{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 54 "Procedures for evaluating the inv erse tangent function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, \+ Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2 007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "load interpolation and func tion approximation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "remez " }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the o ne that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx .m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load numerical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Ma ple m-file " }{TEXT 262 8 "numfcn.m" }{TEXT -1 72 " contains the code \+ for the alternative mathematical functions including " }{TEXT 0 7 "arc tan_" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read \+ into a Maple session by a command similar to the one that follows, whe re the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/numfcn.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "A fixed pre cision procedure to evaluate the inverse tangent function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 101 " In this section we construct a fixed precision version of the inverse \+ tangent function which can use " }{TEXT 259 34 "hardware floating poin t arithmetic" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A rational approximation for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG 6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 69 "When working with fixed precision we can replace the Maclaurin series" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x) = x-x^3/3+x^5/5-x^7/7+` . . . `;" "6#/-%'arc tanG6#%\"xG,,F'\"\"\"*&F'\"\"$F+!\"\"F,*&F'\"\"&F.F,F)*&F'\"\"(F0F,F,% (~.~.~.~GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 62 "by a fixed polynomial or rational function which approximates " }{XPPEDIT 18 0 " arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[-tan(Pi/12), tan(Pi/12)];" "6#7$,$-%$tanG6#*&%#PiG\"\" \"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "We choose a rational function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "s(x) = (a[0]*x+a[1]*x^2+a[2]*x^3+` . . . `+a[m]*x ^m)/(b[0]+b[1]*x+a[2]*x^2+` . . . `+b[n]*x^n);" "6#/-%\"sG6#%\"xG*&,,* &&%\"aG6#\"\"!\"\"\"F'F/F/*&&F,6#F/F/*$F'\"\"#F/F/*&&F,6#F4F/*$F'\"\"$ F/F/%(~.~.~.~GF/*&&F,6#%\"mGF/)F'F>F/F/F/,,&%\"bG6#F.F/*&&FB6#F/F/F'F/ F/*&&F,6#F4F/*$F'F4F/F/F:F/*&&FB6#%\"nGF/)F'FNF/F/!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 6 "Let " }{XPPEDIT 18 0 "f(x) = PIECE WISE([(x-arctan(x))/(x^3), x <> 0],[1/3, x = 0]);" "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$*&,&F'\"\"\"-%'arctanG6#F'!\"\"F.*$F'\"\"$F20F'\"\"!7$* &F.F.F4F2/F'F6" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 36 " has the Maclaurin series expansion " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1/3-x^2/5+x^4/7- x^6/9+` . . . `" "6#/-%\"fG6#%\"xG,,*&\"\"\"F*\"\"$!\"\"F**&F'\"\"#\" \"&F,F,*&F'\"\"%\"\"(F,F**&F'\"\"'\"\"*F,F,%(~.~.~.~GF*" }{TEXT -1 1 " ," }}{PARA 258 "" 0 "" {TEXT -1 35 "and is therefore an even function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We s hall approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 79 " on the interval [-0.26795,0.26795], which is slightly wider than \+ the interval " }{XPPEDIT 18 0 "[-tan(Pi/12), tan(Pi/12)]" "6#7$,$-%$ta nG6#*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 37 ", and then \+ use the rational function " }{XPPEDIT 18 0 "s(x) = x-r(x)*`.`*x^3" "6# /-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$!\"\"" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG " }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 5 "remez" }{TEXT -1 30 " can cope with the expression " } {XPPEDIT 18 0 "(x-arctan(x))/(x^3)" "6#*&,&%\"xG\"\"\"-%'arctanG6#F%! \"\"F&*$F%\"\"$F*" }{TEXT -1 30 ", since it automatically uses " } {XPPEDIT 18 0 "Limit((x-arctan(x))/(x^3),x = 0,right) = 1/3;" "6#/-%&L imitG6%*&,&%\"xG\"\"\"-%'arctanG6#F)!\"\"F**$F)\"\"$F./F)\"\"!%&rightG *&F*F*F0F." }{TEXT -1 21 " for the \"value\" at " }{XPPEDIT 18 0 "x=0 " "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "To ob tain desirable error characteristics for the final function " } {XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 40 ", we use a weight function of the form " }{XPPEDIT 18 0 "w(x)=x^2+delta" "6#/-%\"wG6#% \"xG,&*$F'\"\"#\"\"\"%&deltaGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 12 " is small. " }{XPPEDIT 18 0 "del ta" "6#%&deltaG" }{TEXT -1 39 " prevents the weight from being 0 when \+ " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Calculation \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "evalf(remez((x-arctan(x))/x^3,x=-0.26795..0.26795,[6,6],\n \+ 'maxerr',weight=x^2+1e-40,type=even,errtype=absolute,\n \+ maxgraph=7,info=true),25):\nr := unapply(%,x);\nmaxerr;" }{TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~6G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#*&,*$\"D()*>\"yc*=ZCEFLLLLLLL!#N\"\"\" *&$\"DJ%o\\.y\")HKawklWqBYRF'F()%\"xG\"\"#F(F(*&$\"DRbA+&yKlIx_C.wZ]B5 F'F()F-\"\"%F(F(*&$\"D_kh<&zqM.`i?>'\\Y5/\"!#PF()F-\"\"'F(F(F(,*$F(\" \"!F(*&$\"D!=<$zKafuW;F'R86(Qy\"!#MF(F,F(F(*&$\"Dn+TVd=$))3$=r$fBn-)[* F'F(F2F(F(*&$\"DL\")Q\"\\$H)R>8q]i)z=AT\"F'F(F8F(F(!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%g oal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"A1 8$H\"3$o\"[mPuezPgG!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerror~graph~drawn~for~the~right~half ~of~the~intervalG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%'CURVESG6%7[v7$$\"\"!F)%%FAILG7$$\"AKLLLLLek.a+_i*p#=!#N$\"AIAP ^xbiB'>DFK&QE?!#a7$$\"Almmmmm;H23,/D*Rl$F.$\"AmsI[15!HrLwGnS_5)F17$$\" A(**********\\P4@;gv))4[&F.$\"Al9Prlrl=HhS?kcB=!#`7$$\"AILLLLLLe9;-3]) zI(F.$\"AO0OZ\\YjM7CX(H:;C$F<7$$\"A++++++v=UK?^x>'4\"!#M$\"A(e)f\"3ATr DZ-t,J=H(F<7$$\"Ammmmmmm\"HK/;+(fh9FE$\"Ak5Ga-f\"p?dvuvweH\"!#_7$$\"A* ***********\\P%[1C]&R#>#FE$\"A'=yC]]GHY#>ZU\"RG\"HFM7$$\"ALLLLLLL$ek3K +%>BHFE$\"A7(4IX`@cr/0Uf47<&FM7$$\"A*************\\(oH\"[+\"z%Q%FE$\"A #pGh^oV(p6$H&)[?*e6!#^7$$\"Ammmmmmmm\"H]VSpFgn7$$\"AnmmmmmmT5w,LiSl;Feo $\"Ape>+#p(*p@r@Z)Gu;:!#]7$$\"Anmmmmmm\"HF=Vmu7C#Feo$\"A7^gni7Jq\"zsf& z%o^#F]p7$$\"ALLLLLL$3-`([$41W\"GFeo$\"AeyN6iK(RhGM'fKUJNF]p7$$\"Ammmm mm;/,rD%HsdM$Feo$\"AJ%e]x[4f(3I2>mtZVF]p7$$\"ALLLLLL$3-yRdvr3i$Feo$\"A C[)*3a8l0X4$He]7o%F]p7$$\"A++++++]PfCA<7(f*QFeo$\"At#4FOK'*zIEfZ6gR$\\ F]p7$$\"Ammmmmm;/w?_X`AQSFeo$\"Ap\\;X'eKUgh/8.@'G]F]p7$$\"ALLLLLL$3Fp@ QZz/=%Feo$\"ABbaMX^uo9;:krn'4&F]p7$$\"Ammmmmm;/,:(z`1;D%Feo$\"AT$[fjXT XI+\"3OXI?^F]p7$$\"A************\\P487-OtAVFeo$\"Ak1FF3s]N!*=\\0tzO^F] p7$$\"ALLLLLL$3x6rimgQR%Feo$\"A`Jbg_H%pn!)p_L@g9&F]p7$$\"Ammmmmm;/E4UI x)\\Y%Feo$\"A&GnYGq'46dqG'3cy9&F]p7$$\"ALLLLLL3FWR<*o')e`%Feo$\"AG5ey6 CjJ.$Q<=DA9&F]p7$$\"A**************\\ip#zk&y1YFeo$\"AB;b.qA:$[]_&\\[0H ^F]p7$$\"Ammmmmm\"H2)*zmg%oxYFeo$\"ALxqPX!GL]PHp(yE3^F]p7$$\"ALLLLLL$e *)*HVlNe[ZFeo$\"A'yH\\]Ogf_07@h,)z]F]p7$$\"AmmmmmmmTN!RH[\"Q!*[Feo$\"A $ouj.D(45H2(*)pa'**\\F]p7$$\"A++++++](=2X/Sz@.&Feo$\"A`C(G#*[sv&R$*[== T))[F]p7$$\"AmmmmmmT&Q89^d#*QK&Feo$\"A]pnZqC`Ej]&RMDAc%F]p7$$\"ALLLLLL L$e>$y\\dg:cFeo$\"Aq!QFz%*p)Q)p7t^c&3TF]p7$$\"A************\\(o\\9G(oa seFeo$\"AA$o_qav([.\\[=+q4OF]p7$$\"Ammmmmmm\"zzXe*z[HhFeo$\"A\"HA%3h$* 4I3\"p,]Xk-$F]p7$$\"ALLLLLL$3xcRS%Gu=kFeo$\"A!=]>Cc\"[B.L:EkK#G#F]p7$$ \"A+++++++]PLB#p(*zq'Feo$\"A#\\D\">>*p(3#yASz*fj9F]p7$$\"A++++++++D[i[ -W)*pFeo$\"A)4HY)4^\"f2g`mo.R)eFgn7$$\"A+++++++]7j,0G)))G(Feo$!A\\Z\\x G)y>%R.X@\")3bJFgn7$$\"A+++++++](['zD!y(ovFeo$!AUc*4JvXud\"Gt$)R3*=\"F ]p7$$\"A+++++++]imdYKn[yFeo$!ApvFNwv9@&GAEG;./#F]p7$$\"ALLLLLL3F>k[@\\ %G5)Feo$!AAA%yT*[%p`X*4\"HhKx#F]p7$$\"Ammmmmm;/whR'f;qN)Feo$!A\"e$fj(= DK5S\"*pcB%\\MF]p7$$\"A***********\\P4hu=I^#f')Feo$!A!\\soa\\w.#p5dsB! f:%F]p7$$\"ALLLLLLL$e/`t+'[h*)Feo$!A!=\"o<]@2Q/$**\\_c;t%F]p7$$\"AKLLL LLLek[3Z%=v@*Feo$!A3,L#R$REeO@\"e\"oG+^F]p7$$\"ALLLLLLLL$o;o)3bt%*Feo$ !AsS*3G*zkN3\\*[L-lV5# eR&F]p7$$\"A**********\\Pf3sp$=kCi*Feo$!A_\"H;Ky`h9'[SxiqKaF]p7$$\"ALL LLLeRAru8K3#pp*Feo$!AYUDT))RHj\\m*[TY$eaF]p7$$\"AmmmmmmT&Qtx0[x8x*Feo$ !AEK&G#3#e:\"3[WLKjsaF]p7$$\"A*********\\P%['*z,HT$e%)*Feo$!AEbQw!*4f( *4<\\3D[vaF]p7$$\"ALLLLL$e9\"f#eux!H?**Feo$!A$Gdsly')*\\KGmVW$oY&F]p7$ $\"Ammmmm\"zW<_)*eUZZ***Feo$!AP#>*o%))GL/?@@0amW&F]p7$$\"A++++++vVyQV2 /#p+\"!#K$!A:!RSl$)>O;QOx[J\\T&F]p7$$\"Ammmmm;aja\"fu5$GL5Fi\\l$!AYxF^ CI'*pw+xjMI5_F]p7$$\"ALLLLLLL$3V%[2ekf5Fi\\l$!AKKQc1D-MbBi#RvZ'[F]p7$$ \"Ammmmm;a8U`nn)p&)3\"Fi\\l$!A(*[hMM!He*zpJ^$QCL%F]p7$$\"A++++++vV`i'y #R\\<6Fi\\l$!A([5uDaYSzVuzP2Zl$F]p7$$\"A++++++](oS7JZO]9\"Fi\\l$!A@to$ H@mIpCw')ykJ*GF]p7$$\"A++++++DJg&e$=!zD<\"Fi\\l$!A&[kv'*o&)4x)fZE?!3/# F]p7$$\"Ammmmm;/,*=?\\x;8?\"Fi\\l$!A%e*yKbc6)*prCM=c#3\"F]p7$$\"ALLLLL L$3x\"=[JX0I7Fi\\l$!Am!3U*fgm'e'>\"RUH)>')FM7$$\"ALLLLL$e9m3iHwWkD\"Fi \\l$\"A;J1[-uW&**p^Zj(GB$)Fgn7$$\"ALLLLLL3_bBW%*\\$GG\"Fi\\l$\"A98qQL. 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'\"4`#FA7$$\":^$HCp+h**fxg;@F*$!'4wIFA7$$\":Lj\"GA@/hmlWJ@F*$!'P[OFA7$ $\":9L?`#F*$!'(\\6&FA7$$\":Q7vuQ-#o*fSc ?#F*$!'?=_FA7$$\":HZ%*RT=*)H+gI@#F*$!'5C_FA7$$\":>#Q^SWjH1%z/A#F*$!'b) >&FA7$$\":5A$)\\kbBF*$ \"'vVMF*7$$\":otH**o#*)G4dzpBF*$\"(7dD\"F*7$$\":Mm&\\JteNNk%RQ#F*$\"(% fY@F*7$$\":+fhI(>GUhr4)R#F*$\"(\\f)HF*7$$\":m^FYhw*[()yC7CF*$\"(S3u$F* 7$$\":KW$>c7nb8')RECF*$\"(^sP%F*7$$\":)p$fx*eOiR$\\0W#F*$\"(#=h[F*7$$ \":jHD$R01pl+qaCF*$\"())*f^F*7$$\":wz&4m7,Iyg3eCF*$\"(p1?&F*7$$\":()Hm G*>'444s9Y#F*$\"(c'G_F*7$$\":**zO'>F\">N5e[Y#F*$\"((fV_F*7$$\":5I2kWjG h6W#oCF*$\"(k^C&F*7$$\":LI[***[wMTh,vCF*$\"(Np?&F*7$$\":dI*[`jmcm\")y \"[#F*$\"(m<6&F*7$$\":.Jr0Ep/q@K`\\#F*$\"()3WZF*7$$\":]J`w;sUuEw)3DF*$ \"(IZ8%F*7$$\":tJ%>@O+u$F*7$$\":(>`tu]2)yJ?C_#F*$\"(=sG $F*7$$\":?Kw#Gl(*4VB>HDF*$\"(c*yFF*7$$\":VK<=)z(=$oV'f`#F*$\"(q*=AF*7$ $\":nKe`VzPNROFa#F*$\"(\"=7;F*7$$\":!H$**)))3ov=%3&\\DF*$\"'P['*F*7$$ \":8LSCM#e(RW!GcDF*$\"'mYGF*7$$\":NL\")fz$[>pC0jDF*$!&+>%FA7$$\":[.()y AhR^$*H.d#F*$!'j)=\"FA7$$\":gt#zf'Q%3,ugxDF*$!'Fc>FA7$$\":sV)p\"4;Hq'[ )[e#F*$!'z-FFA7$$\":&QTgBNR(HLi@f#F*$!'*fS$FA7$$\":'R)4b&4(=*)zR%*f#F* $!'ESSFA7$$\":4a:uQ[j[Ee#33t%*Rh#F*$!'(3)\\FA 7$$\":M%pA^KIv'>s7i#F*$!'D;_FA7$$\":Tzzr'>asH4\"\\i#F*$!'Wd_FA7$$\":Zk KJo!ypi'\\&GEF*$!'1S_FA7$$\":a\\&3*R>qcR)=KEF*$!'+e^FA7$$\":fMQ]6eU'Gr #ej#F*$!'\"[+&FA7$$\":s/WpaN(e%f/Jk#F*$!'YdWFA7$$\":$[(\\)yH@`g?Q]EF*$ !'.RNFA7$$\":!*f-[p^/Nz?Sl#F*$!'[?HFA7$$\":'\\av5/pZE&fwl#F*$!'6%=#FA7 $$\":-I3n7H\\%f#)HhEF*$!'k?8FA7$$\":3:hE%y;U#*p$\\m#F*$!&P?$FA7$$\":9+ 9'elSRDddoEF*$\"'8p#)F*7$$\":?&ocu_kOeW@sEF*$\"(=>8#F*7$$\":Eq>0*R)Q8> `en#F*$\"(aeg$F*7$$\":JbskqA6V#>\\zEF*F+-%&COLORG6&%$RGBG$\"\"%!\"\"\" \"!$\"\"*Fdjp-%+AXESLABELSG6$Q\"x6\"Q!F\\[q-%%VIEWG6$;$!:ObskqA6V#>\\z EF*$\":ObskqA6V#>\\zEF*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "evalf(evalf(numapprox[infnorm](1-h(x)/arctan(x),x=-tan(Pi/12)..tan (Pi/12)),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&0E&!#B" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The maxim um relative error in using " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctan G6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-tan(Pi/12 ), tan(Pi/12)];" "6#7$,$-%$tanG6#*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+ F," }{TEXT -1 11 " is about " }{XPPEDIT 18 0 "5*`. `*10^(-19);" "6#*( \"\"&\"\"\"%#.~GF%)\"#5,$\"#>!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "xx := eval f(tan(Pi/12),20);\nevalf(evalf(h(xx),20),17);\nevalf(evalf(arctan(xx), 20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5Z1F7JC>\\zE!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2W\\\"*z(Q*zh#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2W\\\"*z(Q*zh#!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the ap proximation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " fo r " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 35 " wi th random numbers between 0 and " }{XPPEDIT 18 0 "tan(Pi/12);" "6#-%$t anG6#*&%#PiG\"\"\"\"#7!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The printout occurs only when the relative error is greater tha n or equal to the specified \"" }{TEXT 262 3 "eps" }{TEXT -1 3 "\".\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "randomize():\neps := Fl oat(5,-16);\nfor i from 1 to 100 do\n xx := evalf(rand()/Float(1,12) *.267949192431122706,20);\n axx := evalf(evalf(arctan(xx),20),16); \n hxx := evalf(h(xx),16);\n e := evalf(abs((axx-hxx)/axx),20);\n \+ if e>=eps then \n printf(\" trial no. %d, x = %.16f,\\n\",i,x x);\n printf(\" arctan(x) = %.16f, h(x) = %.16f, rel error = %.2 e\\n\\n\",axx,hxx,e); \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"&!#;" }}{PARA 6 "" 1 "" {TEXT -1 39 " tria l no. 11, x = .1837926968567053," }}{PARA 6 "" 1 "" {TEXT -1 79 " ar ctan(x) = .1817641640400988, h(x) = .1817641640400987, rel error = 5.5 0e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 30, x = .1220989350101880," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .1214975488374267, h(x) = .1214975488374266, rel err or = 8.23e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 33, x = .1200273744093992," }}{PARA 6 "" 1 " " {TEXT -1 79 " arctan(x) = .1194559117446262, h(x) = .11945591174462 63, rel error = 8.37e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 36, x = .1108426537503180," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .1103920298757554, h(x) = .110392 0298757555, rel error = 9.06e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 43, x = .1555442963011861," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .1543077826623356, h(x) = .1543077826623357, rel error = 6.48e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 45, x = .104343417246 8945," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .1039671905710525, h(x) = .1039671905710526, rel error = 9.62e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 83, x = .158 1236549868607," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .15682521 89675337, h(x) = .1568252189675336, rel error = 6.38e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 87, x = .1702664211996971," }}{PARA 6 "" 1 "" {TEXT -1 79 " arctan(x) = .1 686490836384884, h(x) = .1686490836384883, rel error = 5.93e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 91, x = .1493614647551838," }}{PARA 6 "" 1 "" {TEXT -1 79 " arcta n(x) = .1482654048411055, h(x) = .1482654048411054, rel error = 6.74e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "arctan16" }}{PARA 0 "" 0 "" {TEXT -1 140 "Here is the c ode for the fixed precision version of the arctangent function, which \+ can be evaluated with hardware floating point arithmetic." }}{PARA 0 " " 0 "" {TEXT 259 4 "Note" }{TEXT -1 61 ": Argument reduction is achiev ed by means of the formulas: \n " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x) = Pi/2-arctan(1/x);" "6#/-%'arctanG6#%\"xG,&* &%#PiG\"\"\"\"\"#!\"\"F+-F%6#*&F+F+F'F-F-" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x)=arctan(x[0]) + ar ctan((x-x[0])/(1+x*x[0]))" "6#/-%'arctanG6#%\"xG,&-F%6#&F'6#\"\"!\"\" \"-F%6#*&,&F'F.&F'6#F-!\"\"F.,&F.F.*&F'F.&F'6#F-F.F.F5F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1353 "arctan16 := proc(xx)\n local piBy2,piBy3,piBy6,a1 ,a2,a3,a4,a5,b1,b2,b3,\n x,z,t,num,den,flag,isneg,val;\n\n # coefficients of numerator in rational approximation\n a1 := .333333 33333333273025;\n a2 := .39463041876833004067;\n a3 := .1023549039 6768294247;\n a4 := .10411605479799355376e-2;\n\n # coefficients o f denominator in rational approximation\n b1 := 1.783891256304325820 7;\n b2 := .94882803719677902342;\n b3 := .14122880827773776789;\n \n piBy2 := 1.5707963267948966192;\n piBy3 := 1.047197551196597746 2;\n piBy6 := .52359877559829887308;\n x := evalf(xx);\n \n # Ha ndle the argument reduction\n if x<0 then\n isneg := true;\n \+ x := -x;\n else isneg := false end if;\n\n if x<0.2679491924311 22 then\n flag := 1;\n elif xx>3.732050807568877 then\n fl ag := 4;\n x := 1/x;\n else\n t := .5773502691896258; # 1/ sqrt(3) \n if x<1 then\n flag := 2;\n x := (x-t )/(1+x*t);\n else\n flag := 3;\n x := (1-x*t)/(x+ t);\n end if;\n end if;\n \n # evaluate the rational approx imation\n z := x*x;\n num := (a1+(a2+(a3+a4*z)*z)*z)*z;\n den := 1+(b1+(b2+b3*z)*z)*z;\n val := (1-num/den)*x;\n\n if flag=2 then \+ val := val+piBy6\n elif flag=3 then val := piBy3-val\n elif flag=4 then val := piBy2-val end if;\n if isneg then val := -val end if;\n val;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Testing the procedure arctan16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "We can use Maple's inverse tang ent function " }{TEXT 262 9 "arctan(x)" }{TEXT -1 40 " to check the ac curacy of the procedure " }{TEXT 0 8 "arctan16" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xx := 2.;\narctan16(xx);\narctan(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+<([r5\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=([r5\"!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf(plot(arctan(x)-'arctan16'(x),x=0..10,color=blue),20);" }} {PARA 13 "" 1 "" {GLPLOT2D 416 224 224 {PLOTDATA 2 "6&-%'CURVESG6#7\\q 7$$\"\"!F)F(7$$\"5lmmmmT&)G\\a!#@$!#CF-7$$\"5LLLLL3x&)*3\"!#?$\"\"%F37 $$\"5++++]ilyM;F3$!\"$F37$$\"5mmmmm;arz@F3$!#6F37$$\"5*****\\7yD&y;CF3 $\"\"*F37$$\"5LLL$e*)4bQl#F3$!\"(F37$$\"5mm\"zW#fD78FF3$\"%uEF37$$\"5* ***\\7`>+RsFF3$\"%oEF37$$\"5LL3x\")zulJGF3$\"%aEF37$$\"5mmmT5S\\#4*GF3 $\"%[EF37$$\"5LL$3x1')f%4IF3$\"%hEF37$$\"5*******\\7y%*z7$F3$\"%tEF37$ $\"5LLLeRAY1lLF3$\"%gEF37$$\"5mmm;ajW8-OF3FW7$$\"5*****\\(o/V?RQF3$\"% lEF37$$\"5LLLL$e9ui2%F3FR7$$\"5*******\\(oMrU^F3$\"%kEF37$$\"5mmmmm\"z _\"4iF3F[o7$$\"5lmmmmm6m#G(F3$\"%nEF37$$\"5lmmmmT&phN)F3$\"%mEF37$$\"5 ILL$3-js.*))F3Feo7$$\"5(******\\(=ddC%*F3$\"%jEF37$$\"5ILL3-jsn\"p*F3$ \"%rEF37$$\"5jmm;H2)y(e**F3$\"%dEF37$$\"5+]PfL>ab-5!#>$!%mEF37$$\"5LL3 F%z&HB45Fir$!%aEF37$$\"5m;z%\\l\\5f,\"Fir$!%eEF37$$\"5++]i:N!)eA5Fir$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(codegen,C):\nC(arctan16,ansi);" }}{PARA 6 "" 1 " " {TEXT -1 17 "#include " }}{PARA 6 "" 1 "" {TEXT -1 26 "doubl e arctan16(double xx)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " doubl e a2;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a3;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a4;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a5; " }}{PARA 6 "" 1 "" {TEXT -1 12 " double b1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b2;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b3; " }}{PARA 6 "" 1 "" {TEXT -1 13 " double den;" }}{PARA 6 "" 1 "" {TEXT -1 11 " int flag;" }}{PARA 6 "" 1 "" {TEXT -1 12 " int isneg; " }}{PARA 6 "" 1 "" {TEXT -1 13 " double num;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double piBy2;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double \+ piBy3;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double piBy6;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double t;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double val;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double x;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 28 " a1 = 0.3333333333333327;" }}{PARA 6 "" 1 " " {TEXT -1 26 " a2 = 0.39463041876833;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a3 = 0.1023549039676829;" }}{PARA 6 "" 1 "" {TEXT -1 31 " a 4 = 0.1041160547979936E-2;" }}{PARA 6 "" 1 "" {TEXT -1 30 " b1 = 0. 1783891256304326E1;" }}{PARA 6 "" 1 "" {TEXT -1 27 " b2 = 0.9488280 37196779;" }}{PARA 6 "" 1 "" {TEXT -1 28 " b3 = 0.1412288082777378; " }}{PARA 6 "" 1 "" {TEXT -1 33 " piBy2 = 0.1570796326794897E1;" }} {PARA 6 "" 1 "" {TEXT -1 33 " piBy3 = 0.1047197551196598E1;" }} {PARA 6 "" 1 "" {TEXT -1 31 " piBy6 = 0.5235987755982989;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }}{PARA 6 "" 1 "" {TEXT -1 17 " if( x < 0.0 )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }} {PARA 6 "" 1 "" {TEXT -1 18 " isneg = 1;" }}{PARA 6 "" 1 "" {TEXT -1 15 " x = -x;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 16 " isneg = 0;" }}{PARA 6 "" 1 "" {TEXT -1 31 " if( x < 0.267 949192431122 )" }}{PARA 6 "" 1 "" {TEXT -1 15 " flag = 1;" }} {PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 37 " if( 0.3732050807568877E1 < xx )" }}{PARA 6 "" 1 "" {TEXT -1 9 " \+ \{" }}{PARA 6 "" 1 "" {TEXT -1 19 " flag = 4;" }} {PARA 6 "" 1 "" {TEXT -1 18 " x = 1/x;" }}{PARA 6 "" 1 "" {TEXT -1 9 " \}" }}{PARA 6 "" 1 "" {TEXT -1 11 " else " }} {PARA 6 "" 1 "" {TEXT -1 9 " \{" }}{PARA 6 "" 1 "" {TEXT -1 33 " t = 0.5773502691896258;" }}{PARA 6 "" 1 "" {TEXT -1 23 " \+ if( x < 1.0 )" }}{PARA 6 "" 1 "" {TEXT -1 13 " \{" } }{PARA 6 "" 1 "" {TEXT -1 23 " flag = 2;" }}{PARA 6 "" 1 "" {TEXT -1 34 " x = (x-t)/(1.0+x*t);" }}{PARA 6 "" 1 "" {TEXT -1 13 " \}" }}{PARA 6 "" 1 "" {TEXT -1 15 " \+ else " }}{PARA 6 "" 1 "" {TEXT -1 13 " \{" }}{PARA 6 "" 1 " " {TEXT -1 23 " flag = 3;" }}{PARA 6 "" 1 "" {TEXT -1 34 " x = (1.0-x*t)/(x+t);" }}{PARA 6 "" 1 "" {TEXT -1 13 " \+ \}" }}{PARA 6 "" 1 "" {TEXT -1 9 " \}" }}{PARA 6 "" 1 "" {TEXT -1 12 " z = x*x;" }}{PARA 6 "" 1 "" {TEXT -1 36 " num = (a1+(a2+(a3+a4*z)*z)*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 33 " den \+ = 1.0+(b1+(b2+b3*z)*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 26 " val = (1 .0-num/den)*x;" }}{PARA 6 "" 1 "" {TEXT -1 21 " if( flag == 2.0 )" }}{PARA 6 "" 1 "" {TEXT -1 19 " val += piBy6;" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 23 " if( flag = = 3.0 )" }}{PARA 6 "" 1 "" {TEXT -1 24 " val = piBy3-val;" }} {PARA 6 "" 1 "" {TEXT -1 11 " else " }}{PARA 6 "" 1 "" {TEXT -1 25 " if( flag == 4.0 )" }}{PARA 6 "" 1 "" {TEXT -1 26 " \+ val = piBy2-val;" }}{PARA 6 "" 1 "" {TEXT -1 17 " if( (isneg) )" }}{PARA 6 "" 1 "" {TEXT -1 17 " val = -val;" }}{PARA 6 "" 1 "" {TEXT -1 16 " return(val);" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }} {PARA 6 "" 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "More fixed precision procedures" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 38 "A 25 digit rational approximation for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[ -tan(Pi/28),tan(Pi/28)]" "6#7$,$-%$tanG6#*&%#PiG\"\"\"\"#G!\"\"F,-F&6# *&F)F*F+F," }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We construct a rational approximat ion for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 77 " on the interval [-0.1127,0.1127], which is slightly wider than th e interval " }{XPPEDIT 18 0 "[-tan(Pi/28), tan(Pi/28)];" "6#7$,$-%$tan G6#*&%#PiG\"\"\"\"#G!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "We shall approximate " }{XPPEDIT 18 0 "f(x) = P IECEWISE([(x-arctan(x))/(x^3), x <> 0], [1/3, x = 0])" "6#/-%\"fG6#%\" xG-%*PIECEWISEG6$7$*&,&F'\"\"\"-%'arctanG6#F'!\"\"F.*$F'\"\"$F20F'\"\" !7$*&F.F.F4F2/F'F6" }{TEXT -1 37 ", and then use the rational function " }{XPPEDIT 18 0 "s(x) = x-r(x)*`.`*x^3" "6#/-%\"sG6#%\"xG,&F'\"\"\"* (-%\"rG6#F'F)%\".GF)F'\"\"$!\"\"" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "evalf(remez((x-arctan(x))/x^3,x=-0.1127..0.1127,[6,8],\n 'max err',weight=x^2+1e-40,type=even,errtype=absolute,\n maxgra ph=7,info=true),35):\nr := unapply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"2(G^)4*R()pQ!#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,*$\" DanP8y,LLLLLLLLLLL$!#N\"\"\"*&$\"D'3dZ:K]#f%Q\"zU4++q/&F0F1)9$\"\"#F1F 1*&$\"Do.3'e^&=M'[^$*ezeYA@F0F1)F6\"\"%F1F1*&$\"DaPD7YjT1*3*Q(H(\\739# !#OF1)F6\"\"'F1F1F1,,$F1\"\"!F1*&$\"DQ3*f&H]MN:u$GG++59@!#MF1F5F1F1*&$ \"DQv38**p.,X$z]ZN$GmZ\"FIF1F;F1F1*&$\"D*>U6Cp)[Z2Q?KF_=\\x$F0F1FAF1F1 *&$\"D_AUMw7oEryEYHN7Fc#F@F1)F6\"\")F1F1!\"\"F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"DQ2V%[L " 0 "" {MPLTEXT 1 0 486 "s:=x->x-(.33333333333333333333333017813376754+.50470 000009427913845925032154757086*x^2+.2122465879589351486341855158608036 8*x^4+.21408124972973890890641634612253754e-1*x^6)/(1.+2.1141000002828 374153534502955990838*x^2+1.4766283354750793450103699913087538*x^4+.37 749185227322038074748869241142199*x^6+.2562712352946267871266812763442 2252e-1*x^8)*x^3:\nh := unapply(evalf(convert(s(x),horner),30),x);\nev alf(plot((1-h(x)/arctan(x)),\n x=-tan(Pi/28)..tan(Pi/28),color=COLOR(R GB,.4,0,.9)),35);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&9$\"\"\"**$F.\"\"!F.,*$\"?M\"y,LLLLLLLLLLL$ !#IF.*&$\"?[:K]#f%Q\"zU4++q/&F5F.)F-\"\"#F.F.*&$\"?he^&=M'[^$*ezeYA@F5 F.)F-\"\"%F.F.*&$\"?BhM;k!*3*Q(H(\\739#!#JF.)F-\"\"'F.F.F.,,$F.F1F.*&$ \"?g&H]MN:u$GG++59@!#HF.F9F.F.*&$\"?J\"**p.,X$z]ZN$GmZ\"FKF.F>F.F.*&$ \"?6Cp)[Z2Q?KF_=\\x$F5F.FDF.F.*&$\"?WjF\"oEryEYHN7Fc#FCF.)F-\"\")F.F.! \"\"F-\"\"$FWF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7a_l7$$!DwjU)4XI`p\\56+*RHn7\"!#N$!(+>f$!#M 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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2019 "arctan25 := proc(xx)\n local t1 ,t2,t3,a1,a2,a3,a4,b1,b2,b3,b4,x,z,piBy2,\n isneg,invert,n,d,v;\n \n # coefficients in rational approximation\n a1 := .3333333333333 33333333330178134;\n a2 := .504700000094279138459250321548;\n a3 : = .212246587958935148634185515861;\n a4 := .214081249729738908906416 346123e-1;\n b1 := 2.11410000028283741535345029560;\n b2 := 1.4766 2833547507934501036999131;\n b3 := .377491852273220380747488692411; \n b4 := .256271235294626787126681276344e-1;\n\n piBy2 := 1.570796 32679489661923132169164;\n x := evalf(xx);\n\n # argument reductio n to interval [0,1]\n if x<0 then\n isneg := true;\n x := \+ -x;\n else isneg := false end if;\n\n if x>1 then\n invert := true;\n x := 1/x;\n else invert := false end if;\n \n if x <=.3499151339 then # tan(3*Pi/28)\n if x<=.1126729399 then # tan( Pi/28)\n z := x*x;\n n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n v := (1-n/d)*x\n \+ else\n t1 := .228243474390149938077611362061; # tan(Pi/14 )\n x := (x-t1)/(1+x*t1);\n z := x*x;\n n := ( a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z)*z)*z)*z; \n v := (1-n/d)*x;\n v := v + .22439947525641380274733 1670234; # Pi/14\n end if;\n else \n if x<=.6283416454 the n # tan(5*Pi/28)\n t2 := .481574618807528644332162353057; # ta n(Pi/7)\n x := (x-t2)/(1+x*t2);\n z := x*x;\n \+ n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z)*z)* z)*z;\n v := (1-n/d)*x;\n v := v + .448798950512827605 494663340469; # Pi/7\n else\n t3 := .7974733888824039614 15688254214; # tan(3*Pi/14)\n x := (x-t3)/(1+x*t3);\n \+ z := x*x;\n n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+( b1+(b2+(b3+b4*z)*z)*z)*z;\n v := (1-n/d)*x;\n v := v + .673198425769241408241995010703; # 3*Pi/7\n end if;\n end if; \n\n if invert then v := piBy2-v end if;\n if isneg then v := -v e nd if;\n v;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Testing the procedure arctan25 " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := evalf(tan(2/3),30) ;\nevalf(evalf(arctan25(xx),30),27);\nevalf(evalf(arctan(xx),30),27); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?=bayo'*Ht(HZ*)G%oy!#I" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 76 "xx := 77.;\nevalf( evalf(arctan25(xx),30),27);\nevalf(evalf(arctan(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#x\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 100 "xx := evalf(tan(Pi/2-0.0001),30); \nevalf(evalf(arctan25(xx),30),27);\nevalf(evalf(Pi/2-0.0001,30),27); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?GBWWWWkmmmm********!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"m*[zEjpq:!#E" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"m*[zEjpq:!#E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "evalf(plot(1-'arc tan25'(x)/arctan(x),x=0..1.2,\n color=COLOR(R GB,.4,0,.9)),35);" }}{PARA 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" }}{PARA 0 "" 0 "" {TEXT -1 21 "We shall approximate " }{XPPEDIT 18 0 "f(x) = PIECEWISE([ (x-arctan(x))/(x^3), x <> 0], [1/3, x = 0])" "6#/-%\"fG6#%\"xG-%*PIECE WISEG6$7$*&,&F'\"\"\"-%'arctanG6#F'!\"\"F.*$F'\"\"$F20F'\"\"!7$*&F.F.F 4F2/F'F6" }{TEXT -1 37 ", and then use the rational function " } {XPPEDIT 18 0 "s(x) = x-r(x)*`.`*x^3" "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-% \"rG6#F'F)%\".GF)F'\"\"$!\"\"" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 2 ". 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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur ve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14 " "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "C urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33 " "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" }} }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%-iteration~27G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------- -----------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisio nal~rational~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,4$ \"fpB(3oKI%ze>_ng<%=0d&[?LLLLLLLLLLLLLLLLLLLLLLLLLL$!#\"*\"\"\"*&$\"fp 3D$o?&yR=%H%*[%Q6W<%oW!39f(z\"*ymUCE!R&pwDAmWM4h&HV!Q8!#!*F()%\"xG\"\" #F(F(*&$\"fpePw&)pV%z2-5lPR7*)H[OOuJ3,=+#p8k(Rqk&y>t1ilj)*H)z-AF,F()F. \"\"%F(F(*&$\"fpF,F 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" }}{PARA 0 "" 0 "" {TEXT -1 38 "The relat ive error graph is drawn for " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG " }{TEXT -1 11 ", which is " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 72 " with the coefficients rounded to 60 digits and arranged \+ in nested form." }}{PARA 0 "" 0 "" {TEXT -1 29 "The values are multipl ied by " }{XPPEDIT 18 0 "10^30" "6#*$\"#5\"#I" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1552 " s:=x->x-(.33333333333333333333333333333333333333333333333333333 20485570518417607+1.33804329561093444662225766953902624426678917975914 0804468417441138449*x^2+2.20279829986365620673197856470397641369200180 1083174363648298912393765*x^4+1.91543892645691060677868393548546347423 8993968118660677402447311302726*x^6+.943575701655984800790764658278602 9007438768632770717249614757581400043*x^8+.262052799245168409244983824 8787386694231810240616438037925291120794985*x^10+.38312535271866582594 23800208143581927766637466520099009640514522568501e-1*x^12+.2492817096 851094157418621160414626126297079159050094472712705119328854e-2*x^14+. 4679904289164285675134214248117712197879573314630553590882527261489555 e-4*x^16)/(1.+4.614129886832803339866773008617078732800367539277377869 422005696799493*x^2+8.948301403119222052687428070710747909327654498425 757006809169242893049*x^4+9.471146717361539811053224978523138759018178 615361781697938491112279441*x^6+5.943744842646016271487718757028267799 210930476051597129331349692831904*x^8+2.248479729550878428044741522514 445948889135405850391463131708547609150*x^10+.498108498495833185840412 8219393935840119220854778261610798191104938660*x^12+.59554889994449490 17881934418548353544119285255167276915519051417211526e-1*x^14+.3222623 128935314342085119203060410504826213521882606259296108546567759e-2*x^1 6+.5088402557363804947954347103724288524341795209751840924668281147785 650e-4*x^18)*x^3:\nh := unapply(evalf(convert(s(x),horner),60),x);\nev alf(plot(10^30*(1-s(x)/arctan(x)),x=-tan(Pi/28)..tan(Pi/28),color=COLO R(RGB,.4,0,.9)),65);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,&9$\"\"\"**$F.\"\"!F.,4$\"gnd&[?LLLLLLLLLL LLLLLLLLLLLLLLLL$!#gF.*&$\"gnUoW!39f(z\"*ymUCE!R&pwDAmWM4h&HV!Q8!#fF.) 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,a5,a6,a7,a8,a9,\n b1,b2,b3,b4,b5,b6,b7,b8,b9,x,z,piBy2,isneg,inv ert,n,d,v;\n\n # coefficients in rational approximation\n a1 :=\n \+ .333333333333333333333333333333333333333333333333333332048557;\n \+ a2 :=\n 1.3380432956109344466222576695390262442667891797591408044 6842;\n a3 :=\n 2.2027982998636562067319785647039764136920018010 8317436364830;\n a4 :=\n 1.9154389264569106067786839354854634742 3899396811866067740245;\n a5 := \n .9435757016559848007907646582 78602900743876863277071724961476;\n a6 := \n .262052799245168409 244983824878738669423181024061643803792529;\n a7 := \n .383125352 718665825942380020814358192776663746652009900964051e-1;\n a8 := \n \+ .249281709685109415741862116041462612629707915905009447271271e-2;\n \+ a9 := \n .4679904289164285675134214248117712197879573314630553590 88253e-4;\n\n b1 := \n 4.6141298868328033398667730086170787328003 6753927737786942201;\n b2 := \n 8.9483014031192220526874280707107 4790932765449842575700680917;\n b3 := \n 9.4711467173615398110532 2497852313875901817861536178169793849;\n b4 := \n 5.9437448426460 1627148771875702826779921093047605159712933135;\n b5 := \n 2.2484 7972955087842804474152251444594888913540585039146313171;\n b6 := \n \+ .498108498495833185840412821939393584011922085477826161079819;\n \+ b7 := \n .595548899944494901788193441854835354411928525516727691551 905e-1;\n b8 := \n .322262312893531434208511920306041050482621352 188260625929611e-2;\n b9 := \n .508840255736380494795434710372428 852434179520975184092466828e-4;\n\n piBy2 := \n 1.570796326794896 61923132169163975144209858469968755291048747;\n x := evalf(xx);\n\n \+ # argument reduction to interval [0,1]\n if x<0 then\n isneg \+ := true;\n x := -x;\n else isneg := false end if;\n\n if x>1 \+ then\n invert := true;\n x := 1/x;\n else invert := false \+ end if;\n \n if x<=.3499151339 then # tan(3*Pi/28)\n if x<=.1 126729399 then # tan(Pi/28)\n z := x*x;\n n := (a1+(a2 +(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n z)*z) *z)*z)*z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+(b4+(b5+(b6+(b7+(b8+b 9*z)*\n z)*z)*z)*z)*z)*z)*z)*z);\n \+ v := (1-n/d)*x;\n else \n t1 := # tan(Pi/14)\n .228 243474390149938077611362061014782738781680980352637979689;\n x := (x-t1)/(1+x*t1);\n z := x*x;\n n := (a1+(a2+(a3+(a 4+(a5+(a6+(a7+(a8+a9*z)*\n z)*z)*z)*z)* z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+(b4+(b5+(b6+(b7+(b8+b9*z)*\n z)*z)*z)*z)*z)*z)*z)*z);\n v : = (1-n/d)*x;\n v := v + \n .224399475256413802747331670234 250206014083528526793272926782;\n end if;\n else \n if x<= .6283416454 then # tan(5*Pi/28)\n t2 := # tan(Pi/7)\n .481 574618807528644332162353056970575219078891752299935554205;\n x := (x-t2)/(1+x*t2);\n z := x*x;\n n := (a1+(a2+(a3+(a 4+(a5+(a6+(a7+(a8+a9*z)*\n z)*z)*z)*z)* z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+(b4+(b5+(b6+(b7+(b8+b9*z)*\n z)*z)*z)*z)*z)*z)*z)*z);\n v : = (1-n/d)*x;\n v := v + \n .448798950512827605494663340468 500412028167057053586545853564;\n else\n t3 := # tan(3*P i/14)\n .797473388882403961415688254214434877743320657436834294264 606;\n x := (x-t3)/(1+x*t3);\n z := x*x;\n n : = (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n \+ z)*z)*z)*z)*z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+(b4+(b5+(b6+ (b7+(b8+b9*z)*\n z)*z)*z)*z)*z)*z)*z)*z );\n v := (1-n/d)*x;\n v := v + \n .6731984257692 41408241995010702750618042250585580379818780345;\n end if;\n e nd if;\n\n if invert then v := piBy2-v end if;\n if isneg then v : = -v end if;\n v;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 31 "Testing the procedure arctan55 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := evalf( tan(2/3),65);\nevalf(evalf(arctan55(xx),60),58);\nevalf(evalf(arctan(x x),60),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\\o6 " 0 "" {MPLTEXT 1 0 76 "xx := 77.;\nevalf(evalf( arctan55(xx),60),58);\nevalf(evalf(arctan(xx),60),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#x\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"en2vG=j:]uYBx,SxY&ec3JNTsuQ/5yb\"!#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"en2vG=j:]uYBx,SxY&ec3JNTsuQ/5yb\"!#d" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "xx := \+ evalf(tan(Pi/2-0.0001),60);\nevalf(evalf(arctan55(xx),60),58);\nevalf( evalf(Pi/2-0.0001,60),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$ \"gnGQ?3g9)yBf#f#fKU!GBWWWWkmmmm********!#c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"en([5Hb(o*p%e)4U9vR;p@8B>m*[zEjpq:!#d" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"en([5Hb(o*p%e)4U9vR;p@8B>m*[zEjpq:!#d" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "evalf(plot(10^30*(1-'arctan55'(x)/arctan(x)),x=0..1.2,\n \+ color=COLOR(RGB,.4,0,.9)),65);" }}{PARA 13 "" 1 "" {GLPLOT2D 658 209 209 {PLOTDATA 2 "6&-%'CURVESG6#7^im7$$\"\\o+++++++++ ++++++++++++++++++D\"G$R<)!#o$!%=&)!#M7$$\"\\o++++++++++++++++++++++++ +++Dc'yM;!#n$!&(GLF-7$$\"\\o++++++++++++++++++++++++++]P%)z@X#F1$!&:?( F-7$$\"\\o+++++++++++++++++++++++++++]7t&pKF1$!'#4@\"F-7$$\"\\o+++++++ 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1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "A proce dure for evaluating the the inverse tangent function: " }{TEXT 0 7 "ar ctan_" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "arctan_: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 20 "Callin g Sequence: " }{TEXT -1 13 "arctan_( x ) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 265 21 "x - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }} {PARA 0 "" 0 "" {TEXT 268 12 "Description:" }{TEXT -1 1 " " }{TEXT 267 14 "The procedure " }{TEXT 0 7 "arctan_" }{TEXT 266 46 " calculate s the arctan(x) for a real number x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How t o activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure ac tive open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "arctan_: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11301 "arctan_ := proc(x::algebraic)\n local t;\n\n if nargs <> 1 then\n error \"expecting 1 argument, got %1\", nargs;\n end if;\n i f type(x,'float') then evalf('arctan_'(x))\n elif (type(x,'numeric') or type(x,'radnum')) \n and signum(0,x,0)<0 \+ then -arctan_(-x)\n elif type(x,`*`) and type(op(1,x),'numeric') \n \+ and signum(0,op(1,x),0)<0 then -arctan_(-x)\n eli f type(x,'function') and nops(x)=1 then\n t := op(1,x);\n if op(0,x)='tan_' or op(0,x)='tan' then t\n else 'arctan_'(x)\n \+ end if;\n elif type(x,'realcons') then\n if x=0 then 0\n \+ elif x='infinity' then Pi/2\n elif x=-'infinity' then -Pi/2\n \+ elif x=1 then Pi/4\n elif x=sqrt(3) then Pi/3\n elif x=sq rt(3)/3 then Pi/6\n elif x=1+sqrt(2) then 3*Pi/8\n elif x=sq rt(2)-1 then Pi/8\n elif x=2+sqrt(3) then 5*Pi/12\n elif x=2 -sqrt(3) then Pi/12\n else 'arctan_'(x)\n end if\n else 'a rctan_'(x)\n end if;\nend proc: # of arctan_\n\n`evalf/arctan_` := p roc(xx)\n local x,z,saveDigits,eps,maxit,isneg,flag,\n even,k,s um,term,pow,t,hfDigits;\n \n if xx=0 then return xx end if;\n if not type(xx,realcons) then return 'arctan_'(xx) end if;\n\n # Use t he fixed precision procedure for low precision values\n hfDigits := \+ trunc(evalhf(Digits));\n if Digits<=hfDigits then\n x := evalf( xx,hfDigits+2);\n return evalf(evalhf(arctan16(x)));\n end if; \n \n\011\011 # Increase precision for the computation by a few digit s\n\011\011 saveDigits := Digits;\n Digits := Digits + length(Digits )+2;\n x := evalf(xx);\n\n if saveDigits<27 then\n sum := arc tan25(x);\n elif saveDigits<57 then\n sum := arctan55(x);\n e lse \n if x<0 then\n isneg := true;\n x := -x;\n \+ else\n isneg := false\n end if;\n\n if x<0.2679 49192431122 then\n flag := 1;\n elif x>3.73205080756888 t hen\n flag := 4;\n x := 1/x;\n else\n t : = evalf(root3)/3; \n if x<1 then\n flag := 2;\n \+ x := (x-t)/(1+x*t);\n else\n flag := 3; \n x := (1-x*t)/(x+t);\n end if;\n end if;\n \+ \n # Initialisation for Maclaurin series loop.\n eps := Fl oat(5,-Digits);\n maxit := Digits*3;\n\n pow := x;\n z \+ := x*x;\n sum := x;\n even := false; \n for k from 3 to maxit by 2 do\n pow := pow*z;\n term := pow/k;\n \+ if even then\n sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs(term)<=eps*ab s(sum) then break end if;\n even := not even;\n end do;\n \n if flag=2 then sum := sum + evalf(Pi)/6\n elif flag=3 the n sum := evalf(Pi)/3-sum\n elif flag=4 then sum := evalf(Pi)*0.5- sum end if;\n if isneg then sum := -sum end if;\n end if;\n\n \+ # Return arctangent rounded to the original precision\n Digits := s aveDigits;\n return evalf(sum)\nend proc: # `evalf/arctan_`\n\n`eval f/constant/root3` := proc()\nlocal d,r;\nglobal _root3;\n if Digits< =55 then evalf(1.73205080756887729352744634150587236694280525381038062 8)\n elif Digits<=length(op(1,_root3)) then evalf(_root3)\n else\n d := length(op(1,_root3));\n r := _root3;\n while d1 then\n invert := true;\n x := 1/ x;\n else invert := false end if;\n \n if x<=.3499151339 then # \+ tan(3*Pi/28)\n if x<=.1126729399 then # tan(Pi/28)\n z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n \+ z)*z)*z)*z)*z)*z)*z)*z;\n d := (1+(b1+( b2+(b3+(b4+(b5+(b6+(b7+(b8+b9*z)*\n z)* z)*z)*z)*z)*z)*z)*z);\n v := (1-n/d)*x;\n else \n \+ t1 := # tan(Pi/14)\n .228243474390149938077611362061014782738781680 980352637979689;\n x := (x-t1)/(1+x*t1);\n z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n \+ z)*z)*z)*z)*z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+ (b4+(b5+(b6+(b7+(b8+b9*z)*\n z)*z)*z)*z )*z)*z)*z)*z);\n v := (1-n/d)*x;\n v := v + \n .22 4399475256413802747331670234250206014083528526793272926782;\n end if;\n else \n if x<=.6283416454 then # tan(5*Pi/28)\n \+ t2 := # tan(Pi/7)\n .481574618807528644332162353056970575219078891 752299935554205;\n x := (x-t2)/(1+x*t2);\n z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n \+ z)*z)*z)*z)*z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+ (b4+(b5+(b6+(b7+(b8+b9*z)*\n z)*z)*z)*z )*z)*z)*z)*z);\n v := (1-n/d)*x;\n v := v + \n .44 8798950512827605494663340468500412028167057053586545853564;\n el se\n t3 := # tan(3*Pi/14)\n .79747338888240396141568825421 4434877743320657436834294264606;\n x := (x-t3)/(1+x*t3);\n \+ z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+a9*z)*\n z)*z)*z)*z)*z)*z)*z)*z;\n d := (1+(b1+(b2+(b3+(b4+(b5+(b6+(b7+(b8+b9*z)*\n \+ z)*z)*z)*z)*z)*z)*z)*z);\n v := (1-n/d)*x;\n v : = v + \n .6731984257692414082419950107027506180422505855803798187 80345;\n end if;\n end if;\n\n if invert then v := piBy2-v e nd if;\n if isneg then v := -v end if;\n v;\nend proc: # of arctan 55\n\narctan25 := proc(xx)\n local t1,t2,t3,a1,a2,a3,a4,b1,b2,b3,b4, x,z,piBy2,\n isneg,invert,n,d,v;\n\n # coefficients in rational approximation\n a1 := .333333333333333333333330178134;\n a2 := .5 04700000094279138459250321548;\n a3 := .2122465879589351486341855158 61;\n a4 := .214081249729738908906416346123e-1;\n b1 := 2.11410000 028283741535345029560;\n b2 := 1.47662833547507934501036999131;\n \+ b3 := .377491852273220380747488692411;\n b4 := .25627123529462678712 6681276344e-1;\n\n piBy2 := 1.57079632679489661923132169164;\n x : = evalf(xx);\n\n # argument reduction to interval [0,1]\n if x<0 t hen\n isneg := true;\n x := -x;\n else isneg := false end \+ if;\n\n if x>1 then\n invert := true;\n x := 1/x;\n else invert := false end if;\n \n if x<=.3499151339 then # tan(3*Pi/28 )\n if x<=.1126729399 then # tan(Pi/28)\n z := x*x;\n \+ n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z )*z)*z)*z;\n v := (1-n/d)*x\n else\n t1 := .22824 3474390149938077611362061; # tan(Pi/14)\n x := (x-t1)/(1+x*t1) ;\n z := x*x;\n n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n \+ d := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n v := (1-n/d)*x;\n \+ v := v + .224399475256413802747331670234; # Pi/14\n end if;\n else \n if x<=.6283416454 then # tan(5*Pi/28)\n t2 := \+ .481574618807528644332162353057; # tan(Pi/7)\n x := (x-t2)/(1+ x*t2);\n z := x*x;\n n := (a1+(a2+(a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n v := (1-n/d)*x;\n v := v + .448798950512827605494663340469; # Pi/7\n else \n t3 := .797473388882403961415688254214; # tan(3*Pi/14)\n \+ x := (x-t3)/(1+x*t3);\n z := x*x;\n n := (a1+(a2+ (a3+a4*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n \+ v := (1-n/d)*x;\n v := v + .673198425769241408241995010703; # 3*Pi/7\n end if;\n end if;\n\n if invert then v := piBy2- v end if;\n if isneg then v := -v end if;\n v;\nend proc: # of arc tan25 \n\n# procedure for default and/or low precision\narctan16 := p roc(xx)\n local piBy2,piBy3,piBy6,a1,a2,a3,a4,a5,b1,b2,b3,\n \+ x,z,t,num,den,flag,isneg,val;\n\n # coefficients of numerator in ra tional approximation\n a1 := .33333333333333273025;\n a2 := .39463 041876833004067;\n a3 := .10235490396768294247;\n a4 := .104116054 79799355376e-2;\n\n # coefficients of denominator in rational approx imation\n b1 := 1.7838912563043258207;\n b2 := .948828037196779023 42;\n b3 := .14122880827773776789;\n\n piBy2 := 1.5707963267948966 192;\n piBy3 := 1.0471975511965977462;\n piBy6 := .523598775598298 87308;\n x := evalf(xx);\n \n # Handle the argument reduction\n \+ if x<0 then\n isneg := true;\n x := -x;\n else isneg := fa lse end if;\n\n if x<0.267949192431122 then\n flag := 1;\n el if xx>3.732050807568877 then\n flag := 4;\n x := 1/x;\n el se\n t := .5773502691896258; # 1/sqrt(3) \n if x<1 then\n \+ flag := 2;\n x := (x-t)/(1+x*t);\n else\n \+ flag := 3;\n x := (1-x*t)/(x+t);\n end if;\n end if;\n \n # evaluate the rational approximation\n z := x*x;\n num := (a1+(a2+(a3+a4*z)*z)*z)*z;\n den := 1+(b1+(b2+b3*z)*z)*z;\n val : = (1-num/den)*x;\n\n if flag=2 then val := val+piBy6\n elif flag=3 then val := piBy3-val\n elif flag=4 then val := piBy2-val end if;\n if isneg then val := -val end if;\n val;\nend proc: # of arctan16 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 0 7 "arctan_" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Testing the procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "xx := evalf(tan(1),65); \nevalf(arctan_(xx),60);\nevalf(evalf(arctan(xx),65),60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\\odgYRQQ+_\"Qs2D(3t,Oeu![(p]IA!\\lCxS d:!#k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gn++++++++++++++++++++++++ +++++5!#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gn++++++++++++++++++++ +++++++++5!#f" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xx := evalf(100);\nevalf(arctan_(xx),10);\nevalf (arctan(xx),10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$+\"\"\"! 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Some exact values are available." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 50:\nxx := 1 +sqrt(2);\narctan_(xx);\nevalf(evalf(%,Digits+3));\narctan_(evalf(xx,D igits+3));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,& \"\"\"F&*$\"\"##F&F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\" \"\"\"\")!\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SdwC&QRd \"e8)H(o7\\BWYsh4Xs4y6!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SdwC& QRd\"e8)H(o7\\BWYsh4Xs4y6!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Digits := 50:\nxx := 2+sqrt (3);\narctan_(xx);\nevalf(evalf(%,Digits+3));\narctan_(evalf(xx,Digits +3));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&\"\"# \"\"\"*$\"\"$#F'F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&\"\" \"\"#7!\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SjS;R:#3N&f kOw!oFp#=Zd**Qp**38!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SjS;R:#3 N&fkOw!oFp#=Zd**Qp**38!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "st := time():\nfor i to 500 0 do arctan_(rand()*Float(i,-11)) end do:\ntime()-st;\nst := time():\n for i to 5000 do arctan(rand()*Float(i,-11)) end do:\ntime()-st;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%,C!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%[B!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Digits := 25:\nst := time():\nfor \+ i to 5000 do arctan_(rand()*Float(i,-11)) end do:\ntime()-st;\nst := t ime():\nfor i to 5000 do arctan(rand()*Float(i,-11)) end do:\ntime()-s t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%ov!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&9<\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "Digits := 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"6#-%$IntG6$-%'arctanG6#%\"xG/F);\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%!\"\"%#PiG\"\"\"F(*&#F(\"\"#F (-%#lnG6#F+F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0wu6tX#)Q%!#:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(Int(arctan_(x),x=0..1),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0wu6tX#)Q%!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 7 " Since " } {XPPEDIT 18 0 "arcsin(x) = arctan(x/sqrt(1-x^2))" "6#/-%'arcsinG6#%\"x G-%'arctanG6#*&F'\"\"\"-%%sqrtG6#,&F,F,*$F'\"\"#!\"\"F3" }{TEXT -1 13 ", we can use " }{TEXT 0 7 "arctan_" }{TEXT -1 36 " to obtain a proce dure to evaluate " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f := x -> arctan_(x/sqrt(1-x^2));\nplot(f(x),x=- 1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 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