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%!PS-Adobe-3.0 %%Title: (Microsoft Word - complex_edit\311) %%Creator: (Microsoft Word: PSPrinter 8.1.1) %%CreationDate: (3:48 PM Tuesday, November 5, 1996) %%For: () %%Pages: 5 %%DocumentFonts: Times-Bold Times-Roman Symbol Times-Italic Courier-Bold %%DocumentNeededFonts: Times-Bold Times-Roman Symbol Times-Italic Courier-Bold %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %ADO_PaperArea: -31 -30 761 582 %ADO_ImageableArea: 0 0 730 552 %%EndComments /md 183 dict def md begin /currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /$m matrix def /av 81 def /por true def /normland false def /psb-nosave{}bd /pse-nosave{}bd /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ false }ifelse def /featurecleanup { stopped cleartomark countdictstack exch sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { /pm save store mT concat }bd /endp { pm restore showpage }def /$c/DeviceRGB def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: adobe_psp_colorspace_level1 %%Copyright: Copyright 1991-1993 Adobe Systems Incorporated. All Rights Reserved. /G/setgray ld /:F/setrgbcolor ld %%EndFile %%BeginFile: adobe_psp_uniform_graphics %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: adobe_psp_basic_text %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg true def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding 39/quotesingle put MacEncoding 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave /Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacEncoding 128 128 getinterval astore pop level2 startnoload /copyfontdict { findfont dup length dict begin { 1 index/FID ne{def}{pop pop}ifelse }forall }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd level2 not endnoload md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff def }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff def }bd /scf { scalefont def }bd /scf-non { $m scale :mf setfont }bd /ps Z /fz{/ps xs}bd /sf/setfont ld /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse def }def %%EndFile %%BeginFile: adobe_psp_derived_styles %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /wi version(23.0)eq { { gS 0 0 0 0 rC stringwidth gR }bind }{ /stringwidth load }ifelse def /$o 1. def /gl{$o G}bd /ms{:M S}bd /condensedmtx[.82 0 0 1 0 0]def /:mc { condensedmtx :mf def }bd /extendedmtx[1.18 0 0 1 0 0]def /:me { extendedmtx :mf def }bd /basefont Z /basefonto Z /dxa Z /dxb Z /dxc Z /dxd Z /dsdx2 Z /bfproc Z /:fbase { dup/FontType get 0 eq{ dup length dict begin dup{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall /FDepVector exch/FDepVector get[exch/:fbase load forall]def }/bfproc load ifelse /customfont currentdict end definefont }bd /:mo { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin basefont setfont ( )dup 0 4 -1 roll put dup wi setcharwidth 0 0 :M gS gl dup show gR basefonto setfont show end }def }store :fbase }bd /:mso { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxa 1 ps div def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxa add exch }if setcharwidth dup 0 0 ms dup dxa 0 ms dup dxa dxa ms dup 0 dxa ms gl dxa 2. div dup ms end end }def }store :fbase }bd /:ms { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxb .05 def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dxb add }if exch setcharwidth dup dxb .01 add 0 ms 0 dxb :T gS gl dup 0 0 ms gR basefonto setfont 0 0 ms end end }def }store :fbase }bd /:mss { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxc 1 ps div def /dsdx2 .05 dxc 2 div add def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dsdx2 add }if exch setcharwidth dup dsdx2 .01 add 0 ms 0 .05 dxc 2 div sub :T dup 0 0 ms dup dxc 0 ms dup dxc dxc ms dup 0 dxc ms gl dxc 2 div dup ms end end }def }store :fbase }bd /:msb { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxd .03 def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxd add exch }if setcharwidth dup 0 0 ms dup dxd 0 ms dup dxd dxd ms 0 dxd ms end end }def }store :fbase }bd /italicmtx[1 0 -.212557 1 0 0]def /:mi { italicmtx :mf def }bd /:v { [exch dup/FontMatrix get exch dup/FontInfo known { /FontInfo get dup/UnderlinePosition known { dup/UnderlinePosition get 2 index 0 3 1 roll transform exch pop }{ .1 }ifelse 3 1 roll dup/UnderlineThickness known { /UnderlineThickness get exch 0 3 1 roll transform exch pop abs }{ pop pop .067 }ifelse }{ pop pop .1 .067 }ifelse ] }bd /$t Z /$p Z /$s Z /:p { aload pop 2 index mul/$t xs 1 index mul/$p xs .012 mul/$s xs }bd /:m {gS 0 $p rm $t lw 0 rl stroke gR }bd /:n { gS 0 $p rm $t lw 0 rl gS gl stroke gR strokepath $s lw /setstrokeadjust where{pop currentstrokeadjust true setstrokeadjust stroke setstrokeadjust }{ stroke }ifelse gR }bd /:o {gS 0 $p rm $t 2 div dup rm $t lw dup 0 rl stroke gR :n }bd %%EndFile /currentpacking where {pop sc_oldpacking setpacking}if end % md %%EndProlog %%BeginSetup md begin countdictstack [ {%stopped %%BeginFeature: *ManualFeed False 1 dict dup /ManualFeed false put setpagedevice %%EndFeature }featurecleanup countdictstack [ {%stopped %%BeginFeature: *PageSize LetterSmall 1 dict dup /Policies 2 dict dup /PageSize 2 put dup /MediaType 0 put put setpagedevice 2 dict dup /PageSize [612 792] put dup /ImagingBBox null put setpagedevice %%EndFeature }featurecleanup ()setjob /pT[1 0 0 -1 30 761]def/mT[1 0 0 -1 30 761]def /sD 16 dict def 300 level2{1 dict dup/WaitTimeout 4 -1 roll put setuserparams}{statusdict/waittimeout 3 -1 roll put}ifelse %%IncludeFont: Times-Bold %%IncludeFont: Times-Roman %%IncludeFont: Symbol %%IncludeFont: Times-Italic %%IncludeFont: Courier-Bold /f0_1/Times-Bold :mre /f0_18 f0_1 18 scf /f0_16 f0_1 16 scf /f0_12 f0_1 12 scf /f1_1/Times-Roman :mre /f1_12 f1_1 12 scf /f1_10 f1_1 10 scf /f1_9 f1_1 9 scf /f2_1/Symbol :bsr 240/apple pd :esr /f2_10 f2_1 10 scf /f3_1/Times-Italic :mre /f3_12 f3_1 12 scf /f3_10 f3_1 10 scf /f4_1 f2_1 :mi /f4_10 f4_1 10 scf /f5_1/Courier-Bold :mre /f5_10 f5_1 10 scf /f5_9 f5_1 9 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (; page: 1 of 5)setjob %%EndPageSetup gS 0 0 552 730 rC 156 55 :M f0_18 sf -.03(Complex Arithmetic Edits\321Update)A 186 80 :M f0_12 sf -.026(WG14/N620 X3J11/96-84 \(Draft 10/15/96\))A 264 104 :M f1_12 sf -.11(Jim Thomas)A 54 104 :M (5)S 249 116 :M -.021(517 Hendon Court)A 241 128 :M -.018(Sunnyvale, CA 94087)A 246 140 :M f5_10 sf (jthomas@best.com)S 48 167 :M f1_12 sf (1)S 54 167 :M (0)S 96 179 :M f3_12 sf -.021(The changes in this update are:)A 78 203 :M .97 .097(1. The words "and exception behavior" were added in three places to clarify that the)J 78 215 :M -.004(exception behavior as well as result is determined by the table.)A 48 227 :M f1_12 sf (1)S 54 227 :M (5)S 78 239 :M f3_12 sf .183 .018(2. The sample multiplication implementation was updated to catch overflow cases which)J 78 251 :M -.01(the previous version allowed to result in NaN+iNaN instead of an infinity.)A 78 275 :M .364 .036(3. The sample division implementation was updated to catch non-zero/zero cases which)J 78 287 :M -.01(the previous version allowed to result in NaN+iNaN instead of an infinity.)A 48 287 :M f1_12 sf (2)S 54 287 :M (0)S 78 326 :M f0_16 sf -.02(Y.5 Binary operators)A 96 354 :M f1_12 sf .258 .026(The following subclauses supplement 6.3 in order to specify the type of the result for)J 48 354 :M (2)S 54 354 :M (5)S 78 366 :M -.016(an operation with an imaginary operand.)A 96 390 :M .12 .012(For most operand types, the value of the result of a binary operator with an imaginary)J 78 402 :M .741 .074(or complex operand is completely determined, with reference to real arithmetic, by the)J 78 414 :M .219 .022(usual mathematical formula. For some operand types, the usual mathematical formula is)J 48 414 :M (3)S 54 414 :M (0)S 78 426 :M 1.659 .166(problematic because of its treatment of infinities and because of undue overflow or)J 78 438 :M 1.507 .151(underflow \(7.X.1\); in these cases the result satisfies certain properties \(specified in)J 78 450 :M -.015(Y.5.1\), but is not completely determined.)A 78 486 :M f0_12 sf -.032(Y.5.1 Multiplicative operators)A 48 486 :M f1_12 sf (3)S 54 486 :M (5)S 78 513 :M f0_12 sf -.123(Semantics)A 96 537 :M f1_12 sf .206 .021(If one operand has real type and the other operand has imaginary type, then the result)J 78 549 :M .659 .066(has imaginary type. If both operands have imaginary type, then the result has complex)J 48 549 :M (4)S 54 549 :M (0)S 78 561 :M -.007(type. \(If either operand has complex type, then the result has complex type.\))A endp %%Page: 2 2 %%BeginPageSetup initializepage (; page: 2 of 5)setjob %%EndPageSetup gS 0 0 552 730 rC 42 14 :M f1_12 sf -.032(WG14/N620 X3J11/96-84)A 403 14 :M -.075(Draft 10/15/96)A 60 722 :M (2)S 303 722 :M -.01(C9X Complex Arithmetic\321Update)A 60 50 :M .372 .037(If the operands are not both complex, then the result and exception behavior of the )J f5_10 sf (*)S 42 62 :M f1_12 sf -.017(operator is defined by the usual mathematical formula:)A 107 94 :M f3_10 sf (*)S 206 94 :M f1_10 sf -.328(real)A 221 94 :M f5_9 sf -.459( )A f3_10 sf (x)S 296 94 :M f1_10 sf -.067(imaginary)A 337 94 :M f5_9 sf -.346( )A f3_10 sf -.409(y*I)A 400 94 :M f1_10 sf -.128(complex)A f5_9 sf -.14( )A f3_10 sf -.125(x + y*I)A 55 77 1 1 rF 55 77 1 1 rF 56 77 106 1 rF 162 77 1 1 rF 163 77 108 1 rF 271 77 1 1 rF 272 77 107 1 rF 379 77 1 1 rF 380 77 107 1 rF 487 77 1 1 rF 487 77 1 1 rF 55 78 1 18 rF 162 78 1 18 rF 164 78 1 18 rF 271 78 1 18 rF 379 78 1 18 rF 487 78 1 18 rF 97 115 :M f1_10 sf -.328(real)A 112 115 :M f5_9 sf -.207( )A f3_10 sf (u)S 211 115 :M -.219(x*u)A 311 115 :M -.07(\(y*u\)*I)A 403 115 :M -.019(\(x*u\) + \(y*u\)*I)A 55 96 1 3 rF 56 96 106 1 rF 56 98 106 1 rF 162 96 1 1 rF 162 98 1 1 rF 164 96 1 1 rF 164 98 1 1 rF 165 96 106 1 rF 165 98 106 1 rF 271 96 1 3 rF 272 96 107 1 rF 272 98 107 1 rF 379 96 1 3 rF 380 96 107 1 rF 380 98 107 1 rF 487 96 1 3 rF 55 99 1 18 rF 162 99 1 18 rF 164 99 1 18 rF 271 99 1 18 rF 379 99 1 18 rF 487 99 1 18 rF 80 134 :M f1_10 sf -.067(imaginary)A 121 134 :M f5_9 sf -.346( )A f3_10 sf -.409(v*I)A 203 134 :M -.143(\(x*v\)*I)A 305 134 :M -.031(-y*v + 0*I)A 402 134 :M -.031(\(-y*v\) + \(x*v\)*I)A 55 117 1 1 rF 56 117 106 1 rF 164 117 1 1 rF 162 117 1 1 rF 165 117 106 1 rF 271 117 1 1 rF 272 117 107 1 rF 379 117 1 1 rF 380 117 107 1 rF 487 117 1 1 rF 55 118 1 18 rF 162 118 1 18 rF 164 118 1 18 rF 271 118 1 18 rF 379 118 1 18 rF 487 118 1 18 rF 75 153 :M f1_10 sf -.095(complex)A f5_9 sf -.105( )A f3_10 sf -.095(u + v*I)A 187 153 :M -.05(\(x*u\) + \(x*v\)*I)A 293 153 :M (\(-y*v\) + \(y*u\)*I)S 55 136 1 1 rF 56 136 106 1 rF 164 136 1 1 rF 162 136 1 1 rF 165 136 106 1 rF 271 136 1 1 rF 272 136 107 1 rF 379 136 1 1 rF 380 136 1 1 rF 381 136 106 1 rF 487 136 1 1 rF 487 136 1 1 rF 55 137 1 18 rF 55 155 1 1 rF 55 155 1 1 rF 56 155 106 1 rF 162 137 1 18 rF 164 137 1 18 rF 162 155 1 1 rF 163 155 108 1 rF 271 137 1 18 rF 271 155 1 1 rF 272 155 107 1 rF 379 137 1 18 rF 379 155 1 1 rF 379 155 1 1 rF 60 177 :M f1_12 sf .307 .031(If the second operand is not complex, then the result and exception behavior of the )J 468 177 :M f5_10 sf (/)S 18 177 :M f1_12 sf (5)S 42 189 :M -.017(operator is defined by the usual mathematical formula:)A 108 221 :M f3_10 sf (/)S 216 221 :M (x)S 319 221 :M -.383(y*I)A 419 221 :M -.159(x + y*I)A 55 204 1 1 rF 55 204 1 1 rF 56 204 106 1 rF 162 204 1 1 rF 163 204 108 1 rF 271 204 1 1 rF 272 204 107 1 rF 379 204 1 1 rF 380 204 107 1 rF 487 204 1 1 rF 487 204 1 1 rF 55 205 1 18 rF 162 205 1 18 rF 164 205 1 18 rF 271 205 1 18 rF 379 205 1 18 rF 487 205 1 18 rF 107 242 :M (u)S 212 242 :M -.108(x/u)A 312 242 :M -.033(\(y/u\)*I)A 405 242 :M -.059(\(x/u\) + \(y/u\)*I)A 55 223 1 3 rF 56 223 106 1 rF 56 225 106 1 rF 162 223 1 1 rF 162 225 1 1 rF 164 223 1 1 rF 164 225 1 1 rF 165 223 106 1 rF 165 225 106 1 rF 271 223 1 3 rF 272 223 107 1 rF 272 225 107 1 rF 379 223 1 3 rF 380 223 107 1 rF 380 225 107 1 rF 487 223 1 3 rF 55 226 1 19 rF 162 226 1 19 rF 164 226 1 19 rF 271 226 1 19 rF 379 226 1 19 rF 487 226 1 19 rF 103 261 :M -.383(v*I)A 203 261 :M -.138(\(-x/v\)*I)A 307 261 :M -.091(y/v + 0*I)A 404 261 :M (\(y/v\) + \(-x/v\)*I)S 55 244 1 1 rF 56 244 106 1 rF 164 244 1 1 rF 162 244 1 1 rF 165 244 106 1 rF 271 244 1 1 rF 272 244 107 1 rF 379 244 1 1 rF 380 244 107 1 rF 487 244 1 1 rF 55 245 1 18 rF 55 263 1 1 rF 55 263 1 1 rF 56 263 106 1 rF 162 245 1 18 rF 164 245 1 18 rF 162 263 1 1 rF 163 263 108 1 rF 271 245 1 18 rF 271 263 1 1 rF 272 263 107 1 rF 379 245 1 18 rF 379 263 1 1 rF 380 263 107 1 rF 487 245 1 18 rF 487 263 1 1 rF 487 263 1 1 rF 60 285 :M f1_12 sf .241 .024(A complex or imaginary value with at least one infinite part is regarded as an )J 440 285 :M f3_12 sf -.142(infinity)A 42 297 :M f1_12 sf .152 .015(\(even if its other part is a NaN\). A complex or imaginary value is a )J 374 297 :M f3_12 sf -.024(finite number )A 442 297 :M f1_12 sf -.051(if each)A 12 297 :M (1)S 18 297 :M (0)S 42 309 :M .032 .003(of its parts is a finite number \(neither infinite nor NaN\). A complex or imaginary value is)J 42 321 :M (a )S f3_12 sf (zero)S 72 321 :M f1_12 sf .641 .064( if each of its parts is a zero. The )J 245 321 :M f5_10 sf .136(*)A f1_12 sf .212 .021( and )J f5_10 sf .136(/)A f1_12 sf .474 .047( operators satisfy the following infinity)J 42 335 :M -.01(properties for all real, imaginary, and complex operands)A 312 332 :M f1_9 sf (1)S 316 335 :M f1_12 sf (:)S 42 359 :M (\245)S 60 359 :M .693 .069(If one operand is an infinity and the other operand is a nonzero finite number or an)J 12 359 :M (1)S 18 359 :M (5)S 60 371 :M -.03(infinity, then the result of the )A 203 371 :M f5_10 sf (*)S f1_12 sf -.025( operator is an infinity.)A 42 395 :M (\245)S 60 395 :M .494 .049(If the first operand is an infinity and the second operand is a finite number, then the)J 60 407 :M -.023(result of the )A f5_10 sf (/)S f1_12 sf -.024( operator is an infinity.)A 12 419 :M (2)S 18 419 :M (0)S 42 431 :M (\245)S 60 431 :M .494 .049(If the first operand is a finite number and the second operand is an infinity, then the)J 60 443 :M -.035(result of the )A f5_10 sf (/)S f1_12 sf -.038( operator is a zero.)A 42 467 :M (\245)S 60 467 :M .115 .011(If the first operand is a nonzero finite number or an infinity and the second operand is)J 60 479 :M -.02(a zero, then the result of the )A 197 479 :M f5_10 sf (/)S f1_12 sf -.025( operator is an infinity.)A 12 479 :M (2)S 18 479 :M (5)S 60 503 :M 1.259 .126(If both operands of the )J f5_10 sf .498(*)A f1_12 sf 1.311 .131( operator are complex or if the second operand of the )J 468 503 :M f5_10 sf (/)S 42 515 :M f1_12 sf .779 .078(operator is complex, the operator raises exceptions if appropriate for the calculation of)J 42 527 :M -.015(the parts of the result, and may raise spurious exceptions.)A 12 539 :M (3)S 18 539 :M (0)S 42 551 :M f0_12 sf -.142(Examples)A 42 575 :M f1_12 sf .47 .047(1. Multiplication of )J 142 575 :M f5_10 sf .144(double\312complex)A f1_12 sf .453 .045( operands could be implemented as follows. Note)J 42 587 :M -.042(that the imaginary unit )A 154 587 :M f5_10 sf (I)S f1_12 sf -.009( has imaginary type \(see Y.6\).)A 12 599 :M (3)S 18 599 :M (5)S 42 611 :M f5_10 sf (#include )S 42 622 :M (#include )S 42 633 :M (#define isinf\(x\) \(fabs\(x\)==INFINITY\))S 42 662 :M f1_12 sf ( )S 42 659.48 -.48 .48 186.48 659 .48 42 659 @a 42 673 :M f1_9 sf (1)S 46 676 :M f1_10 sf -.008( These properties are already implied for those cases covered in the tables, but are required for all cases)A 42 687 :M -.133(\(at least where the state for )A f5_9 sf -.195(CX_LIMITED_RANGE)A f1_10 sf -.14( is )A 244 687 :M f3_10 sf -.278(off)A 255 687 :M f1_10 sf -.828(\).)A endp %%Page: 3 3 %%BeginPageSetup initializepage (; page: 3 of 5)setjob %%EndPageSetup gS 0 0 552 730 rC 78 14 :M f1_12 sf -.075(Draft 10/15/96)A 382 14 :M -.032(WG14/N620 X3J11/96-84)A 96 722 :M -.01(C9X Complex Arithmetic\321Update)A 504 722 :M (3)S 504 713 6 12 rC 504 722 :M (3)S gR gS 0 0 552 730 rC 78 50 :M f5_10 sf (/* Multiply z * w ... */)S 78 61 :M (double complex _Cmultd\(double complex z, double complex w\))S 78 72 :M ({)S 114 83 :M (FP_CONTRACT_OFF;)S 114 94 :M (double a, b, c, d, ac, bd, ad, bc, x, y;)S 54 94 :M f1_12 sf (5)S 114 105 :M f5_10 sf (a = creal\(z\); b = cimag\(z\);)S 114 116 :M (c = creal\(w\); d = cimag\(w\);)S 114 127 :M (ac = a * c; bd = b * d;)S 114 138 :M (ad = a * d; bc = b * c;)S 114 149 :M (x = ac - bd;)S 48 149 :M f1_12 sf (1)S 54 149 :M (0)S 114 160 :M f5_10 sf (y = ad + bc;)S 114 171 :M (/* Recover infinities that computed as NaN+iNaN ... */)S 114 182 :M (if \(isnan\(x\) && isnan\(y\)\))S 114 193 :M ({)S 132 204 :M (int recalc = 0;)S 48 204 :M f1_12 sf (1)S 54 204 :M (5)S 132 215 :M f5_10 sf (if \(isinf\(a\) || isinf\(b\)\) /* z is infinite */)S 132 226 :M ({)S 150 237 :M (/* "Box" the infinity ... */)S 150 248 :M (a = copysign\(isinf\(a\) ? 1.0 : 0.0, a\);)S 150 259 :M (b = copysign\(isinf\(b\) ? 1.0 : 0.0, b\);)S 48 259 :M f1_12 sf (2)S 54 259 :M (0)S 150 270 :M f5_10 sf (/* Change NaNs in the other factor to 0 ... */)S 150 281 :M (if \(isnan\(c\)\) c = copysign\(0.0, c\);)S 150 292 :M (if \(isnan\(d\)\) d = copysign\(0.0, d\);)S 150 303 :M (recalc = 1;)S 132 314 :M (})S 48 314 :M f1_12 sf (2)S 54 314 :M (5)S 132 325 :M f5_10 sf (if \(isinf\(c\) || isinf\(d\)\) /* w is infinite */)S 132 336 :M ({)S 150 347 :M (/* "Box" the infinity ... */)S 150 358 :M (c = copysign\(isinf\(c\) ? 1.0 : 0.0, c\);)S 150 369 :M (d = copysign\(isinf\(d\) ? 1.0 : 0.0, d\);)S 48 369 :M f1_12 sf (3)S 54 369 :M (0)S 150 380 :M f5_10 sf (/* Change NaNs in the other factor to 0 ... */)S 150 391 :M (if \(isnan\(a\)\) a = copysign\(0.0, a\);)S 150 402 :M (if \(isnan\(b\)\) b = copysign\(0.0, b\);)S 150 413 :M (recalc = 1;)S 132 424 :M (})S 48 424 :M f1_12 sf (3)S 54 424 :M (5)S 132 435 :M f5_10 sf (if \(!recalc\))S 132 446 :M ({)S 150 457 :M (/* Recover infinities from overflow cases ... */)S 150 468 :M (if \(isinf\(ac\) || isinf\(bd\) || isinf\(ad\) || isinf\(bc\)\))S 150 479 :M ({)S 48 479 :M f1_12 sf (4)S 54 479 :M (0)S 168 490 :M f5_10 sf (/* Change all NaNs to 0 ... */)S 168 501 :M (if \(isnan\(a\)\) a = copysign\(0.0, a\);)S 168 512 :M (if \(isnan\(b\)\) b = copysign\(0.0, b\);)S 168 523 :M (if \(isnan\(c\)\) c = copysign\(0.0, c\);)S 168 534 :M (if \(isnan\(d\)\) d = copysign\(0.0, d\);)S 48 534 :M f1_12 sf (4)S 54 534 :M (5)S 168 545 :M f5_10 sf (recalc = 1;)S 150 556 :M (})S 132 567 :M (})S 132 578 :M (if \(recalc\))S 132 589 :M ({)S 48 589 :M f1_12 sf (5)S 54 589 :M (0)S 150 600 :M f5_10 sf (x = INFINITY * \(a * c - b * d\);)S 150 611 :M (y = INFINITY * \(a * d + b * c\);)S 132 622 :M (})S 114 633 :M (})S 114 644 :M (return x + I * y;)S 48 644 :M f1_12 sf (5)S 54 644 :M (5)S 78 655 :M f5_10 sf (})S endp %%Page: 4 4 %%BeginPageSetup initializepage (; page: 4 of 5)setjob %%EndPageSetup gS 0 0 552 730 rC 42 14 :M f1_12 sf -.032(WG14/N620 X3J11/96-84)A 403 14 :M -.075(Draft 10/15/96)A 60 722 :M (4)S 303 722 :M -.01(C9X Complex Arithmetic\321Update)A 42 50 :M .058 .006(In ordinary \(finite\) cases, the cost to satisfy the infinity property for the )J f5_10 sf (*)S f1_12 sf .06 .006( operator is only)J 42 62 :M .361 .036(one )J f5_10 sf .149(isnan)A f1_12 sf .499 .05( test. This implementation opts for performance over guarding against undue)J 42 74 :M -.043(overflow and underflow.)A 42 98 :M -.015(2. Division of two )A f5_10 sf -.019(double\312complex)A f1_12 sf -.016( operands could be implemented as follows.)A 18 98 :M (5)S 42 122 :M f5_10 sf (#include )S 42 133 :M (#include )S 42 144 :M (/* isinf is as in example 1 above */)S 12 155 :M f1_12 sf (1)S 18 155 :M (0)S 42 166 :M f5_10 sf (/* Divide z / w ... */)S 42 177 :M (double complex _Cdivd\(double complex z, double complex w\))S 42 188 :M ({)S 78 199 :M (FP_CONTRACT_OFF;)S 78 210 :M (double a, b, c, d, logbw, denom, x, y;)S 12 210 :M f1_12 sf (1)S 18 210 :M (5)S 78 221 :M f5_10 sf (long llogbw = 0;)S 78 232 :M (a = creal\(z\); b = cimag\(z\);)S 78 243 :M (c = creal\(w\); d = cimag\(w\);)S 78 254 :M (logbw = logb\(fmax\(fabs\(c\), fabs\(d\)\)\);)S 78 265 :M (if \(isfinite\(logbw\)\))S 12 265 :M f1_12 sf (2)S 18 265 :M (0)S 78 276 :M f5_10 sf ({)S 96 287 :M (llogbw = \(long\)logbw;)S 96 298 :M (c = scalb\(c, -llogbw\);)S 96 309 :M (d = scalb\(d, -llogbw\);)S 78 320 :M (})S 12 320 :M f1_12 sf (2)S 18 320 :M (5)S 78 331 :M f5_10 sf (denom = c * c + d * d;)S 78 342 :M (x = scalb\(\(a * c + b * d\) / denom, -llogbw\);)S 78 353 :M (y = scalb\(\(b * c - a * d\) / denom, -llogbw\);)S 78 364 :M (/*)S 78 375 :M ( * Recover infinities and zeros that computed as NaN+iNaN;)S 12 375 :M f1_12 sf (3)S 18 375 :M (0)S 78 386 :M f5_10 sf ( * the only cases are infinite/finite, finite/infinite,)S 78 397 :M ( * and non-zero/zero ...)S 78 408 :M ( */)S 78 419 :M (if\312\(isnan\(x\)\312&&\312isnan\(y\)\))S 78 430 :M ({)S 12 430 :M f1_12 sf (3)S 18 430 :M (5)S 96 441 :M f5_10 sf (if \(\(isinf\(a\) || isinf\(b\)\) && isfinite\(c\) && isfinite\(d\)\))S 96 452 :M ({)S 114 463 :M (a = copysign\(isinf\(a\) ? 1.0 : 0.0, a\);)S 114 474 :M (b = copysign\(isinf\(b\) ? 1.0 : 0.0, b\);)S 114 485 :M (x = INFINITY * \(a * c + b * d\);)S 12 485 :M f1_12 sf (4)S 18 485 :M (0)S 114 496 :M f5_10 sf (y = INFINITY * \(b * c - a * d\);)S 96 507 :M (})S 96 518 :M (else if \(\(logbw == INFINITY\) && isfinite\(a\) && isfinite\(b\)\))S 96 529 :M ({)S 114 540 :M (c = copysign\(isinf\(c\) ? 1.0 : 0.0, c\);)S 12 540 :M f1_12 sf (4)S 18 540 :M (5)S 114 551 :M f5_10 sf (d = copysign\(isinf\(d\) ? 1.0 : 0.0, d\);)S 114 562 :M (x = 0.0 * \(a * c + b * d\);)S 114 573 :M (y = 0.0 * \(b * c - a * d\);)S 96 584 :M (})S 96 595 :M (else if \(\(denom == 0.0\) && \(isfinite\(a\) || isfinite\(b\)\)\))S 12 595 :M f1_12 sf (5)S 18 595 :M (0)S 96 606 :M f5_10 sf ({)S 114 617 :M (x = copysign\(INFINITY, c\) * a;)S 114 628 :M (y = copysign\(INFINITY, c\) * b;)S 96 639 :M (})S 78 650 :M (})S 12 650 :M f1_12 sf (5)S 18 650 :M (5)S 78 661 :M f5_10 sf (return x + I * y;)S 42 672 :M (})S endp %%Page: 5 5 %%BeginPageSetup initializepage (; page: 5 of 5)setjob %%EndPageSetup gS 0 0 552 730 rC 78 14 :M f1_12 sf -.075(Draft 10/15/96)A 382 14 :M -.032(WG14/N620 X3J11/96-84)A 96 722 :M -.01(C9X Complex Arithmetic\321Update)A 504 722 :M (5)S 504 713 6 12 rC 504 722 :M (5)S gR gS 0 0 552 730 rC 78 50 :M f1_12 sf .647 .065(Scaling the denominator alleviates the main overflow and underflow problem, which is)J 78 62 :M .724 .072(more serious than for multiplication. In the spirit of the multiplication example above,)J 78 74 :M 1.792 .179(this code does not defend against overflow and underflow in the calculation of the)J 78 86 :M 1.117 .112(numerator. Scaling with the )J 227 86 :M f5_10 sf .276(scalb)A f1_12 sf .911 .091( function, instead of with division, provides better)J 78 98 :M -.025(roundoff characteristics.)A 54 98 :M (5)S 78 137 :M f0_12 sf -.04(Y.5.2 Additive operators)A 78 164 :M -.123(Semantics)A 48 176 :M f1_12 sf (1)S 54 176 :M (0)S 96 188 :M .206 .021(If one operand has real type and the other operand has imaginary type, then the result)J 78 200 :M .659 .066(has complex type. If both operands have imaginary type, then the result has imaginary)J 78 212 :M -.007(type. \(If either operand has complex type, then the result has complex type.\))A 96 236 :M .736 .074(In all cases the result and exception behavior of a )J 348 236 :M f5_10 sf .289(+)A f1_12 sf .321 .032( or )J f5_10 sf .289(-)A f1_12 sf .796 .08( operator is defined by the)J 48 236 :M (1)S 54 236 :M (5)S 78 248 :M -.011(usual mathematical formula:)A 143 280 :M f1_10 sf cF f2_10 sf (\261)S sf 252 280 :M f3_10 sf (x)S 355 280 :M -.383(y*I)A 455 280 :M -.159(x + y*I)A 91 263 1 1 rF 91 263 1 1 rF 92 263 106 1 rF 198 263 1 1 rF 199 263 108 1 rF 307 263 1 1 rF 308 263 107 1 rF 415 263 1 1 rF 416 263 107 1 rF 523 263 1 1 rF 523 263 1 1 rF 91 264 1 18 rF 198 264 1 18 rF 200 264 1 18 rF 307 264 1 18 rF 415 264 1 18 rF 523 264 1 18 rF 143 301 :M (u)S 244 301 :M -.938(x )A 251 301 :M f1_10 sf cF f2_10 sf -.418(\261)A sf f3_10 sf -.571( u)A 344 301 :M cF f4_10 sf (\261)S sf (u + y*I)S 447 301 :M -.766(\(x)A 455 301 :M f1_10 sf cF f2_10 sf -.19(\261)A sf f3_10 sf -.163(u\) + y*I)A 91 282 1 3 rF 92 282 106 1 rF 92 284 106 1 rF 198 282 1 1 rF 198 284 1 1 rF 200 282 1 1 rF 200 284 1 1 rF 201 282 106 1 rF 201 284 106 1 rF 307 282 1 3 rF 308 282 107 1 rF 308 284 107 1 rF 415 282 1 3 rF 416 282 107 1 rF 416 284 107 1 rF 523 282 1 3 rF 91 285 1 18 rF 198 285 1 18 rF 200 285 1 18 rF 307 285 1 18 rF 415 285 1 18 rF 523 285 1 18 rF 139 320 :M -.383(v*I)A 240 320 :M -.938(x )A 247 320 :M f1_10 sf cF f2_10 sf -.199(\261)A sf f3_10 sf -.185( v*I)A 347 320 :M -.766(\(y)A 355 320 :M f1_10 sf cF f2_10 sf -.148(\261)A sf f3_10 sf -.145(v\)*I)A 447 320 :M -.19(x + \(y)A 471 320 :M f1_10 sf cF f2_10 sf -.148(\261)A sf f3_10 sf -.145(v\)*I)A 91 303 1 1 rF 92 303 106 1 rF 200 303 1 1 rF 198 303 1 1 rF 201 303 106 1 rF 307 303 1 1 rF 308 303 107 1 rF 415 303 1 1 rF 416 303 107 1 rF 523 303 1 1 rF 91 304 1 18 rF 198 304 1 18 rF 200 304 1 18 rF 307 304 1 18 rF 415 304 1 18 rF 523 304 1 18 rF 131 339 :M -.086(u + v*I)A 232 339 :M -.766(\(x)A 240 339 :M f1_10 sf cF f2_10 sf -.443(\261)A sf f3_10 sf -.437(u\) )A 256 339 :M f1_10 sf cF f2_10 sf -.199(\261)A sf f3_10 sf -.185( v*I)A 336 339 :M f1_10 sf cF f2_10 sf -.183(\261)A sf f3_10 sf -.164(u + \(y)A 366 339 :M f1_10 sf cF f2_10 sf -.148(\261)A sf f3_10 sf -.145(v\)*I)A 439 339 :M -.766(\(x)A 447 339 :M f1_10 sf cF f2_10 sf -.219(\261)A sf f3_10 sf -.185(u\) + \(y)A 480 339 :M f1_10 sf cF f2_10 sf -.148(\261)A sf f3_10 sf -.145(v\)*I)A 91 322 1 1 rF 92 322 106 1 rF 200 322 1 1 rF 198 322 1 1 rF 201 322 106 1 rF 307 322 1 1 rF 308 322 107 1 rF 415 322 1 1 rF 416 322 107 1 rF 523 322 1 1 rF 91 323 1 18 rF 91 341 1 1 rF 91 341 1 1 rF 92 341 106 1 rF 198 323 1 18 rF 200 323 1 18 rF 198 341 1 1 rF 199 341 108 1 rF 307 323 1 18 rF 307 341 1 1 rF 308 341 107 1 rF 415 323 1 18 rF 415 341 1 1 rF 416 341 107 1 rF 523 323 1 18 rF 523 341 1 1 rF 523 341 1 1 rF endp %%Trailer end % md %%EOF
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