{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 "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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 2 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 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 260 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey \+ Emphasis" -1 271 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 "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 " Maple 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 46 "Procedures for evaluating the tan gent function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }} {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 function approxi mation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 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 c an be read into a Maple session by a command similar to the one that f ollows, 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 num erical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " } {TEXT 271 8 "numfcn.m" }{TEXT -1 72 " contains the code for the altern ative mathematical functions including " }{TEXT 0 4 "tan_" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple sess ion by a command similar to the one that follows, where the file path \+ gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rea d \"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 46 "load numerical integration proced ures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " } {TEXT 271 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 271 8 "gkdata.m" } {TEXT -1 67 " contain the code and data for the numerical integration \+ procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " used in this workshee t. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read into a Maple ses sion by commands similar to those that follow, where the file paths gi ve their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \+ \"K:\\\\Maple/procdrs/intg.m\";\nread \"K:\\\\Maple/procdrs/gkdata.m\" ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Ar gument reduction to the interval " }{XPPEDIT 18 0 "[-Pi/8, Pi/8];" "6# 7$,$*&%#PiG\"\"\"\"\")!\"\"F)*&F&F'F(F)" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 56 "W e can reduce the interval over which we need to define " }{XPPEDIT 18 0 "tan(x)" "6#-%$tanG6#%\"xG" }{TEXT -1 17 " to the interval " } {XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$*&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F (F)" }{TEXT -1 28 " by the periodicity formula " }{XPPEDIT 18 0 "tan(x +Pi) = tan(x);" "6#/-%$tanG6#,&%\"xG\"\"\"%#PiGF)-F%6#F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 64 "This can be achieved by subtracting a suitable integer multiple " }{XPPEDIT 18 0 "k*Pi;" "6#*&%\"kG\"\"\" %#PiGF%" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 6 " from " }{TEXT 272 1 "x" }{TEXT -1 57 ", to ensure that the resulti ng number is in the interval " }{XPPEDIT 18 0 "[-Pi/2, Pi/2];" "6#7$,$ *&%#PiG\"\"\"\"\"#!\"\"F)*&F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 24 "The required integer is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = floor(x/Pi+1/2);" "6#/%\"kG-%&floorG6#,&*&% \"xG\"\"\"%#PiG!\"\"F+*&F+F+\"\"#F-F+" }{TEXT -1 14 " ------- (i), " } }{PARA 257 "" 0 "" {TEXT -1 8 " " }{TEXT 268 9 "_________" } {TEXT -1 24 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{XPPEDIT 18 0 "floor(z)" "6#-%&floorG6#%\"zG" }{TEXT -1 23 " i s the largest integer" }{XPPEDIT 18 0 "`` <= z;" "6#1%!G%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 55 "Then the argument range can b e reduced to the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#P iG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 23 " by using the formulas \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ tan(t heta) = cot(Pi/2-theta) , ``],[ tan(theta) = cot(-Pi/2-theta) , ``])" "6#-%*PIECEWISEG6$7$/-%$tanG6#%&thetaG-%$cotG6#,&*&%#PiG\"\"\"\"\"#!\" \"F2F+F4%!G7$/-F)6#F+-F-6#,&*&F1F2F3F4F4F+F4F5" }{TEXT -1 15 " ------- (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 269 13 "___________ __" }{TEXT -1 25 " " }}{PARA 257 "" 0 "" {TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "The argument can b e reduced further to the interval " }{XPPEDIT 18 0 "[-Pi/8,Pi/8]" "6#7 $,$*&%#PiG\"\"\"\"\")!\"\"F)*&F&F'F(F)" }{TEXT -1 22 " by using the fo rmula " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "tan(2*theta )=2*tan(theta)/(1-tan(theta)^2)" "6#/-%$tanG6#*&\"\"#\"\"\"%&thetaGF)* (F(F)-F%6#F*F),&F)F)*$-F%6#F*F(!\"\"F2" }{TEXT -1 16 " ------- (iii). \+ " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 270 12 "____________" } {TEXT -1 19 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a func tion " }{TEXT 0 4 "tan1" }{TEXT -1 48 " using Maple's tangent function on the interval " }{XPPEDIT 18 0 "[-Pi/8, Pi/8];" "6#7$,$*&%#PiG\"\" \"\"\")!\"\"F)*&F&F'F(F)" }{TEXT -1 58 ", but ensure that it gives no \+ value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Pi/8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=3*p#R!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "tan1 := x -> if abs(x)<=.39 26990818 then tan(x) else FAIL end if;\nplot('tan1(x)',x=-0.5..0.5,thi ckness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tan1Gf*6#%\"xG6\"6$% )operatorG%&arrowGF(@%1-%$absG6#9$$\"+=3*p#R!#5-%$tanGF0%%FAILGF(F(F( " }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$ 7V7$$!3*)******z!Qf#R!#=$!3rzlS>B!49%F*7$$!3?+++9ib8RF*$!3AnXs^ZSETF*7 $$!3))*****pMu6!RF*$!3J$p\"o*)>#>6%F*7$$!3%******\\h5k(QF*$!3%)z2a?1+$ 3%F*7$$!3%******>)ok^QF*$!3MZbEww8aSF*7$$!3++++<%>@!QF*$!3y(QA8&\\e'*R F*7$$!31+++_>f_PF*$!3J'G%p='f#RRF*7$$!3A+++5j-]OF*$!3JD.RO2D@QF*7$$!3! *******o1YZNF*$!3/w#z#)HjTq$F*7$$!3;+++-OJNLF*$!3C'e%HLLvkMF*7$$!3E+++ %*o%Q7$F*$!3'G=0$=!*eHKF*7$$!3!******>RFj!HF*$!3Kz'Ra,?5*HF*7$$!3***** **f4OZr#F*$!3E5\\K^EX$y#F*7$$!3!******\\n\\!*\\#F*$!3#)*))e5&pS_DF*7$$ 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\"37+++l*RRL$F*$\"3Glj.r[@jMF*7$$\"3=+++`<.YNF*$\"3hc#)>4%QDq$F*7$$\"3 9+++K&*>^OF*$\"3,k70K`fAQF*7$$\"3E+++8tOcPF*$\"3VHd0%F*7$$\"39+++ Z^;xQF*$\"37'=%*o'4)Q3%F*7$$\"39+++9ZK,RF*$\"3S+Vn\"y(47TF*7$$\"3\")** ***p\\/M\"RF*$\"3pFXl-sAETF*7$$\"38+++\"G%[DRF*$\"3v+W:<2PSTF*7$%*unde finedGFd[l-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F]\\lF\\\\l-%*THICKNESSG6# \"\"#-%+AXESLABELSG6$Q\"x6\"Q!Ff\\l-%%VIEWG6$;$!\"&F[\\l$\"\"&F[\\l%(D EFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " Now we define a function " }{TEXT 0 4 "tan2" }{TEXT -1 33 " using the \+ formulas (i) and (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 594 "tan2 := proc(x)\n local xx,pi,k, t;\n\n pi := evalf(Pi);\n xx := evalf(x);\n\n # Reduce the range to -Pi/2<=xx<=Pi/2\n if abs(xx)>0.5*pi then \n # Determine the number of multiples of Pi to add or subtract.\n k := floor(xx/pi +1/2);\n xx := xx - k*pi;\n end if;\n\n # Reduce the range to -Pi/8<=xx<=Pi/8 \n if abs(xx)>0.25*pi then\n if xx>0 then\n \+ xx := pi*0.5-xx;\n else\n xx := -pi*0.5-xx;\n \+ end if;\n xx := xx*0.5;\n t := tan1(xx);\n (1-t^2)/(2*t );\n else\n xx := xx*0.5;\n t := tan1(xx);\n 2*t/(1-t ^2); \n end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%tan 2Gf*6#%\"xG6&%#xxG%#piG%\"kG%\"tG6\"F-C&>8%-%&evalfG6#%#PiG>8$-F26#9$@ $2,$*&$\"\"&!\"\"\"\"\"F0FAFA-%$absG6#F6C$>8&-%&floorG6#,&*&F6FAF0F@FA #FA\"\"#FA>F6,&F6FA*&FGFAF0FAF@@%2,$*&$\"#D!\"#FAF0FAFAFBC&@%2\"\"!F6> F6,&*&F>FAF0FAFAF6F@>F6,&*&$F?F@FAF0FAF@F6F@>F6,$*&F>FAF6FAFA>8'-%%tan 1GFD,$*&FMFA*&,&FAFA*$)FboFNFAF@FAFboF@FAFAC%>F6F_o>FboFco,$*(FNFAFboF AFhoF@FAF-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 218 "p1 := plot('tan2(x)',x=-5..8,-3..3,discont=tr ue,color=red,thickness=2):\np2 := plots[implicitplot](\{x=-3*Pi/2,x=-P i/2,x=Pi/2,x=3*Pi/2,x=5*Pi/2\},\n x=-5..8,y=-3..3,linestyle=4,c olor=black):\nplots[display]([p1,p2]);" 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es7$F`gu$\"3!e(oO71Hd;F-7$7$F`guFafsFf^v7$Fj^v7$F`gu$\"3yvoO71H(*=F-7$ 7$F`guFifsF\\_v7$F`_v7$F`gu$\"3,woO71HP@F-7$7$F`guFagsFb_v7$Ff_v7$F`gu $\"3AwoO71HxBF-7$7$F`guFigsFh_v7$F\\`v7$F`gu$\"3WwoO71H " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "An arbitrary precision procedure to evalu ate the tangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "tanAP" }{TEXT -1 12 " calculates " }{XPPEDIT 18 0 "tan(x)=sin(x)/cos(x)" "6#/ -%$tanG6#%\"xG*&-%$sinG6#F'\"\"\"-%$cosG6#F'!\"\"" }{TEXT -1 5 " for \+ " }{TEXT 273 1 "x" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[- Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 46 " by using Maclaurin series approximations for " }{XPPEDIT 18 0 "sin(x) " "6#-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(x)" "6#-% $cosG6#%\"xG" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "t anAP: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2368 "tanAP := proc(xx::realcons)\n local x ,z,term,eps,k,even,maxit,saveDigits,\n extraDigits,sum1,sum2,val,pi, isneg,complement,double;\n\n # increase precision \n saveDigits : = Digits;\n Digits := Digits+length(Digits)+3;\n x := evalf(xx);\n # argument reduction involves loss of precision\n # increase prec ision further when x has large magnitude\n extraDigits := max(ilog10 (x),0);\n if extraDigits>0 then\n Digits := Digits+extraDigits; \n x := evalf(xx)\n end if;\n\n pi := evalf(Pi); # Maple eval uates pi rapidly\n \n # Reduce the range to -Pi/2<=x<=Pi/2 a pproximately\n if abs(x)>1.5707963267948966192 then\n # Determi ne the number of multiples of Pi to add or subtract.\n k := floor (x/pi+0.5);\n x := x - k*pi;\n end if;\n\n Digits := Digits-e xtraDigits;\n\n # If abs(x)>Pi/4, replace x by Pi/2-x or -Pi/2-x\n \+ # and use the formula tan(x)=cot(Pi/2-x)=cot(-Pi/2-x)\n complement \+ := false; \n if abs(x)>.78539816339744830962 then\n if x>0 then \n x := pi*0.5-x;\n else\n x := -pi*0.5-x;\n \+ end if;\n complement := true;\n end if;\n\n # If abs(x)>Pi/8, replace x by x/2 \n # and use the formula tan(x)=2*tan(x/2)/(1-tan( x/2)^2)\n if abs(x)>.39269908169872415481 then\n x := x*0.5;\n \+ double := true;\n else \n double := false;\n end if;\n \n # Initialisation for Maclaurin series loops\n eps := Float(1 ,-saveDigits);\n maxit := Digits*4;\n\n # calculate sin(x)\n ter m := x;\n sum1 := term;\n z := x*x;\n even := false; \n for k \+ from 2 to maxit by 2 do\n term := term*z/(k*(k+1));\n if eve n then\n sum1 := sum1+term;\n else\n sum1 := sum1 -term;\n end if;\n if abs(term)<=eps*abs(sum1) then break en d if;\n even := not even;\n end do;\n\n # calculate cos(x)\n \+ term := 1.0;\n sum2 := term;\n z := x*x;\n even := false; \n \+ for k from 1 to maxit by 2 do\n term := term*z/(k*(k+1));\n \+ if even then\n sum2 := sum2+term;\n else\n sum2 \+ := sum2-term;\n end if;\n if abs(term)<=eps*abs(sum2) then b reak end if;\n even := not even;\n end do;\n\n if complement \+ then\n val := sum2/sum1;\n if double then val := (val*val-1) /(2*val) end if;\n else\n val := sum1/sum2;\n if double th en val := 2*val/(1-val*val) end if;\n end if;\n\n Digits := saveDi gits;\n evalf(val);\nend proc: # of tanAP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xx := 1.2;\ntanAP(xx);\ntan(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +A;:sD!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+A;:sD!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Testing the procedure tanAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "p1 := plot('tanAP'(x),x=-5..8,-3..3,discont=true,color=red,thic kness=2):\np2 := plots[implicitplot](\{x=-3*Pi/2,x=-Pi/2,x=Pi/2,x=3*Pi /2,x=5*Pi/2\},\n x=-5..8,y=-3..3,linestyle=4,color=black):\nplo 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yo\\Q*yk))=\\['z*e)eNrA4xlb5i0z2m]?dY\\zN&)H$Qb2*G()\\*e(GyfesK!$+$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$!g]l&e%Rp_&fJ$Q<4OsA(RAH\\UWs#QrH%R7R;#y.SD$)HoaB&G'zI Nip`Q!ezWw=0yo\\Q*yk))=\\['z*e)eNrA4xlb5i0z2m]?dY\\zN " 0 "" {MPLTEXT 1 0 149 "st := time():\nfor i to 500 do tanAP(rand()*Float(1,-11)) end do: \ntime()-st;\nst := time():\nfor i to 500 do tan(rand()*Float(1,-11)) \+ end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%V:!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$f\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "st := time():\nDi gits := 50:\nfor i to 100 do tanAP(rand()*Float(1,-11)) end do:\ntime( )-st;\nst := time():\nfor i to 100 do tan(rand()*Float(1,-11)) end do: \ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%oU! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$!>!\"$" }}}{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 63 "A fixed precision procedure to evaluate the tangent fun ction " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 93 "In this section we construct a fixed precision versi on of the tangent function which can use " }{TEXT 259 34 "hardware flo ating point arithmetic" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A rational approximation for " }{XPPEDIT 18 0 "tan(x);" " 6#-%$tanG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "We choose a rational functio n " }}{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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "t o approximate " }{XPPEDIT 18 0 "tan(x)" "6#-%$tanG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\" \"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 77 "Initially, we construct a rational approximation for the contin uous function " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x )=PIECEWISE([(tan(x)-x)/x^3,x<>0],[1/3,x=0])" "6#/-%\"fG6#%\"xG-%*PIEC EWISEG6$7$*&,&-%$tanG6#F'\"\"\"F'!\"\"F1*$F'\"\"$F20F'\"\"!7$*&F1F1F4F 2/F'F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Limit((tan(x)-x)/(x^3),x=0);\nvalue (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&-%$tanG6#%\"xG \"\"\"F+!\"\"F,F+!\"$/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\" \"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "evalf(remez((tan(x)-x)/x^3,x=-Pi/4..Pi/4,[4,6],'maxe rr',\n type=even,errtype=absolute,weight=x^2+1e-30,info=true),25):\n r := unapply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,itera tion~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G------------------------- -------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerror~graph~drawn~for~the~right~half~of~the~inte rvalG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "63-%'CURV ESG6%7]u7$$\"\"!F)%%FAILG7$$\"@![nzqIm/f_2UjA&RAUU2c*fioF;#!#]7$ $\"@#)RPmXIPs?gO2TTG%F5$\"@C&zP..m99hd4`-WQFH7$$\"@t4c\\o&f&3J!\\5;@Ek F5$\"@1UFE)zmRYRBxTHV')FH7$$\"@lzuK\"4YZ9/KZ@Go&)F5$\"@].#y*))zXsP_b3_ ^`\"!#\\7$$\"@&>7*p8>r@1)4ABC&G\"!#K$\"@rA7TMom6)*fYG()[W$FX7$$\"@$f\\ l#=#\\*G3k%Hkl8j&z%Qh<5#H89X7!#[7$$\"@O5$\\5J`PW'pes*p/KFfn$\"@h:DwUdmJ!G5!oQ&) 3#Fco7$$\"@iF;Qh^G\"oz-cQ`\")[Ffn$\"@VcMj=H**Hc2rklbl%Fco7$$\"@b42_\"[ \\6gPFJJ[plFfn$\"@C'y5EJ9VWrHP@[bzFco7$$\"@_Ah][wYasf#y)4%\\#)Ffn$\"@' *Q\"\\'eJhK;Ur__<;\"!#Z7$$\"@ma]D\"fKVryI=q\"p!)*Ffn$\"@*\\uaj()Hl3_Ci lR,:Fhp7$$\"@kW_!=,M5M$\\Zai>9\"!#J$\"@1#ft'z=]QGKy:\\!==Fhp7$$\"@UXNr uz&G?g>\"3\\(38Faq$\"@]R>#QX!>+\"4(H*yzr?Fhp7$$\"@cR@dIXwXa]h!\\(=R\"F aq$\"@O'o-7O'okQ3\"z=rf@Fhp7$$\"@rL2V'3r')o]5J2+v9Faq$\"@$o5iy[Y%=&Qo2 J8`H`x^\"Faq$\"@$=pYiiX4!zvEm2QB#Fhp7$$\"@*3'*4P )[rsZL&fe]g:Faq$\"@^YzzOQ[GR'*[B%HTAFhp7$$\"@on(HI$es$z0kT@)=e\"Faq$\" @]!4qk*>?yJND`+:C#Fhp7$$\"@[u&\\ByOZ\"oZPUeKg\"Faq$\"@*e*zSr\">B:IYciI RAFhp7$$\"@G\"Qp;tZd$yaeqMYi\"Faq$\"@#f)eoF8>4atBjW#e3Q)>Fhp7$$\"@h]g\"4ebqa _zLg?m>Faq$\"@D>#**R^[M7iMrB$>#=Fhp7$$\"@%*\\/K>K)[@qfE(Q80#Faq$\"@WO? 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[$)fjRgBW3#F3F()F.\"\"'F(F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-6c;v@B!#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(*&,($!:2dD[>JLLLLLL$!#D\"\"\"*&$\":A55Yl_*QK;)Q0#!#EF1)9$\"\"#F 1F1*&$\":7'*=k:]pWL#F5F1F=F1F?*&$\":'>i![$)fjRgBW3# F " 0 " " {MPLTEXT 1 0 226 "r := x -> (-.3333333333333311948255707+.2053881632 389526546101022e-1*x^2-.2009591097985621764189612e-3*x^4)/(-1.+.461616 4489718728631652782*x^2-.2334469501519571218676459e-1*x^4+.20844236039 63598348062196e-3*x^6):\n'r(x)'=r(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"rG6#%\"xG*&,($\":2dD[>JLLLLLL$!#D!\"\"*&$\":A55Yl_*QK;)Q0#!#E \"\"\")F'\"\"#F2F2*&$\":7'*=k:]pWL#F1F2F9F2F-*&$\":'>i ![$)fjRgBW3#F8F2)F'\"\"'F2F2F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "The rational function " }{XPPEDIT 18 0 "s (x)=x+r(x)*`.`*x^3" "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F' \"\"$F)" }{TEXT -1 14 " approximates " }{XPPEDIT 18 0 "tan(x)" "6#-%$t anG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/4, P i/4];" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 38 "The relative error graph is drawn for " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 11 ", which is " } {XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 72 " with the coeffic ients rounded to 20 digits and arranged in nested form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 355 "s:=x-> x+(-.3333333333333311948255707+.2053881632389526546101022e-1*x^2-.2009 591097985621764189612e-3*x^4)/(-1.+.4616164489718728631652782*x^2-.233 4469501519571218676459e-1*x^4+.2084423603963598348062196e-3*x^6)*x^3; \nh := unapply(evalf(convert(s(x),horner),20),x);\nevalf(plot(1-s(x)/t an(x),x=-Pi/4..Pi/4,\n color=COLOR(RGB,.4,0,.9)),2 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,&9$\"\"\"*(,($!:2dD[>JLLLLLL$!#DF.*&$\":A55Yl_*QK;)Q0#!#EF. )F-\"\"#F.F.*&$\":7'*=k:]pWL#F7F.F>F.F@*&$\":'>i![$ )fjRgBW3#F=F.)F-\"\"'F.F.F@F-\"\"$F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 523 233 233 {PLOTDATA 2 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.desf#35F-7$$\":cyMbuL;Y.A$zxF*$!:0*>VllMp3X&\\&yF87$$\":L0K=6]n;p()** y(F*$!:sLcyCvN6K'>)H&F87$$\":4KH\"yk'=([Ll+yF*$!:lkD;$ofGlJk!R#F87$$\" :')eEW%G)pd+>8\"yF*$\":1]KY\"GNr<&*>.*)FC7$$\":j&Qs5#*4#Gm%)>#yF*$\":= (38#*o([^ts#oXF87$$\":R7@qd:s)>.lKyF*$\":iT@zXlEi\"3zn')F87$$\":;R=L%> L#p(fJVyF*$\":]8ct\"e*p%HxW@8F-7$$\":$fch4$[uRj\")R&yF*F+-%+AXESLABELS G6$Q\"x6\"Q!Fh^p-%&COLORG6&%$RGBG$\"\"%!\"\"\"\"!$\"\"*F`_p-%%VIEWG6$; $!:3m:'4$[uRj\")R&yF*$\":3m:'4$[uRj\")R&yF*%(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 67 "evalf(evalf(numapprox[infnorm](1-s(x)/tan(x),x=- Pi/4..Pi/4),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&BI#!#@" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The maxim um relative error in using " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "tan(x);" "6#-%$tanG6#% \"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-Pi/4, Pi/4]; " "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 11 " is about " }{XPPEDIT 18 0 "2.3*`. `*10^(-17);" "6#*(-%&FloatG6$\"#B!\"\"\"\"\" %#.~GF))\"#5,$\"# " 0 "" {MPLTEXT 1 0 77 "xx := evalf(Pi/4,20);\ne valf(evalf(h(xx),20),16);\nevalf(evalf(tan(xx),20),16);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#xxG$\"5i4$[uRj\")R&y!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1+++++++5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1 +++++++5!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approximation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "tan(x);" "6#-%$tanG6#%\"xG" }{TEXT -1 35 " with random numbers between 0 and " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 92 "The printout occurs only when the relativ e error is greater than or equal to the specified \"" }{TEXT 271 3 "ep s" }{TEXT -1 3 "\".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "ra ndomize():\neps := Float(4,-16);\nfor i from 1 to 200 do\n xx := eva lf(rand()/Float(1,12)*.78539816339744830962,20);\n axx := evalf(tan (xx),20);\n hxx := evalf(h(xx),16);\n e := evalf(abs((axx-hxx)/axx ),20);\n if e>=eps then \n printf(\" trial no. %d, x = %.16f, \\n\",i,xx);\n printf(\" tan(x) = %.16f, h(x) = %.16f, rel error = %.2e\\n\\n\",axx,hxx,e); \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"%!#;" }}{PARA 6 "" 1 "" {TEXT -1 39 " tria l no. 25, x = .1500303559008248," }}{PARA 6 "" 1 "" {TEXT -1 76 " ta n(x) = .1511662674866383, h(x) = .1511662674866382, rel error = 4.50e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " t rial no. 33, x = .1227988374308856," }}{PARA 6 "" 1 "" {TEXT -1 76 " \+ tan(x) = .1234198340269806, h(x) = .1234198340269807, rel error = 6.2 7e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 36, x = .1908630167014937," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1932149233708471, h(x) = .1932149233708472, rel error \+ = 4.56e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 63, x = .1544763239752737," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1557169198344583, h(x) = .1557169198344582, r el error = 4.46e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 39 " trial no. 66, x = .1306073817940755," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1313551304526640, h(x) = .13135513045266 39, rel error = 4.63e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 73, x = .1756847009124702," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1775148082754275, h(x) = .177514808 2754276, rel error = 4.03e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 77, x = .1854739679491243," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1876304513551478, h(x) = .1 876304513551477, rel error = 4.60e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 91, x = .132825841817320 2," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1336125296861783, h(x) \+ = .1336125296861784, rel error = 6.10e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 198, x = .14880017441 33885," }}{PARA 6 "" 1 "" {TEXT -1 76 " tan(x) = .1499082081427994, h (x) = .1499082081427995, rel error = 4.23e-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 5 "tan16 " }}{PARA 0 "" 0 "" {TEXT -1 138 "Here is the code for the fixed preci sion version of the tangent function, which can be evaluated with hard ware floating point arithmetic.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1014 "tan16 := proc(xx)\n local a1,a2,a3,b1,b2,b3,pi,x, z,num,den,k,complement; \n\n # coefficients of numerator in rational approximation\n a1 := .33333333333333119483;\n a2 := -.2053881632 3895265461e-1;\n a3 := .20095910979856217642e-3;\n\n # coefficient s of denominator in rational approximation\n b1 := -.461616448971872 86317;\n b2 := .23344695015195712187e-1;\n b3 := -.208442360396359 83481e-3;\n\n pi := 3.1415926535897932385;\n\n x := evalf(xx);\n \+ \n # Reduce the argument range to -Pi/4<=x<=Pi/4\n if abs(x)>1.570 796326794897 then\n k := floor(x/pi+0.5);\n x := x - k*pi;\n end if;\n\n complement := false;\n if abs(x)>.7853981633974483 \+ then\n if x>0 then\n x := pi*0.5-x;\n else\n \+ x := -pi*0.5-x;\n end if;\n complement := true;\n end if; \n \n # evaluate the rational approximation\n z := x*x;\n n um := (a1+(a2+a3*z)*z)*z;\n den := 1+(b1+(b2+b3*z)*z)*z;\n\n if co mplement then\n den/((den+num)*x)\n else\n (1+num/den)*x\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 27 "Testing the procedure tan16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "p1 := plot('tan16 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{XPPMATH 20 "6#$\"1w6o8tW9\\!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 1w6o8tW9\\!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "xx := 27;\ne valf(evalhf(tan16(xx)),15);\nevalf(evalhf(tan(xx)),15);\nevalf(tan(xx) ,15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#F" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!06G/!QqtK!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0 7G/!QqtK!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!07G/!QqtK!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Translating from Maple \+ code to C code " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "We can translate this program into C using the Maple pro cedure C in the code generating package codegen. \n see " }{HYPERLNK 17 "codegen[C]" 2 "codegen[C]" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "c odegen/C/procedure" 2 "codegen/C/procedure" "" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(codegen,C):\nC(tan16,ansi);" }}{PARA 6 "" 1 "" {TEXT -1 17 "# include " }}{PARA 6 "" 1 "" {TEXT -1 23 "double tan16(double x x)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 12 " \+ double a1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a2;" }}{PARA 6 " " 1 "" {TEXT -1 12 " double a3;" }}{PARA 6 "" 1 "" {TEXT -1 12 " dou ble b1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b2;" }}{PARA 6 "" 1 " " {TEXT -1 12 " double b3;" }}{PARA 6 "" 1 "" {TEXT -1 17 " int comp lement;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double den;" }}{PARA 6 "" 1 "" {TEXT -1 8 " int k;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double num; " }}{PARA 6 "" 1 "" {TEXT -1 12 " double pi;" }}{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 = 0.3333333333333312;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a2 = -0. 2053881632389527E-1;" }}{PARA 6 "" 1 "" {TEXT -1 31 " a3 = 0.200959 1097985622E-3;" }}{PARA 6 "" 1 "" {TEXT -1 29 " b1 = -0.46161644897 18729;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b2 = 0.2334469501519571E-1; " }}{PARA 6 "" 1 "" {TEXT -1 32 " b3 = -0.2084423603963598E-3;" }} {PARA 6 "" 1 "" {TEXT -1 30 " pi = 0.3141592653589793E1;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }}{PARA 6 "" 1 "" {TEXT -1 40 " if( 0.1570796326794897E1 < fabs(x) )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 28 " k = floor(x/pi+ 0.5);" }}{PARA 6 "" 1 "" {TEXT -1 19 " x += -k*pi;" }}{PARA 6 " " 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 19 " comple ment = 0;" }}{PARA 6 "" 1 "" {TEXT -1 38 " if( 0.7853981633974483 < fabs(x) )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 21 " if( 0.0 < x )" }}{PARA 6 "" 1 "" {TEXT -1 23 " \+ x = 0.5*pi-x;" }}{PARA 6 "" 1 "" {TEXT -1 13 " else " }} {PARA 6 "" 1 "" {TEXT -1 24 " x = -0.5*pi-x;" }}{PARA 6 "" 1 "" {TEXT -1 23 " complement = 1;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 12 " z = x*x;" }}{PARA 6 "" 1 "" {TEXT -1 29 " num = (a1+(a2+a3*z)*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 33 " den = 1.0+(b1+(b2+b3*z)*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 22 " if( (complement) )" }}{PARA 6 "" 1 "" {TEXT -1 30 " \+ return(den/(den+num)/x);" }}{PARA 6 "" 1 "" {TEXT -1 9 " else \+ " }}{PARA 6 "" 1 "" {TEXT -1 30 " return((1.0+num/den)*x);" }} {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 fi xed precision procedures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 25 digit rational approxim ation for " }{XPPEDIT 18 0 "tan(x)" "6#-%$tanG6#%\"xG" }{TEXT -1 4 " o n " }{XPPEDIT 18 0 "[-Pi/8,Pi/8]" "6#7$,$*&%#PiG\"\"\"\"\")!\"\"F)*&F& F'F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "evalf(remez(tan(x)/x^3-1/x^2,x=-Pi /8..Pi/8,[8,6],\n type=even,errtype=absolute,weight=x^2+1e-50),30): \nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~erro r:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0.8g!)\\X2(!#V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(*&,,$!?3pGLLLLLLLLLLLL!#I\"\"\"*&$\"? IrKr,%)f&*e%QmIE=#!#JF1)9$\"\"#F1F1*&$\"?tRof9dhR8j8MpRE!#LF1)F7\"\"%F 1!\"\"*&$\"?ZMc5tLeJ(\\8rE2y#!#PF1)F7\"\"'F1F1*&$\"?HAX3na$>[nP@=j0*!# SF1)F7\"\")F1F1F1,*$F?\"\"!F1*&$\"?M]C!3_z'oP:*>*yaYF0F1F6F1F1*&$\"?w4 LUo0VCya39(y]#F5F1F=F1F?*&$\"?l/_*HO)40UtQ'4yw#F " 0 "" {MPLTEXT 1 0 444 "s:=x->x+(-.333333333333333333333333286908+.218263066 384589559840171327130e-1*x^2-.263969341363133961571459683973e-3*x^4+.2 78072671134973158337310563447e-7*x^6+.905631821376748193546708452229e- 10*x^8)/(-1.+.465478919915376867952080245034*x^2-.25078714085478244305 6842330976e-1*x^4+.276780963873420509836299520465e-3*x^6)*x^3:\nh := u napply(evalf(convert(s(x),horner),30),x);\nevalf(plot((1-s(x)/tan(x)), x=-Pi/8..Pi/8,color=COLOR(RGB,.4,0,.9)),35);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*(,, $!?3pGLLLLLLLLLLLL!#IF.*&$\"?IrKr,%)f&*e%QmIE=#!#JF.)F-\"\"#F.F.*&$\"? tRof9dhR8j8MpRE!#LF.)F-\"\"%F.!\"\"*&$\"?ZMc5tLeJ(\\8rE2y#!#PF.)F-\"\" 'F.F.*&$\"?HAX3na$>[nP@=j0*!#SF.)F-\"\")F.F.F.,*$F@\"\"!F.*&$\"?M]C!3_ z'oP:*>*yaYF3F.F8F.F.*&$\"?w4LUo0VCya39(y]#F7F.F>F.F@*&$\"?l/_*HO)40Ut Q'4yw#F=F.FEF.F.F@F-\"\"$F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 459 280 280 {PLOTDATA 2 "6&-%'CURVESG6#7[`l7$$!DyP*4HUIy![:C()p\"3*p#R!#N$ \"($**3nF*7$$!DdWcX!3-J([=or/*4k@RF*$\"(,p$RF*7$$!DN^8+QPPQ\\@7cR;\"H; RF*$\"(3*o:F*7$$!D9eqa&RXO+Xi0WP8%4\"RF*$!&MI%!#M7$$!D\"\\w#4`q\"*o]F+ D4^\"f0RF*$!'*R4#F<7$$!Dqr%Q1r)=M^IW4WoT-!RF*$!'9`MF<7$$!D[yT=o.Y*>N$) Q*y&=*[*QF*$!'*o`%F<7$$!DE&))Hd-KZElB$y8.U&*)QF*$!'Xs`F<7$$!D0#fvKo.+L &Rwi[?#>%)QF*$!'C&)fF<7$$!D#))H@3Mv_RD/sMyB%)yQF*$!'(*)R'F<7$$!Dh0qO)* pagaXkJ=b#\\tQF*$!'tNmF<7$$!DR7F\"fl=e_&[3;`sU\"oQF*$!'3;nF<7$$!D=>%eM J!4\"f:D0!))*GziQF*$!'6fmF<7$$!DvK)\\&GOj@ddSpdC$4_QF*$!'m-iF<7$$!DLY7 kVp<_ejGQFf$RTQF*$!'n#R&F<7$$!D*)fEte-s#)fp;2(RRpIQF*$!']OVF<7$$!DYtS# QdjK6cZgn'G%**>QF*$!'mCJF<7$$!D.([:*))o!QC;G\\kLYH4QF*$!'eK=F<7$$!Dg+p +/-Nuj(3Qh!)\\f)z$F*$!&QA&F<7$$!D<9$)4>N*[]O*o#eF`*yy$F*$\"'(ob(F*7$$! 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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "xx := evalf(10^(-1),30);\nevalf(evalf(h(xx) ,30),25);\nevalf(evalf(tan(xx),30),25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?++++++++++++++5!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\":+3e]a]a3sYL+\"!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":+3e]a]a3s YL+\"!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 25 digit rational approximation for " }{XPPEDIT 18 0 "1/x^4-1/(3*x^2)-cot(x)/x ^3" "6#,(*&\"\"\"F%*$%\"xG\"\"%!\"\"F%*&F%F%*&\"\"$F%*$F'\"\"#F%F)F)*& -%$cotG6#F'F%*$F'F,F)F)" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-3/(10^3) , 3/(10^3)];" "6#7$,$*&\"\"$\"\"\"*$\"#5F&!\"\"F**&F&F'*$F)F&F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "evalf(remez(1/x^4-1/(3*x^2)-cot(x)/x^3,x=-0.003 ..0.003,4,\n type=even,errtype=absolute,variation=0.2,weight=x^2+1e -60,info=true),40);\nr := unapply(%,x);" 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++DcI;[$HF*$\"#tFV7$$\"D+++++++++](=UnU'H%HF*FO7$$\"D++++++++++v=#HA6^ HF*FJ7$$\"D+++++++++]i:5>g#fHF*FE7$$\"D++++++++++]7G:3u'HF*F@7$$\"D+++ ++++++]P4Y6cb(HF*F;7$$\"D++++++++++D1k2/P)HF*F67$$\"D+++++++++]7.#Q?&= *HF*F17$$\"D+++++++++++++++++$F*F+-%&COLORG6&%$RGBG$F``lF`jlFe`l$\"\"* F`jl-%+AXESLABELSG6$Q\"x6\"Q!F_cm-%%VIEWG6$;F(Fbbm%(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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "tan25" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1926 "tan25 := proc(xx )\n local a1,a2,a3,a4,a5,b1,b2,b3,x,z,n,d,v,\n flag,isneg,piBy4 ,piBy2;\n\n piBy2 := 1.5707963267948966192313216916397514; \n\n i f xx<0 then\n x := -xx;\n isneg := true;\n else\n x : = xx;\n isneg := false;\n end if;\n \n if x>1.5677965 then \n # use an approximation for cot(Pi/2-x) when x is near Pi/2\n \+ a1 := .333333333333333333333333333333333;\n a2 := .2222222222 22222222250817091037e-1;\n a3 := .211640211640018830285954067623e -2;\n a4 := .211640581216006167720708852107e-3;\n x := evalf (piBy2-x,Digits+5);\n z := x*x;\n v := 1/x-(a1+(a2+(a3+a4*z) *z)*z)*x;\n else\n # use an approximation for tan(x)\n a1 \+ := .333333333333333333333333286908;\n a2 := -.2182630663845895598 40171327130e-1;\n a3 := .263969341363133961571459683973e-3;\n \+ a4 := -.278072671134973158337310563447e-7;\n a5 := -.9056318213 76748193546708452229e-10;\n b1 := -.46547891991537686795208024503 4;\n b2 := .250787140854782443056842330976e-1;\n b3 := -.276 780963873420509836299520465e-3;\n piBy4 := .785398163397448309615 660845820;\n\n if x>piBy4 then\n if x>1.17809724509617246 4423491 then\n x := piBy2-x;\n flag := 3;\n \+ else\n x := x-piBy4;\n flag := 2;\n \+ end if;\n else\n if x>.3926990816987241548078304 then\n \+ x := piBy4-x;\n flag := 1;\n else\n \+ flag := 0;\n end if;\n end if;\n \n # e valuate the rational approximation\n z := x*x;\n n := (a1+(a 2+(a3+(a4+a5*z)*z)*z)*z)*z;\n d := 1+(b1+(b2+b3*z)*z)*z;\n v := (1+n/d)*x;\n if flag=1 then\n v := (1+n/d)*x;\n \+ v := (1-v)/(1+v);\n elif flag=2 then\n v := (1+n/d)*x; \n v := (1+v)/(1-v);\n elif flag=3 then\n v := d/ ((d+n)*x);\n end if;\n end if;\n if isneg then -v else v end \+ if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing the procedure tan25 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xx := evalf(2*Pi/7,30);\neva lf(evalf(tan25(xx),30),27);\nevalf(evalf(tan(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?O4oE$*)4@bc-,zf(*)!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<)4\"4dPQqiwLgRD\"!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<)4\"4dPQqiwLgRD\"!#E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "xx := evalf(1.57,30) ;\nprintlevel := 1:\nevalf(evalf(tan25(xx),30),27);\nprintlevel := 1: \nevalf(evalf(tan(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$\"$d\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<,V0m/;p+:fldD\"!# B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<,V0m/;p+:fldD\"!#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "x x := evalf(1.570796,30);\nprintlevel := 1:\nevalf(evalf(tan25(xx),30), 27);\nprintlevel := 1:\nevalf(evalf(tan(xx),35),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"('zq:!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"<$f;x(zYc$>1L-gI!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<$f;x(z Yc$>1L-gI!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "evalf(plot(1-'tan25'(x)/tan(x),x=0..1.57096,colo r=COLOR(RGB,.4,0,.9)),35);" }}{PARA 13 "" 1 "" {GLPLOT2D 658 209 209 {PLOTDATA 2 "6&-%'CURVESG6#7`[m7$$\"DLLLLLLLLLL3F4zw+2\"!#P$\"%<`!#N7$ $\"Dmmmmmmmmmm;a=e`,9#F*$\"&T7#F-7$$\"D*******************\\7ys.B5KF*$ \"&Wx%F-7$$\"DKLLLLLLLLLL3P;2.G%F*$\"&RZ)F-7$$\"D)******************* 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1171 "s:=x->x+(.33333333333333333333333333333333333333333 33333333333341522111487639526-.270249893976815865451032057424083120203 1116508872039722433793196557995e-1*x^2+.624079891275778054335262710462 9237557407158458300590350162576687104680e-3*x^4-.548310961870316849334 1759981771317538272368239906942886507010809531319e-5*x^6+.186592887480 2033970703717760914442222075616341608615399611224272753192e-7*x^8-.191 1733087062447006624221037168618881073086973809490638448100709191482e-1 0*x^10+.51504239310050065015835877472527917055025843601829933412186145 38248044e-16*x^12)/(1.-.4810749681930447596353096172272249360609334952 661561689132595355857835*x^2+.3239746504628333325522487311751684092969 078373560002726340978478814266e-1*x^4-.6955727721992330465092640129145 739040436956808877241957172956380075346e-3*x^6+.5865138190061974450888 043508331175632981361270513017204480225695504440e-5*x^8-.1939766016487 759500704684849228360253601771895769346063430467161619217e-7*x^10+.193 9848664663895478697033992429578809091418044932519377907293026427267e-1 0*x^12)*x^3:\nh := unapply(evalf(convert(s(x),horner),60),x);\nevalf(p lot(10^20*(1-s(x)/tan(x)),x=-Pi/16..Pi/16,color=COLOR(RGB,.4,0,.9)),65 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,&9$\"\"\"*(,0$\"gn6A:MLLLLLLLLLLLLLLLLLLLLLLLLLL!#gF.*&$\"gn zLCsR?()3l6J??J3Cu0K5Xle\"o(R*)\\-F!#hF.)F-\"\"#F.!\"\"*&$\"gnei,N!f+$ e%erSdvBHY5FENV0ydF\"*)zSi!#jF.)F-\"\"%F.F.*&$\"gn,2l)G%p!*R#oBFQvJr<) *f!#qF.)F- \"#5F.F:*&$\"gnh=7M$*H=gVe-bq\"z_su(e$e,l+05$RU]^!#wF.)F-\"#7F.F.F.,0$ F.\"\"!F.*&$\"gngK\"*oh:m_\\L41O\\AFsh4`jfZ/$>o\\2\"[F3F.F8F.F:*&$\"gn )4MEF+gNPy!pH4%o^Cx)3o&pVS!Rd9H, k#4l/L#*>sFd&pF>F.FEF.F:*&$\"gnB![/s,80Fh8)Hjv6L3N/))3Xu>1!>Q^'eFDF.FK F.F.*&$\"gnZIM1Y$pd*=x,ODg$G#\\[o/2]fx[;gwR>FJF.FQF.F:*&$\"gnH2zP>D$\\ /=944)y&HC*R.(pya*Qmk'[)R>FPF.FWF.F.F:F-\"\"$F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 455 192 192 {PLOTDATA 2 "6&-%'CURVESG6#7cil7$$!\\o)p.M4 \"Q6W4Y(3BBEI*o\\X6_\"RSx?O\\3a\\j>!#l$\")`T`7!#X7$$!\\o'*[g\"p2%*=VT' \\l6GDKAyAS5bOC4%eB&\\?3'>F*$\"(B*o8F-7$$!\\o$41)*GMu'>>#=,5Rz9dn+!po= pu51y>eX\"e>F*$!'%o'f!#W7$$!\\o!H2!))3YX2C+uMqfq5HNx(pA=]A\"G?(o1Zb>F* $!(ev-\"F87$$!\\o)[3i[([B&*G=Op\\D$*\\#QYl_eW`P,DYbdz_>F*$!(_RA\"F87$$ !\\o'o4W39:IQj$)R!H\"f\"*eB>`N%4nD:s/A%37]>F*$!(aMC\"F87$$!\\o$)3h#oS& zqQagQ3dQG*3#4%=I(*fnTp%*GfWZ>F*$!(&GM6F87$$!\\o!373Gnv&eVsAt(G7wE%\\' G,mBj#=;*o:5xW>F*$!'Uk$*F87$$!\\oyK,zQfNY[!\\yq')QogzP;%=+lw>QJC5'4U>F *$!'HFoF87$$!\\owW@x/i8M`3ZUYa1Y\\1Tqwj(p7-O<*=@%R>F*$!'\"o*RF87$$!\\o tcTvqk\">#eE4xD?H&G]$=*\\t-tFAe\"fFYn$>F*$!'4%3\"F87$$!\\oqohtOnp4jWr6 0'=XiNczK4HwUU!eEOrS$>F*$\"(+mu\"F-7$$!\\oo!==F+xuzEOjW=XP'4#Hn:Xbzdi- S\\kRJ>F*$\"(t5P%F-7$$!\\om#>+(osD&G2e4QwrHI1-b)4=GGF[Uh`@(G>F*$\"(s#* p'F-7$$!\\ok/AoMv.tx)zbJM)>U;\\F9o\"3'yGq%)GiYg#>F*$\"(;*p')F-7$$!\\oi ;Um+y\"3Eo,-D#\\U\")px/VEX$*GI#pi4F*$\")7eC5F-7$$!\\ofGikm!)f[([B[ =]^1Ki?=Z)3EzJ9pjz'p?>F*$\")y'49\"F-7$$!\\ocS#GELyjBHX%>\"3y)fwMf+VseH LO6J)=-=>F*$\")T/;7F-7$$!\\oa_-h)feTs4nS0m/\"**HjOH,O\"*zMe`)ppM:>F*$ \")X+^7F-7$$!\\o_kAfk)Q>@!*)o))R7LQ$=R\"ef*R-j.efc?n7>F*$\")81[7F-7$$! 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" }}{PARA 0 "" 0 " " {TEXT -1 38 "The relative 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 multiplied by " }{XPPEDIT 18 0 "10^30" "6#*$\"#5\"#I" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1019 "s:=x->1/x-x/3- (-.22222222222222222222222222 22222222222222222222222222223575877939945737e-1+.113146320950384187731 1004019878158598202193893613191712700668997415426e-2*x^2-.166104326063 2606696906023645098548920637539081582983830292567936670808e-4*x^4+.850 8336955747494244346637772020798916629222029833749924842303298159974e-7 *x^6-.1254380209587186841584732317893716377166525111094788775191406135 226472e-9*x^8+.1621771812793836330288227530391368153532817071159559152 525784515388035e-14*x^10)/(-1.+.14615393966576812257423327613261237501 43368204506833508753409266461627*x^2-.51430827687863989730584988434002 87491842780231328242072840421492419333e-2*x^4+.63704837651009012956258 97091637612146579956031603466653374264255706725e-4*x^6-.29306937358039 06355234030205382845165266548048099527381886715003292689e-6*x^8+.39623 68061385728967916574394975496404639169260239114728243404022019862e-9*x ^10)*x^3:\nh := unapply(evalf(convert(s(x),horner),60),x);\nevalf(plot (10^30*(1-s(x)/cot(x)),x=-2/25..2/25,color=COLOR(RGB,.4,0,.9)),65);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,(*&\"\"\"F.9$!\"\"F.*&$\"gnLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL!#gF.F/F .F0**$F.\"\"!F.,.$!gn)edBAAAAAAAAAAAAAAAAAAAAAAAAAA#!#hF.*&$\"gnn+Fr\" >8O*Q>-#)fe\"y)>S+6t(=%Q]4KYJ6!#iF.)F/\"\"#F.F.*&$\"gnd#HIQ)He\"3Rvj?* [&)4XO-1pp1EjgK/h;!#kF.)F/\"\"%F.F0*&$\"gnIU[#*\\P$)H?AHm\"*)z??xPmMWU \\Zd&pL3&)!#nF.)F/\"\"'F.F.*&$\"gnT\">v()y%46^_mrP;P*yJKZeTo=(e4-Qa7!# pF.)F/\"\")F.F0*&$\"gnyDD:f&f62!zF*$\"$X#F87$$!\\ommmmmmmmmmmmmmmmmmmm mmmmmmm\"HU,\"*yF*$\"$1#F87$$!\\oLLLLLLLLLLLLLLLLLLLLLLLLLL$3_c6!)yF*$ \"$e\"F87$$!\\o****************************************************** \\2<#pyF*$\"$.\"F87$$!\\ommmmmmmmmmmmmmmmmmmmmmmmmm;z\\=$eyF*$\"#jF87$ $!\\oLLLLLLLLLLLLLLLLLLLLLLLLLLL3#*>u%yF*$\"#;F87$$!\\o*************** *************************************\\PM@l$yF*$!\"&F-7$$!\\ommmmmmmmm mmmmmmmmmmmmmmmmmmmwAc#yF*$!\"*F-7$$!\\oLLLLLLLLLLLLLLLLLLLLLLLLLL$e*= CZ\"yF*$!#8F-7$$!\\o************************************************** ***\\7c#Q!yF*$!#;F-7$$!\\ommmmmmmmmmmmmmmmmmmmmmmmmm;a.FHz(F*$!#>F-7$$ !\\oLLLLLLLLLLLLLLLLLLLLLLLLLLL$e%G?y(F*$!#AF-7$$!\\ommmmmmmmmmmmmmmmm mmmmmmmmmmTIJ-w(F*$!#EF-7$$!\\o*************************************** ****************\\TVQxF*$!#GF-7$$!\\ommmmmmmmmmmmmmmmmmmmmmmmmm;HdNvs( F*$!#HF-7$$!\\oLLLLLLLLLLLLLLLLLLLLLLLLLLLe*pj;xF*Fjr7$$!\\o********** ******************************************\\(=%Qdq(F*Fer7$$!\\ommmmmmm mmmmmmmmmmmmmmmmmmmm;%)R[p(F*F`r7$$!\\o******************************* **********************\\(oUIn(F*$!#BF-7$$!\\oLLLLLLLLLLLLLLLLLLLLLLLLL LLL`X7l(F*$!#=F-7$$!\\ommmmmmmmmmmmmmmmmmmmmmmmmmTg%*zAj(F*$!#:F-7$$! 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\\olmmmmmmmmmmmmmmmmmmmmmmmm;/,*>]i[F*Ff\\m7$$\"\\o)****************** *********************************\\i&)oK([F*Ff\\m7$$\"\\oJLLLLLLLLLLLL LLLLLLLLLLLL$eRAd.%)[F*Ff\\m7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmT&)e-[*[ F*Fa\\m7$$\"\\o)*************************************************\\(oa %pb!\\F*Fa\\m7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLLL3KOj\"\\F*$\"$3\"F87$$ \"\\o***************************************************\\7`+(y$\\F*F^ [l7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm;ay.%f\\F*F^[l7$$\"\\o************ **************************************\\i:lq,(\\F*$\"#**F87$$\"\\oLLLL LLLLLLLLLLLLLLLLLLLLLL3x^P4)\\F*$\"$5\"F87$$\"\\ommmmmmmmmmmmmmmmmmmmm mmmm;aQQ/<*\\F*$\"$0\"F87$$\"\\o++++++++++++++++++++++++++++DrC+&F*Fi[ m7$$\"\\oommmmmmmmmmmmmmmmmmmmmmmmmmTUHa3&F*Fd[m7$$\"\\oNLLLLLLLLLLLLL LLLLLLLLLLLLL$)f(Qo^F*Fiam7$$\"\\o.++++++++++++++++++++++++++DxX8D&F*F jjl7$$\"\\oqmmmmmmmmmmmmmmmmmmmmmmmmmmm%RIM`F*F\\bl7$$\"\\oqmmmmmmmmmm mmmmmmmmmmmmmmmm;mS\">aF*F\\il7$$\"\\oqmmmmmmmmmmmmmmmmmmmmmmmmmmmPxR] &F*F_jl7$$\"\\oqmmmmmmmmmmmmmmmmmmmmmmmmmm;49))e&F*Fdil7$$\"\\oqmmmmmm mmmmmmmmmmmmmmmmmmmmm!3ltcF*Fbt7$$\"\\oNLLLLLLLLLLLLLLLLLLLLLLLLLL3.&y ddF*Fdil7$$\"\\o+++++++++++++++++++++++++++]D>>%eF*F_]l7$$\"\\oKLLLLLL LLLLLLLLLLLLLLLLLLL$3nj)R)eF*F\\il7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmm \"zM0EfF*Fg\\l7$$\"\\o(*********************************************** ****\\7f?\"ofF*Fb\\l7$$\"\\oILLLLLLLLLLLLLLLLLLLLLLLLLLLq(=5gF*F]u7$$ \"\\ojmmmmmmmmmmmmmmmmmmmmmmmmm;/,V)[gF*Fdam7$$\"\\o(***************** ***********************************\\<$)\\(3'F*Fggl7$$\"\\oILLLLLLLLLL LLLLLLLLLLLLLLL$eCO:EhF*Fbgl7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmm;$*3[;' F*F]gl7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLLLea>@C'F*$\"$p\"F87$$\"\\o++++ ++++++++++++++++++++++++;I%>jF*F^fl7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLLL $[o.3kF*$\"$!=F87$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmmm`Vm\\'F*$\"$V\"F87 $$\"\\oILLLLLLLLLLLLLLLLLLLLLLLLLL3)o%4a'F*Fd\\p7$$\"\\o(************* ****************************************\\A]_e'F*$\"#mF87$$\"\\ojmmmmm mmmmmmmmmmmmmmmmmmmmm\"pNbHmF*Fa`o7$$\"\\oILLLLLLLLLLLLLLLLLLLLLLLLLLL \"p&Qn'F*Fdhl7$$\"\\o(************************************************ ****\\_B[8nF*F\\il7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmm;z2Jv'F*F_]l7$$\" \\oKLLLLLLLLLLLLLLLLLLLLLLLLLL3BLFz'F*Faq7$$\"\\o+++++++++++++++++++++ +++++++neB$oF*Ffq7$$\"\\oommmmmmmmmmmmmmmmmmmmmmmmmm\"4T)>(oF*F[r7$$\" \\oNLLLLLLLLLLLLLLLLLLLLLLLLLL$[&4;\"pF*Fhs7$$\"\\o.++++++++++++++++++ +++++++D\"3f^@pF*Fhs7$$\"\\oqmmmmmmmmmmmmmmmmmmmmmmmmm;zEA9$pF*F[r7$$ \"\\oPLLLLLLLLLLLLLLLLLLLLLLLLL3xiG8%pF*F[r7$$\"\\o.++++++++++++++++++ ++++++++v)\\B^pF*F[r7$$\"\\oPLLLLLLLLLLLLLLLLLLLLLLLLL$32x/rpF*F[r7$$ \"\\oqmmmmmmmmmmmmmmmmmmmmmmmmmmmUg3*pF*F[r7$$\"\\oPLLLLLLLLLLLLLLLLLL LLLLLLL$e4%4J.(F*F]t7$$\"\\o.++++++++++++++++++++++++++DRe`2(F*Fdil7$$ \"\\oPLLLLLLLLLLLLLLLLLLLLLLLLLeR)G['4(F*F_]l7$$\"\\oqmmmmmmmmmmmmmmmm mmmmmmmmm;aP2w6(F*Fgp7$$\"\\o.+++++++++++++++++++++++++vo'=tQrF*Fg\\l7 $$\"\\oNLLLLLLLLLLLLLLLLLLLLLLLLLL$ej&)frF*Fb\\l7$$\"\\opmmmmmmmmmmmmm mmmmmmmmmmmm\"z\\3)4=(F*$\"#9F87$$\"\\o-+++++++++++++++++++++++++]7M0@ ?(F*$\"#^F87$$\"\\oNLLLLLLLLLLLLLLLLLLLLLLLLL3F$)HKA(F*Fc[l7$$\"\\oomm mmmmmmmmmmmmmmmmmmmmmmmmTKaVC(F*$\"$;\"F87$$\"\\o,++++++++++++++++++++ +++++Dc\")yaE(F*$\"$`\"F87$$\"\\oMLLLLLLLLLLLLLLLLLLLLLLLLL$32Lg'G(F*F dz7$$\"\\onmmmmmmmmmmmmmmmmmmmmmmmmmT&)zFxI(F*$\"$7#F87$$\"\\o++++++++ ++++++++++++++++++++H_)GtF*Fjy7$$\"\\o++++++++++++++++++++++++++]P7r! \\tF*$\"$V#F87$$\"\\o+++++++++++++++++++++++++++v&**GptF*F[y7$$\"\\o++ ++++++++++++++++++++++++vVP*RztF*F`x7$$\"\\o++++++++++++++++++++++++++ ]7z3&*Q(F*F`x7$$\"\\o++++++++++++++++++++++++++D\"3#='*R(F*$\"$_#F87$$ \"\\o+++++++++++++++++++++++++++]iF(4uF*Fhhp7$$\"\\o++++++++++++++++++ +++++++++DHl,X(F*$\"$J#F87$$\"\\o++++++++++++++++++++++++++++'Hg!\\(F* Fj`p7$$\"\\o++++++++++++++++++++++++++]Pz@3^(F*$\"$N\"F87$$\"\\o++++++ +++++++++++++++++++++viS5`(F*$\"$1\"F87$$\"\\o++++++++++++++++++++++++ ++]7Yf7b(F*$\"#`F87$$\"\\o+++++++++++++++++++++++++++]Hy9d(F*F]u7$$\" \\o++++++++++++++++++++++++++v=r(e\"e(F*Fb\\l7$$\"\\o+++++++++++++++++ +++++++++](Grp\"f(F*Fbp7$$\"\\o++++++++++++++++++++++++++Dca1=g(F*F\\i l7$$\"\\o+++++++++++++++++++++++++++D'f\">h(F*F_]l7$$\"\\o++++++++++++ ++++++++++++++]izM@j(F*Fbt7$$\"\\o++++++++++++++++++++++++++++j`Bl(F*F fq7$$\"\\o++++++++++++++++++++++++++D\"G:3uwF*Fhs7$$\"\\o+++++++++++++ +++++++++++++]iU4ep(F*$!#FF-7$$\"\\o+++++++++++++++++++++++++]7`Ptmq(F *F_\\q7$$\"\\o++++++++++++++++++++++++++vVKPvr(F*Fjr7$$\"\\o++++++++++ +++++++++++++++]PMF,%GxF*Fjr7$$\"\\o+++++++++++++++++++++++++++DAl#RxF *Fer7$$\"\\o++++++++++++++++++++++++++D17$*4w(F*F`r7$$\"\\o+++++++++++ +++++++++++++++](=5s#y(F*Fhs7$$\"\\o+++++++++++++++++++++++++]7y'\\e$z (F*Fc^l7$$\"\\o++++++++++++++++++++++++++vo\"*[W!yF*Faq7$$\"\\o+++++++ ++++++++++++++++++]Pf'GJ:yF*F\\q7$$\"\\o+++++++++++++++++++++++++++]\" o " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "tan55" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4676 "tan55 := proc(xx)\n local a0,a1,a2,a3,a4, a5,a6,a7,b1,b2,b3,b4,b5,b6,\n x,z,n,d,v,u,t,flag,p1,p2,p3,p4,tn1, tn2,isneg,double; \n\n if xx<0 then\n x := -xx;\n isneg := true;\n else\n x := xx;\n isneg := false;\n end if;\n\n if x>1.490796 then\n # use an approximation for cot(Pi/2-x) wh en x is near Pi/2\n a0 :=\n .33333333333333333333333333333333333 333333333333333333333333333333; \n a1 := \n .22222222222222 2222222222222222222222222222222222222222357588e-1;\n a2 := \n - .113146320950384187731100401987815859820219389361319171270067e-2;\n \+ a3 := \n .166104326063260669690602364509854892063753908158298383 029257e-4;\n a4 := \n -.850833695574749424434663777202079891662 922202983374992484230e-7;\n a5 := \n .125438020958718684158473 231789371637716652511109478877519141e-9;\n a6 := \n -.1621771812 79383633028822753039136815353281707115955915252578e-14;\n b1 := \+ \n -.146153939665768122574233276132612375014336820450683350875341; \n b2 := \n .5143082768786398973058498843400287491842780231328 24207284042e-2;\n b3 := \n -.6370483765100901295625897091637612 14657995603160346665337426e-4;\n b4 := \n .2930693735803906355 23403020538284516526654804809952738188672e-6;\n b5 := \n -.3962 36806138572896791657439497549640463916926023911472824340e-9;\n p4 := # Pi/2\n 1.570796326794896619231321691639751442098584699687552910 4874722962;\n x := evalf(p4-x,Digits+5);\n z := x*x;\n \+ n := (a1+(a2+(a3+(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n d := 1+(b1+(b2+(b 3+(b4+b5*z)*z)*z)*z)*z;\n v := 1/x-(a0+n/d)*x;\n else\n # \+ use an approximation for tan(x)\n a1 := \n .333333333333333333 333333333333333333333333333333333334152211;\n a2 := \n -.270249 893976815865451032057424083120203111650887203972243379e-1;\n a3 : = \n .624079891275778054335262710462923755740715845830059035016258e- 3;\n a4 := \n -.54831096187031684933417599817713175382723682399 0694288650701e-5;\n a5 := \n .186592887480203397070371776091444 222207561634160861539961122e-7;\n a6 := \n -.1911733087062447006 62422103716861888107308697380949063844810e-10;\n a7 := \n .5150 42393100500650158358774725279170550258436018299334121861e-16;\n b 1 := \n -.481074968193044759635309617227224936060933495266156168913 260;\n b2 := \n .323974650462833332552248731175168409296907837 356000272634098e-1;\n b3 := \n -.695572772199233046509264012914 573904043695680887724195717296e-3;\n b4 := \n .586513819006197 445088804350833117563298136127051301720448023e-5;\n b5 := \n -. 193976601648775950070468484922836025360177189576934606343047e-7;\n \+ b6 := \n .19398486646638954786970339924295788090914180449325193779 0729e-10;\n p1 := # Pi/4\n .7853981633974483096156608458198 75721049292349843776455243736;\n p2 := # 3*Pi/8\n 1.1780972 4509617246442349126872981358157393852476566468286560;\n p3 := # 7 *Pi/16\n 1.37444678594553454182740648018478251183626161222660879 667654;\n p4 := # Pi/2\n 1.57079632679489661923132169163975 1442098584699687552910487;\n tn1 := # tan(3*Pi/8)\n 2.41421 356237309504880168872420969807856967187537694807317668;\n tn2 := \+ # tan(7*Pi/16)\n 5.027339492125848104514975071064072385737194252 07548712827448;\n\n if x>p1 then\n if x>p3 then\n \+ x := x-p3;\n flag := 4;\n elif x>p2 then\n \+ x := x-p2;\n flag := 3;\n else\n \+ x := x-p1;\n flag := 2;\n end if;\n else\n \+ if x>.3926990816987241548078304 then\n x := p1-x;\n \+ flag := 1;\n else\n flag := 0;\n e nd if;\n end if;\n\n if abs(x)>.1963495408493620774 then\n \+ x := x*0.5;\n double := true;\n else\n dou ble := false;\n end if;\n \n # evaluate the rational app roximation\n z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+a7*z)* z)*z)*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+(b4+(b5+b6*z)*z)*z)*z)*z)* z;\n v := (1+n/d)*x;\n \n if flag=0 then \n i f double then\n v := 2*v/(1-v^2)\n end if;\n e lif flag=1 then\n if double then\n t := 1-v^2;\n \+ u := v+v;\n v := (t-u)/(t+u);\n else \n \+ v := (1-v)/(1+v);\n end if;\n elif flag=2 then\n if double then\n t := 1-v^2;\n u := v+v ;\n v := (t+u)/(t-u);\n else \n v := (1+ v)/(1-v);\n end if;\n elif flag=3 then \n v \+ := (tn1+v)/(1-tn1*v);\n else # flag=4 \n v := (tn2+v) /(1-tn2*v);\n end if;\n end if;\n if isneg then -v else v end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing the procedure tan55 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "xx := evalf(15/10+1/9 0,60);\nevalf(evalf(tan55(xx),60),57);\nevalf(evalf(tan(xx),60),57);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gn6666666666666666666666666 6666:!#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ZNKnn)4bcHh[P,Ubepd)z)y sd<7oYt;!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ZNKnn)4bcHh[P,Ubepd) z)ysd<7oYt;!#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "xx := evalf(1.57075,60);\nevalf(evalf(tan55(xx ),60),58);\nevalf(evalf(tan(xx),65),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"'vq:!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 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7bZ6f$z[#R#3*>dV)45!\\k>7*4t=>*3gc\"F^ex$\"$i*Fbt7$$\"\\oo1G`G)e\"3<:h DeX]Yw$[!=n^&*)e1#*3Yd?n:F^ex$\"%#G\"Fbt7$$\"\\oJ9w$>!H?dSU(>T.5e$4p3N &)QrlK5\"*Hd-%o:F^ex$\"%F>Fbt7$$\"\\o7=+kQ\\sJ-c:0sFY!e<1OWC$4/;0Uk)4+ p:F^ex$\"%nDFbt7$$\"\\o%>UU`(pC1kpL)*4b6DUa7_.EZU***H*)*R*fp:F^ex$\"%^ QFbt7$$\"\\owD[/7!p2eK=:zCo(p3Zkgi>&3G[RM8y>q:F^ex$\"%(p(Fbt7$$\"\\oeH su[5Hb(o*p%e)4U9vR;p@8B>m*[zEizq:F^ex$\"('z\"4'Fbt-%+AXESLABELSG6$Q\"x 6\"Q!Fid_l-%&COLORG6&%$RGBG$\"\"%!\"\"F)$\"\"*Fae_l-%%VIEWG6$;F($\"\\o iHsu[5Hb(o*p%e)4U9vR;p@8B>m*[zEizq:F^ex%(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 0 129 "Digits := 55:\nxx := evalf[Digits+15](Pi/2-7*10^( -10));\ntan55(xx);\nevalf(tan(xx));\nevalf(evalf[Digits+25](tan(xx))); \nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"ao3R:'Hsu[ 5Hb(o*p%e)4U9vR;p@8B>m*[4Ejzq:!#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"X:I(e,teh(3Q_4Q_4Q_>r&G9dG9dG9!#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"X&fJ5-seh(3Q_4Q_4Q_>r&G9dG9dG9!#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"X;I(e,teh(3Q_4Q_4Q_>r&G9dG9dG9!#X" }}}{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 49 " A procedure for evaluating the tangent function: " }{TEXT 0 4 "tan_" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "tan_: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 20 "Calling Sequence: " }{TEXT -1 10 "tan_( x ) " }{TEXT 263 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 15 "Parameters: " }{TEXT 264 21 "x - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 267 12 "D escription:" }{TEXT -1 1 " " }{TEXT 266 14 "The procedure " }{TEXT 0 4 "tan_" }{TEXT 265 40 " calculates the sine of a real number x." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure active 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 20 " tan_: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12121 "tan_ := proc(x::algebraic)\n local t ,n;\n\n if nargs <> 1 then\n error \"expecting 1 argument, got \+ %1\", nargs;\n end if;\n if type(x,'float') then evalf('tan_'(x)) \n elif type(x,'numeric') and signum(0,x,0)<0 then -tan_(-x)\n eli f type(x,`*`) and type(op(1,x),'numeric') and signum(0,op(1,x),0)<0 th en -tan_(-x)\n elif type(x,'function') and nops(x)=1 then\n n : = op(0,x);\n t := op(1,x);\n if n='arcsin_' or n='arcsin' th en t/sqrt(1-t^2)\n elif n='arccos_' or n='arccos' then sqrt(1-t^2 )/t\n elif n='arctan_' or n='arctan' then t\n else 'tan_'(x) \n end if;\n elif type(x,'realcons') then\n if x=0 then 0 \n elif x='infinity' then 'undefined'\n elif x=-'infinity' t hen 'undefined'\n elif type(x,`*`) and type(x,'`&*`(rational,iden tical(Pi))') then\n t := op(1,x);\n if t<=1/2 then \n \+ if t=1/2 then 'undefined'\n elif t=1/3 then sqrt (3)\n elif t=1/4 then 1\n elif t=1/6 then sqrt(3 )/3\n elif t=3/8 then 1+sqrt(2) \n elif t=1/8 th en sqrt(2)-1\n elif t=5/12 then 2+sqrt(3)\n elif t=1/12 then 2-sqrt(3)\n else 'tan_'(x)\n end if \n else tan_((t-floor(t+1/2))*Pi)\n end if\n else 'tan_'(x)\n end if \n else 'tan_'(x)\n end if;\nend proc:\n \n`evalf/tan_` := proc(xx)\n local x,z,term,eps,k,even,maxit,saveDig its,\n extraDigits,sum1,sum2,val,pi,complement,double,hfDigits;\n\n \+ if xx=0 then return xx end if;\n if not type(xx,realcons) then ret urn 'tan_'(xx) end if;\n\n # Use the fixed precision procedure for l ow precision values\n hfDigits := trunc(evalhf(Digits));\n if Digi ts<=hfDigits then\n x := evalf(xx,hfDigits+2);\n if max(ilog 10(x),0)<=3 then\n return evalf(evalhf(tan16(x)))\n end i f;\n end if;\n\n # increase precision \n saveDigits := Digits; \n Digits := Digits+length(Digits)+3;\n x := evalf(xx);\n # argu ment reduction involves loss of precision\n # increase precision fur ther when x has large magnitude\n extraDigits := max(ilog10(x),0);\n if extraDigits>0 then\n Digits := Digits+extraDigits;\n x := evalf(xx)\n end if;\n\n pi := evalf(Pi); # Maple evaluates pi \+ rapidly\n \n # Reduce the range to -Pi/2<=x<=Pi/2 approximat ely\n if abs(x)>1.5707963267948966192 then\n # Determine the nu mber of multiples of Pi to add or subtract.\n k := floor(x/pi+0.5 );\n x := x - k*pi;\n end if;\n\n Digits := Digits-extraDigit s;\n\n if saveDigits<=hfDigits then \n return evalf(evalhf(ta n16(x)))\n elif saveDigits<26 and abs(x)<1.57 then\n val := tan 25(x)\n elif saveDigits<56 then\n val := tan55(x)\n else\n \+ # Use the Maclaurin series for sin(x) and cos(x)\n\n # If abs( x)>Pi/4, replace x by Pi/2-x or -Pi/2-x\n # and use the formula t an(x)=cot(Pi/2-x)=cot(-Pi/2-x)\n complement := false; \n if \+ abs(x)>.78539816339744830962 then\n if x>0 then\n x := pi*0.5-x;\n else\n x := -pi*0.5-x;\n end if;\n complement := true;\n end if;\n \n # if abs (x)>Pi/8, replace x by x/2 \n # and use the formula tan(x)=2*tan( x/2)/(1-tan(x/2)^2)\n if abs(x)>.39269908169872415481 then\n \+ x := x*0.5;\n double := true;\n else\n double := false;\n end if;\n \n eps := Float(1,-saveDigits);\n \+ maxit := Digits*4;\n\n # calculate sin(x)\n term := x;\n sum1 := term;\n z := x*x;\n even := false; \n for k from 2 to maxit by 2 do\n term := term*z/(k*(k+1));\n \+ if even then\n sum1 := sum1+term;\n else\n \+ sum1 := sum1-term;\n end if;\n if abs(term)<=eps *abs(sum1) then break end if;\n even := not even;\n end d o;\n\n # calculate cos(x)\n term := 1.0;\n sum2 := term ;\n z := x*x;\n even := false; \n for k from 1 to maxit by 2 do\n term := term*z/(k*(k+1));\n if even then\n \+ sum2 := sum2+term;\n else\n sum2 := sum2 -term;\n end if;\n if abs(term)<=eps*abs(sum2) then br eak end if;\n even := not even;\n end do;\n\n if com plement then\n val := sum2/sum1;\n if double then val \+ := (val*val-1)/(2*val) end if;\n else\n val := sum1/sum2; \n if double then val := 2*val/(1-val*val) end if;\n end \+ if;\n end if;\n\n Digits := saveDigits;\n evalf(val);\nend proc: # of tanAP\n\ntan55 := proc(xx)\n local a0,a1,a2,a3,a4,a5,a6,a7,b1, b2,b3,b4,b5,b6,\n x,z,n,d,v,u,t,flag,p1,p2,p3,p4,tn1,tn2,isneg,do uble; \n\n if xx<0 then\n x := -xx;\n isneg := true;\n e lse\n x := xx;\n isneg := false;\n end if;\n\n if x>1.49 0796 then\n # use an approximation for cot(Pi/2-x) when x is near Pi/2\n a0 :=\n .33333333333333333333333333333333333333333333333 333333333333333333; \n a1 := \n .22222222222222222222222222 2222222222222222222222222222357588e-1;\n a2 := \n -.11314632095 0384187731100401987815859820219389361319171270067e-2;\n a3 := \n \+ .166104326063260669690602364509854892063753908158298383029257e-4;\n a4 := \n -.850833695574749424434663777202079891662922202983374 992484230e-7;\n a5 := \n .125438020958718684158473231789371637 716652511109478877519141e-9;\n a6 := \n -.1621771812793836330288 22753039136815353281707115955915252578e-14;\n b1 := \n -.14615 3939665768122574233276132612375014336820450683350875341;\n b2 := \+ \n .514308276878639897305849884340028749184278023132824207284042e-2 ;\n b3 := \n -.637048376510090129562589709163761214657995603160 346665337426e-4;\n b4 := \n .293069373580390635523403020538284 516526654804809952738188672e-6;\n b5 := \n -.396236806138572896 791657439497549640463916926023911472824340e-9;\n p4 := # Pi/2\n \+ 1.5707963267948966192313216916397514420985846996875529104874722962;\n \+ x := evalf(p4-x,Digits+5);\n z := x*x;\n n := (a1+(a2+( a3+(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+(b4+b5*z)*z) *z)*z)*z;\n v := 1/x-(a0+n/d)*x;\n else\n # use an approxi mation for tan(x)\n a1 := \n .33333333333333333333333333333333 3333333333333333333334152211;\n a2 := \n -.27024989397681586545 1032057424083120203111650887203972243379e-1;\n a3 := \n .624079 891275778054335262710462923755740715845830059035016258e-3;\n a4 : = \n -.548310961870316849334175998177131753827236823990694288650701e -5;\n a5 := \n .18659288748020339707037177609144422220756163416 0861539961122e-7;\n a6 := \n -.191173308706244700662422103716861 888107308697380949063844810e-10;\n a7 := \n .515042393100500650 158358774725279170550258436018299334121861e-16;\n b1 := \n -.4 81074968193044759635309617227224936060933495266156168913260;\n b2 := \n .32397465046283333255224873117516840929690783735600027263409 8e-1;\n b3 := \n -.69557277219923304650926401291457390404369568 0887724195717296e-3;\n b4 := \n .58651381900619744508880435083 3117563298136127051301720448023e-5;\n b5 := \n -.19397660164877 5950070468484922836025360177189576934606343047e-7;\n b6 := \n . 193984866466389547869703399242957880909141804493251937790729e-10;\n \+ p1 := # Pi/4\n .785398163397448309615660845819875721049292349 843776455243736;\n p2 := # 3*Pi/8\n 1.178097245096172464423 49126872981358157393852476566468286560;\n p3 := # 7*Pi/16\n \+ 1.37444678594553454182740648018478251183626161222660879667654;\n \+ p4 := # Pi/2\n 1.5707963267948966192313216916397514420985846996 87552910487;\n tn1 := # tan(3*Pi/8)\n 2.4142135623730950488 0168872420969807856967187537694807317668;\n tn2 := # tan(7*Pi/16) \n 5.02733949212584810451497507106407238573719425207548712827448 ;\n\n if x>p1 then\n if x>p3 then\n x := x-p3; \n flag := 4;\n elif x>p2 then\n x := x- p2;\n flag := 3;\n else\n x := x-p1;\n \+ flag := 2;\n end if;\n else\n if x>.392 6990816987241548078304 then\n x := p1-x;\n flag \+ := 1;\n else\n flag := 0;\n end if;\n \+ end if;\n\n if abs(x)>.1963495408493620774 then\n x := x* 0.5;\n double := true;\n else\n double := false; \n end if;\n \n # evaluate the rational approximation\n \+ z := x*x;\n n := (a1+(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z)*z)*z)*z )*z;\n d := 1+(b1+(b2+(b3+(b4+(b5+b6*z)*z)*z)*z)*z)*z;\n v : = (1+n/d)*x;\n \n if flag=0 then \n if double then \n v := 2*v/(1-v^2)\n end if;\n elif flag=1 th en\n if double then\n t := 1-v^2;\n u := v+v;\n v := (t-u)/(t+u);\n else \n v := (1-v)/(1+v);\n end if;\n elif flag=2 then\n if d ouble then\n t := 1-v^2;\n u := v+v;\n \+ v := (t+u)/(t-u);\n else \n v := (1+v)/(1-v);\n \+ end if;\n elif flag=3 then \n v := (tn1+v)/(1 -tn1*v);\n else # flag=4 \n v := (tn2+v)/(1-tn2*v);\n end if;\n end if;\n if isneg then -v else v end if;\nend pro c: # of tan55\n\ntan25 := proc(xx)\n local a1,a2,a3,a4,a5,b1,b2,b3,x ,z,n,d,v,\n flag,isneg,piBy4,piBy2;\n\n piBy2 := 1.570796326794 8966192313216916397514; \n\n if xx<0 then\n x := -xx;\n i sneg := true;\n else\n x := xx;\n isneg := false;\n end \+ if;\n \n if x>1.5677965 then\n # use an approximation for cot (Pi/2-x) when x is near Pi/2\n a1 := .333333333333333333333333333 333333;\n a2 := .222222222222222222250817091037e-1;\n a3 := \+ .211640211640018830285954067623e-2;\n a4 := .21164058121600616772 0708852107e-3;\n x := evalf(piBy2-x,Digits+5);\n z := x*x;\n v := 1/x-(a1+(a2+(a3+a4*z)*z)*z)*x;\n else\n # use an app roximation for tan(x)\n a1 := .333333333333333333333333286908;\n \+ a2 := -.218263066384589559840171327130e-1;\n a3 := .26396934 1363133961571459683973e-3;\n a4 := -.2780726711349731583373105634 47e-7;\n a5 := -.905631821376748193546708452229e-10;\n b1 := -.465478919915376867952080245034;\n b2 := .250787140854782443056 842330976e-1;\n b3 := -.276780963873420509836299520465e-3;\n \+ piBy4 := .785398163397448309615660845820;\n\n if x>piBy4 then\n \+ if x>1.178097245096172464423491 then\n x := piBy2-x ;\n flag := 3;\n else\n x := x-piBy4;\n \+ flag := 2;\n end if;\n else\n if x>.39 26990816987241548078304 then\n x := piBy4-x;\n f lag := 1;\n else\n flag := 0;\n end if;\n \+ end if;\n \n # evaluate the rational approximation\n \+ z := x*x;\n n := (a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z;\n d : = 1+(b1+(b2+b3*z)*z)*z;\n v := (1+n/d)*x;\n if flag=1 then\n v := (1+n/d)*x;\n v := (1-v)/(1+v);\n elif flag= 2 then\n v := (1+n/d)*x;\n v := (1+v)/(1-v);\n el if flag=3 then\n v := d/((d+n)*x);\n end if;\n end if; \n if isneg then -v else v end if;\nend proc: # of tan25 \n\n# proc edure for default and/or low precision\ntan16 := proc(xx)\n local a1 ,a2,a3,b1,b2,b3,pi,x,z,num,den,k,complement; \n\n # coefficients of \+ numerator in rational approximation\n a1 := .33333333333333119483;\n a2 := -.20538816323895265461e-1;\n a3 := .20095910979856217642e-3 ;\n\n # coefficients of denominator in rational approximation\n b1 := -.46161644897187286317;\n b2 := .23344695015195712187e-1;\n b3 := -.20844236039635983481e-3;\n\n pi := 3.1415926535897932385;\n\n \+ x := evalf(xx);\n \n # Reduce the argument range to -Pi/4<=x<=Pi/4 \n if abs(x)>1.570796326794897 then\n k := floor(x/pi+0.5);\n \+ x := x - k*pi;\n end if;\n\n complement := false;\n if abs(x )>.7853981633974483 then\n if x>0 then\n x := pi*0.5-x;\n else\n x := -pi*0.5-x;\n end if;\n complement \+ := true;\n end if;\n \n # evaluate the rational approximation \n z := x*x;\n num := (a1+(a2+a3*z)*z)*z;\n den := 1+(b1+(b2+b3* z)*z)*z;\n\n if complement then\n den/((den+num)*x)\n else\n \+ (1+num/den)*x\n end if;\nend proc: # of tan16" }}}{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 4 "tan_" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Testing t he procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "z := evalf(570000*Pi/7,40);\nevalf(tan(z),20);\nev alf(evalf(tan(z),25),20);\nevalf(tan_(z),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG$\"Ig&[BXq1/\"e>8N" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$!5D2B[`niG\"Q%!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!5C2B[`niG\"Q %!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "z := evalf(10^300);\nevalf(tan(z),10);\nevalf(tan_(z) ,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG$\"+++++5\"$\"H" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E>3ge!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+E>3ge!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "p1 := plot(tan_(x),x=-5..8, -3..3,discont=true,color=red,thickness=2):\np2 := plots[implicitplot]( \{x=-3*Pi/2,x=-Pi/2,x=Pi/2,x=3*Pi/2,x=5*Pi/2\},\n x=-5..8,y=-3. .3,linestyle=4,color=black):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 275 275 {PLOTDATA 2 "6*-%'CURVESG6*7hn7$$!\"&\"\"!$ \"+1]^!Q$!\"*7$$!+W4u$*\\F-$\"+IV)*fMF-7$$!+u\\H))\\F-$\"+&3%)>`$F-7$$ !+5/<#)\\F-$\"+EyL;OF-7$$!+[_+w\\F-$\"+*p5^q$F-7$$!+*Qp)p\\F-$\"+!4#e( z$F-7$$!+i1=k\\F-$\"+B'\\s)QF-7$$!+9.He\\F-$\"+uAP%)RF-7$$!+4&)>_\\F-$ \"+@9t*3%F-7$$!+Ri7Y\\F-$\"+%R5,?%F-7$$!+k,))R\\F-$\"+I9o>VF-7$$!+='yV 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