{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 "" -1 268 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "Grey Emphasis" -1 269 "Times" 1 12 96 52 84 1 0 1 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 "Hea ding 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 "Co urier" 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 Out put" -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 55 "Procedures for evaluating the sin e and cosine functions" }}{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 func tion approximation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 269 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 269 8 "numfcn.m" }{TEXT -1 72 " contains the code \+ for the alternative mathematical functions including " }{TEXT 0 4 "sin _" }{TEXT -1 6 " and " }{TEXT 0 4 "cos_" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 123 "It can be read into a Maple session by a command \+ similar to the one that follows, where the file path gives its locatio n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/p rocdrs/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 procedures and data" }} {PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 269 6 "intg.m " }{TEXT -1 5 " and " }{TEXT 269 8 "gkdata.m" }{TEXT -1 67 " contain t he code and data for the numerical integration procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read into a Maple session by commands simila r to those that follow, where the file paths give their location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/i ntg.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 73 "Arbitrary precision proc edures to evaluate the sine and cosine functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "The procedu res " }{TEXT 0 5 "sinAP" }{TEXT -1 5 " and " }{TEXT 0 5 "cosAP" } {TEXT -1 25 " calculate the values of " }{XPPEDIT 18 0 "sin(x)" "6#-%$ sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#% \"xG" }{TEXT -1 46 " by using Maclaurin series approximations for " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 66 ".\nEach proced ure makes use of both series along with the formulas " }{XPPEDIT 18 0 "sin(theta)=cos(Pi/2-theta)" "6#/-%$sinG6#%&thetaG-%$cosG6#,&*&%#PiG\" \"\"\"\"#!\"\"F.F'F0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(theta)=s in(Pi/2-theta)" "6#/-%$cosG6#%&thetaG-%$sinG6#,&*&%#PiG\"\"\"\"\"#!\" \"F.F'F0" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "sinAP , cosAP: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4236 "sinAP := proc(x::realcons)\n lo cal z,zs,term,eps,k,even,maxit,saveDigits,\n extraDigits,sum,val,pi, pi2,neg;\n\n # argument reduction involves loss of precision\n # w hen x has large magnitude \n extraDigits := length(max(floor(abs( x))-1,0));\n \n # Increase precision\n saveDigits := Digits;\n \+ Digits := Digits+length(Digits)+extraDigits;\n z := evalf(x);\n p i := evalf(Pi); # Maple evaluates Pi rapidly\n \n # Reduce the \+ range to -Pi/2<=z<=3*Pi/2\n if z>4.7123889803846898577 or z<-1.57079 63267948966192 then\n pi2 := pi+pi;\n\n # Determine the numb er of multiples of 2*Pi to add or subtract.\n k := floor(z/pi2+0. 25);\n z := z - k*pi2;\n end if;\n\n # If z>Pi/2, replace z b y Pi-z so that abs(z)<=Pi/2\n if z>1.5707963267948966192 then z := p i-z end if;\n\n # Use the fact that sine is an odd function\n neg \+ := false;\n if z<0 then\n z := -z;\n neg := true;\n end \+ if;\n \n # Initialisation for Maclaurin series loops\n eps := Fl oat(5,-Digits);\n maxit := Digits*4;\n\n if z<.7853981633974483096 2 then \n # use the sine series to calculate sin(z)\n Digits := Digits - extraDigits;\n term := z;\n sum := term;\n \+ zs := z*z;\n even := false; \n for k from 2 to maxit by 2 d o\n term := term*zs/(k*(k+1));\n if even then\n \+ sum := sum+term;\n else\n sum := sum-term;\n \+ end if;\n if abs(term)<=eps*abs(sum) then break end if; \n even := not even;\n end do;\n else\n z := pi*0. 5-z;\n Digits := Digits-extraDigits; \n # use the cosine se ries to calculate cos(z)\n term := 1.0;\n sum := term;\n \+ zs := z*z;\n even := false; \n for k from 1 to maxit by 2 \+ do\n term := term*zs/(k*(k+1));\n if even then\n \+ sum := sum+term;\n else\n sum := sum-term;\n \+ end if;\n if abs(term)<=eps*abs(sum) then break end if ;\n even := not even;\n end do;\n end if;\n\n Digits \+ := saveDigits;\n if neg then\n -evalf(sum);\n else\n eva lf(sum);\n end if;\nend proc: # of sinAP\n\ncosAP := proc(x::realcon s)\n local z,zs,term,eps,k,even,maxit,saveDigits,\n extraDigits,su m,val,pi,pi2,reflect;\n\n # argument reduction involves loss of prec ision\n # when x has large magnitude \n extraDigits := length(m ax(floor(abs(x))-1,0));\n \n # Increase precision\n saveDigits : = Digits;\n Digits := Digits+length(Digits)+extraDigits;\n z := ev alf(x);\n pi := evalf(Pi); # Maple evaluates Pi rapidly\n \n \+ # Reduce the range to -Pi<=z<=Pi\n if z>3.1415926535897932385 or z<- 3.1415926535897932385 then\n pi2 := pi+pi;\n\n # Determine t he number of multiples of 2*Pi to add or subtract.\n k := floor(z /pi2+0.5);\n z := z - k*pi2;\n end if;\n\n # If z<0, replace \+ z by - z so that 0<=z<=Pi\n if z<0 then z := -z end if;\n\n # If z >Pi/2, replace z by Pi - z so that 0<=z<=Pi/2\n if z>1.5707963267948 966192 then\n z := pi-z;\n reflect := true;\n else\n \+ reflect := false;\n end if;\n \n # Initialisation for Maclaurin \+ series loops\n eps := Float(5,-Digits);\n maxit := Digits*4;\n\n \+ if z<.78539816339744830962 then\n Digits := Digits-extraDigits; \n # use the cosine series to calculate cos(z)\n term := 1.0 ;\n sum := term;\n zs := z*z;\n even := false; \n \+ for k from 1 to maxit by 2 do\n term := term*zs/(k*(k+1));\n \+ if even then\n sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs(term)<=eps *abs(sum) then break end if;\n even := not even;\n end do ; \n else\n z := pi*0.5-z;\n Digits := Digits-extraDigits; \n # use the sine series to calculate sin(z)\n term := z;\n \+ sum := term;\n zs := z*z;\n even := false; \n for \+ k from 2 to maxit by 2 do\n term := term*zs/(k*(k+1));\n \+ if even then\n sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs(term)<=eps*abs (sum) then break end if;\n even := not even;\n end do;\n \+ end if;\n\n Digits := saveDigits;\n Digits := saveDigits;\n \+ if reflect then\n -evalf(sum);\n else\n evalf(sum);\n e nd if;\nend proc: # of cosAP" }}}{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 sinAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot('sinAP'(x),x= -2*Pi..5*Pi,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 499 189 189 {PLOTDATA 2 "6'-%'CURVESG6#7eu7$$!3)****>YH&=$G'!#<$\"3)fq*)fi efD\"!#D7$$!3Ex7\"4=\\L;'F*$\"3'QBiU5&\\&>\"!#=7$$!3VbD?nI^VgF*$\"3C,4 3zC%QP#F37$$!3hLQ\\`pnBfF*$\"3KSd%\\gS\"=NF37$$!3y6^yR3%Q!eF*$\"3KuurA i(>h%F37$$!3SI'HiJ.`f&F*$\"3/AKj'\\A!\\jF37$$!3+\\Tn#zlnQ&F*$\"3&Hjl$y A'4\"yF37$$!3ots'**F37$$!3:--vlOmSZF*$\"3+ac8xH+'***F37$$!3;O,WgWk\"o%F*$\"3/BlsWUF& ***F37$$!31r+8b_iAYF*$\"3Kq@V7%R(f**F37$$!310+#)\\ggjXF*$\"3Eg>i@A_*)) *F37$$!32R*4X%oe/XF*$\"3oDI]'=n[y*F37$$!3'R()*>RwcXWF*$\"3;)GfEuQhk*F3 7$$!3h<=))\\Gh5UF*$\"3_$Qryb'Hn()F37$$!3QgPcg!ec(RF*$\"3TsW*35*o1uF37$ $!3%oQfX*[#yv$F*$\"3?x%GqXX'zdF37$$!3K8]bG<**RNF*$\"35=w-VHVzQF37$$!3: [4b:\\@FMF*$\"3\"y/`wdXv\"GF37$$!3V$)oa-\"QWJ$F*$\"344,J:4')>7$$!3a`(QlZ%)))3$F*$!3yap-AvPo_F\\s7$$!3K^ 5?N0DsHF*$!3e\"GvS9R`o\"F37$$!35\\L'Qf;c&GF*$!3'*oV3V+%4#GF37$$!3)okDD l#)*QFF*$!3O$>^P$)4#=RF37$$!35Xz=6([Bi#F*$!3][b2Y(QA'\\F37$$!3;x1d@*G) *Q#F*$!3Gv\\OA?GHoF37$$!3y3M&>84t:#F*$!3LdcB@wvG$)F37$$!3<;p,[R8=>F*$! 3'ybqN1BGS*F37$$!3aB/3k(e*y;F*$!3WkyTU9cT**F37$$!3=m6.cBHE;F*$!3IWkY0 \\g%)**F37$$!3-4>)z%fit:F*$!3E-kCm*f*****F37$$!3(=lK*R&f4_\"F*$!3/2N\\ VSe()**F37$$!3]%R$)=8$Ho9F*$!3oeh*pX6v%**F37$$!3'*z[y:.'HO\"F*$!3;*eeJ H(z%y*F37$$!3Yljo*\\FwD\"F*$!3d&\\%=?6i8&*F37$$!3#3z^Q*yY?5F*$!3E!HvP3 GN_)F37$$!3#>;s,)G3LyF3$!3)R?s.tui0(F37$$!3OvBO]eTUmF3$!3Q]5ce9ikhF37$ $!3!))e_0#)[C7ILTF37$$! 3o:I$4w9/2$F3$!3xZi_Q!)RAIF37$$!3&*edZ_k)H#>F3$!3E!pNTtc6\">F37$$!31A] =S9ebxF\\s$!3gdWMJ*3yu(F\\s7$$\"3OXvQW;q=PF\\s$\"3V\"=#*3iWyr$F\\s7$$ \"3G6g*GZ)H>:F3$\"39%Gq>IgM^\"F37$$\"3'QlrDIt7c#F3$\"3d%3$HU7OLDF37$$ \"3X'HZA8[Kg$F3$\"3GD&>N1#yDNF37$$\"3/RH#>'HAXYF3$\"31_KJ$od*zWF37$$\" 3k\"e)f\"z(>(o&F3$\"3Q5eNwy`&Q&F37$$\"3AVP\\+b@l\")F3$\"3S1\"4!*Q$o(G( F37$$\"3[!*)Q4KBV1\"F*$\"3D1/QS(3Yu)F37$$\"3W5T4DRCu7F*$\"3mWbTXk\\j&* F37$$\"3SI$\\#HX;%[\"F*$\"3!3^(e@\")\\i**F37$$\"3/)yk4U6_a\"F*$\"3g1(* \\Tss'***F37$$\"3WX-o7$eig\"F*$\"3ux-p'*Gr$***F37$$\"3%Gq&R/_In;F*$\"3 ?J!*HF*$\"3x%*e3wq!Q?*F37$$\"3[0mjNqo)=#F*$\"3K! 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:RdGJFlL*4rjZ!G+z!4$p-6$!$+$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "xx:= Pi/4:\nDigits := 100:\n sinAP(xx);\nevalf(sin(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qky?Q;nP>D%>N0J#RiO&**o)R$)eOSZ)oPf$[GR!\\[5iV%3SCv a'=\"y1rq!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qly?Q;nP>D%>N0J# RiO&**o)R$)eOSZ)oPf$[GR!\\[5iV%3SCva'=\"y1rq!$+\"" }}}{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 cosAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot ('cosAP'(x),x=-2*Pi..5*Pi,color=red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 499 189 189 {PLOTDATA 2 "6'-%'CURVESG6#7gu7$$!3)****>YH&=$G' !#<$\"2))***************F*7$$!37$$!3;O,WgWk \"o%F*$!3YrZgvw'R2$Fio7$$!310+#)\\ggjXF*$!3C%4].>YB[\"F27$$!3'R()*>Rwc XWF*$!3N\"\\J[&fmOEF27$$!3zX3aW-4GVF*$!3#)[LNR64\\PF27$$!3h<=))\\Gh5UF *$!3[[XMF[$)4[F27$$!3U*yA_XNJ4%F*$!3S$)*4#*euU!eF27$$!3QgPcg!ec(RF*$!3 ^bF%)**>q=nF27$$!3%oQfX*[#yv$F*$!3)3K-0v=1;)F27$$!3K8]bG<**RNF*$!3r%\\ $))HC$o@*F27$$!3:[4b:\\@FMF*$!3%*=R'G8l[f*F27$$!3V$)oa-\"QWJ$F*$!3UbJ! *yP*4&)*F27$$!3%3&[/'p\\!eKF*$!3))3)>5RlA$**F27$$!3q=Ga*Gh;?$F*$!3%)pE (*=U'>)**F27$$!37'ySI)GFXJF*$!3=uBjzA$*****F27$$!3a`(QlZ%)))3$F*$!3W7( [fX7h)**F27$$!35\\L'Qf;c&GF*$!3m*RZ5pnQf*F27$$!35Xz=6([Bi#F*$!3kwr7;g% >o)F27$$!3;x1d@*G)*Q#F*$!3p24SL]&[I(F27$$!3y3M&>84t:#F*$!3wsVF*$!3pj\"4;HcRS$F27$$!3') p'[gNY&)z\"F*$!3;rNDuA'yD#F27$$!3aB/3k(e*y;F*$!3;s&*e)f;&z5F27$$!3-4>) z%fit:F*$!3:X&**4T<'HG!#?7$$!3]%R$)=8$Ho9F*$\"3:@+K1zBB5F27$$!3'*z[y:. 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3'>)\\8kJ'fr*F27$$\"3+.*R)>!yGb\"F^cm$!3))eM&=>'*)R)*F27$$\"3)>gwxy])e :F^cm$!39-^g]otG**F27$$\"3'4I8dbB[c\"F^cm$!3/jS\"[Io@)**F27$$\"3'**** \\OK'zq:F^cm$!2c***************F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\" !FeanFdan-%+AXESLABELSG6$Q\"x6\"Q!Fjan-%*THICKNESSG6#\"\"#-%%VIEWG6$;$ !+3`=$G'!\"*$\"+Fjzq:Fcan%(DEFAULTG" 1 2 0 1 10 2 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 72 "xx:= sqrt(24)/5;\nDigits := \+ 300:\ncosAP(xx);\nevalf(cos(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*(\"\"#\"\"\"\"\"&!\"\"\"\"'#F(F'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]l$R9A4*zI\"yresLI`DI=)*[q9si'ofI-G<\\k* HnWFDr1/YrY%Q5OnB()3;e&)>Ta*>wb1\"H7h$>:s+\\&QAMd&!$+$" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"g]l#R9A4*zI\"yresLI`DI=)*[q9si'ofI -G<\\k*HnWFDr1/YrY%Q5OnB()3;e&)>Ta*>wb1\"H7h$>:s+\\&QAMd& !$+$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "xx:= 10^500:\nDigits := \+ 300:\ncosAP(xx);\nevalf(cos(xx));\nDigits := 10:" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$!g]l#pa!)*4E%3.)y+:lEjzK>fdn,f$*Giz)Rnt6x%pS^^h\"z:Z[0&oH)4>RP*>b#)eX#*yhb4xefhdjE0m<)[([hp5)\\ h3;UXw7vfdn,f$*Giz)Rnt6x%pS^^h\"z:Z[0&oH)4>RP*>b#)eX#*yhb4xefhdjE0m<)[( [hp5)\\h3;UXw7v " 0 "" {MPLTEXT 1 0 75 "xx:= Pi/2-10^(-10):\nDigits := 100:\ncosAP(xx);\nevalf(cos(xx));\n Digits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q*RA>tb_\"GoRDoRD oRDokmmmmmmmmm;MLLLLLLLLL$)********************!$5\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q*RA>tb_\"GoRDoRDoRDokmmmmmmmmm;MLLLLLLLLL$)**** ****************!$5\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Speed comparisons for sinAP " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " }{TEXT 0 3 "sin " }{TEXT -1 16 " is faster than " }{TEXT 0 5 "sinAP" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "st := time():\nfor i to 500 do sinAP(rand()*Float(1,-11)) end d o:\ntime()-st;\nst := time():\nfor i to 500 do sin(rand()*Float(1,-11) ) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$.*!\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$[\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "st := time( ):\nDigits := 50:\nfor i to 100 do sinAP(rand()*Float(1,-11)) end do: \ntime()-st;\nst := time():\nfor i to 100 do sin(rand()*Float(1,-11)) \+ end do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"$8)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$[$!\"$" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Speed comparisons for cosAP " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's f unction " }{TEXT 0 3 "cos" }{TEXT -1 16 " is faster than " }{TEXT 0 5 "cosAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "st := time():\nfor i to 500 do cos AP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 5 00 do cos(rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"$u*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$V\" !\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "st := time():\nDigits := 50:\nfor i to 100 do cosAP( rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 100 \+ do cos(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$9)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$o\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 71 "Fixed prec ision procedures to evaluate the sine and cosine functions " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 101 "In this section we construct fixed precision versions of the s ine and cosine functions which can use " }{TEXT 259 34 "hardware float ing point arithmetic" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 " Each of the procedures " }{TEXT 0 5 "sin16" }{TEXT -1 5 " and " } {TEXT 0 5 "cos16" }{TEXT -1 39 " uses rational approximations for both " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 17 " on the interv al " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F &F'F(F)" }{TEXT -1 29 ", together with the formulas " }{XPPEDIT 18 0 " sin(theta)=cos(Pi/2-theta)" "6#/-%$sinG6#%&thetaG-%$cosG6#,&*&%#PiG\" \"\"\"\"#!\"\"F.F'F0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(theta)=s in(Pi/2-theta)" "6#/-%$cosG6#%&thetaG-%$sinG6#,&*&%#PiG\"\"\"\"\"#!\" \"F.F'F0" }{TEXT -1 3 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "A r ational approximations for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG " }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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'F NF/F/!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "to approxim ate " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\" \"F)*&F&F'F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "First we find a rational approximation fo r the function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f (x) = PIECEWISE([(x-sin(x))/(x^3), x <> 0],[1/6, x = 0]);" "6#/-%\"fG6 #%\"xG-%*PIECEWISEG6$7$*&,&F'\"\"\"-%$sinG6#F'!\"\"F.*$F'\"\"$F20F'\" \"!7$*&F.F.\"\"'F2/F'F6" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 156 "evalf(remez((x-sin(x))/x^3,x=-Pi/4..Pi/4,[6,4],'ma xerr',\n type=even,errtype=absolute,weight=x^2+1e-50,maxgraph=7,info= true),25):\nr := unapply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~minimax~error~estimate~by~solvin g~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M--~minimis ing~the~weighted~ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,*$\"Cs^FPKuS([bmmmmmmm\"!#M\"\" \"*&$\"Cg7jialt%o,\"yB<0#['f!#OF()%\"xG\"\"#F(!\"\"*&$\"Cmp.S<$f(3%\\( )G*zv'R=*!#QF()F.\"\"%F(F(*&$\"C;gE\"Qddc80O^yPXqG&!#SF()F.\"\"'F(F0F( ,($F(\"\"!F(*&$\"C$RdD*H<]o&R@b'*o26U\"!#NF(F-F(F(*&$\"Ch%pCUNs.:E!Q(p $4d6rF4F(F5F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%goal~for~relative~difference :G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"@$H<*RB>d1PC2d`K-(!#T" }} {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 "63-%'CURVESG6%7\\u7$$\"\"!F) %%FAILG7$$\"@![nzqIm/f_2Uj&\\f?$!#`7$$\"@'\\$fT hK4=0:%o_.r5!#L$\"@xy(=-&=[;!QE__M#G\"!#_7$$\"@W-R7#*)RrxDi-Hb1;F5$\"@ d)z/z$H+ImR*Hs9&)GF87$$\"@\"*p=$G_'=O5Io`q?9#F5$\"@[\\MWaA/9p&3G\"G)G^ F87$$\"@([!yC%yzUb^C0e58KF5$\"@L(p'fvoO$QchA!zP:\"!#^7$$\"@#)RPmXIPs?g O2TTG%F5$\"@PY9kI8'o>6U?ok]?FH7$$\"@t4c\\o&f&3J!\\5;@EkF5$\"@ajKu25@)> FxA+k5YFH7$$\"@lzuK\"4YZ9/KZ@Go&)F5$\"@EsI)f]2%y6>GsY%)=)FH7$$\"@&>7*p 8>r@1)4ABC&G\"!#K$\"@dtM')yb/]'p#411r$=!#]7$$\"@$f\\l#=#\\*G3k%Hkl8GD$Fhn7$$\"@9.ulk7NO'omx!y\"fCFen$\"@rUll'yS))4%e& f)zMj'Fhn7$$\"@O5$\\5J`PW'pes*p/KFen$\"@k\\*e+WUh^,ZW&*e66!#\\7$$\"@iF ;Qh^G\"oz-cQ`\")[Fen$\"@NzgQ)p7UME,kr3qCFho7$$\"@b42_\"[\\6gPFJJ[plFen $\"@0kx=*y0s_gN^!)z,UFho7$$\"@_Ah][wYasf#y)4%\\#)Fen$\"@W[`Kp\\(4Bcs4p U*4'Fho7$$\"@ma]D\"fKVryI=q\"p!)*Fen$\"@$)zi!=c:!*4p`A,=GyFho7$$\"@kW_ !=,M5M$\\Zai>9\"!#J$\"@Z**eKA#=\\slN(\\jnR*Fho7$$\"@.&RfK*f%>xE(H\"eND 7F`q$\"@I\\z$4^GOMI)\\i$)f+\"!#[7$$\"@UXNruz&G?g>\"3\\(38F`q$\"@Yw\\b' zSw)Ga#R%)>f5Fhq7$$\"@cR@dIXwXa]h!\\(=R\"F`q$\"@xS8^=@*=@I*3,^s4\"Fhq7 $$\"@rL2V'3r')o]5J2+v9F`q$\"@cul/a^oYNkT([p=6Fhq7$$\"@]S0vN?o4<7K,xj\\ \"F`q$\"@D/(o9Hh(Q*G.1/O@6Fhq7$$\"@IZ.2&)HpIF>`H`x^\"F`q$\"@J&Q571M\"p \"y-OM#G7\"Fhq7$$\"@5a,RMRq^PEudH\"R:F`q$\"@8`^Z\"H&pv$p?j\"oI7\"Fhq7$ $\"@*3'*4P)[rsZL&fe]g:F`q$\"@Uo,?NWF%f06Fhq7$$\"@[3xk)z)*[RIi^RK@%G:pj'z\"F`q$\"@CE,$[)fRD:(fU34F5Fhq7$$\"@h]g\"4ebqa_zLg?m>F `q$\"@28W%4f@os7DSv!Rl)Fho7$$\"@%*\\/K>K)[@qfE(Q80#F`q$\"@h%Q:L_h3_(G@ s)\\vvFho7$$\"@E\\[sd3r#)y)R>9ZO@F`q$\"@5;7gjqC!>KEDS7VjFho7$$\"@cc^\" H7IAiwhVD^=AF`q$\"@7)Hxz$\\+4A(o4u>K]Fho7$$\"@'QY0\")Q\\>o0FEsQBfDOFho7$$\"@4%fp,Lr1OuIo[0vBF`q$\"@!)=rq\"[mXB&edqQ*)G #Fho7$$\"@JCPBsKffLy(ogb\\CF`q$\"@L%z()*yW%3v/&\\\")GU>*Fhn7$$\"@b#)** 42C6?i[??X\"QDF`q$!@X0jt2m!ylNk193$=(Fhn7$$\"@ySi'>aJ13*=`LMni#F`q$!@5 V%p@CdK'f.*p:0EBFho7$$\"@\\)G!y@0-?ea9&4y,FF`q$!@xx5![2#*zM@xFasKOFho7 $$\"@?OVf,&4%fD!fnv#ox#F`q$!@5Hw=;S!3/Ag!*RHh[Fho7$$\"@(eB#[DI.6'eCy;U ^HF`q$!@D=[c%RHGt33#e]-G(Fho7$$\"@%*eSPFTn7uFTO%pGIF`q$!@.c?'GW-Kyf*ow Q&)4)Fho7$$\"@*>)eEH_J9i4+0nf5$F`q$!@S!*exY$F`q$!@! 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#M$\"C]dG))GPu#yPwcF\">))4LF%$\"Cv\\Qzu8LtGIha;GbP=*!#HF()F.\"\"%F(F(*&$\":#[F*)f69 [Div'G&!#JF()F.\"\"'F(F0F(,($F(\"\"!F(*&$\":=(QQ*HW7!*4V6U\"!#EF(F-F(F (*&$\":!Qox#GkSG5n?6(F4F(F5F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"-Sb'G_:\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operat orG%&arrowGF(*&,*$\":^cQ+bmmmmmmm\"!#D\"\"\"*&$\":$p`_wWAk,:wkf!#FF1)9 $\"\"#F1!\"\"*&$\":/*p)yk2*4'>bP=*!#HF1)F7\"\"%F1F1*&$\":#[F*)f69[Div' G&!#JF1)F7\"\"'F1F9F1,($F1\"\"!F1*&$\":=(QQ*HW7!*4V6U\"!#EF1F6F1F1*&$ \":!Qox#GkSG5n?6(F=F1F>F1F1F9F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\":Twb`78(Rb'G_:\"!#U" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "The rational function " }{XPPEDIT 18 0 "s(x) = x-r (x)*`.`*x^3;" "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$! \"\"" }{TEXT -1 31 " provides an approximation for " }{XPPEDIT 18 0 "s in(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "The relative error graph is drawn for " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 11 " , which is " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 74 " wi th the coefficients rounded to 20 digits and arranged in nested form. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "s:=x->x-(.1666666666666665500385651-.596476150164224 4765253693e-2*x^2+.9183755196099076478869904e-4*x^4-.52867562254814115 98927482e-6*x^6)/(1.+.1421143099012442993838718e-1*x^2+.71120671028406 42827768380e-4*x^4)*x^3:\nh := unapply(evalf(convert(s(x),horner),20), x);\nevalf(plot((1-h(x)/sin(x)),\n x=-Pi/4..Pi/4,color=COLOR(RGB,.4,0, .9)),25);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&9$\"\"\"**$F.\"\"!F.,*$\"5/]lmmmmmm;!#?F.*&$\"5`wWAk ,:wkf!#AF.)F-\"\"#F.!\"\"*&$\"5*yk2*4'>bP=*!#CF.)F-\"\"%F.F.*&$\"5*)f6 9[Div'G&!#EF.)F-\"\"'F.FGVyF*$!'M)e*F-7$$!:euw)HWA\"zK#eKy F*$!'WOmF-7$$!:\"*G2+\\7\")[n#)=#yF*$!'C^RF-7$$!:D$y8]0+&=-$=6yF*$!'k= :F-7$$!:ePo-h))=)oL[+yF*$\"';\\nF*7$$!:\">*)Rqmxy:Py*y(F*$\"(!\\UEF*7$ $!:DYH0tkcF1%3zxF*$\"(dmR%F*7$$!:e+g1z_D(4WQoxF*$\"(4&\\fF*7$$!:\"\\0z ]3WpcZodxF*$\"($o7tF*7$$!:D4@4\"*GjO5&)pu(F*$\"(ft\\)F*7$$!:ej^5(p@j]a GOxF*$\"((G9&*F*7$$!:\"z@=J]5g(z&eDxF*$\")!zt.\"F*7$$!:eEV9:\")Q:\\'=/ xF*$\")k(f;\"F*7$$!:BN/RL9%zyQhwF*$\")mDv7F*7$$!:B)f4_9KQE#)o]wF*$\")Hgw7F *7$$!:c_EA^4_Ld))*RwF*$\")-3p7F*7$$!:Ah([Kc)*Gn#*e=wF*$\")$3-B\"F*7$$! 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[:F-7$$\":j&Qs5#*4#Gm%)>#yF*$!'ivRF-7$$\":R7@qd:s)>.lKyF*$!'MamF-7$$\" :;R=L%>L#p(fJVyF*$!';)f*F-7$$\":$fch4$[uRj\")R&yF*F+-%&COLORG6&%$RGBG$ \"\"%!\"\"\"\"!$\"\"*Fhap-%+AXESLABELSG6$Q\"x6\"Q!F`bp-%%VIEWG6$;$!:3m :'4$[uRj\")R&yF*$\":3m:'4$[uRj\")R&yF*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalf(evalf( numapprox[infnorm](1-h(x)/sin(x),x=-Pi/4..Pi/4),25),5);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"&AG\"!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in using " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " in the in terval " }{XPPEDIT 18 0 "[-Pi/4, Pi/4];" "6#7$,$*&%#PiG\"\"\"\"\"%!\" \"F)*&F&F'F(F)" }{TEXT -1 11 " is about " }{XPPEDIT 18 0 "1.3*`. `*10 ^(-17);" "6#*(-%&FloatG6$\"#8!\"\"\"\"\"%#.~GF))\"#5,$\"# " 0 "" {MPLTEXT 1 0 77 "xx := evalf(Pi/4,20);\nevalf(evalf(h(xx),20),17) ;\nevalf(evalf(sin(xx),20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# xxG$\"5i4$[uRj\")R&y!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2`Zl=\"y 1rq!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2_Zl=\"y1rq!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approximation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"x G" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"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 relative error is great er than or equal to the specified \"" }{TEXT 269 3 "eps" }{TEXT -1 3 " \".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "randomize():\neps \+ := Float(5,-16);\nfor i from 1 to 200 do\n xx := evalf(rand()/Float( 1,12)*.78539816339744830962,20);\n axx := evalf(sin(xx),20);\n hx x := 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(\" sin(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 " trial no. 33, x = . 0103495858909910," }}{PARA 6 "" 1 "" {TEXT -1 76 " sin(x) = .01034940 11278473, h(x) = .0103494011278473, rel error = 7.55e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 82, x = .1418959492215318," }}{PARA 6 "" 1 "" {TEXT -1 76 " sin(x) = .1414 202619642462, h(x) = .1414202619642461, rel error = 5.81e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 101 , x = .1194241456945338," }}{PARA 6 "" 1 "" {TEXT -1 76 " sin(x) = . 1191404743442833, h(x) = .1191404743442832, rel error = 6.83e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial n o. 186, x = .1042488943610687," }}{PARA 6 "" 1 "" {TEXT -1 76 " sin( x) = .1040601703633738, h(x) = .1040601703633737, rel error = 9.07e-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 30 "A rational approximations for " }{XPPEDIT 18 0 "cos(x) " "6#-%$cosG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "We choose a rationa l function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "s(x) \+ = (a[0]*x+a[1]*x^2+a[2]*x^3+` . . . `+a[m]*x^m)/(b[0]+b[1]*x+a[2]*x^2+ ` . . . `+b[n]*x^n);" "6#/-%\"sG6#%\"xG*&,,*&&%\"aG6#\"\"!\"\"\"F'F/F/ *&&F,6#F/F/*$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/*$F'F4F/F/F:F /*&&FB6#%\"nGF/)F'FNF/F/!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"x G" }{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "First we find a rati onal approximation for the function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([(1-cos(x))/(x^2), x <> 0],[1/2, \+ x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&,&\"\"\"F.-%$cosG6#F'!\" \"F.*$F'\"\"#F20F'\"\"!7$*&F.F.F4F2/F'F6" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "e valf(remez((1-cos(x))/x^2,x=-Pi/4..Pi/4,[6,6],\n 'maxerr',type=even,e rrtype=absolute,weight=x^2+1e-50,maxgraph=7,info=true),25):\nr := unap ply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorit hm:~calculating~minimax~error~estimate~by~solving~a~rational~equationG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M--~minimising~the~weighted~ABSOL UTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G --------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,*$\"D\">n%3*3$[%G-+++++++]!#N\"\"\"*&$\"DFf8,K)>`ltq %pDu\"*R:$!#OF()%\"xG\"\"#F(!\"\"*&$\"D!eLE_)H5@f!Q$)3y:U&R'!#QF()F.\" \"%F(F(*&$\"Dm))RGd\\LMS^m)\\&[G,7%!#SF()F.\"\"'F(F0F(,*$F(\"\"!F(*&$ \"DqVDrdlE.W8T%>[)\\`-#F,F(F-F(F(*&$\"DhB!obEolsuM7**z2)4*=F4F(F5F(F(* &$\"D+3eq*ymEz_jK,>\"GPh)!#TF(F;F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4cw4;&HRsr#!#a " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%a[B6!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5re lative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"A[r6G,'3jYVj r8.$\\q!#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~diff erence:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"A18$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%7jt7 $$\"\"!F)%%FAILG7$$\"A]ZnzqIm/f_2Uj@y0-toD'pi*=%FC7$$\"A.(4c\\o&f&3J!\\5;@EkF5$!A0gW=7fKk (R,NNMuT*FC7$$\"A0'zuK\"4YZ9/KZ@Go&)F5$!A.\"H'*R(yv6oQ9Gg\">n\"!#`7$$ \"AT>7*p8>r@1)4ABC&G\"!#L$!Ag&>#\\*zBKY&)4#3m3ZPFX7$$\"A@f\\l#=#\\*G3k %Hkl8R3)[C# Fco7$$\"A/wi\"Qh^G\"oz-cQ`\")[Ffn$!AU377*)zWp*oCxhV)3\\Fco7$$\"AK&42_ \"[\\6gPFJJ[plFfn$!A]a/=^jJ=c4\"o/SD:)Fco7$$\"A,D71&[wYasf#y)4%\\#)Ffn $!A2eIsFO!)z-`@mJ8X6!#^7$$\"AQY0b7fKVryI=q\"p!)*Ffn$!An8Ol#e))yVj*)H(e a69Fhp7$$\"A]]PlajLig+HBrKh5!#K$!A1\">Kj9\"Faq$!AblJjM&*)[\"3ex&y3Ug\"Fhp7$$\"AJ)>B`-+\\c+h)y\"fO=\"F aq$!Av6WbX&Q9iz-6Sfcj\"Fhp7$$\"AE]RfK*f%>xE(H\"eND7Faq$!A#o`6&fq=<)f%p nC]e;Fhp7$$\"ACE$Hi))RnH^G+8/iC\"Faq$!Aml>lz/#y$48^2`bm;Fhp7$$\"A@-Z') R)>S([V3ZC0n7Faq$!A3\\zXf2\"Qx(\\\\+`Hs;Fhp7$$\"A=y+]$z*H^%=STw+zG\"Fa q$!A:t[6HS(evVc@Vxcn\"Fhp7$$\"A;aa8Z(z&G?g>\"3\\(38Faq$!Aw9V4sejNs7[^5 mw;Fhp7$$\"A`R>yOh%e8lMu`I&H8Faq$!A+YIX`327fOvi&=_n\"Fhp7$$\"A!\\UGk_7 JCGtO*>J]8Faq$!AYRF7UMJt#GbUnD8n\"Fhp7$$\"AF5\\2;*y.N\">\"*\\M4r8Faq$! 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$\":,j(p?t&RC#>,MvF*$\"&Pk$F*7$$\":a;#H`+>aOKMbvF*$\"&8M(F*7$$\":2q')e yAW1bumd(F*$\"'C*4\"F*7$$\":fB\"[=bluke+)f(F*$\"'1\\9F*7$$\":7xv5D))[) yrL>wF*$\"'8rm( F*$\"'krAF*7$$\":qPf)[ke:@6L$o(F*$\"'I9CF*7$$\":Zkc^\"Gq?yn*Rp(F*$\"'y \\CF*7$$\":B\"RX\"=>e_ViYq(F*$\"'HdCF*7$$\":+=^xaN4B4G`r(F*$\"'qLCF*7$ $\":vW[S\">0O\\P*fs(F*$\"'pvBF*7$$\":G)HkYYGYj]KZxF*$\"'!>9#F*7$$\":\" =vBzt^cxjloxF*$\"'sC8\" yF*$\"&K'>F*7$$\":j&Qs5#*4#Gm%)>#yF*$!%*f$F-7$$\":R7@qd:s)>.lKyF*$!%s* *F-7$$\":;R=L%>L#p(fJVyF*$!&;s\"F-7$$\":$fch4$[uRj\")R&yF*F+-%&COLORG6 &%$RGBG$\"\"%!\"\"\"\"!$\"\"*F\\dp-%+AXESLABELSG6$Q\"x6\"Q!Fddp-%%VIEW G6$;$!:3m:'4$[uRj\")R&yF*$\":3m:'4$[uRj\")R&yF*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "evalf(evalf(numapprox[infnorm](1-h(x)/cos(x), x=-Pi/4..Pi/4),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&%RD!#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The ma ximum relative error in using " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG " }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6 #%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-Pi/4, Pi/4] ;" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 12 " is abou t " }{XPPEDIT 18 0 "2.5*`.`*10^(-19);" "6#*(-%&FloatG6$\"#D!\"\"\"\" \"%\".GF))\"#5,$\"#>F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "xx := evalf(Pi/4,20); \nevalf(evalf(h(xx),20),17);\nevalf(evalf(cos(xx),20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5i4$[uRj\")R&y!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2_Zl=\"y1rq!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"2_Zl=\"y1rq!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "We can test the accuracy of the approximation " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 35 " with random numbers b etween 0 and " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The printout occurs only when the relative error is greater than or equal to the specified \" " }{TEXT 269 3 "eps" }{TEXT -1 3 "\".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "randomize():\neps := Float(3,-16);\nfor i from 1 to \+ 200 do\n xx := evalf(rand()/Float(1,12)*.78539816339744830962,20);\n axx := evalf(cos(xx),20);\n hxx := evalf(h(xx),16);\n e := eva lf(abs((axx-hxx)/axx),20);\n if e>=eps then \n printf(\" trial no. %d, x = %.16f,\\n\",i,xx);\n printf(\" cos(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 " trial no. 13, x = .7695898231074810," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7181961477994204, h(x) = .71819614779942 02, rel error = 3.34e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 15, x = .7710841575882734," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7171555275421453, h(x) = .717155527 5421457, rel error = 5.31e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 40 " trial no. 107, x = .7744805680036549, " }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7147843925786553, h(x) = \+ .7147843925786556, rel error = 3.51e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 121, x = .7776100596806 054," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7125923019497168, h(x ) = .7125923019497165, rel error = 3.75e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 133, x = .74368925 28868423," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7359759202585065 , h(x) = .7359759202585062, rel error = 4.03e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 150, x = .74 52386863081996," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .7349260588 549739, h(x) = .7349260588549735, rel error = 5.18e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 181, x \+ = .7821054610626047," }}{PARA 6 "" 1 "" {TEXT -1 76 " cos(x) = .70943 12359457308, h(x) = .7094312359457305, rel error = 4.09e-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 12 "sin16, cos16" }}{PARA 0 "" 0 "" {TEXT -1 147 "Here is the code for the fixed precision versions of the sine and cosine function s, which can be evaluated with hardware floating point arithmetic." }} {PARA 0 "" 0 "" {TEXT -1 32 "There are subsidiary procedures " }{TEXT 0 9 "sin_apprx" }{TEXT -1 5 " and " }{TEXT 0 9 "cos_apprx" }{TEXT -1 47 " for evaluating the rational approximations to " }{XPPEDIT 18 0 "s in(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[- Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3022 "sin16 := proc(xx)\n local x,pi,pi2,piBy2,k,neg,val; \n \+ \n pi := 3.141592653589793;\n pi2 := 6.283185307179586;\n piBy2 \+ := 1.570796326794897;\n\n x := evalf(xx);\n \n # Reduce the r ange to -Pi/2<=x <=3*Pi/2 approximately\n if x>4.712388980384690 or \+ x<-1.570796326794897 then\n\n # Determine the number of multiples of 2*Pi to add or subtract.\n k := floor(x/pi2+0.25);\n x : = x-k*pi2;\n end if;\n\n # If x>Pi/2, replace x by Pi-x so that ab s(x)<=Pi/2\n if x>1.570796326794897 then x := pi-x end if;\n\n # U se the fact that sine is an odd function\n if x<0 then\n x := - x;\n neg := true;\n else neg := false end if;\n \n if x<.78 53981633974483 then \n # use the rational approximate for sin\n \+ val := sin_apprx(x);\n else\n x := piBy2-x;\n # use th e rational approximate for cos\n val := cos_apprx(x);\n end if; \n\n if neg then -val else val end if;\nend proc: # sin16\n\ncos16 : = proc(xx)\n local x,pi,pi2,piBy2,k,reflect,val; \n \n pi := 3.1 41592653589793;\n pi2 := 6.283185307179586;\n piBy2 := 1.570796326 794897;\n\n x := evalf(xx);\n \n # Reduce the range to -Pi<=x x<=Pi\n if x>pi or x<-pi then\n # Determine the number of multi ples of 2*Pi to add or subtract.\n k := floor(x/pi2+0.5);\n \+ x := x-k*pi2;\n end if;\n\n # If x<0, replace x by -x so that 0<=x <=Pi\n if x<0 then x := -x end if;\n\n # If x>Pi/2, replace x by P i-x so that 0<=x<=Pi/2\n if x>1.570796326794897 then\n x := pi- x;\n reflect := true;\n else\n reflect := false;\n end i f;\n \n if x<.7853981633974483 then \n # use the rational app roximate for cos\n val := cos_apprx(x);\n else\n x := piBy 2-x;\n # use the rational approximate for sin\n val := sin_a pprx(x);\n end if;\n\n if reflect then -val else val end if;\nend \+ proc: # cos16\n\nsin_apprx := proc(x)\n local z,a1,a2,a3,a4,b1,b2,nu m,den;\n\n # coefficients of numerator in rational approximation\n \+ a1 := .1666666666666665500385651;\n a2 := -.59647615016422447652536 93e-2;\n a3 := .9183755196099076478869904e-4;\n a4 := -.5286756225 481411598927482e-6;\n\n # coefficients of denominator in rational ap proximation\n b1 := .1421143099012442993838718e-1;\n b2 := .711206 7102840642827768380e-4;\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* z)*z;\n\n (1-num/den)*x;\nend proc: # sin_apprx \n\ncos_apprx := pro c(x)\n local z,a1,a2,a3,a4,b1,b2,b3,num,den;\n\n # coefficients of numerator in rational approximation\n a1 := .5000000000000000023503 724;\n a2 := -.3154004216236836154796330e-1;\n a3 := .639549337563 8683593037923e-3;\n a4 := -.4120239986012705773813651e-5;\n\n # co efficients of denominator in rational approximation\n b1 := .2025324 900859680520646671e-1;\n b2 := .1890916480636244018744239e-3;\n b3 := .8613069640975447170742372e-6;\n\n # evaluate the rational appro ximation\n z := x*x;\n num := (a1+(a2+(a3+a4*z)*z)*z)*z;\n den : = 1+(b1+(b2+b3*z)*z)*z;\n\n 1-num/den;\nend proc: # of cos_apprx \+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Testing th e procedures sin16, cos16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 85 "We can use Maple's sine and cosine functions to ch eck the accuracy of the procedures " }{TEXT 0 5 "sin16" }{TEXT -1 5 " \+ and " }{TEXT 0 5 "cos16" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "xx := 0.15;\ncos_apprx(xx);\ncos(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#:!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+z2r())*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z2r())*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf(plot(sin(x)-'sin16'(x),x=0..10*Pi,color=blue),20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7[\\l7$$\" \"!F)F(7$$\"5;e2#e*p*))QU$!#?$\"#QF-7$$\"5J;:k\"*RzxZoF-$!#aF-7$$\"5'3 )R@evj;?sF-$!#lF-7$$\"5RXkyC6[b#f(F-$\"#zF-7$$\"5mxE23H!\\(yxF-$\"#JF- 7$$\"5$*4*e8pCV\\'zF-$\"&Dm#F-7$$\"5>U^kuku8^\")F-$\"&9h#F-7$$\"5Yu8$z DoJtL)F-$\"&!fDF-7$$\"5`.j2\"Rb3@3*F-$\"&DM#F-7$$\"5hK7ACDa)o#)*F-$\"& C6#F-7$$\"53460z;RkJ6!#>$\"&\">;F-7$$\"5*[4!o0\"H*f!G\"Fjn$\"&$*3\"F-7 $$\"5)zx:;'RsNk8Fjn$\"%0yF-7$$\"51h9b<)=:\"[9Fjn$\"%hYF-7$$\"5g-$>bC;% ***[\"Fjn$\"%vIF-7$$\"59Wr[tOJ(=`\"Fjn$\"%#[\"F-7$$\"5\"\\1ruQi7Gb\"Fj n$\"$%oF-7$$\"5o&)\\X,6@vt:Fjn$\"$$=F-7$$\"5X1*Qa\")f\"p%f\"Fjn$\"%![ \"F-7$$\"5AFGUH&3Jch\"Fjn$\"%vFF-7$$\"5Q$>%HT#)p9$y\"Fjn$\"&^I\"F-7$$ \"5afb;`zGm]>Fjn$\"&iH#F-7$$\"5\"Q9,b2=*G>@Fjn$\"&&GKF-7$$\"54Gn$y>[:z G#Fjn$\"&%pSF-7$$\"57w'GJ'pP**3BFjn$\"&q;%F-7$$\"5:C1UGd?2IBFjn$\"&HE% F-7$$\"5>sDr$\\M]6N#Fjn$\"&lN%F-7$$\"5A?X+fK'GAP#Fjn$\"&9s\"F-7$$\"5H; 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')o5\"F-7$$\"5(4\"pdnyFdTIFi[m$!'n$\\\"F-7$$\"54G8C7'p7&\\IFi[m$!'gt;F -7$$\"5AXd!pNh_u0$Fi[m$!'*G%=F-7$$\"5^\\=2$HfP%fIFi[m$!'Y$)=F-7$$\"5z` zBHsDUhIFi[m$!'JB>F-7$$\"53eSSl^vSjIFi[m$!'%>p\"F-7$$\"5Ni,d,JDRlIFi[m $!'.E " 0 "" {MPLTEXT 1 0 57 "xx := 3.;\nevalhf(sin16 (xx));\nevalf(evalf(sin(xx),20),18);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#xxG$\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3-r')f!3+7T \"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3As')f!3+7T\"!#=" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "xx := 28.5;\nevalhf(sin16(xx));\nevalf(evalf(sin(xx),20),18);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$&G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3+vz'=ScvB#!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3 Ckz'=ScvB#!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "cos16" }{TEXT -1 36 " has sim ilar error characteristics. 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'z7'y2)HFhhl$!(eTw\"F]p7$$\"5tG7))f6D&G)HFhhl$!(^^w\"F]p7$$\"5 yHs!)p.k#\\)HFhhl$!):PlFF]q7$$\"5*=Bf'*y=u!*)HFhhl$!(HDw#F]p7$$\"5,M7^ 4s>A$*HFhhl$!(v\\v#F]p7$$\"5BQ_@\\Sv^,IFhhl$!'qDFF-7$$\"5YU#>*))3J\")4 IFhhl$!'KxEF-7$$\"5rw![#yVHpDIFhhl$!'AMDF-7$$\"5(4\"pdnyFdTIFhhl$!'*pK #F-7$$\"54G8C7'p7&\\IFhhl$!'=,AF-7$$\"5AXd!pNh_u0$Fhhl$!'Ei?F-7$$\"5Ni ,d,JDRlIFhhl$!'FY;F-7$$\"5[zXBY[CLtIFhhl$!'C/:F-7$$\"5cl2Uu*\\(R!4$Fhh l$!'>o6F-7$$\"5j^pg-^DY2JFhhl$!&=)zF-7$$\"5rPJzI-w_CJFhhl$!&'\\SF-7$$ \"5yB$z*e`EfTJFhhlF(-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG 6$Q\"x6\"Q!F`ip-%%VIEWG6$;F($\"5&QKz*e`EfTJFhhl%(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 38 "Translating from Maple code to C \+ code " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "We can translate the programs above into C using the Maple procedure \+ C in the code generating package " }{TEXT 269 7 "codegen" }{TEXT -1 8 ". \n see " }{HYPERLNK 17 "codegen[C]" 2 "codegen[C]" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "codegen/C/procedure" 2 "codegen/C/procedure" "" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "with(codegen,C):\nC(sin16,ansi);\nC(cos16,ansi );\nC(sin_apprx,ansi);\nC(cos_apprx,ansi);" }}{PARA 6 "" 1 "" {TEXT -1 17 "#include " }}{PARA 6 "" 1 "" {TEXT -1 23 "double sin16( double xx)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 8 " int k;" }}{PARA 6 "" 1 "" {TEXT -1 10 " int neg;" }}{PARA 6 " " 1 "" {TEXT -1 12 " double pi;" }}{PARA 6 "" 1 "" {TEXT -1 13 " dou ble pi2;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double piBy2;" }}{PARA 6 " " 1 "" {TEXT -1 13 " double val;" }}{PARA 6 "" 1 "" {TEXT -1 11 " do uble x;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 30 " pi = 0.3141592653589793E1;" }}{PARA 6 "" 1 "" {TEXT -1 31 " pi2 = 0.6283185307179586E1;" }}{PARA 6 "" 1 "" {TEXT -1 33 " pi By2 = 0.1570796326794897E1;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0. 1E1*xx;" }}{PARA 6 "" 1 "" {TEXT -1 62 " if( 0.471238898038469E1 < \+ x || x < -0.1570796326794897E1 )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \+ \{" }}{PARA 6 "" 1 "" {TEXT -1 30 " k = floor(x/pi2+0.25);" }} {PARA 6 "" 1 "" {TEXT -1 20 " x += -k*pi2;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 34 " if( 0.1570796 326794897E1 < x )" }}{PARA 6 "" 1 "" {TEXT -1 15 " x = pi-x;" }} {PARA 6 "" 1 "" {TEXT -1 17 " if( x < 0.0 )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 15 " x = -x;" }}{PARA 6 "" 1 "" {TEXT -1 16 " neg = 1;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }} {PARA 6 "" 1 "" {TEXT -1 14 " neg = 0;" }}{PARA 6 "" 1 "" {TEXT -1 32 " if( x < 0.7853981633974483 )" }}{PARA 6 "" 1 "" {TEXT -1 25 " val = sin_apprx(x);" }}{PARA 6 "" 1 "" {TEXT -1 9 " else \+ " }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 20 " x = piBy2-x;" }}{PARA 6 "" 1 "" {TEXT -1 27 " val = cos_apprx(x);" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 15 " if( (neg) )" }}{PARA 6 "" 1 "" {TEXT -1 19 " \+ return(-val);" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 " " 1 "" {TEXT -1 18 " return(val);" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }}{PARA 6 "" 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 17 "#include " }}{PARA 6 "" 1 "" {TEXT -1 23 "double cos16(double xx)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{ " }}{PARA 6 "" 1 "" {TEXT -1 8 " int k;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double pi;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double pi2;" }} {PARA 6 "" 1 "" {TEXT -1 15 " double piBy2;" }}{PARA 6 "" 1 "" {TEXT -1 14 " int reflect;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double val;" } }{PARA 6 "" 1 "" {TEXT -1 11 " double x;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 30 " pi = 0.3141592653589793E1; " }}{PARA 6 "" 1 "" {TEXT -1 31 " pi2 = 0.6283185307179586E1;" }} {PARA 6 "" 1 "" {TEXT -1 33 " piBy2 = 0.1570796326794897E1;" }} {PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }}{PARA 6 "" 1 "" {TEXT -1 27 " if( pi < x || x < -pi )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 29 " k = floor(x/pi2+0. 5);" }}{PARA 6 "" 1 "" {TEXT -1 20 " x += -k*pi2;" }}{PARA 6 " " 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 17 " if( x \+ < 0.0 )" }}{PARA 6 "" 1 "" {TEXT -1 13 " x = -x;" }}{PARA 6 "" 1 "" {TEXT -1 34 " if( 0.1570796326794897E1 < x )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = pi-x; " }}{PARA 6 "" 1 "" {TEXT -1 20 " reflect = 1;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }} {PARA 6 "" 1 "" {TEXT -1 18 " reflect = 0;" }}{PARA 6 "" 1 "" {TEXT -1 32 " if( x < 0.7853981633974483 )" }}{PARA 6 "" 1 "" {TEXT -1 25 " val = cos_apprx(x);" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 20 " x = piBy2-x;" }}{PARA 6 "" 1 "" {TEXT -1 27 " \+ val = sin_apprx(x);" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }} {PARA 6 "" 1 "" {TEXT -1 19 " if( (reflect) )" }}{PARA 6 "" 1 "" {TEXT -1 19 " return(-val);" }}{PARA 6 "" 1 "" {TEXT -1 9 " el se " }}{PARA 6 "" 1 "" {TEXT -1 18 " return(val);" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }}{PARA 6 "" 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 26 "double sin_apprx(dou ble 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 " double a4;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b1;" }} {PARA 6 "" 1 "" {TEXT -1 12 " double b2;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double den;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double num;" }} {PARA 6 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 28 " a1 = 0.1666666666666666;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a2 = -0.5964761501642245E-2;" }} {PARA 6 "" 1 "" {TEXT -1 31 " a3 = 0.9183755196099076E-4;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a4 = -0.5286756225481412E-6;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b1 = 0.1421143099012443E-1;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b2 = 0.7112067102840643E-4;" }}{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 26 " den = 1.0+(b1+b 2*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 28 " 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 6 "" 1 "" {TEXT -1 26 "double \+ cos_apprx(double 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 " double a4;" }}{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 13 " double num;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 13 " a1 = 0.5;" }}{PARA 6 "" 1 "" {TEXT -1 32 " \+ a2 = -0.3154004216236836E-1;" }}{PARA 6 "" 1 "" {TEXT -1 31 " a3 = 0.6395493375638684E-3;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a4 = -0. 4120239986012706E-5;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b1 = 0.202532 4900859681E-1;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b2 = 0.189091648063 6244E-3;" }}{PARA 6 "" 1 "" {TEXT -1 31 " b3 = 0.8613069640975447E- 6;" }}{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 24 " return(1.0-num/den);" }}{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 32 "More fixed precision procedures \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "More fixed precision procedures for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 25 dig it rational approximation for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#% \"xG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-Pi/12,Pi/12]" "6#7$,$*&%#Pi G\"\"\"\"#7!\"\"F)*&F&F'F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "evalf(remez((sin( x)-x)/x^3,x=-Pi/12..Pi/12,[6,6],\n type=even,errtype=absolute,weig ht=x^2+1e-50),35):\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2cOy))) pFo%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,*$!DJ!>Gwom mmmmmmmmmm;!#N\"\"\"*&$\"DDV+'y^uZicS'oSW())oa!#PF1)9$\"\"#F1F1*&$\"D8 5WlUCPgDgc\")eN`n'= " 0 "" {MPLTEXT 1 0 443 "s:=x->x+ (-.16666666666666666666666668762819031+.546888744406864056624774517860 04325e-2*x^2-.77307815424271150028223895617441013e-4*x^4+.371138060444 79548913425564133996137e-6*x^6)/(1.+.171866753355881566025603724426519 23e-1*x^2+.13270446884884424820587604594623362e-3*x^4+.482458833703235 10749763539802704236e-6*x^6)*x^3:\nh := unapply(evalf(convert(s(x),hor ner),30),x);\nevalf(plot((1-s(x)/sin(x)),x=-Pi/12..Pi/12,color=COLOR(R GB,.4,0,.9)),35);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&9$\"\"\"*(,*$!?Gwommmmmmmmmmm;!#IF.*&$\"?gy ^uZicS'oSW())oa!#KF.)F-\"\"#F.F.*&$\"?uh&*QAG+:rUU:yIx!#MF.)F-\"\"%F.! \"\"*&$\"?S8kbU8*[&zW/1Q6P!#OF.)F-\"\"'F.F.F.,*$F.\"\"!F.*&$\"?FWs.c-m :)eN`n'=F.F.*&$\"?F!)RNw\\2 ^B.P$)eC[FDF.FEF.F.F@F-\"\"$F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 459 280 280 {PLOTDATA 2 "6&-%'CURVESG6#7_]l7$$!D&=Ht_h`&QlV\\\"*z(Q*zh #!#N$!'k)y\"!#M7$$!D.jPI?Zt[K77\")pKFWh#F*$!'e07F-7$$!DAMULDe\"*e*4[2( fxg3h#F*$!&q)oF-7$$!DT0ZOIp4pm\\Pg\\A%H2EF*$!&\\L#F-7$$!Dgw^RN!y#zL=+] RnFPg#F*$\"'1V;F*7$$!DzZcUS\"f%*3qG'RH7h,g#F*$\"'r(3&F*7$$!D)*=hXX-k*z cb#H>d%f'f#F*$\"'iP!)F*7$$!DvDF*$\"(N$G(y\"F*7$$!Dpf.t!>q7>P(*e$Gh&\\kDF*$ \"(e_.#zJQ1&*zTCF*$!'&yj\"F-7$$!D\\2ef@<@b$=4k_elQHCF*$!'h[Cz(**Qm2CF*$!'*yw\"F-7$$!Dp'GyLI*\\&Q(3O:GmgXS#F*$! 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\n\n # coefficients of numerator in rational approxima tion\n a1 := .166666666666666666666666687628;\n a2 := -.5468887444 06864056624774517860e-2;\n a3 := .773078154242711500282238956174e-4; \n a4 := -.371138060444795489134255641340e-6;\n\n # coefficients o f denominator in rational approximation\n b1 := .1718667533558815660 25603724427e-1;\n b2 := .132704468848844248205876045946e-3;\n b3 : = .482458833703235107497635398027e-6;\n\n if abs(xx)>.261799 then\n \+ x := xx/3;\n triple := true;\n else x := xx;\n triple := false;\n end if;\n \n # evaluate the rational approximati on\n z := x*x;\n num := (a1+(a2+(a3+a4*z)*z)*z)*z;\n den := 1+(b 1+(b2+b3*z)*z)*z;\n val := (1-num/den)*x;\n if triple then val := \+ val*(3-4*val^2) end if;\n val;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := evalf(Pi/4,30);\nevalf(evalf(sin25(xx),30),2 7);\nevalf(evalf(sin(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#xxG$\"??e%3m:'4$[uRj\")R&y!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 " " {MPLTEXT 1 0 91 "evalf(plot(1-'sin25'(x)/sin(x),x=-Pi/4..Pi/4,\n \+ color=COLOR(RGB,.4,0,.9)),35);" }}{PARA 13 "" 1 "" {GLPLOT2D 490 207 207 {PLOTDATA 2 "6&-%'CURVESG6#7_[m7$$!Dcv)>e%3m:'4$ [uRj\")R&y!#N$!'&)Q9!#M7$$!D8*G64;/iupjL%4)>GVyF*$!&'3(*F-7$$!Dq-F+wuu w)HWA\"zK#eKyF*$!&Xb&F-7$$!DF;T4\"z!H2+\\7\")[n#)=#yF*$!&:*=F-7$$!D%)H b=1T$y8]0+&=-$=6yF*$\"'-98F*7$$!DTVpF@uPo-h))=)oL[+yF*$\"'w$4%F*7$$!D) pNojt?*)Rqmxy:Py*y(F*$\"'0ykF*7$$!Dbq(f90k%H0tkcF1%3zxF*$\"'z&\\)F*7$$ !D7%=^lO2+m!z_D(4WQoxF*$\"(Qu,\"F*7$$!DE6St'*R4@4\"*GjO5&)pu(F*$\"(SD3?%Rs784$*pX9')[r( F*$\"(p\\V\"F*7$$!Dal'*4dsEV9:\")Q:\\'=/xF*$\"(v(\\9F*7$$!D6z5>s0\"Qd6 #p2&Qo[$p(F*$\"(ejW\"F*7$$!Dn#\\#G()QN/`Y<0Wl>RL9%zyQhwF*$\"(;UM\"F*7$$!D'p9[w9FlA7&4_Ld))*RwF*$\"(!G:7F*7$$ !D5u4$yx8w[Kc)*Gn#*e=wF*$\"(%o]5F*7$$!DB,Q,3/q[FvhF7'**=(f(F*$\"'+,')F *7$$!DQGm>Qqy4I(y`;b1zvvF*$\"'-?lF*7$$!D^b%z$oO(3F$*RJ5\\8RavF*$\"'_PV F*7$$!Dl#Gi&)Hg>`8,4/V?*H`(F*$\"'/<@F*7$$!Dy4^uGp/$zLi'ypt#f6vF*$!$r)F -7$$!D

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\"\"*&$\"aoK0ce5CQv$zVz%Q\\Yise\">t[W^%z7K11L(y\\dB\"!#vF1)F7\"\"'F1F1 *&$\"ao!e%3U!Qfan*)>?(f%zr**[+/8\"Q.T+Qf,0'4\\H/v'!#yF1)F7\"\")F1F?*&$ \"ao3#fV5#R3wgYMT.C20`k5eQ#f5L/mo]Dz-4D7#!#!)F1)F7\"#5F1F1*&$\"aoL2/\\ 6\"H'41/mhVK@4wv;\")RV?1m%fO%[bl+^[O!#$)F1)F7\"#7F1F?*&$\"ao4J95VDNY_+ >CeEk.*f3PSwy=ctkT$o$\\Cz%=F!#')F1)F7\"#9F1F1F1,0$F1\"\"!F1*&$\"ao`!)f ?\"\\C`\"o['oazs,Dz!\\uq&y'=v\"4SJ05D9\\L*F5F1F6F1F1*&$\"aoN\"\\is\"\\ )>'e^P)))3XJ#\\F5t<%)>'zw!=#>H#!#zF1FJF1F1*&$\"aoCmOYdf)4B+JK*zaP\"zN$f<5dX[rA!)Gk=B&HU#G !##)F1FPF1F1*&$\"ao,E&Hv]D#\\&ecT?1EJ+P<&41Podx)R_YaaVk()y\"!#&)F1FVF1 F1F?F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The rational function " }{XPPEDIT 18 0 "s(x)=x+r(x)*`.`*x^3" "6 #/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$F)" }{TEXT -1 14 " approximates " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/12, Pi/12];" "6#7$,$*&%#Pi G\"\"\"\"#7!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "The following graph gives the relative error (multiplied \+ by " }{XPPEDIT 18 0 "10^30" "6#*$\"#5\"#I" }{TEXT -1 20 ") involved in using " }{XPPEDIT 18 0 "s(x);" "6#-%\"sG6#%\"xG" }{TEXT -1 16 " to ap proximate " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1257 "s:=x->x+(-.1666666666666666666666666666666666666666 666666666666666664101093938469+.67775142914991143318041355357154251534 58304534088558213269758233241279e-2*x^2-.12766592976850961739467142850 54318541628665002651082527569440620623886e-3*x^4+.12357497873306063212 79451444873191587262464938479437937538241058560532e-5*x^6-.67504294909 60501593800410338113040048997179459720198967545938042084580e-8*x^8+.21 22509027925506866043310592385810645305072403413446607608392104359208e- 10*x^10-.3648510065554843659466062043398116757609213243616604060962911 149040733e-13*x^12+.27184792449368341647356187876403708599036426582419 00524635254310143109e-16*x^14)/(1.+.9334914251005314009175186785707449 079250172795468648681532449120598053e-2*x^2+.4226510068513292863631171 984177310274923145088883751586198491726249135e-4*x^4+.1201546887058357 418932845885416958634566828028977562287340320285650802e-6*x^6+.2291921 807679619727747910951594417375921069456118716502354038351331614e-9*x^8 +.28242295231864288022714845571017593357913754799323100230985957463666 24e-12*x^10+.178876443545446523987757683706095173700312606204156585492 2550752952601e-15*x^12)*x^3:\nh := unapply(evalf(convert(s(x),horner), 60),x);\nevalf(plot(10^30*(1-s(x)/sin(x)),x=-Pi/12..Pi/12,color=COLOR( RGB,.4,0,.9)),65);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&9$\"\"\"*(,2$!gn5kmmmmmmmmmmmmmmmmmmmmmmmmm mm;!#gF.*&$\"gnwpK@e&)3MXIeM:DarNb8/=L9\"*\\\"H9vx'!#iF.)F-\"\"#F.F.*& $\"gnWpv_#3^E+l'G;a=V0&G9n%R<'4&o(HfmF\"!#jF.)F-\"\"%F.!\"\"*&$\"gnCQv $zVz%Q\\Yise\">t[W^%z7K11L(y\\dB\"!#lF.)F-\"\"'F.F.*&$\"gn%fan*)>?(f%z r**[+/8\"Q.T+Qf,0'4\\H/v'!#oF.)F-\"\")F.F@*&$\"gnR3wgYMT.C20`k5eQ#f5L/ mo]Dz-4D7#!#qF.)F-\"#5F.F.*&$\"gn\"H'41/mhVK@4wv;\")RV?1m%fO%[bl+^[O!# tF.)F-\"#7F.F@*&$\"gnDNY_+>CeEk.*f3PSwy=ctkT$o$\\Cz%=F!#wF.)F-\"#9F.F. 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a1,a2,a3,a4,a5,a6,a7,a8,b1,b2,b3,b4,b5,b6,\n x,z,num,den ,val,triple; \n\n # coefficients of numerator in rational approxim ation\n a1 := \n .16666666666666666666666666666666666666666666666 6666666666410;\n a2 := \n -.67775142914991143318041355357154251534 5830453408855821326976e-2;\n a3 := \n .12766592976850961739467142 8505431854162866500265108252756944e-3;\n a4 := \n -.12357497873306 0632127945144487319158726246493847943793753824e-5;\n a5 := \n .67 5042949096050159380041033811304004899717945972019896754594e-8;\n a6 \+ := \n -.212250902792550686604331059238581064530507240341344660760839e -10;\n a7 := \n .3648510065554843659466062043398116757609213243616 60406096291e-13;\n a8 := \n -.2718479244936834164735618787640370859 90364265824190052463525e-16;\n\n # coefficients of denominator in ra tional approximation\n b1 := \n .93349142510053140091751867857074 4907925017279546864868153245e-2;\n b2 := \n .42265100685132928636 3117198417731027492314508888375158619849e-4;\n b3 := \n .12015468 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"6#>%#xxG$\"?5HUIy![:C()p\"3*p #R!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<*=$=Ghv'G6D`zQ#*!#F" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<*=$=Ghv'G6D`zQ#*!#F" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "xx := evalf(10^(-1),30);\nevalf(evalf(h(xx) ,30),27);\nevalf(evalf(cos(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?++++++++++++++5!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"<))>c&4md-y_;/]**!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"<))>c&4m d-y_;/]**!#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "cos25 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 901 "cos25 := proc(xx)\n local a1,a2,a3,a4,a5,b1,b2,b3,x,z,num,d en,t,double; \n\n # coefficients of numerator in rational approxima tion\n a1 := .500000000000000000000000012316;\n a2 := -.3426368873 45311509740784786598e-1;\n a3 := .820207706023133020456081233929e-3; \n a4 := -.811319660314636758451306687466e-5;\n a5 := .30341539473 332860025130494359240953e-7;\n\n # coefficients of denominator in ra tional approximation\n b1 := .148059558642710313851815554012e-1;\n \+ b2 := .964672896244075448606175611013e-4;\n b3 := .2880670203852002 77935785351089e-6;\n\n if abs(xx)>.392699 then\n x := xx*0.5;\n double := true;\n else\n x := xx;\n double := false; \n end if;\n \n # evaluate the rational approximation\n z : = x*x;\n num := (a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z;\n den := 1+(b1+( b2+b3*z)*z)*z;\n t := num/den;\n if double then 1-(2-t)*(t+t) else 1-t end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := evalf(Pi/4,30);\nevalf(evalf(cos25(xx),30),27);\nevalf(evalf (cos(xx),30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"??e%3m:' 4$[uRj\")R&y!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 104 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'-pMFF7$$\"DC/z37$*G#\\mFq^x9;PwF*$!'u.XFF7$$\"DX_h%H.+v$H'=cHPFr]wF*$ !'?#f&FF7$$\"Dm+W!Qv5FQf4U2(*REkwF*$!'E8mFF7$$\"D()[Emu9#z#e0!G&oD:yn( F*$!'aQvFF7$$\"D2(*3_&>KJF_\"RJm^m8p(F*$!'DO$)FF7$$\"DGX\"zj\"HM=([#)* 4kx<\\q(F*$!''3(*)FF7$$\"D\\$RPsj`N;Xt&)=O!p%=xF*$!')GS*FF7$$\"De0(\\%zxF*$!'AavFF7$$\"D_M'G:C2'*QFG:3N`A'y(F *$!'yBoFF7$$\"Diex&>g7AhvB3(\\'4+$z(F*$!';efFF7$$\"Dt#)oQiz\"[$Q#>,'[f w(*z(F*$!'*z%\\FF7$$\"D#o+;GKBu0s9%\\ZA_l!yF*$!'Y$y$FF7$$\"D$48XKoG+G? 5(QY&yK8yF*$!'PaCFF7$$\"D.bUnVSj-&o0!GX[.,#yF*$!&1]*FF7$$\"D9zL5/%R_s; ,tT9\"zo#yF*$\"''fS(F*7$$\"DD.D`kZ%y%\\mf1VuaO$yF*$\"(\">HEF*7$$\"DNF; '\\7]/<8#*e>u.VSyF*$\"(5ys%F*7$$\"DY^2R&[bIRh(=&3/g?ZyF*$\"(>!\\qF*7$$ \"Dcv)>e%3m:'4$[uRj\")R&yF*F+-%+AXESLABELSG6$Q\"x6\"Q!F\\_r-%&COLORG6& %$RGBG$\"\"%!\"\"\"\"!$\"\"*Fd_r-%%VIEWG6$;$!Dsv)>e%3m:'4$[uRj\")R&yF* $\"Dsv)>e%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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "st := time():\nfo r i to 5000 do evalf(cos25(rand()*Float(1,-11)*.7853981633974483),27) \+ end do:\ntime()-st;\nst := time():\nfor i to 5000 do evalf(cos(rand()* Float(1,-11)*.7853981633974483),27) end do:\ntime()-st;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"%$)p!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"%v*)!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 55 dig it rational approximation for " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#% \"xG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-Pi/16,Pi/16]" "6#7$,$*&%#Pi G\"\"\"\"#;!\"\"F)*&F&F'F(F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "evalf(remez((1-co s(x))/x^2,x=-Pi/16..Pi/16,[12,12],\n type=even,errtype=absolute,wei ght=x^2+1e-100),70):\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D`(3;N# )RZ^\"*>><62pN\"*!#$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&, 0$\"aoXA\"HR[7oIRB4%eEt'e,gN!#rF1)9$\"\"#F1 !\"\"*&$\"aozt7bGv*)z#z%yo!fYF9#z!HkFD96f5!H4)e4:x>*!#tF1)F7\"\"%F1F1* &$\"ao)[dx'*ohh\"\\6=jB=F@'[K/Q!=Y&*f2?$*H(Q#[]%3\"!#uF1)F7\"\"'F1F9*& $\"aoRpe4&[\"opa()*GQb]!z/$RGAx]\\59L7;6Frh(\\k!#xF1)F7\"\")F1F1*&$\"a oy#*H<[\"3:zlR9algD*>sH(\\,o%G=vsdH3DsH5>!#zF1)F7\"#5F1F9*&$\"aoPP.dkx%)Hfe)zcpjSA)G#!##)F1)F7\"#7F1F1F1,0$F1\"\"!F1*&$\"ao qrIDe7r;ao,()*)o)z3'f%\\l$3(>Zl[^;!o)f,L@\"F5F1F6F1F1*&$\"ao-m*GYk[%f6 'o=()phg2RxeKY(H[/')[J7_'p!*\\G(FCF1F>F1F1*&$\"aom`Jyz[!*\\%pe#G-cg3uE 9rKA#G\\a]rb*)e?:3\"G!#wF1FDF1F1*&$\"aoK@lYsSgLiN5R,;No$odzP\"=6'zT(H2 '3())fnvuFOF1FJF1F1*&$\"ao+s4u!p!HRH?O==E)3o]Rhot+$pe(=T]^q)=,E8!#\")F 1FPF1F1*&$\"aowM69?'ySq+xh*>5>g$4gmsC#[k&oKTq)p^:6c7!#%)F1FVF1F1F9F(F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Th e rational function " }{XPPEDIT 18 0 "s(x) = 1-r(x)*`.`*x^2;" "6#/-%\" sG6#%\"xG,&\"\"\"F)*(-%\"rG6#F'F)%\".GF)F'\"\"#!\"\"" }{TEXT -1 14 " a pproximates " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/16, Pi/16];" "6#7$,$*&%#PiG\" \"\"\"#;!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "The following graph gives the relative error (multiplied by " } {XPPEDIT 18 0 "10^30" "6#*$\"#5\"#I" }{TEXT -1 20 ") involved in using " }{XPPEDIT 18 0 "s(x);" "6#-%\"sG6#%\"xG" }{TEXT -1 16 " to approxim ate " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1174 "s:=x->1-(.499999999999999999999999999999999999999999999999999999 4092290172912245-.3560015867326584092339306812483919368622673221731461 303388314224794063e-1*x^2+.9197715095880929010591114252764290792142746 590687847927989752855127379e-3*x^4-.1084504823872993200759954618038043 248621271823631811491616168967757488e-4*x^6+.6449761712711161233141049 507722283930479050553828987546968148509586939e-7*x^8-.1910297225082957 727518284680149729721992560655414396579150814817299278e-9*x^10+.228822 4063695679858592984776457033719446630006584605794416143539311737e-12*x ^12)/(1.+.121330159868016514865471970836549459608798688987016854167112 5825307170e-1*x^2+.728499069652123148860448297463258773907606169871868 6115944864462896602e-4*x^4+.281081520588955715054492822232711426740860 5602282586944990487978315366e-6*x^6+.747567598870860729741796111813779 5768368351601391035623360407246652132e-9*x^8+.132601188705150411875869 3007368613950680882618183620293929069074097200e-11*x^10+.1256111551698 704132685644822472666009360191019961770070407862014113476e-14*x^12)*x^ 2:\nh := unapply(evalf(convert(s(x),horner),60),x);\nevalf(plot(10^20* (1-s(x)/cos(x)),x=-Pi/16..Pi/16,color=COLOR(RGB,.4,0,.9)),65);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&$\"\"\"\"\"!F.**$F.F/F.,0$\"gnH#4%******************************* *********************\\!#gF.*&$\"gnJ)Q.8YJR[7oIRB4%eEt'e,gN!#h F.)9$\"\"#F.!\"\"*&$\"gnv*)z#z%yo!fYF9#z!HkFD96f5!H4)e4:x>*!#jF.)F;\" \"%F.F.*&$\"gn<;;\\6=jB=F@'[K/Q!=Y&*f2?$*H(Q#[]%3\"!#kF.)F;\"\"'F.F=*& $\"gn:opa()*GQb]!z/$RGAx]\\59L7;6Frh(\\k!#nF.)F;\"\")F.F.*&$\"gn\"3:zl R9algD*>sH(\\,o%G=vsdH3DsH5>!#pF.)F;\"#5F.F=*&$\"gn9;Wz0Ye1+jY%>P.dkx% )Hfe)zcpjSA)G#!#sF.)F;\"#7F.F.F.,0F-F.*&$\"gn8r;ao,()*)o)z3'f%\\l$3(>Z l[^;!o)f,L@\"F9F.F:F.F.*&$\"gn'[%f6'o=()phg2RxeKY(H[/')[J7_'p!*\\G(FGF .FBF.F.*&$\"gn\\!*\\%pe#G-cg3uE9rKA#G\\a]rb*)e?:3\"G!#mF.FHF.F.*&$\"gn TgLiN5R,;No$odzP\"=6'zT(H2'3())fnvuFSF.FNF.F.*&$\"gn2HRH?O==E)3o]Rhot+ $pe(=T]^q)=,E8!#rF.FTF.F.*&$\"gn'ySq+xh*>5>g$4gmsC#[k&oKTq)p^:6c7!#uF. 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" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "sin_, cos_: implementation" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18044 "sin_ := 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('sin_'(x))\n elif type(x,'numeri c') and signum(0,x,0)<0 then -sin_(-x)\n elif type(x,`*`) and type(o p(1,x),'numeric') and signum(0,op(1,x),0)<0 then -sin_(-x)\n elif ty pe(x,'function') and nops(x)=1 then\n n := op(0,x);\n t := o p(1,x);\n if n='arcsin_' or n='arcsin' then t\n elif n='arcc os_' or n='arccos' then sqrt(1-t^2)\n elif n='arctan_' or n='arct an' then t/sqrt(1+t^2)\n else 'sin_'(x)\n end if;\n elif t ype(x,'realcons') then\n if x=0 or x=Pi then 0\n elif x='inf inity' then 'undefined'\n elif x=-'infinity' then 'undefined'\n \+ elif type(x,`*`) and type(x,'`&*`(rational,identical(Pi))') then\n t := op(1,x);\n if t<=1/2 then \n if t=1/2 then 1\n elif t=1/3 then sqrt(3)/2\n elif t=1/4 then sqrt(2)/2\n elif t=1/6 then 1/2\n elif t=1 /8 then sqrt(2-sqrt(2))/2\n elif t=3/8 then sqrt(2+sqrt(2)) /2\n elif t=1/12 then sqrt(2)*(sqrt(3)-1)/4\n el if t=5/12 then sqrt(2)*(sqrt(3)+1)/4\n else 'sin_'(x)\n \+ end if\n elif t<=1 then\n if t=1 then 0\n \+ else sin_((1-t)*Pi)\n end if\n elif t<2 t hen -sin_((2-t)*Pi)\n else sin_((t-2*floor(t/2))*Pi)\n \+ end if\n else 'sin_'(x)\n end if\n else 'sin_'(x)\n end if;\nend proc: # sin_\n\ncos_ := 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('cos_'(x))\n eli f type(x,'numeric') and signum(0,x,0)<0 then cos_(-x)\n elif type(x, `*`) and type(op(1,x),'numeric') and signum(0,op(1,x),0)<0 then cos_(- x)\n elif type(x,'function') and nops(x)=1 then\n n := op(0,x); \n t := op(1,x);\n if n='arccos_' or n='arccos' then t\n \+ elif n='arcsin_' or n='arcsin' then sqrt(1-t^2)\n elif n='arcta n_' or n='arctan' then 1/sqrt(1+t^2)\n else 'cos_'(x)\n end \+ if\n elif type(x,'realcons') then\n if x=0 then 1\n elif x ='infinity' then 'undefined'\n elif x=-'infinity' then 'undefined '\n elif type(x,`*`) and type(x,'`&*`(rational,identical(Pi))') t hen\n t := op(1,x);\n if t<=1/2 then \n if \+ t=1/2 then 0\n elif t=1/3 then 1/2\n elif t=1/4 \+ then sqrt(2)/2\n elif t=1/6 then sqrt(3)/2\n eli f t=1/8 then sqrt(2+sqrt(2))/2\n elif t=3/8 then sqrt(2-sqr t(2))/2\n elif t=1/12 then sqrt(2)*(sqrt(3)+1)/4\n \+ elif t=5/12 then sqrt(2)*(sqrt(3)-1)/4\n else 'cos_'(x) \n end if\n elif t<=1 then\n if t=1 then -1\n else -cos_((1-t)*Pi)\n end if\n el if t<2 then cos_((2-t)*Pi)\n else cos_((t-2*floor(t/2))*Pi)\n \+ end if\n else 'cos_'(x)\n end if\n else 'cos_'(x) \n end if;\nend proc: # cos_\n\n`evalf/sin_` := proc(xx)\n local x ,z,term,eps,k,even,maxit,saveDigits,\n extraDigits,sum,pi,pi2,isneg, hfDigits;\n\n if xx=0 then return xx end if;\n if not type(xx,real cons) then return 'sin_'(xx) end if;\n\n # Use the fixed precision p rocedure for low precision values\n hfDigits := trunc(evalhf(Digits) );\n if Digits<=hfDigits then\n x := evalf(xx,hfDigits+2);\n \+ if max(ilog10(x),0)<=3 then\n return evalf(evalhf(sin16(x)) )\n end if;\n end if;\n\n # increase precision \n saveDigi ts := Digits;\n Digits := Digits+length(Digits)+1;\n x := evalf(xx );\n # argument reduction involves loss of precision\n # increase \+ precision further when x has large magnitude\n extraDigits := max(il og10(x),0);\n if extraDigits>0 then\n Digits := Digits+extraDig its;\n x := evalf(xx)\n end if;\n \n pi := evalf(Pi); # Map le evaluates Pi rapidly\n \n # Reduce the range to -Pi/2<=x<=3* Pi/2\n if x>4.7123889803846898577 or x<-1.5707963267948966192 then\n pi2 := pi+pi;\n\n # Determine the number of multiples of 2* Pi to add or subtract.\n k := floor(x/pi2+0.25);\n x := x - \+ k*pi2;\n end if;\n\n # If x > Pi/2, replace x by Pi - x so that ab s(x) <= Pi/2\n if x>1.5707963267948966192 then x := pi-x end if;\n\n if saveDigits<=evalhf(Digits-extraDigits) then\n # Use the fix ed precision procedure\n return evalf(evalhf(sin16(x)));\n end \+ if;\n\n # sine is an odd function\n if x<0 then\n x := -x;\n \+ isneg := true;\n else\n isneg := false;\n end if;\n\n \+ if saveDigits<26 then\n if x<.78539816339744830962 then\n \+ Digits := Digits-extraDigits;\n sum := sin25(x)\n else\n x := pi*0.5-x;\n Digits := Digits-extraDigits; \n \+ sum := cos25(x)\n end if;\n elif saveDigits<56 then\n \+ if x<.78539816339744830962 then\n Digits := Digits-extraDig its;\n sum := sin55(x)\n else\n x := pi*0.5-x;\n \+ Digits := Digits-extraDigits; \n sum := cos55(x)\n \+ end if; \n else\n # use the Maclaurin series \n ep s := Float(1,-saveDigits);\n maxit := Digits*4;\n\n if x<.78 539816339744830962 then \n # use the sine series to calculate \+ sin(x)\n Digits := Digits-extraDigits;\n term := x;\n \+ sum := 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 sum := sum+term;\n \+ else\n sum := sum-term;\n end if;\n \+ if abs(term)<=eps*abs(sum) then break end if;\n e ven := not even;\n end do;\n else\n x := pi*0.5-x ;\n Digits := Digits-extraDigits; \n # use the cosine series to calculate cos(x)\n term := 1.0;\n sum := te rm;\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 sum := sum+term;\n else\n \+ sum := sum-term;\n end if;\n if abs( term)<=eps*abs(sum) then break end if;\n even := not even; \n end do;\n end if;\n end if;\n\n Digits := saveDigi ts;\n if isneg then\n -evalf(sum);\n else\n evalf(sum); \n end if;end proc: # of `evalf/sin_`\n\n`evalf/cos_` := proc(xx)\n \+ local x,z,term,eps,k,even,maxit,saveDigits,\n extraDigits,sum,pi,p i2,reflect,hfDigits;\n\n if not type(xx,realcons) then return 'cos_' (xx) end if;\n\n # Use the fixed precision procedure for low precisi on values\n hfDigits := trunc(evalhf(Digits));\n if Digits<=hfDigi ts then\n x := evalf(xx,hfDigits+2);\n if max(ilog10(x),0)<= 3 then\n return evalf(evalhf(cos16(x)))\n end if;\n end if;\n\n # increase precision \n saveDigits := Digits;\n Digits := Digits+length(Digits);\n x := evalf(xx);\n # argument reductio n involves loss of precision\n # increase precision further when x h as large magnitude\n extraDigits := max(ilog10(x),0);\n if extraDi gits>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<=x<=Pi\n if x>3.1415926535897932385 or x<-3.1415926535897932385 then\n pi2 := pi+pi;\n\n # Dete rmine the number of multiples of 2*Pi to add or subtract.\n k := \+ floor(x/pi2+0.5);\n x := x - k*pi2;\n end if;\n\n # If x<0, r eplace x by -x so that 0<=x<=Pi\n if x<0 then x := -x end if;\n\n \+ # If x>Pi/2, replace x by Pi-x so that 0<=x<=Pi/2\n if x>1.570796326 7948966192 then\n x := pi-x;\n reflect := true;\n else\n \+ reflect := false;\n end if;\n\n if saveDigits<=evalhf(Digits-e xtraDigits) then\n # Use the fixed precision procedure\n if \+ reflect then \n return -evalf(evalhf(cos16(x)));\n else\n return evalf(evalhf(cos16(x)));\n end if;\n end if;\n \n if saveDigits<26 then\n if x<.78539816339744830962 then\n \+ Digits := Digits-extraDigits;\n sum := cos25(x)\n e lse\n x := pi*0.5-x;\n Digits := Digits-extraDigits;\n sum := sin25(x)\n end if;\n elif saveDigits<56 then\n \+ if x<.78539816339744830962 then\n Digits := Digits-extr aDigits;\n sum := cos55(x)\n else\n x := pi*0.5-x ;\n Digits := Digits-extraDigits; \n sum := sin55(x)\n end if;\n else\n # use the Maclaurin series\n eps := Float(1,-saveDigits);\n maxit := Digits*4;\n\n if x<.785398 16339744830962 then\n Digits := Digits-extraDigits;\n \+ # use the cosine series to calculate cos(x)\n term := 1.0;\n \+ sum := 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 sum := sum+term;\n \+ else\n sum := sum-term;\n end if;\n \+ if abs(term)<=eps*abs(sum) then break end if;\n ev en := not even;\n end do; \n else\n x := pi*0.5-x ;\n Digits := Digits-extraDigits;\n # use the sine ser ies to calculate sin(x)\n term := x;\n sum := 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 e ven then\n sum := sum + term;\n else\n \+ sum := sum - term;\n end if;\n if abs(te rm)<=eps*abs(sum) then break end if;\n even := not even;\n \+ end do;\n end if;\n end if;\n\n Digits := saveDigits; \n if reflect then\n -evalf(sum);\n else\n evalf(sum);\n end if;\nend proc: # of `evalf/cos_`\n\nsin25 := proc(xx)\n local a1,a2,a3,a4,b1,b2,b3,x,z,num,den,val,triple; \n\n # coefficients \+ of numerator in rational approximation\n a1 := .16666666666666666666 6666687628;\n a2 := -.546888744406864056624774517860e-2;\n a3 := . 773078154242711500282238956174e-4;\n a4 := -.37113806044479548913425 5641340e-6;\n\n # coefficients of denominator in rational approximat ion\n b1 := .171866753355881566025603724427e-1;\n b2 := .132704468 848844248205876045946e-3;\n b3 := .482458833703235107497635398027e-6 ;\n\n if abs(xx)>.261799 then\n x := xx/3;\n triple := tru e;\n else x := xx;\n triple := false;\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 if triple then val := val*(3-4*val^2) end if;\n val;\nen d proc: # of sin25\n\ncos25 := proc(xx)\n local a1,a2,a3,a4,a5,b1,b2 ,b3,x,z,num,den,t,double; \n\n # coefficients of numerator in ratio nal approximation\n a1 := .500000000000000000000000012316;\n a2 := -.342636887345311509740784786598e-1;\n a3 := .820207706023133020456 081233929e-3;\n a4 := -.811319660314636758451306687466e-5;\n a5 := .30341539473332860025130494359240953e-7;\n\n # coefficients of deno minator in rational approximation\n b1 := .1480595586427103138518155 54012e-1;\n b2 := .964672896244075448606175611013e-4;\n b3 := .288 067020385200277935785351089e-6;\n\n if abs(xx)>.392699 then\n x := xx*0.5;\n double := true;\n else\n x := xx;\n dou ble := false;\n end if;\n \n # evaluate the rational approxim ation\n z := x*x;\n num := (a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z;\n d en := 1+(b1+(b2+b3*z)*z)*z;\n t := num/den;\n if double then 1-(2- t)*(t+t) else 1-t end if;\nend proc: # of cos25\n\nsin55 := proc(xx)\n local a1,a2,a3,a4,a5,a6,a7,a8,b1,b2,b3,b4,b5,b6,\n x,z,num,den ,val,triple; \n\n # coefficients of numerator in rational approxim ation\n a1 := \n .16666666666666666666666666666666666666666666666 6666666666410;\n a2 := \n -.67775142914991143318041355357154251534 5830453408855821326976e-2;\n a3 := \n .12766592976850961739467142 8505431854162866500265108252756944e-3;\n a4 := \n -.12357497873306 0632127945144487319158726246493847943793753824e-5;\n a5 := \n .67 5042949096050159380041033811304004899717945972019896754594e-8;\n a6 \+ := \n -.212250902792550686604331059238581064530507240341344660760839e -10;\n a7 := \n .3648510065554843659466062043398116757609213243616 60406096291e-13;\n a8 := \n -.2718479244936834164735618787640370859 90364265824190052463525e-16;\n\n # coefficients of denominator in ra tional approximation\n b1 := \n .93349142510053140091751867857074 4907925017279546864868153245e-2;\n b2 := \n .42265100685132928636 3117198417731027492314508888375158619849e-4;\n b3 := \n .12015468 8705835741893284588541695863456682802897756228734032e-6;\n b4 := \n \+ .229192180767961972774791095159441737592106945611871650235404e-9;\n b5 := \n .2824229523186428802271484557101759335791375479932310023 09860e-12;\n b6 := \n .1788764435454465239877576837060951737003126 06204156585492255e-15; \n\n if abs(xx)>.261799 then\n x := xx/3 ;\n triple := true;\n else x := xx;\n triple := false;\n \+ end if;\n \n # evaluate the rational approximation\n z := x* x;\n num := (a1+(a2+(a3+(a4+(a5+(a6+(a7+a8*z)*z)*z)*z)*z)*z)*z)*z;\n den := 1+(b1+(b2+(b3+(b4+(b5+b6*z)*z)*z)*z)*z)*z;\n val := (1-num /den)*x;\n if triple then val := val*(3-4*val^2) end if;\n val;\ne nd proc: # of sin55 \n\ncos55 := proc(xx)\n local a1,a2,a3,a4,a5,a6, a7,b1,b2,b3,b4,b5,b6,\n x,z,num,den,val,flag; \n\n # coefficie nts of numerator in rational approximation\n a1 := \n .4999999999 99999999999999999999999999999999999999999999409229;\n a2 := \n -.3 56001586732658409233930681248391936862267322173146130338831e-1;\n a3 := \n .919771509588092901059111425276429079214274659068784792798975 e-3;\n a4 := \n -.108450482387299320075995461803804324862127182363 181149161617e-4;\n a5 := \n .6449761712711161233141049507722283930 47905055382898754696815e-7;\n a6 := \n -.1910297225082957727518284 68014972972199256065541439657915081e-9;\n a7 := \n .22882240636956 7985859298477645703371944663000658460579441614e-12;\n\n # coefficien ts of denominator in rational approximation\n b1 := \n .121330159 868016514865471970836549459608798688987016854167113e-1;\n b2 := \n \+ .728499069652123148860448297463258773907606169871868611594486e-4;\n \+ b3 := \n .2810815205889557150544928222327114267408605602282586944 99049e-6;\n b4 := \n .7475675988708607297417961118137795768368351 60139103562336041e-9;\n b5 := \n .13260118870515041187586930073686 1395068088261818362029392907e-11;\n b6 := \n .12561115516987041326 8564482247266600936019101996177007040786e-14;\n \n if abs(xx)>.39269 9 then\n x := xx*0.25;\n flag := 2;\n elif abs(xx)>.196349 54 then\n x := xx*0.5;\n flag := 1;\n else\n x := xx; \n flag := 0;\n end if;\n \n # evaluate the rational appr oximation\n z := x*x;\n num := (a1+(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z) *z)*z)*z)*z;\n den := 1+(b1+(b2+(b3+(b4+(b5+b6*z)*z)*z)*z)*z)*z;\n \+ val := 1-num/den;\n if flag=1 then\n val := 2*val^2-1\n elif flag=2 then \n z := val^2;\n val := 8*z*(z-1)+1\n end if; \n val;\nend proc: # of cos55 \n\n# procedures for default and/or lo w precision\nsin16 := proc(xx)\n local x,pi,pi2,piBy2,k,neg,val; \n \+ \n pi := 3.141592653589793;\n pi2 := 6.283185307179586;\n piBy 2 := 1.570796326794897;\n\n x := evalf(xx);\n \n # Reduce the range to -Pi/2<=x <=3*Pi/2 approximately\n if x>4.712388980384690 o r x<-1.570796326794897 then\n\n # Determine the number of multipl es of 2*Pi to add or subtract.\n k := floor(x/pi2+0.25);\n x := x - k*pi2;\n end if;\n\n # If x>Pi/2, replace x by Pi-x so tha t abs(x)<=Pi/2\n if x>1.570796326794897 then x := pi-x end if;\n\n \+ # Use the fact that sine is an odd function\n if x < 0 then\n \+ x := -x;\n neg := true;\n else neg := false end if;\n \n if x<.7853981633974483 then \n # use the rational approximate for s in\n val := sin_apprx(x);\n else\n x := piBy2-x;\n # \+ use the rational approximate for cos\n val := cos_apprx(x);\n e nd if;\n\n if neg then -val else val end if;\nend proc: # sin16\n\nc os16 := proc(xx)\n local x,pi,pi2,piBy2,k,reflect,val; \n \n pi \+ := 3.141592653589793;\n pi2 := 6.283185307179586;\n piBy2 := 1.570 796326794897;\n\n x := evalf(xx);\n \n # Reduce the range to \+ -Pi<=xx<=Pi\n if x>pi or x<-pi then\n # Determine the number of multiples of 2*Pi to add or subtract.\n k := floor(x/pi2+0.5);\n x := x - k*pi2;\n end if;\n\n # If x<0, replace x by -x so t hat 0<=x<=Pi\n if x < 0 then x := -x end if;\n\n # If x>Pi/2, repl ace x by Pi-x so that 0<=x<=Pi/2\n if x>1.570796326794897 then\n \+ x := pi-x;\n reflect := true;\n else\n reflect := false; \n end if;\n \n if x<.7853981633974483 then \n # use the ra tional approximate for cos\n val := cos_apprx(x);\n else\n \+ x := piBy2-x;\n # use the rational approximate for sin\n va l := sin_apprx(x);\n end if;\n\n if reflect then -val else val end if;\nend proc: # cos16\n\nsin_apprx := proc(x)\n local z,a1,a2,a3,a 4,b1,b2,num,den;\n\n # coefficients of numerator in rational approxi mation\n a1 := .1666666666666665500385651;\n a2 := -.5964761501642 244765253693e-2;\n a3 := .9183755196099076478869904e-4;\n a4 := -. 5286756225481411598927482e-6;\n\n # coefficients of denominator in r ational approximation\n b1 := .1421143099012442993838718e-1;\n b2 \+ := .7112067102840642827768380e-4;\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*z)*z;\n\n (1-num/den)*x;\nend proc: # sin_apprx \n\ncos_ap prx := proc(x)\n local z,a1,a2,a3,a4,b1,b2,b3,num,den;\n\n # coeff icients of numerator in rational approximation\n a1 := .500000000000 0000023503724;\n a2 := -.3154004216236836154796330e-1;\n a3 := .63 95493375638683593037923e-3;\n a4 := -.4120239986012705773813651e-5; \n\n # coefficients of denominator in rational approximation\n b1 \+ := .2025324900859680520646671e-1;\n b2 := .1890916480636244018744239 e-3;\n b3 := .8613069640975447170742372e-6;\n\n # evaluate the rat ional 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\n 1-num/den;\nend proc: # of co s_apprx " }}}{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 9 "sin_,cos_" }{TEXT -1 10 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Testing the procedures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "z := evalf(570000*Pi /7,40);\nevalf(sin_(z),20);\nevalf(evalf(sin(z),25),20);\nevalf(cos_(z ),20);\nevalf(evalf(cos(z),25),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"zG$\"Ig&[BXq1/\"e>8N " 0 "" {MPLTEXT 1 0 81 "z := evalf(1 0^300);\nevalf(sin_(z));\nevalf(sin(z));\nevalf(cos_(z));\nevalf(cos(z ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG$\"+++++5\"$\"H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+_U]d)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+_U]d)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+WW9#o\"!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+WW9#o\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(sin_(x) ,x=-10..20);" }}{PARA 13 "" 1 "" {GLPLOT2D 454 164 164 {PLOTDATA 2 "6% -%'CURVESG6$7ex7$$!#5\"\"!$\"+46@SaF)7$$!+W8_O)*!\"*$\"+Xh2-SF)7$$!+)o UIn*F0$\"+.EAdCF)7$$!+KSc4&*F0$\"+WCXo%)!#67$$!+v`3Y$*F0$!+I$R6'yF=7$$ !+rM%Q?*F0$!+Q\\T\">#F)7$$!+m:gh!*F0$!+*G_Cb$F)7$$!+h'f$>*)F0$!+L[tT[F )7$$!+cx6x()F0$!+y2ALgF)7$$!+e=:<')F0$!+nL[EsF)7$$!+ff=d%)F0$!+H3AN#)F )7$$!+h+A(H)F0$!+vanL!*F)7$$!+iTDP\")F0$!+@\"f9g*F)7$$!+M5uc!)F0$!+\"R Y^z*F)7$$!+1zAwzF0$!+_APD**F)7$$!+yZr&*yF0$!+()HH\"***F)7$$!+];?:yF0$! +2:[#***F)7$$!+%RvTl(F0$!+O00,)*F)7$$!+Q\"\\J\\(F0$!+DB.c$*F)7$$!+6#)) GL(F0$!+4Ovs')F)7$$!+&GFE<(F0$!+2S?nxF)7$$!+ejO7qF0$!+::fimF)7$$!+Ja5_ oF0$!+beA(Q&F)7$$!+QI_.nF0$!+\"3)o!3%F)7$$!+W1%\\b'F0$!+uwA%o#F)7$$!+] #ejS'F0$!+WshG7F)7$$!+cexdiF0$\"+/y%e\"F)7$$!+c'y1#GF0$!+!GSV:$F)7$$!+SQ\"*eEF0$!+*4O:k%F)7$$!+D! \\r\\#F0$!+#\\Pv+'F)7$$!+()*>ZL#F0$!+YMF@sF)7$$!+]4Hs@F0$!+cw!\\C)F)7$ $!+7>')4?F0$!+BF\\^!*F)7$$!+vGVZ=F0$!+$4&z>'*F)7$$!+Qw;pPM:F0$!+[ qO$***F)7$$!+]9%yP\"F0$!+T*=W\")*F)7$$!+v4J@7F0$!+-1\\&R*F)7$$!+3j;z5F 0$!+xbk:))F)7$$!+0k@q$*F)$!+0#zz0)F)7$$!+K(p([zF)$!+dWxPrF)7$$!*1Bt_'F 0$!+')GftgF)7$$!+&\\&3P[F)$!+rXl]YF)7$$!+Iz%o9$F)$!+#en^4$F)7$$!+l.hc9 F)$!+w\\Y^9F)7$$\"(siL#!\")$\"+w%fgL#F=7$$\"+],[l;F)$\"+S7zd;F)7$$\"++ JL(4$F)$\"+yk/[IF)7$$\"+]g=HXF)$\"+ss\"fP%F)7$$\")!R5'fFjbl$\"+fgA9cF) 7$$\"+]ViEwF)$\"+*H)[3pF)7$$\"++(4AH*F)$\"+r'R:,)F)7$$\"+0&zd4\"F0$\"+ #G]G*))F)7$$\"*/QBE\"Fjbl$\"+3v-G&*F)7$$\"+/V0O8F0$\"+sPuD(*F)7$$\"+o0 x49F0$\"+?Mjq)*F)7$$\"++(GmW\"F0$\"+%*4,B**F)7$$\"+Ko[$[\"F0$\"+n%4>'* *F)7$$\"+k\\M?:F0$\"+**fF()**F)7$$\"+&4.sb\"F0$\"+Mh2****F)7$$\"+e$>4j \"F0$\"+\\:$>)**F)7$$\"+Acj/*fF)7$$\" +7j7`EF0$\"+!Q@Fp%F)7$$\"+&3dr!GF0$\"+F1O#G$F)7$$\"+dy=hHF0$\"+`\"yUz \"F)7$$\"*j=_6$Fjbl$\"+7m4PEF=7$$\"+$eLfF$F0$!+z?PR8F)7$$\"+N&[mV$F0$! +rJ$z!HF)7$$\"+)[jtf$F0$!+Pfa,WF)7$$\"*Wy!ePFjbl$!+fVr\"y&F)7$$\"+N\\m 0RF0$!+3Yn=pF)7$$\"+I9D`SF0$!++s?0zF)7$$\"+Dz$3?%F0$!+#oi)>()F)7$$\"*U C%[VFjbl$!+)fG\\M*F)7$$\"+#R;w]%F0$!+FD2\"z*F)7$$\"+l$3om%F0$!+OQh*)** F)7$$\"+]VSYZF0$!+J`@%***F)7$$\"+P.+E[F0$!+(oJb$**F)7$$\"+Bjf0\\F0$!++ X$R\")*F)7$$\"*J#>&)\\Fjbl$!+UP>I'*F)7$$\"+I![0:&F0$!+JEMb!*F)7$$\"+]P !fJ&F0$!+H$fMB)F)7$$\"+q%f7[&F0$!+N]'p=(F)7$$\"*>:mk&Fjbl$!+'R3W%fF)7$ $\"+Lyb!z&F0$!+ycUHZF)7$$\"+v/]MfF0$!+&y?mT$F)7$$\"+=JWygF0$!+1u9L?F)7 $$\"*w&QAiFjbl$!+l&4i2'F=7$$\"+bx%yP'F0$\"+7L6_%*F=7$$\"+](4L`'F0$\"+: `CvCF)7$$\"+Xu?`F)7$$\"+#>T[+( F0$\"+6nG1mF)7$$\"+X'[a;(F0$\"+=**y@xF)7$$\"+(4cgK(F0$\"+N#R&Q')F)7$$ \"*bjm[(Fjbl$\"+P\"QHL*F)7$$\"+5myVwF0$\"+9C!*z(*F)7$$\"+q'44!yF0$\"+Q ,#f)**F)7$$\"++7ZzyF0$\"+'e^n***F)7$$\"+IF.ezF0$\"+pc\"f%**F)7$$\"+gUf O!)F0$\"+#)fsL)*F)7$$\"*zb^6)Fjbl$\"+%fu3m*F)7$$\"+F/=n#)F0$\"+T1Te\"* F)7$$\"+l]?>%)F0$\"+3&*oW%)F)7$$\"+-(H7d)F0$\"+>YF)7$$\"*6W%)R*Fjbl$\"+C23LEF=7$$\"+?h6]&*F0$!+rC5]7F )7$$\"+I\")y,(*F0$!+X-\"[t#F)7$$\"+S,Y`)*F0$!+pescTF)7$$\"+:K^+5Fjbl$! +&o,K[&F)7$$\"+Suq;5Fjbl$!+.)>)fnF)7$$\"+k;!H.\"Fjbl$!+-nafyF)7$$\"+)) e4\\5Fjbl$!+%y/Ov)F)7$$\"+7,Hl5Fjbl$!+8$)f=%*F)7$$\"+'[k*z5Fjbl$!+R:n3 )*F)7$$\"+g)QY4\"Fjbl$!+zi!z)**F)7$$\"+Zg(>5\"Fjbl$!+&Gvq***F)7$$\"+MK J46Fjbl$!+j)\\C&**F)7$$\"+A/l;6Fjbl$!+E,Fa)*F)7$$\"+4w)R7\"Fjbl$!+2W1. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(sin_(cos_(x)),x=0..Pi/2);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%%sin_G6#-%%cos_G6#%\"xG/F,; \"\"!,$*&\"\"#!\"\"%#PiG\"\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+5uVK*)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(sin_(cos_(x)),x=0..Pi/2);\nquad(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%%sin_G6#-%%cos_G6#%\"xG/F,; \"\"!,$*&\"\"#!\"\"%#PiG\"\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+5uVK*)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(sin(cos(x)),x=0..Pi/2);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#-%$cosG6#%\"xG/F,;\"\"!,$* &\"\"#!\"\"%#PiG\"\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5uVK*) !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(sin(cos(x)),x=0..Pi/2);\nvalue(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#-%$cosG6#%\"xG/F,;\" \"!,$*&\"\"#!\"\"%#PiG\"\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& #\"\"\"\"\"#F&*&%#PiGF&-%(StruveHG6$\"\"!F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5uVK*)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }