071C00 0E0E00 0E0E00 0FFE00 0FFE00
 1FFF00 1C0700 1C0700 1C0700 7F1FC0 FF1FE0 7F1FC0>
PXLC RP
846 2208 XY F65(A)S
XP /F65 83 21 2 0 19 25 25 24 0
<07E300 1FFF00 7FFF00 781F00 F00F00 E00700 E00700 E00000 F00000
 780000 7F8000 1FF000 07FC00 00FE00 000F00 000780 000380 000380
 E00380 E00380 F00780 F80F00 FFFE00 FFFC00 C7F000>
PXLC RP
861 2208 XY F65(S)S
XP /F65 69 21 1 0 20 25 25 24 0
<7FFFC0 FFFFC0 7FFFC0 1C01C0 1C01C0 1C01C0 1C01C0 1C0000 1C0000
 1C1C00 1C1C00 1FFC00 1FFC00 1FFC00 1C1C00 1C1C00 1C0000 1C00E0
 1C00E0 1C00E0 1C00E0 1C00E0 7FFFE0 FFFFE0 7FFFE0>
PXLC RP
876 2208 XY F65(E)S -10 y 9 x F48(X)S 2342 Y 383 X F2(Usage:)S 2485 Y 
-61 x F48(Z)S 22 x(:=)S 21 x(LOG\(X,)S 21 x(10.0\);)S 64 x(--)S
XP /F48 98 22 0 0 20 25 25 24 0
<7E0000 FE0000 7E0000 0E0000 0E0000 0E0000 0E0000 0E3E00 0EFF80
 0FFFC0 0F83E0 0F01E0 0E00F0 0E00F0 0E0070 0E0070 0E0070 0E0070
 0E00F0 0F00E0 0F01E0 0F83C0 0FFFC0 0EFF00 063C00>
PXLC RP
989 2485 XY F48(base)S 21 x(10)S 21 x(logarithm)S 50 y 466 X(Z)S 
22 x(:=)S 21 x(LOG\(X,)S 21 x(2.0\);)S 924 X(--)S 21 x(base)S 21 x
(2)S 22 x(logarithm)S 50 y 466 X(Z)S 22 x(:=)S 21 x(LOG\(X,)S 21 x
(BASE\);)S 64 x(--)S 21 x(base)S 21 x(BASE)S 21 x(logarithm)S 2728 Y 
383 X F2(Domain:)S 2842 Y 466 X F6(\(a\))S 37 x F48(X)S 20 x F22
(\025)S 21 x F48(0.0)S 62 y 466 X F6(\(b\))S 35 x F48(BASE)S
XP /F21 62 32 3 -2 28 22 24 32 0
<E0000000 78000000 1E000000 07800000 01E00000 00780000 001E0000
 00078000 0001E000 00007800 00001E00 00000780 00000780 00001E00
 00007800 0001E000 00078000 001E0000 00780000 01E00000 07800000
 1E000000 78000000 E0000000>
PXLC RP
656 2904 XY F21(>)S 21 x F48(0.0)S 62 y 466 X F6(\(c\))S 37 x F48
(BASE)S
XP /F22 54 0 6 -10 25 30 40 24 0
<000060 000060 0000E0 0000C0 0001C0 000180 000380 000300 000700
 000600 000600 000E00 000C00 001C00 001800 003800 003000 007000
 006000 00E000 00C000 00C000 01C000 018000 038000 030000 070000
 060000 0E0000 0C0000 0C0000 1C0000 180000 380000 300000 700000
 600000 E00000 C00000 C00000>
PXLC RP
656 2966 XY F22(6)S F20(=)S 21 x F48(1.0)S 3128 Y 1254 X F6(12)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 474 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Range:)S 453 Y -69 x F6(Mathematically)S 20 x
(unbounded)S 592 Y 383 X F2(Accuracy:)S 704 Y 466 X F6(\(a\))S 37 x
(Maximum)S 17 x(relative)S 17 x(error)S 17 x F20(=)S 15 x F48(4.0)S
10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 466 X 
F6(\(b\))S 35 x F48(LOG\(1.0,)S 19 x(BASE\))S 18 x F20(=)S 19 x F48
(0.0)S 906 Y 383 X F2(Notes:)S 1017 Y -53 x F6(\(a\))S 37 x(When)S
16 x F48(BASE)S 13 x F21(>)S 14 x F48(1.0)S F6(,)S 16 x(the)S 16 x
(reachable)S 16 x(range)S 16 x(of)S 16 x F48(LOG)S 15 x F6(is)S 16 x
(approximately)S 16 x(given)S 16 x(by)S 1109 Y 708 X F20(log)S 10 y 
-1 x F65(BASE)S -10 y 10 x F48(FLOAT_TYPE\015SAFE_SMALL)S 7 x F22
(\024)S 12 x F48(LOG\(X,)S 20 x(BASE\))S 63 y 1300 X F22(\024)S 12 x 
F20(log)S 10 y F65(BASE)S -10 y 10 x F48(FLOAT_TYPE\015SAFE_LARGE)S
1275 Y 466 X F6(\(b\))S 35 x(When)S 15 x F48(0.0)S 13 x F21(<)S 13 x 
F48(BASE)S 13 x F21(<)S 13 x F48(1.0)S -1 x F6(,)S 15 x(the)S 15 x
(reachable)S 16 x(range)S 15 x(of)S 15 x F48(LOG)S 15 x F6(is)S 16 x
(approximately)S 16 x(given)S 15 x(by)S 1367 Y 708 X F20(log)S 10 y 
-1 x F65(BASE)S -10 y 10 x F48(FLOAT_TYPE\015SAFE_LARGE)S 7 x F22
(\024)S 12 x F48(LOG\(X,)S 20 x(BASE\))S 63 y 1300 X F22(\024)S 12 x 
F20(log)S 11 y F65(BASE)S -11 y 10 x F48(FLOAT_TYPE\015SAFE_SMALL)S
1533 Y 466 X F6(\(c\))S 37 x(When)S 20 x F48(X)S 18 x F20(=)S 18 x 
F48(0.0)S F6(,)S 21 x(see)S 20 x(Section)S 20 x(11.)S 1682 Y 300 X 
F50(13.4.)S 33 x(EXP)S 26 x(\320)S 27 x(Exponential)S 27 x(Function)S
1821 Y 383 X F2(Declaration:)S 1962 Y 466 X F48(function)S 20 x(EXP)S
21 x(\(X)S 22 x(:)S 21 x(FLOAT_TYPE\))S 20 x(return)S 20 x
(FLOAT_TYPE;)S 2101 Y 383 X F2(De\014nition:)S 2213 Y 466 X F48
(EXP\(X\))S 19 x F22(\031)S 21 x F21(e)S
XP /F65 88 21 1 0 20 25 25 24 0
<7F1F80 7F3F80 7F1F80 0E1E00 0E1C00 073C00 073800 03B800 03F000
 01F000 01E000 00E000 01E000 01F000 03F000 03B800 07B800 071C00
 071C00 0E0E00 0E0E00 1C0700 7F1FC0 FF1FE0 7F1FC0>
PXLC RP
690 2198 XY F65(X)S 2352 Y 383 X F2(Usage:)S 2493 Y -61 x F48(Z)S 
22 x(:=)S 21 x(EXP\(X\);)S 64 x(--)S 21 x(e)S 22 x(raised)S 20 x(to)S
22 x(the)S 21 x(power)S 21 x(X)S 2632 Y 383 X F2(Domain:)S 2743 Y 
466 X F6(Mathematically)S 20 x(unbounded)S 2883 Y 383 X F2(Range:)S
2994 Y -69 x F48(EXP\(X\))S 19 x F22(\025)S 21 x F48(0.0)S 3128 Y 
1254 X F6(13)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 468 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Accuracy:)S 450 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x
(relative)S 17 x(error)S 17 x F20(=)S 15 x F48(4.0)S 10 x F22(\001)S
10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x 
F48(EXP\(0.0\))S 18 x F20(=)S 20 x F48(1.0)S 647 Y 383 X F2(Notes:)S
755 Y -53 x F6(The)S 18 x(usable)S 18 x(domain)S 19 x(of)S 18 x F48
(EXP)S 18 x F6(is)S 19 x(approximately)S 19 x(given)S 18 x(by)S 
840 Y 968 X F48(X)S 12 x F22(\024)S 12 x F20(log)S 10 y -1 x F55(e)S
-10 y 9 x F48(FLOAT_TYPE\015SAFE_LARGE)S 986 Y 300 X F50(13.5.)S 
33 x(\201**\201)S 26 x(\320)S 26 x(Exponentiation)S 26 x(Operator)S
1120 Y 383 X F2(Declaration:)S 1258 Y 466 X F48(function)S 20 x
("**")S 21 x(\(X,)S 21 x(Y)S 22 x(:)S 21 x(FLOAT_TYPE\))S 20 x
(return)S 21 x(FLOAT_TYPE;)S 1392 Y 383 X F2(De\014nition:)S 1500 Y 
466 X F48(X)S 21 x(**)S 20 x(Y)S 20 x F22(\031)S 21 x F48(X)S
XP /F65 89 21 1 0 20 25 25 24 0
<FE0FE0 FF1FE0 FE0FE0 1C0700 1C0700 0E0E00 0E0E00 071C00 071C00
 071C00 03B800 03B800 01F000 01F000 00E000 00E000 00E000 00E000
 00E000 00E000 00E000 00E000 03F800 07FC00 03F800>
PXLC RP
690 1485 XY F65(Y)S 1635 Y 383 X F2(Usage:)S 1772 Y -61 x F48(Z)S 
22 x(:=)S 21 x(X)S 22 x(**)S 21 x(Y;)S 65 x(--)S 21 x(X)S 22 x
(raised)S 20 x(to)S 22 x(the)S 21 x(power)S 21 x(Y)S 1907 Y 383 X 
F2(Domain:)S 2015 Y 466 X F6(\(a\))S 37 x F48(X)S 20 x F22(\025)S 
21 x F48(0.0)S 63 y 466 X F6(\(b\))S 35 x F48(Y)S 19 x F22(6)S F20
(=)S 19 x F48(0.0)S 20 x F6(when)S 20 x F48(X)S 19 x F20(=)S 19 x 
F48(0.0)S 2212 Y 383 X F2(Range:)S 2321 Y -69 x F48(X)S 21 x(**)S 
20 x(Y)S 20 x F22(\025)S 21 x F48(0.0)S 2455 Y 383 X F2(Accuracy:)S
2564 Y 466 X F6(\(a\))S 37 x(Maximum)S 19 x(relative)S 19 x(error)S
19 x(\(when)S 18 x F48(X)S 17 x F21(>)S 17 x F48(0.0)S -1 x F6(\))S
19 x F20(=)S
XP /F23 18 31 9 -97 29 2 99 24 0
<000030 000060 0000C0 000180 000300 000700 000E00 000C00 001800
 003800 003000 007000 00E000 00C000 01C000 018000 038000 038000
 030000 070000 060000 0E0000 0E0000 0C0000 1C0000 1C0000 1C0000
 180000 380000 380000 380000 380000 700000 700000 700000 700000
 700000 700000 700000 E00000 E00000 E00000 E00000 E00000 E00000
 E00000 E00000 E00000 E00000 E00000 E00000 E00000 E00000 E00000
 E00000 E00000 E00000 E00000 E00000 E00000 700000 700000 700000
 700000 700000 700000 700000 380000 380000 380000 380000 180000
 1C0000 1C0000 1C0000 0C0000 0E0000 0E0000 060000 070000 030000
 038000 038000 018000 01C000 00C000 00E000 007000 003000 003800
 001800 000C00 000E00 000700 000300 000180 0000C0 000060 000030>
PXLC RP
870 2608 XY F23(\022)S 59 y -1 x F48(4.0)S 9 x F20(+)S
XP /F22 106 12 5 -10 7 31 41 8 0
<C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0
 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0 C0>
PXLC RP
1021 2639 XY F22(j)S F48(Y)S 9 x F22(\001)S 9 x F20(log)S 10 y -1 x 
F55(e)S -10 y 9 x F48(X)S F22(j)S 1021 2657 XY 175 2 R
XP /F48 51 22 2 0 19 25 25 24 0
<07E000 1FFC00 7FFE00 781F00 780F00 780700 000700 000F00 000F00
 003E00 07FC00 07F800 07FC00 001E00 000700 000780 000380 000380
 F00380 F00780 F00700 F81F00 7FFE00 3FFC00 07E000>
PXLC RP
1065 2695 XY F48(32.0)S
XP /F23 19 31 1 -97 21 2 99 24 0
<C00000 600000 300000 180000 0C0000 0E0000 070000 030000 018000
 01C000 00C000 00E000 007000 003000 003800 001800 001C00 001C00
 000C00 000E00 000600 000700 000700 000300 000380 000380 000380
 000180 0001C0 0001C0 0001C0 0001C0 0000E0 0000E0 0000E0 0000E0
 0000E0 0000E0 0000E0 000070 000070 000070 000070 000070 000070
 000070 000070 000070 000070 000070 000070 000070 000070 000070
 000070 000070 000070 000070 000070 000070 0000E0 0000E0 0000E0
 0000E0 0000E0 0000E0 0000E0 0001C0 0001C0 0001C0 0001C0 000180
 000380 000380 000380 000300 000700 000700 000600 000E00 000C00
 001C00 001C00 001800 003800 003000 007000 00E000 00C000 01C000
 018000 030000 070000 0E0000 0C0000 180000 300000 600000 C00000>
PXLC RP
1201 2608 XY F23(\023)S 59 y 9 x F22(\001)S 9 x F48
(FLOAT_TYPE\015BASE)S
(\015EPSILON)S 2783 Y 466 X F6(\(b\))S 35 x F48(X)S 21 x(**)S 20 x
(0.0)S 17 x F20(=)S 19 x F48(1.0)S 19 x F6(when)S 20 x F48(X)S 17 x 
F21(>)S 19 x F48(0.0)S 62 y 466 X F6(\(c\))S 37 x F48(0.0)S 20 x(**)S
21 x(Y)S 17 x F20(=)S 19 x F48(0.0)S 19 x F6(when)S 20 x F48(Y)S 
17 x F21(>)S 19 x F48(0.0)S 62 y 466 X F6(\(d\))S 35 x F48(X)S 21 x
(**)S 20 x(1.0)S 19 x F20(=)S 21 x F48(X)S 62 y 466 X F6(\(e\))S 
37 x F48(1.0)S 20 x(**)S 21 x(Y)S 19 x F20(=)S 20 x F48(1.0)S 3128 Y 
1254 X F6(14)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 423 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Notes:)S 456 Y -53 x F6(\(a\))S 37 x(The)S 11 x
(usable)S 11 x(domain)S 11 x(of)S 11 x F48("**")S F6(,)S 11 x(when)S
11 x F48(X)S 12 x F21(>)S 12 x F48(0.0)S -1 x F6(,)S 12 x(is)S 11 x
(approximately)S 12 x(the)S 12 x(set)S 11 x(of)S 12 x(values)S 11 x
(for)S 11 x F48(X)S 11 x F6(and)S 11 x F48(Y)S 11 x F6(satisfying)S
557 Y 945 X F48(Y)S 9 x F22(\001)S 9 x F20(log)S 10 y -1 x F55(e)S
-10 y 9 x F48(X)S 11 x F22(\024)S 12 x F20(log)S 10 y F55(e)S -10 y 
9 x F48(FLOAT_TYPE\015SAFE_LARGE)S 657 Y 549 X F6(This)S 14 x
(imposes)S 13 x(a)S 14 x(positive)S 14 x(upper)S 14 x(bound)S 14 x
(on)S 14 x F48(Y)S 14 x F6(\(as)S 14 x(a)S 14 x(function)S 14 x(of)S
13 x F48(X)S F6(\))S 14 x(when)S 13 x F48(X)S 12 x F21(>)S 12 x F48
(1.0)S -1 x F6(,)S 14 x(and)S 14 x(a)S 14 x(negative)S 13 x(lower)S
50 y 549 X(bound)S 18 x(on)S 18 x F48(Y)S 17 x F6(\(as)S 17 x(a)S 
17 x(function)S 17 x(of)S 18 x F48(X)S F6(\))S 17 x(when)S 17 x F48
(0.0)S 16 x F21(<)S 16 x F48(X)S 15 x F21(<)S 16 x F48(1.0)S F6(.)S
63 y 466 X(\(b\))S 35 x(When)S 18 x F48(X)S 15 x F20(=)S 16 x F48
(0.0)S 17 x F6(and)S 18 x F48(Y)S 16 x F21(<)S 16 x F48(0.0)S 17 x 
F6(\(together\),)S 18 x(see)S 17 x(Section)S 18 x(11.)S 921 Y 300 X 
F50(13.6.)S 33 x(SIN)S 20 x(\320)S 19 x(T)S -3 x(rigonometric)S 19 x
(Sine)S 19 x(Function,)S 71 y 656 X(Natural)S 25 x(Cycle)S 24 x
(\(Angle)S 24 x(in)S 24 x(Radians\))S 1136 Y 383 X F2(Declaration:)S
1280 Y 466 X F48(function)S 20 x(SIN)S 21 x(\(X)S 22 x(:)S 21 x
(FLOAT_TYPE\))S 20 x(return)S 20 x(FLOAT_TYPE;)S 1425 Y 383 X F2(De)S
(\014nition:)S 1539 Y 466 X F48(SIN\(X\))S 19 x F22(\031)S
XP /F20 115 16 1 0 14 18 18 16 0
<1F90 3070 4030 C010 C010 E010 F800 7F80 3FE0 0FF0 00F8 8038 8018
 C018 C018 E010 D060 8FC0>
PXLC RP
670 1539 XY F20(s)S
XP /F20 105 12 1 0 10 29 29 16 0
<1800 3C00 3C00 1800 0000 0000 0000 0000 0000 0000 0000 FC00 1C00
 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00 1C00
 1C00 1C00 FF80>
PXLC RP
686 1539 XY F20(i)S
XP /F20 110 23 1 0 21 18 18 24 0
<FC7C00 1C8700 1D0300 1E0380 1C0380 1C0380 1C0380 1C0380 1C0380
 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 FF9FF0>
PXLC RP
698 1539 XY F20(n)S 7 x F48(X)S 1683 Y 383 X F2(Usage:)S 1827 Y -61 x 
F48(Z)S 22 x(:=)S 21 x(SIN\(X\);)S 64 x(--)S 21 x(X)S 22 x(in)S 21 x
(radians)S 1972 Y 383 X F2(Domain:)S 2086 Y 466 X F6(Mathematically)S
20 x(unbounded)S 2230 Y 383 X F2(Range:)S 2345 Y -69 x F22(j)S F48
(SIN\(X\))S -2 x F22(j)S 20 x(\024)S 21 x F48(1.0)S 2489 Y 383 X F2
(Accuracy:)S 2603 Y 466 X F6(\(a\))S 37 x(Maximum)S 12 x(relative)S
12 x(error)S 12 x F20(=)S 12 x F48(2.0)S 6 x F22(\001)S 6 x F48
(FLOAT_TYPE)S
(\015BASE\015EPSILON)S 7 x F6(when)S 12 x F22(j)S F48(X)S F22(j)S 
11 x F6(is)S 12 x(less)S 12 x(than)S 13 x(or)S 13 x(equal)S 13 x(to)S
12 x(some)S 50 y 549 X(documented)S 16 x(implementation-dependent)S
15 x(threshold,)S 16 x(which)S 16 x(must)S 15 x(not)S 16 x(be)S 15 x
(less)S 15 x(than)S 2754 Y 832 X F48(FLOAT_TYPE\015MACHINE_RADIX)S
XP /F65 70 21 1 0 20 25 25 24 0
<FFFFE0 FFFFE0 FFFFE0 1C00E0 1C00E0 1C00E0 1C00E0 1C0000 1C0000
 1C1C00 1C1C00 1FFC00 1FFC00 1FFC00 1C1C00 1C1C00 1C0000 1C0000
 1C0000 1C0000 1C0000 1C0000 FF8000 FFC000 FF8000>
PXLC RP
1355 2737 XY F65(F)S
XP /F65 76 21 1 0 20 25 25 24 0
<FFC000 FFC000 FFC000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000
 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0040
 1C00E0 1C00E0 1C00E0 1C00E0 FFFFE0 FFFFE0 FFFFE0>
PXLC RP
1370 2737 XY F65(L)S
XP /F65 79 21 2 0 19 25 25 24 0
<1FFC00 3FFE00 7FFF00 780F00 F00780 E00380 E00380 E00380 E00380
 E00380 E00380 E00380 E00380 E00380 E00380 E00380 E00380 E00380
 E00380 F00780 F00780 780F00 7FFF00 3FFE00 1FFC00>
PXLC RP
1385 2737 XY F65(OA)S
XP /F65 84 21 1 0 20 25 25 24 0
<7FFFE0 FFFFE0 FFFFE0 E0E0E0 E0E0E0 E0E0E0 E0E0E0 00E000 00E000
 00E000 00E000 00E000 00E000 00E000 00E000 00E000 00E000 00E000
 00E000 00E000 00E000 00E000 07FC00 0FFE00 07FC00>
PXLC RP
1416 2737 XY F65(T)S
XP /F65 95 21 2 -4 19 -1 3 24 0
<FFFF80 FFFF80 FFFF80>
PXLC RP
1431 2737 XY F65(_TY)S
XP /F65 80 21 1 0 19 25 25 24 0
<7FF800 FFFE00 7FFF00 1C0F80 1C0380 1C03C0 1C01C0 1C01C0 1C01C0
 1C03C0 1C0380 1C0F80 1FFF00 1FFE00 1FF800 1C0000 1C0000 1C0000
 1C0000 1C0000 1C0000 1C0000 7F0000 FF8000 7F0000>
PXLC RP
1477 2737 XY F65(PE)S
XP /F65 13 21 8 14 13 26 12 8 0
<20 70 F8 70 70 70 70 70 70 70 70 70>
PXLC RP
1507 2737 XY F65(\015)S
XP /F65 77 21 1 0 20 25 25 24 0
<FC07E0 FE0FE0 FE0FE0 3A0B80 3B1B80 3B1B80 3B1B80 3B1B80 3B1B80
 3BBB80 39B380 39B380 39B380 39B380 39F380 38E380 38E380 380380
 380380 380380 380380 380380 FE0FE0 FE0FE0 FE0FE0>
PXLC RP
1523 2737 XY F65(MA)S
XP /F65 67 21 2 0 19 25 25 24 0
<01F180 07FF80 0FFF80 1F0F80 3C0780 780780 780380 700380 F00000
 E00000 E00000 E00000 E00000 E00000 E00000 E00000 F00000 700380
 780380 780380 3C0780 1F0F00 0FFE00 07FC00 01F000>
PXLC RP
1553 2737 XY F65(C)S
XP /F65 72 21 1 0 20 25 25 24 0
<7F1FC0 FFBFE0 7F1FC0 1C0700 1C0700 1C0700 1C0700 1C0700 1C0700
 1C0700 1FFF00 1FFF00 1FFF00 1C0700 1C0700 1C0700 1C0700 1C0700
 1C0700 1C0700 1C0700 1C0700 7F1FC0 FFBFE0 7F1FC0>
PXLC RP
1568 2737 XY F65(H)S
XP /F65 73 21 3 0 18 25 25 16 0
<FFFE FFFE FFFE 0380 0380 0380 0380 0380 0380 0380 0380 0380 0380
 0380 0380 0380 0380 0380 0380 0380 0380 0380 FFFE FFFE FFFE>
PXLC RP
1584 2737 XY F65(I)S
XP /F65 78 21 1 0 20 25 25 24 0
<7E1FC0 FF3FE0 7F1FC0 1D0700 1D8700 1D8700 1D8700 1DC700 1DC700
 1CC700 1CC700 1CE700 1CE700 1CE700 1C6700 1C6700 1C7700 1C7700
 1C3700 1C3700 1C3700 1C1700 7F1F00 FF9F00 7F0F00>
PXLC RP
1599 2737 XY F65(NE_MANTISSA)S
XP /F55 61 17 2 -7 13 22 29 16 0
<0060 0060 00E0 00C0 00C0 01C0 0180 0180 0380 0300 0300 0700 0600
 0600 0E00 0C00 0C00 1C00 1800 1800 3800 3000 3000 7000 6000 6000
 E000 C000 C000>
PXLC RP
1767 2737 XY F55(=)S
XP /F65 50 21 2 0 19 25 25 24 0
<07E000 1FF800 7FFE00 783F00 F00F00 F00780 F00380 F00380 000380
 000380 000780 000700 000F00 001E00 003C00 007800 00F000 03E000
 078000 0F0000 1E0380 3C0380 7FFF80 FFFF80 7FFF80>
PXLC RP
1784 2737 XY F65(2)S 2855 Y 549 X F6(For)S 21 x(lar)S -1 x(ger)S 
21 x(values)S 21 x(of)S 21 x F22(j)S -1 x F48(X)S F22(j)S F6(,)S 
22 x(degraded)S 21 x(accuracy)S 21 x(is)S 21 x(allowed.)S 33 x(An)S
21 x(implementation)S 21 x(must)S 21 x(document)S 21 x(its)S 50 y 
549 X(behavior)S 21 x(for)S 20 x(lar)S -1 x(ge)S 20 x F22(j)S F48(X)S
F22(j)S -1 x F6(.)S 62 y 466 X(\(b\))S 35 x F48(SIN\(0.0\))S 18 x 
F20(=)S 20 x F48(0.0)S 3128 Y 1254 X F6(15)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 540 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY
%%EndPageSetup
300 343 XY F50(13.7.)S 33 x(SIN)S 20 x(\320)S 19 x(T)S -3 x
(rigonometric)S 19 x(Sine)S 19 x(Function,)S 71 y 656 X(Arbitrary)S
22 x(Cycle)S 23 x(\(Angle)S 22 x(in)S 23 x(Arbitrary)S 23 x(Units\))S
566 Y 383 X F2(Declaration:)S 717 Y 466 X F48(function)S 20 x(SIN)S
21 x(\(X,)S 22 x(CYCLE)S 20 x(:)S 22 x(FLOAT_TYPE\))S 19 x(return)S
21 x(FLOAT_TYPE;)S 869 Y 383 X F2(De\014nition:)S 989 Y 466 X F48
(SIN\(X,)S 20 x(CYCLE\))S 16 x F22(\031)S 19 x F20(sin)S 7 x(\(2)S
F21(\031)S 10 x F22(\001)S 9 x F48(X)S F21(=)S F48(CYCLE)S -2 x F20
(\))S 1140 Y 383 X F2(Usage:)S 1292 Y -61 x F48(Z)S 22 x(:=)S 21 x
(SIN\(X,)S 21 x(3)S
XP /F48 54 22 2 0 19 25 25 24 0
<00F800 03FE00 0FFF00 1F0F00 3E0F00 3C0F00 780600 700000 F00000
 F3F800 EFFE00 FFFE00 F80F00 F00780 F00780 E00380 F00380 F00380
 700380 780780 780F00 3C1F00 1FFE00 0FF800 03E000>
PXLC RP
749 1292 XY F48(60.0\);)S 64 x(--)S 22 x(X)S 21 x(in)S 22 x(degrees)S
50 y 466 X(Z)S 22 x(:=)S 21 x(SIN\(X,)S 21 x(CYCLE\);)S 63 x(--)S 
22 x(X)S 21 x(in)S 22 x(units)S 21 x(such)S 21 x(that)S 21 x(one)S
21 x(complete)S 50 y 945 X(--)S 22 x(cycle)S 21 x(of)S 21 x
(rotation)S 20 x(corresponds)S 20 x(to)S 49 y 945 X(--)S 22 x(X)S 
21 x(=)S 22 x(CYCLE)S 1593 Y 383 X F2(Domain:)S 1713 Y 466 X F6
(\(a\))S 37 x F48(X)S 20 x F6(mathematically)S 20 x(unbounded)S 62 y 
466 X(\(b\))S 35 x F48(CYCLE)S 19 x F21(>)S 21 x F48(0.0)S 1927 Y 
383 X F2(Range:)S 2047 Y -69 x F22(j)S F48(SIN\(X,)S 19 x(CYCLE\))S
-1 x F22(j)S 19 x(\024)S 20 x F48(1.0)S 2199 Y 383 X F2(Accuracy:)S
2318 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x(relative)S 17 x(error)S
17 x F20(=)S 15 x F48(2.0)S 10 x F22(\001)S 10 x F48
(FLOAT_TYPE\015BASE)S
(\015EPSILON)S 2425 Y 466 X F6(\(b\))S 35 x(For)S 16 x(integer)S 
16 x F21(k)S 1 x F6(,)S 16 x F48(SIN\(X,)S 20 x(CYCLE\))S 13 x F20
(=)S
XP /F23 56 37 16 -38 29 -1 37 16 0
<0018 0078 00F0 01E0 03C0 0780 0F00 1F00 1E00 3E00 3C00 7C00 7C00
 7800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800
 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800>
PXLC RP
1147 2340 XY F23(8)S
XP /F23 60 37 7 -76 21 1 77 16 0
<007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C
 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 00F8 00F8
 00F8 00F0 01F0 01E0 03E0 03C0 0780 0700 0E00 1C00 3800 F000 C000
 F000 3800 1C00 0E00 0700 0780 03C0 03E0 01E0 01F0 00F0 00F8 00F8
 00F8 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C
 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C 007C>
PXLC RP
1147 2377 XY F23(<)S
XP /F23 58 37 16 -36 29 1 37 16 0
<F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800
 F800 F800 F800 F800 F800 F800 F800 F800 F800 F800 7800 7C00 7C00
 3C00 3E00 1E00 1F00 0F00 0780 03C0 01E0 00F0 0078 0018>
PXLC RP
1147 2452 XY F23(:)S 2375 Y 7 x F48(0.0)S
XP /F21 59 12 4 -8 8 4 12 8 0
<60 F0 F0 70 10 10 10 10 20 20 40 80>
PXLC RP
1257 2375 XY F21(;)S 73 x F48(X)S 11 x F20(=)S 12 x F21(k)S 10 x F22
(\001)S 9 x F48(CYCLE)S -1 x F21(=)S F48(2.0)S 50 y 1191 X(1.0)S F21
(;)S 73 x F48(X)S 11 x F20(=)S 12 x(\()S
XP /F20 52 21 1 0 19 28 28 24 0
<000C00 000C00 001C00 003C00 003C00 005C00 00DC00 009C00 011C00
 031C00 021C00 041C00 0C1C00 081C00 101C00 301C00 201C00 401C00
 C01C00 FFFFC0 001C00 001C00 001C00 001C00 001C00 001C00 001C00
 01FFC0>
PXLC RP
1435 2425 XY F20(4)S F21(k)S 10 x F20(+)S 10 x(1)S -1 x(\))S 10 x 
F22(\001)S 8 x F48(CYCLE)S -1 x F21(=)S F48(4.0)S 50 y 1191 X F22
(\000)S F48(1.0)S F21(;)S 41 x F48(X)S 11 x F20(=)S 12 x(\(4)S F21
(k)S 10 x F20(+)S
XP /F20 51 21 2 -1 18 28 29 16 0
<07E0 1830 201C 201C 781E 780E 781E 381E 001C 001C 0018 0030 0060
 07E0 0030 001C 001C 000E 000F 000F 700F F80F F80F F80F F00E 401C
 201C 1830 07E0>
PXLC RP
1530 2475 XY F20(3)S -1 x(\))S 10 x F22(\001)S 8 x F48(CYCLE)S -1 x 
F21(=)S F48(4.0)S 2620 Y 300 X F50(13.8.)S 33 x(COS)S 20 x(\320)S 
19 x(T)S -3 x(rigonometric)S 19 x(Cosine)S 19 x(Function,)S 71 y 
685 X(Natural)S 25 x(Cycle)S 24 x(\(Angle)S 24 x(in)S 24 x
(Radians\))S 2843 Y 383 X F2(Declaration:)S 2994 Y 466 X F48
(function)S 20 x(COS)S 21 x(\(X)S 22 x(:)S 21 x(FLOAT_TYPE\))S 20 x
(return)S 20 x(FLOAT_TYPE;)S 3128 Y 1254 X F6(16)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 435 717

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(De\014nition:)S 452 Y 466 X F48(COS\(X\))S 19 x F22
(\031)S
XP /F20 99 18 2 0 16 18 18 16 0
<07E0 0C30 1878 3078 7030 6000 E000 E000 E000 E000 E000 E000 6000
 7004 3004 1808 0C30 07C0>
PXLC RP
670 452 XY F20(cos)S 7 x F48(X)S 591 Y 383 X F2(Usage:)S 730 Y -61 x 
F48(Z)S 22 x(:=)S 21 x(COS\(X\);)S 64 x(--)S 21 x(X)S 22 x(in)S 21 x
(radians)S 869 Y 383 X F2(Domain:)S 979 Y 466 X F6(Mathematically)S
20 x(unbounded)S 1118 Y 383 X F2(Range:)S 1229 Y -69 x F22(j)S F48
(COS\(X\))S -2 x F22(j)S 20 x(\024)S 21 x F48(1.0)S 1367 Y 383 X F2
(Accuracy:)S 1478 Y 466 X F6(\(a\))S 37 x(Maximum)S 12 x(relative)S
12 x(error)S 12 x F20(=)S 12 x F48(2.0)S 6 x F22(\001)S 6 x F48
(FLOAT_TYPE)S
(\015BASE\015EPSILON)S 7 x F6(when)S 12 x F22(j)S F48(X)S F22(j)S 
11 x F6(is)S 12 x(less)S 12 x(than)S 13 x(or)S 13 x(equal)S 13 x(to)S
12 x(some)S 50 y 549 X(documented)S 16 x(implementation-dependent)S
15 x(threshold,)S 16 x(which)S 16 x(must)S 15 x(not)S 16 x(be)S 15 x
(less)S 15 x(than)S 1621 Y 832 X F48(FLOAT_TYPE\015MACHINE_RADIX)S
-17 y F65(FLOAT_TYPE\015MACHINE_MANTISSA)S F55(=)S F65(2)S 1714 Y 
549 X F6(For)S 21 x(lar)S -1 x(ger)S 21 x(values)S 21 x(of)S 21 x 
F22(j)S -1 x F48(X)S F22(j)S F6(,)S 22 x(degraded)S 21 x(accuracy)S
21 x(is)S 21 x(allowed.)S 33 x(An)S 21 x(implementation)S 21 x(must)S
21 x(document)S 21 x(its)S 50 y 549 X(behavior)S 21 x(for)S 20 x
(lar)S -1 x(ge)S 20 x F22(j)S F48(X)S F22(j)S -1 x F6(.)S 62 y 466 X
(\(b\))S 35 x F48(COS\(0.0\))S 18 x F20(=)S 20 x F48(1.0)S 1975 Y 
300 X F50(13.9.)S 33 x(COS)S 20 x(\320)S 19 x(T)S -3 x(rigonometric)S
19 x(Cosine)S 19 x(Function,)S 70 y 685 X(Arbitrary)S 22 x(Cycle)S
23 x(\(Angle)S 23 x(in)S 23 x(Arbitrary)S 23 x(Units\))S 2184 Y 
383 X F2(Declaration:)S 2323 Y 466 X F48(function)S 20 x(COS)S 21 x
(\(X,)S 22 x(CYCLE)S 20 x(:)S 22 x(FLOAT_TYPE\))S 19 x(return)S 21 x
(FLOAT_TYPE;)S 2462 Y 383 X F2(De\014nition:)S 2572 Y 466 X F48
(COS\(X,)S 20 x(CYCLE\))S 16 x F22(\031)S 19 x F20(cos)S 7 x(\(2)S
F21(\031)S 11 x F22(\001)S 8 x F48(X)S F21(=)S F48(CYCLE)S -1 x F20
(\))S 2711 Y 383 X F2(Usage:)S 2851 Y -61 x F48(Z)S 22 x(:=)S 21 x
(COS\(X,)S 21 x(360.0\);)S 63 x(--)S 22 x(X)S 21 x(in)S 22 x
(degrees)S 49 y 466 X(Z)S 22 x(:=)S 21 x(COS\(X,)S 21 x(CYCLE\);)S
63 x(--)S 22 x(X)S 21 x(in)S 22 x(units)S 21 x(such)S 21 x(that)S 
21 x(one)S 21 x(complete)S 50 y 945 X(--)S 22 x(cycle)S 21 x(of)S 
21 x(rotation)S 20 x(corresponds)S 20 x(to)S 50 y 945 X(--)S 22 x(X)S
21 x(=)S 22 x(CYCLE)S 3128 Y 46 x F6(17)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 540 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Domain:)S 468 Y 466 X F6(\(a\))S 37 x F48(X)S 20 x F6
(mathematically)S 20 x(unbounded)S 63 y 466 X(\(b\))S 35 x F48
(CYCLE)S 19 x F21(>)S 21 x F48(0.0)S 692 Y 383 X F2(Range:)S 818 Y 
-69 x F22(j)S F48(COS\(X,)S 19 x(CYCLE\))S -1 x F22(j)S 19 x(\024)S
20 x F48(1.0)S 979 Y 383 X F2(Accuracy:)S 1106 Y 466 X F6(\(a\))S 
37 x(Maximum)S 17 x(relative)S 17 x(error)S 17 x F20(=)S 15 x F48
(2.0)S 10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 
1212 Y 466 X F6(\(b\))S 35 x(For)S 16 x(integer)S 16 x F21(k)S 1 x 
F6(,)S 16 x F48(COS\(X,)S 20 x(CYCLE\))S 13 x F20(=)S 1127 Y 14 x 
F23(8)S 38 y -37 x(<)S 74 y -37 x(:)S 1161 Y 7 x F48(1.0)S F21(;)S
73 x F48(X)S 11 x F20(=)S 12 x F21(k)S 10 x F22(\001)S 9 x F48
(CYCLE)S 50 y 1191 X(0.0)S F21(;)S 73 x F48(X)S 11 x F20(=)S 12 x
(\(2)S F21(k)S 10 x F20(+)S 10 x(1)S -1 x(\))S 10 x F22(\001)S 8 x 
F48(CYCLE)S -1 x F21(=)S F48(4.0)S 50 y 1191 X F22(\000)S F48(1.0)S
F21(;)S 41 x F48(X)S 11 x F20(=)S 12 x(\(2)S F21(k)S 10 x F20(+)S 
10 x(1)S -1 x(\))S 10 x F22(\001)S 8 x F48(CYCLE)S -1 x F21(=)S F48
(2.0)S 1414 Y 300 X F50(13.10.)S 33 x(T)S -4 x(AN)S 19 x(\320)S 20 x
(T)S -3 x(rigonometric)S 20 x(T)S -4 x(angent)S 20 x(Function,)S 
70 y 707 X(Natural)S 24 x(Cycle)S 24 x(\(Angle)S 24 x(in)S 24 x
(Radians\))S 1645 Y 383 X F2(Declaration:)S 1806 Y 466 X F48
(function)S 20 x(TAN)S 21 x(\(X)S 22 x(:)S 21 x(FLOAT_TYPE\))S 20 x
(return)S 20 x(FLOAT_TYPE;)S 1967 Y 383 X F2(De\014nition:)S 2094 Y 
466 X F48(TAN\(X\))S 19 x F22(\031)S
XP /F20 116 16 1 0 13 26 26 16 0
<0400 0400 0400 0400 0C00 0C00 1C00 3C00 FFE0 1C00 1C00 1C00 1C00
 1C00 1C00 1C00 1C00 1C00 1C10 1C10 1C10 1C10 1C10 0C10 0E20 03C0>
PXLC RP
670 2094 XY F20(t)S
XP /F20 97 21 2 0 20 18 18 24 0
<1FC000 307000 783800 781C00 301C00 001C00 001C00 01FC00 0F1C00
 381C00 701C00 601C00 E01C40 E01C40 E01C40 603C40 304E80 1F8700>
PXLC RP
686 2094 XY F20(an)S 7 x F48(X)S 2254 Y 383 X F2(Usage:)S 2415 Y 
-61 x F48(Z)S 22 x(:=)S 21 x(TAN\(X\);)S 64 x(--)S 21 x(X)S 22 x(in)S
21 x(radians)S 2576 Y 383 X F2(Domain:)S 2703 Y 466 X F6
(Mathematically)S 20 x(unbounded)S 2864 Y 383 X F2(Range:)S 2991 Y 
-69 x F6(Mathematically)S 20 x(unbounded)S 3128 Y 1254 X(18)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 430 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Accuracy:)S 458 Y 466 X F6(\(a\))S 37 x(Maximum)S 12 x
(relative)S 12 x(error)S 12 x F20(=)S 12 x F48(4.0)S 6 x F22(\001)S
6 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 7 x F6(when)S 12 x F22(j)S
F48(X)S F22(j)S 11 x F6(is)S 12 x(less)S 12 x(than)S 13 x(or)S 13 x
(equal)S 13 x(to)S 12 x(some)S 50 y 549 X(documented)S 16 x
(implementation-dependent)S 15 x(threshold,)S 16 x(which)S 16 x
(must)S 15 x(not)S 16 x(be)S 15 x(less)S 15 x(than)S 610 Y 832 X F48
(FLOAT_TYPE)S
(\015MACHINE_RADIX)S -18 y F65(FLOAT_TYPE\015MACHINE_MANTISSA)S F55
(=)S F65(2)S 711 Y 549 X F6(For)S 21 x(lar)S -1 x(ger)S 21 x(values)S
21 x(of)S 21 x F22(j)S -1 x F48(X)S F22(j)S F6(,)S 22 x(degraded)S
21 x(accuracy)S 21 x(is)S 21 x(allowed.)S 33 x(An)S 21 x
(implementation)S 21 x(must)S 21 x(document)S 21 x(its)S 50 y 549 X
(behavior)S 21 x(for)S 20 x(lar)S -1 x(ge)S 20 x F22(j)S F48(X)S F22
(j)S -1 x F6(.)S 62 y 466 X(\(b\))S 35 x F48(TAN\(0.0\))S 18 x F20
(=)S 20 x F48(0.0)S 977 Y 300 X F50(13.11.)S 33 x(T)S -4 x(AN)S 19 x
(\320)S 20 x(T)S -3 x(rigonometric)S 20 x(T)S -4 x(angent)S 20 x
(Function,)S 71 y 707 X(Arbitrary)S 22 x(Cycle)S 23 x(\(Angle)S 22 x
(in)S 23 x(Arbitrary)S 22 x(Units\))S 1195 Y 383 X F2(Declaration:)S
1342 Y 466 X F48(function)S 20 x(TAN)S 21 x(\(X,)S 22 x(CYCLE)S 20 x
(:)S 22 x(FLOAT_TYPE\))S 19 x(return)S 21 x(FLOAT_TYPE;)S 1489 Y 
383 X F2(De\014nition:)S 1605 Y 466 X F48(TAN\(X,)S 20 x(CYCLE\))S
16 x F22(\031)S 19 x F20(tan)S 7 x(\(2)S F21(\031)S 10 x F22(\001)S
9 x F48(X)S F21(=)S F48(CYCLE)S -2 x F20(\))S 1752 Y 383 X F2
(Usage:)S 1899 Y -61 x F48(Z)S 22 x(:=)S 21 x(TAN\(X,)S 21 x
(360.0\);)S 63 x(--)S 22 x(X)S 21 x(in)S 22 x(degrees)S 50 y 466 X
(Z)S 22 x(:=)S 21 x(TAN\(X,)S 21 x(CYCLE\);)S 63 x(--)S 22 x(X)S 
21 x(in)S 22 x(units)S 21 x(such)S 21 x(that)S 21 x(one)S 21 x
(complete)S 50 y 945 X(--)S 22 x(cycle)S 21 x(of)S 21 x(rotation)S
20 x(corresponds)S 20 x(to)S 50 y 945 X(--)S 22 x(X)S 21 x(=)S 22 x
(CYCLE)S 2195 Y 383 X F2(Domain:)S 2312 Y 466 X F6(\(a\))S 37 x F48
(X)S 20 x F6(mathematically)S 20 x(unbounded)S 62 y 466 X(\(b\))S 
35 x F48(CYCLE)S 19 x F21(>)S 21 x F48(0.0)S 2521 Y 383 X F2(Range:)S
2638 Y -69 x F6(Mathematically)S 20 x(unbounded)S 2785 Y 383 X F2
(Accuracy:)S 2901 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x(relative)S
17 x(error)S 17 x F20(=)S 15 x F48(4.0)S 10 x F22(\001)S 10 x F48
(FLOAT_TYPE)S
(\015BASE\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x F48(TAN\(X,)S 20 x
(CYCLE\))S 13 x F20(=)S 16 x F48(0.0)S 16 x F6(when)S 17 x F48(X)S
15 x F20(=)S 15 x F21(k)S 11 x F22(\001)S 10 x F48(CYCLE)S -1 x F21
(=)S F48(2.0)S F6(,)S 17 x(for)S 17 x(integer)S 17 x F21(k)S 3128 Y 
1254 X F6(19)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 540 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Notes:)S 463 Y -53 x F6(When)S 18 x F48(X)S 15 x F20
(=)S 16 x(\(2)S F21(k)S 10 x F20(+)S 9 x(1\))S 11 x F22(\001)S 10 x 
F48(CYCLE)S -1 x F21(=)S F48(4.0)S -1 x F6(,)S 17 x(for)S 17 x
(integer)S 18 x F21(k)S 1 x F6(,)S 18 x(see)S 17 x(Section)S 17 x
(11.)S 622 Y 300 X F50(13.12.)S 33 x(COT)S 20 x(\320)S 19 x(T)S -3 x
(rigonometric)S 19 x(Cotangent)S 19 x(Function,)S 71 y 714 X
(Natural)S 23 x(Cycle)S 24 x(\(Angle)S 24 x(in)S 24 x(Radians\))S 
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<F8 F8 F8 F8 F8 F8 F8 F8 F8 F8 F8 F8 F8 F8>
PXLC RP
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XP /F48 113 22 2 -9 22 18 27 24 0
<07C700 1FF700 3FFF00 7C3F00 781F00 F00F00 F00700 E00700 E00700
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PXLC RP
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PXLC RP
1381 2991 XY F48(/X)S 3128 Y 1254 X F6(25)S
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PXLC RP
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549 X(within)S 16 x(narrow)S 16 x(limits;)S 16 x(cf.)S 13 x(Section)S
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549 X(\(ii\))S 32 x(in)S 16 x(an)S 16 x(implementation)S 16 x(not)S
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F6(must)S 15 x(deliver)S 15 x F21(\031)S 1 x F6(.)S 2472 Y 300 X F50
(13.19.)S 33 x(ARCT)S -4 x(AN)S 19 x(\320)S 20 x(Inverse)S 20 x(T)S
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466 X F48(function)S 20 x(ARCTAN)S 21 x(\(Y)S 967 X(:)S 22 x
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F48(ARCTAN\(Y,)S 19 x(X,)S 20 x(CYCLE\))S 12 x F22(\031)S 515 Y 13 x 
F23(8)S 38 y -37 x(>)S 12 y -37 x(<)S 75 y -37 x(>)S 12 y -37 x(:)S
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21 x(containing)S 20 x(the)S 21 x(point)S 21 x(\(X,Y\)\))S 50 y 
1207 X(--)S 21 x(whose)S 21 x(tangent)S 20 x(is)S 22 x(Y/X)S 50 y 
466 X(Z)S 22 x(:=)S 21 x(ARCTAN\(Y,)S 20 x(X,)S 21 x(CYCLE\);)S 
1207 X(--)S 21 x(Z,)S 22 x(in)S 21 x(units)S 21 x(such)S 21 x(that)S
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21 x(angle)S 21 x(\(in)S 21 x(the)S 50 y 1207 X(--)S 21 x(quadrant)S
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21 x(Y/X)S 2258 Y 383 X F2(Domain:)S 2399 Y 466 X F6(\(a\))S 37 x 
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549 X(at)S 22 x F48(Y)S 20 x F20(=)S 20 x F48(0.0)S F6(\))S 3128 Y 
1254 X(27)S
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F48(ARCTAN\(0.0,)S 19 x(CYCLE)S 19 x(=>)S 21 x(CYCLE\))S 16 x F20(=)S
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1104 X F22(\006)S F48(CYCLE)S -1 x F21(=)S F48(2.0)S 20 x F20(\()S
F6(see)S 14 x(note)S 14 x(b)S F20(\))S F21(;)S 41 x F48(X)S 12 x F21
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19 x(0.0,)S 20 x(CYCLE\))S 14 x F20(=)S -58 y 16 x F23(\032)S 33 y 
8 x F48(CYCLE)S -2 x F21(=)S F48(4.0)S F21(;)S 73 x F48(Y)S 11 x F21
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860 Y 383 X F2(Notes:)S 971 Y -53 x F6(\(a\))S 37 x F22(\000)S F48
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549 X(slightly;)S 27 x(cf.)S 14 x(Section)S 23 x(9)S 24 x(for)S 24 x
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549 X(requirement)S 13 x(\(c\))S 14 x(or)S 13 x(\(d\))S 13 x
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18 x F6(and)S 18 x F48(Y)S 17 x F20(=)S 17 x F48(0.0)S -1 x F6(:)S
74 y 549 X(\(i\))S 44 x(in)S 12 x(an)S 12 x(implementation)S 11 x
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12 x(4\),)S 12 x F48(ARCTAN)S 10 x F6(must)S 12 x(deliver)S 12 x F22
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F48(Y)S 15 x F6(is)S 15 x(a)S 15 x(negatively)S 14 x(signed)S 14 x
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<FC0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000 1C0000
 1C0000 1C0000 1C7C00 1C8700 1D0300 1E0380 1C0380 1C0380 1C0380
 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380 1C0380
 1C0380 FF9FF0>
PXLC RP
742 2190 XY F20(h)S 7 x F48(X)S 2334 Y 383 X F2(Usage:)S 2477 Y -61 x 
F48(Z)S 22 x(:=)S 21 x(SINH\(X\);)S 2620 Y 383 X F2(Domain:)S 2734 Y 
466 X F6(Mathematically)S 20 x(unbounded)S 2877 Y 383 X F2(Range:)S
2991 Y -69 x F6(Mathematically)S 20 x(unbounded)S 3128 Y 1254 X(31)S
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/paper-automatic true SPS 1 PP EP
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1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
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383 342 XY F2(Accuracy:)S 453 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x
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XP /F48 56 22 2 0 19 25 25 24 0
<07F000 1FFC00 3FFE00 7C1F00 F00780 E00380 E00380 E00380 700700
 7C1F00 1FFC00 07F000 1FFC00 3C1E00 700700 F00780 E00380 E00380
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PXLC RP
1026 453 XY F48(8.0)S 10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE)S
(\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x F48(SINH\(0.0\))S 18 x F20
(=)S 20 x F48(0.0)S 655 Y 383 X F2(Notes:)S 767 Y -53 x F6(The)S 
18 x(usable)S 18 x(domain)S 18 x(of)S 18 x F48(SINH)S 17 x F6(is)S
18 x(approximately)S 18 x(given)S 18 x(by)S 864 Y 860 X F22(j)S -1 x 
F48(X)S F22(j)S 11 x(\024)S 12 x F20(log)S 10 y -1 x F55(e)S -10 y 
9 x F48(FLOAT_TYPE\015SAFE_LARGE)S 5 x F20(+)S 10 x(log)S 10 y -1 x 
F55(e)S -10 y 9 x F48(2.0)S 1012 Y 300 X F50(13.23.)S 33 x(COSH)S 
19 x(\320)S 20 x(Hyperbolic)S 19 x(Cosine)S 19 x(Function)S 1152 Y 
383 X F2(Declaration:)S 1293 Y 466 X F48(function)S 20 x(COSH)S 21 x
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1433 Y 383 X F2(De\014nition:)S 1544 Y 466 X F48(COSH\(X\))S 19 x 
F22(\031)S 20 x F20(cosh)S 8 x F48(X)S 1684 Y 383 X F2(Usage:)S 
1825 Y -61 x F48(Z)S 22 x(:=)S 21 x(COSH\(X\);)S 1965 Y 383 X F2
(Domain:)S 2076 Y 466 X F6(Mathematically)S 20 x(unbounded)S 2216 Y 
383 X F2(Range:)S 2328 Y -69 x F48(COSH\(X\))S 19 x F22(\025)S 21 x 
F48(1.0)S 2468 Y 383 X F2(Accuracy:)S 2579 Y 466 X F6(\(a\))S 37 x
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10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 466 X 
F6(\(b\))S 35 x F48(COSH\(0.0\))S 18 x F20(=)S 20 x F48(1.0)S 2781 Y 
383 X F2(Notes:)S 2893 Y -53 x F6(The)S 18 x(usable)S 18 x(domain)S
18 x(of)S 18 x F48(COSH)S 17 x F6(is)S 18 x(approximately)S 18 x
(given)S 18 x(by)S 2990 Y 860 X F22(j)S -1 x F48(X)S F22(j)S 11 x
(\024)S 12 x F20(log)S 10 y -1 x F55(e)S -10 y 9 x F48(FLOAT_TYPE)S
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9 x F48(2.0)S 3128 Y 1254 X F6(32)S
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/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 406 718

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1000 BP 3300 2550 PM /paper-automatic false SPS 300 0 XY
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300 342 XY F50(13.24.)S 33 x(T)S -4 x(ANH)S 19 x(\320)S 20 x
(Hyperbolic)S 19 x(T)S -4 x(angent)S 20 x(Function)S 486 Y 383 X F2
(Declaration:)S 629 Y 466 X F48(function)S 20 x(TANH)S 21 x(\(X)S 
22 x(:)S 21 x(FLOAT_TYPE\))S 20 x(return)S 20 x(FLOAT_TYPE;)S 773 Y 
383 X F2(De\014nition:)S 887 Y 466 X F48(TANH\(X\))S 19 x F22(\031)S
20 x F20(tanh)S 7 x F48(X)S 1031 Y 383 X F2(Usage:)S 1175 Y -61 x 
F48(Z)S 22 x(:=)S 21 x(TANH\(X\);)S 1318 Y 383 X F2(Domain:)S 1432 Y 
466 X F6(Mathematically)S 20 x(unbounded)S 1576 Y 383 X F2(Range:)S
1690 Y -69 x F22(j)S F48(TANH\(X\))S -2 x F22(j)S 20 x(\024)S 20 x 
F48(1.0)S 1834 Y 383 X F2(Accuracy:)S 1948 Y 466 X F6(\(a\))S 37 x
(Maximum)S 17 x(relative)S 17 x(error)S 17 x F20(=)S 15 x F48(8.0)S
10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 466 X 
F6(\(b\))S 35 x F48(TANH\(0.0\))S 18 x F20(=)S 20 x F48(0.0)S 2162 Y 
300 X F50(13.25.)S 33 x(COTH)S 20 x(\320)S 19 x(Hyperbolic)S 19 x
(Cotangent)S 20 x(Function)S 2305 Y 383 X F2(Declaration:)S 2449 Y 
466 X F48(function)S 20 x(COTH)S 21 x(\(X)S 22 x(:)S 21 x
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(\014nition:)S 2707 Y 466 X F48(COTH\(X\))S 19 x F22(\031)S 20 x F20
(coth)S 7 x F48(X)S 2850 Y 383 X F2(Usage:)S 2994 Y -61 x F48(Z)S 
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%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 391 717

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
%%EndPageSetup
383 342 XY F2(Domain:)S 452 Y 466 X F6(Mathematically)S 20 x
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(COTH\(X\))S -2 x F22(j)S 20 x(\025)S 20 x F48(1.0)S 841 Y 383 X F2
(Accuracy:)S 952 Y 466 X F6(Maximum)S 18 x(relative)S 18 x(error)S
18 x F20(=)S 16 x F48(8.0)S 10 x F22(\001)S 11 x F48
(FLOAT_TYPE\015BASE)S
(\015EPSILON)S 1091 Y 383 X F2(Notes:)S 1201 Y -53 x F6(When)S 20 x 
F48(X)S 19 x F20(=)S 19 x F48(0.0)S -1 x F6(,)S 21 x(see)S 20 x
(Section)S 21 x(11.)S 1350 Y 300 X F50(13.26.)S 33 x(ARCSINH)S 20 x
(\320)S 19 x(Inverse)S 19 x(Hyperbolic)S 19 x(Sine)S 19 x(Function)S
1489 Y 383 X F2(Declaration:)S 1628 Y 466 X F48(function)S 20 x
(ARCSINH)S 21 x(\(X)S 21 x(:)S 21 x(FLOAT_TYPE\))S 20 x(return)S 
21 x(FLOAT_TYPE;)S 1767 Y 383 X F2(De\014nition:)S 1878 Y 466 X F48
(ARCSINH\(X\))S 17 x F22(\031)S 20 x F20(arcsinh)S 8 x F48(X)S 2017 Y 
383 X F2(Usage:)S 2157 Y -61 x F48(Z)S 22 x(:=)S 21 x(ARCSINH\(X\);)S
2296 Y 383 X F2(Domain:)S 2407 Y 466 X F6(Mathematically)S 20 x
(unbounded)S 2545 Y 383 X F2(Range:)S 2656 Y -69 x F6
(Mathematically)S 20 x(unbounded)S 2795 Y 383 X F2(Accuracy:)S 2906 Y 
466 X F6(\(a\))S 37 x(Maximum)S 17 x(relative)S 17 x(error)S 17 x 
F20(=)S 15 x F48(8.0)S 10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE)S
(\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x F48(ARCSINH\(0.0\))S 17 x 
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%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 429 718

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1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
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383 342 XY F2(Notes:)S 450 Y -53 x F6(The)S 17 x(reachable)S 18 x
(range)S 18 x(of)S 18 x F48(ARCSINH)S 16 x F6(is)S 17 x
(approximately)S 18 x(given)S 17 x(by)S 533 Y 762 X F22(j)S -1 x F48
(ARCSINH\(X\))S -2 x F22(j)S 11 x(\024)S 12 x F20(log)S 10 y F55(e)S
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(ARCCOSH)S 19 x(\320)S 19 x(Inverse)S 19 x(Hyperbolic)S 19 x(Cosine)S
19 x(Function)S 812 Y 383 X F2(Declaration:)S 950 Y 466 X F48
(function)S 20 x(ARCCOSH)S 21 x(\(X)S 21 x(:)S 21 x(FLOAT_TYPE\))S
20 x(return)S 21 x(FLOAT_TYPE;)S 1084 Y 383 X F2(De\014nition:)S 
1192 Y 466 X F48(ARCCOSH\(X\))S 17 x F22(\031)S 20 x F20(arccosh)S
8 x F48(X)S 1326 Y 383 X F2(Usage:)S 1463 Y -61 x F48(Z)S 22 x(:=)S
21 x(ARCCOSH\(X\);)S 1597 Y 383 X F2(Domain:)S 1705 Y 466 X F48(X)S
21 x F22(\025)S 21 x F48(1.0)S 1839 Y 383 X F2(Range:)S 1948 Y -69 x 
F48(ARCCOSH\(X\))S 18 x F22(\025)S 20 x F48(0.0)S 2081 Y 383 X F2
(Accuracy:)S 2190 Y 466 X F6(\(a\))S 37 x(Maximum)S 17 x(relative)S
17 x(error)S 17 x F20(=)S 15 x F48(8.0)S 10 x F22(\001)S 10 x F48
(FLOAT_TYPE)S
(\015BASE\015EPSILON)S 62 y 466 X F6(\(b\))S 35 x F48
(ARCCOSH\(1.0\))S 17 x F20(=)S 19 x F48(0.0)S 2386 Y 383 X F2
(Notes:)S 2494 Y -53 x F6(The)S 15 x(upper)S 16 x(bound)S 15 x(of)S
16 x(the)S 15 x(reachable)S 16 x(range)S 16 x(of)S 16 x F48(ARCCOSH)S
14 x F6(is)S 16 x(approximately)S 16 x(given)S 16 x(by)S 2577 Y 
773 X F48(ARCCOSH\(X\))S 10 x F22(\024)S 11 x F20(log)S 10 y F55(e)S
-10 y 9 x F48(FLOAT_TYPE\015SAFE_LARGE)S 5 x F20(+)S 9 x(log)S 10 y 
F55(e)S -10 y 9 x F48(2.0)S 2723 Y 300 X F50(13.28.)S 33 x(ARCT)S 
-4 x(ANH)S 19 x(\320)S 20 x(Inverse)S 19 x(Hyperbolic)S 19 x(T)S 
-4 x(angent)S 18 x(Function)S 2857 Y 383 X F2(Declaration:)S 2994 Y 
466 X F48(function)S 20 x(ARCTANH)S 21 x(\(X)S 21 x(:)S 21 x
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(35)S
%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 457 718

%%PageBoundingBox: (atend)
%%BeginPageSetup
1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
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383 342 XY F2(De\014nition:)S 452 Y 466 X F48(ARCTANH\(X\))S 17 x 
F22(\031)S 20 x F20(arctanh)S 7 x F48(X)S 590 Y 383 X F2(Usage:)S 
730 Y -61 x F48(Z)S 22 x(:=)S 21 x(ARCTANH\(X\);)S 868 Y 383 X F2
(Domain:)S 979 Y 466 X F22(j)S F48(X)S F22(j)S 20 x(\024)S 21 x F48
(1.0)S 1117 Y 383 X F2(Range:)S 1228 Y -69 x F6(Mathematically)S 
20 x(unbounded)S 1366 Y 383 X F2(Accuracy:)S 1477 Y 466 X F6(\(a\))S
37 x(Maximum)S 17 x(relative)S 17 x(error)S 17 x F20(=)S 15 x F48
(8.0)S 10 x F22(\001)S 10 x F48(FLOAT_TYPE\015BASE\015EPSILON)S 62 y 
466 X F6(\(b\))S 35 x F48(ARCTANH\(0.0\))S 17 x F20(=)S 19 x F48
(0.0)S 1677 Y 383 X F2(Notes:)S 1788 Y -53 x F6(When)S 20 x F48(X)S
19 x F20(=)S 18 x F22(\006)S F48(1.0)S F6(,)S 21 x(see)S 20 x
(Section)S 20 x(11.)S 1936 Y 300 X F50(13.29.)S 33 x(ARCCOTH)S 20 x
(\320)S 19 x(Inverse)S 19 x(Hyperbolic)S 20 x(Cotangent)S 19 x
(Function)S 2074 Y 383 X F2(Declaration:)S 2214 Y 466 X F48
(function)S 20 x(ARCCOTH)S 21 x(\(X)S 21 x(:)S 21 x(FLOAT_TYPE\))S
20 x(return)S 21 x(FLOAT_TYPE;)S 2352 Y 383 X F2(De\014nition:)S 
2463 Y 466 X F48(ARCCOTH\(X\))S 17 x F22(\031)S 20 x F20(arccoth)S
8 x F48(X)S 2601 Y 383 X F2(Usage:)S 2741 Y -61 x F48(Z)S 22 x(:=)S
21 x(ARCCOTH\(X\);)S 2879 Y 383 X F2(Domain:)S 2990 Y 466 X F22(j)S
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%%PageTrailer
/paper-automatic true SPS 1 PP EP
%%PageBoundingBox: 72 39 474 718

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1000 BP 3300 2550 PM /paper-automatic false SPS 383 0 XY
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383 342 XY F2(Range:)S 454 Y -69 x F6(Mathematically)S 20 x
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18 x(relative)S 18 x(error)S 18 x F20(=)S 16 x F48(8.0)S 10 x F22
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(Notes:)S 960 Y -53 x F6(When)S 20 x F48(X)S 19 x F20(=)S 18 x F22
(\006)S F48(1.0)S F6(,)S 21 x(see)S 20 x(Section)S 20 x(11.)S 3128 Y 
1254 X(37)S
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DataMuseum.dk

Rational R1000/400 Tapes

⟦c609329b4⟧ TextFile

Derivation

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