{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 "Procedures for evaluating the inv erse sine function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nan aimo, 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 a pproximation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "remez " }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the o ne that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx .m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "load numerical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Ma ple m-file " }{TEXT 262 8 "numfcn.m" }{TEXT -1 72 " contains the code \+ for the alternative mathematical functions including " }{TEXT 0 7 "arc sin_" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read i nto a Maple session by a command similar to the one that follows, wher e the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/numfcn.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "load numeri cal integration procedures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "Th e Maple m-files " }{TEXT 262 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 262 8 "gkdata.m" }{TEXT -1 67 " contain the code and data for the nume rical integration procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " use d in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be rea d into a Maple session by commands similar to those that follow, where the file paths give their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/intg.m\";\nread \"K:\\\\Ma ple/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 65 "A fixed precision procedure to evaluate the invers e sine function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 98 "In this section we construct a fixed prec ision version of the inverse sine function which can use " }{TEXT 259 34 "hardware floating point arithmetic" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 59 ": Argument reduction is achieved by means of the formulas: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "arcsin(x)=Pi/2-arcsin(sqrt(1-x^2))" "6#/-%'arcsinG6#%\" xG,&*&%#PiG\"\"\"\"\"#!\"\"F+-F%6#-%%sqrtG6#,&F+F+*$F'F,F-F-" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x) = arcsin(x[0])+arcsin(x*sqrt(1-x[0]^2)-x[0]*sqrt(1-x^2));" "6#/-%'arc sinG6#%\"xG,&-F%6#&F'6#\"\"!\"\"\"-F%6#,&*&F'F.-%%sqrtG6#,&F.F.*$&F'6# F-\"\"#!\"\"F.F.*&&F'6#F-F.-F46#,&F.F.*$F'F:F;F.F;F." }{TEXT -1 2 ". \+ " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A rational approximation for " }{XPPEDIT 18 0 "arcsin(x)" "6#-% 'arcsinG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 71 "When working with fixed prec ision we can replace the Maclaurin series: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "arcsin(x) = x+``(1/2)*``(x^3/3)+``(1/2) *``(3/4)*``(x^5/5)+``(1/2)*``(3/4)*``(5/6)*``(x^7/7)+` . . . `;" "6#/- %'arcsinG6#%\"xG,,F'\"\"\"*&-%!G6#*&F)F)\"\"#!\"\"F)-F,6#*&F'\"\"$F4F0 F)F)*(-F,6#*&F)F)F/F0F)-F,6#*&F4F)\"\"%F0F)-F,6#*&F'\"\"&F@F0F)F)**-F, 6#*&F)F)F/F0F)-F,6#*&F4F)FF/F/F/,,&%\"bG6#F.F/*&&FB6#F/F/F'F/F/*&&F,6#F4F/*$F'F4F/F/F:F/*&&F B6#%\"nGF/)F'FNF/F/!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 143 "so that the degree is as small as possible subject to the conditi on that the theoretical error does not exceed half a unit in the last \+ decimal." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = P IECEWISE([(arcsin(x)-x)/(x^3), x <> 0],[1/6, x = 0]" "6#/-%\"fG6#%\"xG -%*PIECEWISEG6$7$*&,&-%'arcsinG6#F'\"\"\"F'!\"\"F1*$F'\"\"$F20F'\"\"!7 $*&F1F1\"\"'F2/F'F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 36 " has the Maclaurin series expansion " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/6+3/40*x^2+5/112*x^ 4+35/1152*x^6+` . . . `" "6#,,*&\"\"\"F%\"\"'!\"\"F%*(\"\"$F%\"#SF'%\" xG\"\"#F%*(\"\"&F%\"$7\"F'F+\"\"%F%*(\"#NF%\"%_6F'F+F&F%%(~.~.~.~GF%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "and is therefore an eve n function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We shall approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 79 " on the interval [-0.25882,0.25882], which is slightly w ider than the interval " }{XPPEDIT 18 0 "[-sin(Pi/12), sin(Pi/12)];" " 6#7$,$-%$sinG6#*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 37 " , and then use the rational function " }{XPPEDIT 18 0 "s(x) = x+r(x)*` .`*x^3;" "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$F)" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsin G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 69 "Such a rational approximation can be constructed u sing the procedure " }{TEXT 0 5 "remez" }{TEXT -1 50 ", which can be l oaded from the relevant worksheet." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "remez" }{TEXT -1 30 " can cope with the expre ssion " }{XPPEDIT 18 0 "(arcsin(x)-x)/(x^3);" "6#*&,&-%'arcsinG6#%\"xG \"\"\"F(!\"\"F)*$F(\"\"$F*" }{TEXT -1 30 ", since it automatically use s " }{XPPEDIT 18 0 "Limit((arcsin(x)-x)/(x^3),x = 0,right) = 1/6;" "6# /-%&LimitG6%*&,&-%'arcsinG6#%\"xG\"\"\"F,!\"\"F-*$F,\"\"$F./F,\"\"!%&r ightG*&F-F-\"\"'F." }{TEXT -1 21 " for the \"value\" at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "To obtain desirable error characteristics for the final function \+ " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 40 ", we use a wei ght function of the form " }{XPPEDIT 18 0 "w(x)=x^2+delta" "6#/-%\"wG 6#%\"xG,&*$F'\"\"#\"\"\"%&deltaGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 12 " is small. " }{XPPEDIT 18 0 " delta" "6#%&deltaG" }{TEXT -1 39 " prevents the weight from being 0 wh en " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Calculati on " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "evalf(remez((arcsin(x)-x)/x^3,x=-0.25882..0.25882,[6 ,6],\n 'maxerr',weight=x^2+1e-40,type=even,errtype=absolute,\n \+ maxgraph=7,info=true),25):\nr := unapply(%,x);\nmaxerr;" }} {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#*&,*$!DAsqg;#H%RS)ommmmmm;!#N\"\"\"*&$\"DL3+ o>or\\r(=(fn,p>,#F'F()%\"xG\"\"#F(F(*&$\"DDKQD!Q`-dGFu-`AUoa!#OF()F-\" \"%F(!\"\"*&$\"DNh-7$*Qczj*3wrL:P)R*!#QF()F-\"\"'F(F(F(,*$F5\"\"!F(*&$ \"D_\\'3H!G\"*QwgIb+T\"=d;!#MF(F,F(F(*&$\"Db*Q]]M6*z')4O6yE)zf!)F'F(F3 F(F5*&$\"D\"e7;>``waOBk;)HPt1\"F'F(F:F(F(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%HW&)Qv52\" >d4R!#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%FR\"!#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~ relative~difference: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~inte rvalG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%'CURV ESG6%7it7$$\"\"!F)%%FAILG7$$\"Aommmmm;/^\"=l0WZw\"!#N$!AhmEE*HC!*G+_EM -'pn!#b7$$\"ANLLLLLL3-j.8\")[HNF.$!ASY&R!=e7^!*RZt%Rxq#!#a7$$\"A.+++++ ]7`Wbp@B%H&F.$!A`#>uWY(fAJdqrL.#4'F77$$\"Aqmmmmmm;/E2Ei(*eqF.$!A'f$=$* [F,)f'[vrM$H3\"!#`7$$\"A++++++]i!*3\"RVY)e5!#M$!AJ9e>wx\"p4%4C+<*fV#FB 7$$\"AMLLLLLL$3_9_C&z69FF$!Am/4])e[,U4boVP\"HVFB7$$\"A,++++++D\"y@y'Gp <@FF$!Av#)*Ge8-C^!H%RaF3t*FB7$$\"AnmmmmmmmT!H/\\!fBGFF$!A\\l0@J0FG>tU` Y]F(R5)FF$!Ae+d6k@%4T d53y+.R\"!#^7$$\"Anmmmmmm\"z+*p&Hwg0\"!#L$!A`8@al%eD,8$=B#puJ#Fbo7$$\" ALLLLLLL$eqk%y*f'3;Ffo$!A@)Rb^4[TJWq_5d01&Fbo7$$\"ALLLLLLL$3%\\dWl!\\; #Ffo$!AKjw*=4!yd\"[.0@%\\&Q)Fbo7$$\"Anmmmmmm\"H'Q&yO4&=FFfo$!A-*3N?5mT 'f:1GJtt6!#]7$$\"AMLLLLLLeMv7^+xJKFfo$!AX/@^Mt/A(*>RkWmR9Fhp7$$\"Anmmm mmm\"H'HG0f\\(\\$Ffo$!A\"R81@:(4U-'=S%GyX:Fhp7$$\"A+++++++D\"RQ%f< 2l\"Fhp7$$\"Anmmmmmm\"zpIz\"e.QSFfo$!A1h9\"y&>IYgBZD$p'o;Fhp7$$\"A,+++ +++D^o\"yW)4quO08<Hx!o_dmPNz9XWn\"Fhp7$$\"A,+++++++:PeRi\")\\WFfo$!A:hB0Rs@8c'=1yp:m \"Fhp7$$\"AnmmmmmmTDWu-Ey'e%Ffo$!A9\"Hw]T.+%3e],x$yj\"Fhp7$$\"AMLLLLLL $e80f'*[Ps%Ffo$!A2^l)pV[)o]\"R\"=b1.;Fhp7$$\"A+++++++DYe1H`rg[Ffo$!A1 \"oXz:d:Fhp7$$\"AMLLLLL$eR`*p3))[U^Ffo$!Auw#[!z_x!eo;.9%oF9Fh p7$$\"Anmmmmmmm@KL)GiUU&Ffo$!AK$=QZ>Q3B*)pal\"4_7Fhp7$$\"AMLLLLLL$3$4! 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?CF*7$$\":;xNM4@>(e4;\"e#F*$\"'tw#)F*7$$\":]L#*y#4A([qvYe#F*$\"(eb\\\" F*7$$\":$)*)[Bw?D5X!>)e#F*F+-%&COLORG6&%$RGBG$\"\"%!\"\"\"\"!$\"\"*F^_ p-%+AXESLABELSG6$Q\"x6\"Q!Ff_p-%%VIEWG6$;$!:))*)[Bw?D5X!>)e#F*$\":))*) [Bw?D5X!>)e#F*%(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 82 "evalf(evalf(numapprox[infnorm](1-h( x)/arcsin(x),x=-sin(Pi/12)..sin(Pi/12)),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&?G#!#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in using " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }{TEXT -1 17 " in the \+ interval " }{XPPEDIT 18 0 "[-sin(Pi/12), sin(Pi/12)];" "6#7$,$-%$sinG6 #*&%#PiG\"\"\"\"#7!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 11 " is about " } {XPPEDIT 18 0 "2.3*`. `*10^(-19);" "6#*(-%&FloatG6$\"#B!\"\"\"\"\"%#.~ GF))\"#5,$\"#>F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "xx := evalf(sin(Pi/12),20); \nevalf(evalf(h(xx),20),17);\nevalf(evalf(arcsin(xx),20),17);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5Ni2_-^/>)e#!#?" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"2W\\\"*z(Q*zh#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2W\\\"*z(Q*zh#!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approxi mation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }{TEXT -1 35 " with ra ndom numbers between 0 and " }{XPPEDIT 18 0 "sin(Pi/12);" "6#-%$sinG6# *&%#PiG\"\"\"\"#7!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The printout occurs only when the relative error is greater than or e qual to the specified \"" }{TEXT 262 3 "eps" }{TEXT -1 3 "\".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "randomize():\neps := Float( 5,-16);\nfor i from 1 to 100 do\n xx := evalf(rand()/Float(1,12)*.25 881904510252076235,20);\n axx := evalf(evalf(arcsin(xx),20),16);\n \+ hxx := evalf(h(xx),16);\n e := evalf(abs((axx-hxx)/axx),20);\n i f e>=eps then \n printf(\" trial no. %d, x = %.16f,\\n\",i,xx); \n printf(\" arcsin(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 = .1224302629795898," }}{PARA 6 "" 1 "" {TEXT -1 79 " arcsin(x) = .1227381992607006, h(x) = .1227381992607005, rel error = 8.15e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 18, x = .1199946349254508," }}{PARA 6 "" 1 "" {TEXT -1 79 " arcsi n(x) = .1202844782712087, h(x) = .1202844782712088, rel error = 8.31e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " t rial no. 74, x = .1477528284880431," }}{PARA 6 "" 1 "" {TEXT -1 79 " \+ arcsin(x) = .1482957756084494, h(x) = .1482957756084495, rel error = \+ 6.74e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 79, x = .1016211432785715," }}{PARA 6 "" 1 "" {TEXT -1 79 " arcsin(x) = .1017968656047534, h(x) = .1017968656047533, rel \+ error = 9.82e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 85, x = .1165094140626329," }}{PARA 6 "" 1 " " {TEXT -1 79 " arcsin(x) = .1167746290884914, h(x) = .11677462908849 13, rel error = 8.56e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 95, x = .1038802289894921," }}{PARA 6 "" 1 "" {TEXT -1 79 " arcsin(x) = .1040679724649460, h(x) = .104067 9724649461, rel error = 9.61e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "arcsin16" }}{PARA 0 "" 0 "" {TEXT -1 137 "Here is the code for the fixed precision version of the arcsine function, which can be evaluated with hardware floating p oint arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1495 "arcsin16 := proc(xx)\n local sinPiBy3,a1,a 2,a3,a4,b1,b2,b3,\n x,z,s,num,den,flag,isneg,val; \n\n # coe fficients of numerator in rational approximation\n a1 := .1666666666 6666693455;\n a2 := -.20123068350905740173;\n a3 := .5470750116934 5017021e-1;\n a4 := -.94079204806169027974e-3;\n\n # coefficients \+ of denominator in rational approximation\n b1 := -1.6573841010537498 478;\n b2 := .80621070955878185407;\n b3 := -.10678906394409199148 ;\n\n sinPiBy3 := .86602540378443864676;\n x := evalf(xx);\n \n \+ if abs(x)>1 then\n error \"argument must be in the interval [-1,1 ]\"\n end if;\n\n # Handle the argument reduction\n if x<0 then \n isneg := true;\n x := -x;\n else isneg := false end if; \n\n if x<.2588190451025208 then\n flag := 1;\n else\n s := sqrt(1-x*x); # use Maple's square root\n if x>.96592582628906 83 then\n flag := 4;\n x := s;\n else \n \+ if x<.70710678118654752440 then\n flag := 2;\n \+ x := x*sinPiBy3-s*0.5;\n else\n flag := 3;\n \+ x := s*sinPiBy3-x*0.5;\n end if;\n end if;\n end \+ if;\n \n # evaluate the rational approximation\n z := x*x;\n n um := (a1+(a2+(a3+a4*z)*z)*z)*z;\n den := 1+(b1+(b2+b3*z)*z)*z;\n \+ val := (1+num/den)*x;\n\n if flag=2 then val := val+.523598775598298 87308;\n elif flag=3 then val := 1.0471975511965977462-val\n elif \+ flag=4 then val := 1.5707963267948966192-val end if;\n if isneg then -val else val 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 30 "Testing the procedure arcsin16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can use Maple's inv erse sine function to check the accuracy of the procedure " }{TEXT 0 8 "arcsin16" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot('arcsin16(x)',x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$!3c'*[zEjzq:!#<7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "with(codegen,C):\nC(arcsin16,ansi);" }}{PARA 6 "" 1 " " {TEXT -1 17 "#include " }}{PARA 6 "" 1 "" {TEXT -1 26 "doubl e arcsin16(double xx)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " doubl e a2;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a3;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a4;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b1; " }}{PARA 6 "" 1 "" {TEXT -1 12 " double b2;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double b3;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double den ;" }}{PARA 6 "" 1 "" {TEXT -1 11 " int flag;" }}{PARA 6 "" 1 "" {TEXT -1 12 " int isneg;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double num ;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double piBy2;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double piBy3;" }}{PARA 6 "" 1 "" {TEXT -1 15 " double \+ piBy6;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double s;" }}{PARA 6 "" 1 "" {TEXT -1 18 " double sinPiBy3;" }}{PARA 6 "" 1 "" {TEXT -1 13 " doub le val;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double x;" }}{PARA 6 "" 1 " " {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }} {PARA 6 "" 1 "" {TEXT -1 28 " a1 = 0.1666666666666669;" }}{PARA 6 " " 1 "" {TEXT -1 29 " a2 = -0.2012306835090574;" }}{PARA 6 "" 1 "" {TEXT -1 31 " a3 = 0.5470750116934502E-1;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a4 = -0.9407920480616903E-3;" }}{PARA 6 "" 1 "" {TEXT -1 30 " b1 = -0.165738410105375E1;" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ b2 = 0.8062107095587819;" }}{PARA 6 "" 1 "" {TEXT -1 28 " b3 = -0. 106789063944092;" }}{PARA 6 "" 1 "" {TEXT -1 34 " sinPiBy3 = 0.8660 254037844386;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }} {PARA 6 "" 1 "" {TEXT -1 23 " if( 1.0 < fabs(x) )" }}{PARA 6 "" 1 " " {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 68 " fprintf (stderr, \"argument must be in the interval [-1,1]\" );" }}{PARA 6 "" 1 "" {TEXT -1 16 " exit(1);" }}{PARA 6 "" 1 "" {TEXT -1 7 " \+ \}" }}{PARA 6 "" 1 "" {TEXT -1 17 " if( x < 0.0 )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 18 " isne g = 1;" }}{PARA 6 "" 1 "" {TEXT -1 15 " x = -x;" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }} {PARA 6 "" 1 "" {TEXT -1 16 " isneg = 0;" }}{PARA 6 "" 1 "" {TEXT -1 32 " if( x < 0.2588190451025208 )" }}{PARA 6 "" 1 "" {TEXT -1 15 " flag = 1;" }}{PARA 6 "" 1 "" {TEXT -1 9 " else \+ " }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 26 " s = sqrt(1.0-x*x);" }}{PARA 6 "" 1 "" {TEXT -1 36 " \+ if( 0.9659258262890683 < x )" }}{PARA 6 "" 1 "" {TEXT -1 11 " \+ \{" }}{PARA 6 "" 1 "" {TEXT -1 21 " flag = 4;" }}{PARA 6 "" 1 "" {TEXT -1 18 " x = s;" }}{PARA 6 "" 1 "" {TEXT -1 11 " \}" }}{PARA 6 "" 1 "" {TEXT -1 13 " else " }} {PARA 6 "" 1 "" {TEXT -1 38 " if( x < 0.7071067811865475 )" } }{PARA 6 "" 1 "" {TEXT -1 13 " \{" }}{PARA 6 "" 1 "" {TEXT -1 23 " flag = 2;" }}{PARA 6 "" 1 "" {TEXT -1 35 " \+ x = x*sinPiBy3-0.5*s;" }}{PARA 6 "" 1 "" {TEXT -1 13 " \+ \}" }}{PARA 6 "" 1 "" {TEXT -1 15 " else " }}{PARA 6 "" 1 "" {TEXT -1 13 " \{" }}{PARA 6 "" 1 "" {TEXT -1 23 " \+ flag = 3;" }}{PARA 6 "" 1 "" {TEXT -1 35 " x = s*s inPiBy3-0.5*x;" }}{PARA 6 "" 1 "" {TEXT -1 13 " \}" }} {PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 12 " \+ z = x*x;" }}{PARA 6 "" 1 "" {TEXT -1 36 " num = (a1+(a2+(a3+a4*z )*z)*z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 33 " den = 1.0+(b1+(b2+b3*z) *z)*z;" }}{PARA 6 "" 1 "" {TEXT -1 26 " val = (1.0+num/den)*x;" }} {PARA 6 "" 1 "" {TEXT -1 21 " if( flag == 2.0 )" }}{PARA 6 "" 1 "" {TEXT -1 32 " val += 0.5235987755982989;" }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 23 " if( flag = = 3.0 )" }}{PARA 6 "" 1 "" {TEXT -1 39 " val = 0.10471975511965 98E1-val;" }}{PARA 6 "" 1 "" {TEXT -1 11 " else " }}{PARA 6 "" 1 "" {TEXT -1 25 " if( flag == 4.0 )" }}{PARA 6 "" 1 "" {TEXT -1 41 " val = 0.1570796326794897E1-val;" }}{PARA 6 "" 1 "" {TEXT -1 17 " if( (isneg) )" }}{PARA 6 "" 1 "" {TEXT -1 17 " v al = -val;" }}{PARA 6 "" 1 "" {TEXT -1 16 " return(val);" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }}{PARA 6 "" 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "More fixed precision procedures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 25 digit rational approximati on for " }{XPPEDIT 18 0 "arcsin(x)" "6#-%'arcsinG6#%\"xG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-sin(Pi/36), sin(Pi/36)];" "6#7$,$-%$sinG6#*&% #PiG\"\"\"\"#O!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "evalf(rem ez((arcsin(x)-x)/x^3,x=-0.087156...0.087156,[6,8],\ntype=even,errtype= absolute,weight=x^2+1e-50),35):\nr := unapply(%,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%-iteration~18G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G --------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,*$\"O=!Grmu0Y8iSmmmmmmmmmmmm\"!#Y\"\"\"*&$\"O8%**=(G 3yY:8rI')RW0:IzTO!\\#F'F()%\"xG\"\"#F(!\"\"*&$\"O'=sWooWEja/uRk#Hj\\Kh TGI5F'F()F-\"\"%F(F(*&$\"O#GnEq(pvz*HZ0**47Bf=#*HPH,\"!#ZF()F-\"\"'F(F /F(,,$F(\"\"!F(*&$\"O')**z0;sZf25)[CRmK!4e2&=U%>!#XF(F,F(F/*&$\"O3f\"e eEk>`)3w-'G*eT#\\Bo6_A\"FAF(F3F(F(*&$\"OS?da,sev;bm\\_35DMYrMSOFF'F(F9 F(F/*&$\"O%*o5,!f0(4#Roc?+[cU]_JLQ^\"F8F()F-\"\")F(F(F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"A*Hl`$p-7d5d\"3>K&Gl!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"L*o!R&R+\"4dO?d7!Qkh5e6(>Q$!#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"LqZ%3#pHXDMYW4xPX'4\"[`K4&!#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Scritical~point s~in~the~right~half~of~the~interval:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\"O6x#RO#*)*)eZFyu*3C\")R8T,(R;8!#Z$\"O!>A7F#)fx.(=4jfPA:6y&)RH)*G F%$\"O>FNQ$HF(zWtqh!=E!QhrW%>NJ%F%$\"O3B]qie\\ll*R*=/'yd&F%$\"O [qrm$H:M;Q$\\a\"z)*[Pwk4oOm'F%$\"O!=!#MF(F,F(F/*&$\"Dk>`)3w-'G*eT#\\Bo6_A\"FAF(F3F(F(*& $\"D(ev;bm\\_35DMYrMSOFF'F(F9F(F/*&$\"D1(4#Roc?+[cU]_JLQ^\"F8F()F-\"\" )F(F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"2u&))zFERI>!#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,*$\" D1Y8iSmmmmmmmmmmmm\"!#N\"\"\"*&$\"D\"yY:8rI')RW0:IzTO!\\#F0F1)9$\"\"#F 1!\"\"*&$\"DXEja/uRk#Hj\\KhTGI5F0F1)F6\"\"%F1F1*&$\"Dd(z*HZ0**47Bf=#*H PH,\"!#OF1)F6\"\"'F1F8F1,,$F1\"\"!F1*&$\"Dx%f25)[CRmK!4e2&=U%>!#MF1F5F 1F8*&$\"Dk>`)3w-'G*eT#\\Bo6_A\"FJF1F " 0 "" {MPLTEXT 1 0 484 "s:=x->x+(.1666666 6666666666666666664062134606-.24903641793015054439863071131546781*x^2+ .10302841613249632926439740454632645*x^4-.1012937299218592312099905472 9979757e-1*x^6)/(1.-1.9442185075809032663924488100759477*x^2+1.2252116 823492415892860276088531964*x^4-.27364034714634251008524966551675587*x ^6+.15138333152504256480020566839209706e-1*x^8)*x^3:\nh := unapply(eva lf(convert(s(x),horner),30),x);\nevalf(plot((1-s(x)/arcsin(x)),x=-sin( Pi/36)..sin(Pi/36),color=COLOR(RGB,.4,0,.9)),35);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*(,* $\"?@1kmmmmmmmmmmm;!#IF.*&$\"?:8rI')RW0:IzTO!\\#F3F.)F-\"\"#F.!\"\"*&$ \"?YXS(Rk#Hj\\KhTGI5F3F.)F-\"\"%F.F.*&$\"?+ta!**47Bf=#*HPH,\"!#JF.)F- \"\"'F.F9F.,,$F.\"\"!F.*&$\"?35)[CRmK!4e2&=U%>!#HF.F7F.F9*&$\"?&)3w-'G *eT#\\Bo6_A\"FKF.F=F.F.*&$\"?!#N7$$!DskdMR#4+;#eV3]&3q.()F*$\"'))Q6F-7$$!D?%z;.xv$ R(Ga5DNu#=p)F*$\"&5l%F-7$$!DoByG,Bu=`Fn$\\:S&*z')F*$!%]5!#M7$$!D;`)eA$ 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\n\n # coefficient s for rational approximation\n a1 := .166666666666666666666666640621 ;\n a2 := -.249036417930150544398630711315;\n a3 := .1030284161324 96329264397404546;\n a4 := -.101293729921859231209990547300e-1;\n \+ b1 := -1.94421850758090326639244881008;\n b2 := 1.225211682349241589 28602760885;\n b3 := -.273640347146342510085249665517;\n b4 := .15 1383331525042564800205668392e-1;\n\n s1 := .173648177666930348851716 626769; # sin(Pi/18)\n s2 := .342020143325668733044099614682; # sin( Pi/9)\n s3 := .5; # sin(Pi/6)\n s4 := .642787609686539326322643409 907; # sin(2*Pi/9)\n c1 := .984807753012208059366743024590; # cos(Pi /18)\n c2 := .939692620785908384054109277325; # cos(Pi/9)\n c3 := \+ .866025403784438646763723170753; # cos(Pi/6)\n c4 := .76604444311897 8035202392650555; # cos(2*Pi/9)\n p1 := .174532925199432957692369076 849; # Pi/18\n\n x := xx;\n if x<0 then\n x := -x;\n isn eg := true;\n else\n isneg := false;\n end if;\n\n t := sqr t(1-x*x);\n if x<.7071067812 then \n if x<.4226182617 then # x " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Testing the procedure arcsin25 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := evalf(sin(2/ 3),30);\nevalf(evalf(arcsin25(xx),30),27);\nevalf(evalf(arcsin(xx),30) ,27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?2AY'))\\x+P(pI!)p$= '!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 76 "xx := 0 .5;\nevalf(evalf(arcsin25(xx),30),27);\nevalf(evalf(arcsin(xx),30),27) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 100 "xx := evalf(sin(Pi/2 -0.0001),30);\nevalf(evalf(arcsin25(xx),30),27);\nevalf(evalf(Pi/2-0.0 001,30),27);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?y_mmm;/+++]* *******!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"m*[zEjpq:!#E" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"m*[zEjpq:!#E" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "evalf( plot(1-'arcsin25'(x)/arcsin(x),x=0..1,\n colo r=COLOR(RGB,.4,0,.9)),35);" }}{PARA 13 "" 1 "" {GLPLOT2D 658 209 209 {PLOTDATA 2 "6&-%'CURVESG6#7j[m7$$\"DILLLLLLLLLL$3x1h6o!#Q$\"%07!#N7$$ \"Dmmmmmmmmmmm;a8ABO\"!#P$\"%$z%F-7$$\"D*********************\\7.K[V?F 1$\"&)o5F-7$$\"DKLLLLLLLLLLL3FWYs#F1$\"&H(=F-7$$\"D)****************** ***\\iSmp3%F1$\"&([SF-7$$\"Dlmmmmmmmmmmm;a)G\\aF1$\"&))z'F-7$$\"D(**** ******************\\7G$R<)F1$\"'y\"H\"F-7$$\"DLLLLLLLLLLLL$3x&)*3\"!#O $\"'lzF-7$$\"DLLLLLLLLLL$ekGg?%H\"FP$\"'&)G>F-7$$F0FP$\"'rB>F-7 $$\"DLLLLLLLLLLLe*[Vb)\\\"FP$\"'\\B=F-7$$\"D++++++++++++Dc'yM;FP$\"'8. ;F-7$$\"DmmmmmmmmmmmTgx=5x\"FP$\"'%3F\"F-7$$\"DLLLLLLLLLLL$e*)4D2>FP$ \"&NX)F-7$$\"D+++++++++++DJ?$[V?FP$\"&$QNF-7$$\"DmmmmmmmmmmmmmT:(z@FP$ !%&p\"!#M7$$\"DLLLLLLLLLLeRsL]#)H#FP$!%/iF^q7$$\"D******************** \\7yD&y;CFP$!&&R5F^q7$$\"DmmmmmmmmmmT&Qy,KNDFP$!&6S\"F^q7$$\"DLLLLLLLL LLLe*)4bQl#FP$!&+RsFFP$!&C'=F^q7$ $\"Dmmmmmmmmmm;/,%\\#4*GFP$!&'H>F^q7$$\"DLLLLLLLLLL3x1')f%4IFP$!&k(=F^ q7$$\"D**********************\\7y%*z7$FP$!&Kq\"F^q7$$\"Dmmmmmmmmmm\"H# =qHlC$FP$!&'=9F^q7$$\"DLLLLLLLLLL$eRAY1lLFP$!&)Q5F^q7$$\"D************ *******\\(oHa*f$[$FP$!%meF^q7$$\"DmmmmmmmmmmmTNYM@g$FP$!$5*F^q7$$\"DLL LLLLLLLLe9TQp1s$FP$\"&v9%F-7$$\"D********************\\(o/V?RQFP$\"&p& *)F-7$$\"DmmmmmmmmmmTg_ARx&RFP$\"'(pJ\"F-7$$\"DLLLLLLLLLLLLe9ui2%FP$\" 'UY;F-7$$\"Dmmmmmmmmmm\"z>h!z&4UFP$\"'Au=F-7$$\"D******************** \\il(R)GM%FP$\"''e#>F-7$$\"DLLLLLLLLLL3F>*))=wWFP$\"'h!z\"F-7$$\"Dmmmm mmmmmmm\"H2Q\\4YFP$\"'nv9F-7$$\"D*******************\\ilA()zUZFP$\"'[1 5F-7$$\"DLLLLLLLLLL$3-QO5w[FP$\"&_E%F-7$$\"DmmmmmmmmmmT&Q`&3%4]FP$!%m? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1474 "s:=x->x+(.1666666666666666 666666666666666666666666666666666666668072173718095709-.62021589571967 03407009977417699836930001606674404945867482164542202350*x^2+.93114740 70945793034758749540062397423626015308806319920176929175252844*x^4-.72 14184582226858941049989794663882520187591560078832924271415923389924*x ^6+.305798328235117705810005361986819476668537017346688072054384432651 4874*x^8-.690435301836559427891216828125421989867332797072114511859818 8412164509e-1*x^10+.73821160877769395255085431305632424964191950344276 74903653299675158001e-2*x^12-.2716402274034969233119407060333300734185 948839213071124768809063214394e-3*x^14+.439824339966316840205971876955 3756437461879764066899526331450264577934e-6*x^16)/(1.-4.17129537431802 2044205986450619902158000964004642953039975020552626980*x^2+7.19611021 8153442883605086483959251568133185844428499810832911286757423*x^4-6.63 1740753193789709935203090544899877831753890996466819495906667079967*x^ 6+3.517703217026174293052396222835358825192292541897136595286328813648 381*x^8-1.076850707790136028398683690701069900856338969802202460035776 482986393*x^10+.180101494201357694314041010378272493777935506939215788 3663786259834717*x^12-.14319495893194836959198282295313885991356861973 31505710674487532199387e-1*x^14+.3792205639627848143813721148204574481 223156453419411370097667875699751e-3*x^16)*x^3:\nh := unapply(evalf(co nvert(s(x),horner),60),x);\nevalf(plot(10^30*(1-s(x)/arcsin(x)),x=-sin (Pi/36)..sin(Pi/36),color=COLOR(RGB,.4,0,.9)),65);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"*( ,4$\"gnd*e@?'F3F.)F-\"\"#F.!\"\"*&$\"gn$pj!)3`,EOU(Ri+a\\(eZ.$ zX42u9J*F3F.)F-\"\"%F.F.*&$\"gnUrU#H$)y+c\"f(=?D)Qm%z*)*\\5%*eoA#e%=9s F3F.)F-\"\"'F.F9*&$\"gn%Qa?2)oYt,P&omZ>o)>O0+\"eq<^BG$)z0$F3F.)F-\"\") F.F.*&$\"gn>)f=^9@2(zKt')*)>UD\"Go@\"*yUfl$=IN/p!#hF.)F-\"#5F.F9*&$\"g nI`O!\\nFW.&>>k\\UKcIJa3b_Rpx(3;@Q(!#iF.)F-\"#7F.F.*&$\"gn\")oZ7rI@R)[ f=M2IL.12%>JBp\\.uAS;F!#jF.)F-\"#9F.F9*&$\"gnXJj_**o1k(z=YPkv`&p(=(f?S oJm*RV#)R%!#mF.)F-\"#;F.F.F.,4$F.\"\"!F.*&$\"gn-v*RI&Hk/S'4+e@!*>1X')f ?W?-=VP&HrT!#fF.F7F.F9*&$\"gn\"H$3\")*\\GW%e=L\"o:DfR['30O)GW`\"=-6'>( FaoF.F=F.F.*&$\"gn\"f\\>oY'*4*QvJy()**[a!4._$*4(*y$>`2uJmFaoF.FBF.F9*& $\"gnL'G&fOr*=a#H#>D)e`$GA'R_IHuh-'oN\"*f)QJ&H#G)>fp$[>$*e\\>V\"FLF.FY F.F9*&$\"gnn(4q8T>MXcJA\"[uX?[6s8Q9[yiRc?#z$FXF.FinF.F.F9F-\"\"$F.F(F( F(" }}{PARA 13 "" 1 "" {GLPLOT2D 555 223 223 {PLOTDATA 2 "6&-%'CURVESG 6#7ebm7$$!\\oUvOVPhsEq\\h:,xP^NZP3Fk!eN_q;>\"3wf>aR#yG,Bu=`Fn$\\:S &*z')F*$!&(zDF37$$!\\o4&>C@b?^))*3@4jNVV`)eA$)3\")*=7HOdf!3o')F*$!&06# F37$$!\\o+Dozb\">(*4))4w&GvW\"H))HKOvuZo4*yf<2il)F*$!&TD\"F37$$!\\o#\\ Xp%fxJ9j)3gS\\h%R74?%*=%o0:G:AcPLW')F*$!%UEF37$$!\\o%[3UJO;*GXySafaZ(= %>x+%of)F*$\"'MdCF-7$$!\\o^/E $yx5tQx.![@8`zfg0\\Xn`z/iMulm\\e)F*$\"'R%f#F-7$$!\\oUM_]\"Q4>gv-kpGXv# *3F+3TI`%*QsnPK4t&)F*$\"'0@DF-7$$!\\oLky<&)z];Qz \")>7c)F*$\"'%zE#F-7$$!\\oD%\\]))e16.s+Kz@tN#[\"p>9u0dCq*fT$=(erx,%HnS6FM%40%yHZP&)F*$\"'!GP\"F-7$$! 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\\oPF6C:)fz5*=LYe*pc@9)GKX1')Hp,!oO;Q&e)F*$\"'J$f#F-7$$\"\\oZ/f)**Q?^b id`Kvz_:uobFN>AL`90Jt@(f)F*$\"'G\\CF-7$$\"\\oe\"oIZ'4G-gLQ/[&*)[4M\\)= g!yXt*)GUDI04')F*$\"'iw?F-7$$\"\\ooeaZR:W\\%44MGM*\\MS*H@wwOp8EVz>())3 i)F*$\"'Zu9F-7$$\"\\oyN-A9@g'*G[ViP\"4T(R0T0vaHRDwlTTCFj)F*$\"&on'F-7$ $\"\\o*G,l*)oiPMcg9C$*=P\"R6p[#=a;%*)>P&3,cW')F*$!%JGF37$$\"\\o***y4PE B4zH'[?F(GL&Q<(>**)G,W`j3H!eRc')F*$!&$p7F37$$\"\\o5nXXQQ3QK?^*>_QHzL__ tfrjur+G(\\J#o')F*$!&&>@F37$$\"\\o@W$*>8WC&owP&y;$[Dt$H`y/.t[\"3:l\">n +o)F*Fa`l7$$\"\\oJ@T%z)\\SK,Ncd6\"e@n`8=A,*3^X%H-')G!>p)F*$!&'*H#F37$$ \"\\oU)*))oibczN#*eO1zw6OT4l>xW`4Q%R!eQPq)F*$!%8zF37$$\"\\o^vOVPhsEq\\ h:,xP^NZP3Fk!eN " 0 "" {MPLTEXT 1 0 86 "xx := evalf(sin(Pi/36),65);\nevalf(evalf(h(xx),60),58);\nevalf(eva lf(arcsin(xx),60),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\\o pvOVPhsEq\\h:,xP^NZP3Fk!eN " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "a rcsin55" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3652 "arcsin55 := proc(xx)\n local a1,a2,a3,a4,a5,a6,a7 ,a8,a9,b1,b2,b3,b4,b5,b6,b7,b8,\n s1,s2,s3,s4,c1,c2,c3,c4,t,x,z,n,d, v,isneg,k,p1; \n\n # coefficients for rational approximation\n a1 := \n .1666666666666666666666666666666666666666666666666666668072 17;\n a2 := \n -.620215895719670340700997741769983693000160667440 494586748216;\n a3 := \n .93114740709457930347587495400623974236 2601530880631992017693;\n a4 := \n -.7214184582226858941049989794 66388252018759156007883292427142;\n a5 := \n .305798328235117705 810005361986819476668537017346688072054384;\n a6 := \n -.690435301 836559427891216828125421989867332797072114511859819e-1;\n a7 := \n \+ .738211608777693952550854313056324249641919503442767490365330e-2;\n \+ a8 := \n -.2716402274034969233119407060333300734185948839213071124 76881e-3;\n a9 := \n .4398243399663168402059718769553756437461879 76406689952633145e-6;\n\n b1 := \n -4.171295374318022044205986450 61990215800096400464295303997502;\n b2 := \n 7.19611021815344288 360508648395925156813318584442849981083291;\n b3 := \n -6.6317407 5319378970993520309054489987783175389099646681949591;\n b4 := \n \+ 3.51770321702617429305239622283535882519229254189713659528633;\n b5 := \n -1.076850707790136028398683690701069900856338969802202460035 78;\n b6 := \n .180101494201357694314041010378272493777935506939 215788366379;\n b7 := \n -.143194958931948369591982822953138859913 568619733150571067449e-1;\n b8 := \n .379220563962784814381372114 820457448122315645341941137009767e-3;\n\n s1 := \n .17364817766693 0348851716626769314796000375677184069387236241;\n s2 := \n .3420201 43325668733044099614682259580763083367514160628465048;\n s3 := .5;\n s4 := \n .6427876096865393263226434099072634329075598842056817903 24977;\n c1 := \n .98480775301220805936674302458952301367064325171 9842418790026;\n c2 := \n .939692620785908384054109277324731469936 208134264464633090287;\n c3 := \n .8660254037844386467637231707529 36183471402626905190314027903;\n c4 := \n .76604444311897803520239 2650555416673935832457080395245854045;\n p1 := # Pi/18\n .17453292 5199432957692369076848861271344287188854172545609719;\n\n x := xx;\n if x<0 then\n x := -x;\n isneg := true;\n else\n i sneg := false;\n end if;\n\n t := sqrt(1-x*x);\n if x<.707106781 2 then \n if x<.4226182617 then # x \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Tes ting the procedure arcsin55 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := evalf(sin(2/3),65);\ne valf(evalf(arcsin55(xx),60),58);\nevalf(evalf(arcsin(xx),60),58);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\\o(=tM8&o4**)HUrgaK!zpo?ik)) \\x+P(pI!)p$='!#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ennmmmmmmmmmmm mmmmmmmmmmmmmmmmm!#e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ennmmmmmmmm mmmmmmmmmmmmmmmmmmmm!#e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "xx := 0.5;\nevalf(evalf(arcsin55(xx ),60),58);\nevalf(evalf(arcsin(xx),60),58);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"en#Hoj " 0 "" {MPLTEXT 1 0 100 "xx := evalf (sin(Pi/2-0.0001),60);\nevalf(evalf(arcsin55(xx),60),57);\nevalf(evalf (Pi/2-0.0001,60),57);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gn&* )GWZ$4m2l$zD!yxxx_mmm;/+++]********!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Z\\5Hb(o*p%e)4U9vR;p@8B>m*[zEjpq:!#c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Z\\5Hb(o*p%e)4U9vR;p@8B>m*[zEjpq:!#c" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "evalf(p lot(10^30*(1-'arcsin55'(x)/arcsin(x)),x=0..1,\n color=CO LOR(RGB,.4,0,.9)),65);" }}{PARA 13 "" 1 "" {GLPLOT2D 559 243 243 {PLOTDATA 2 "6&-%'CURVESG6#7fjm7$$\"\\oILLLLLLLLLLLLLLLLLLLLLLLLL$3x1h 6o!#o$!$Z'!#M7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm;a8ABO\"!#n$!%FDF-7$$\" \\o***************************************************\\7.K[V?F1$!%faF -7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLLL3FWYs#F1$!%i\"*F-7$$\"\\olmmmmmmmm 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$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLek.R5#eZF_[n$\"'lW7Fdp7$$\"\\oLLLLLLLLL LLLLLLLLLLLLLLLLL$eR\"3Gy%F_[n$\"'%[R\"Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmm mmmmm;H28se$[F_[n$\"'6Q8Fdp7$$\"\\o*********************************** ****************\\(=7O*))[F_[n$\"'#yE\"Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLL LLLL$e9m>@?-\\F_[n$\"'l+8Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmm\"zW<\"oa \"\\F_[n$\"'HG8Fdp7$$\"\\o******************************************** ****\\PM_6M(G\\F_[n$\"'(4M\"Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLL$3-8,+ U\\F_[n$\"'lP8Fdp7$$\"\\o********************************************* ****\\Pf3@`o\\F_[n$\"'548Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmmmT5k]*\\ F_[n$\"'\"QH\"Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm;zRQb@&F_[n$\"'B27F dp7$$\"\\o****************************************************\\(=>Y2a F_[n$\"'GD6Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmmm\"zXu9cF_[n$\"&:o*Fdp 7$$\"\\o*************************************************\\PfL&G\"GcF_ [n$\"'>I8Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLL$3-)[7:k&F_[n$\"'40:Fdp7$ 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=03(oF_[n$\"&'osFdp7$$\"\\o******************************************* ***********\\P+V)oF_[n$\"&=s&Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLLek`md (3pF_[n$\"&%p!)Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmm;Hd&\\@LpF_[n$\"&,T *Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL$e9\"4gVa%pF_[n$\"&:*pFdp7$$\"\\o* ************************************************\\P4YAndpF_[n$\"&UG&Fd p7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm;/w7*3!*ppF_[n$\"&=s'Fdp7$$\"\\oLLLLL LLLLLLLLLLLLLLLLLLLLLek`H@)pF_[n$\"&]n*Fdp7$$\"\\o******************** ****************************\\iS;=eV*pF_[n$\"&i]*Fdp7$$\"\\ommmmmmmmmm mmmmmmmmmmmmmmm\"H#o#oe1qF_[n$\"&i#fFdp7$$\"\\o*********************** ************************\\iST\\6q7qF_[n$\"&&3]Fdp7$$\"\\oLLLLLLLLLLLLL LLLLLLLLLLL$3_+sa\")=qF_[n$\"&i#eFdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmTN 'f%zH\\-(F_[n$\"&.1)Fdp7$$\"\\o*************************************** ***********\\(=,9(F_[n$\"&.'*)Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL$3-)QR/o9(F _[n$\"&D1(Fdp7$$\"\\o***********************************************\\ 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LLLLLLLLLLL$ekGggRduF_[n$\"&yw(Fdp7$$\"\\o**************************** *******************\\PfL>dojuF_[n$\"&=k(Fdp7$$\"\\ommmmmmmmmmmmmmmmmmm mmmmmmT&Qy`(*puF_[n$\"&Ln(Fdp7$$\"\\o********************************* ***************\\P%['paD[(F_[n$\"&b6)Fdp7$$\"\\oKLLLLLLLLLLLLLLLLLLLLL LLLL$e9S8&\\(F_[n$\"&[u)Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmm\"Hd?&3<]( F_[n$\"&d+*Fdp7$$\"\\o************************************************ **\\il-$G3vF_[n$\"&<;*Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLL3_D`d[^(F_[n $\"&P>*Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmmT&Q?V@vF_[n$\"&q5*Fdp7$$\" \\oLLLLLLLLLLLLLLLLLLLLLLLLL$3_]5eMvF_[n$\"&Hp)Fdp7$$\"\\o************ *****************************************\\i+tZvF_[n$\"&$Q#)Fdp7$$\"\\ oLLLLLLLLLLLLLLLLLLLLLLLLLe*[oXIavF_[n$\"&34)Fdp7$$\"\\ommmmmmmmmmmmmm mmmmmmmmmmm;zW2z3c(F_[n$\"&M-)Fdp7$$\"\\o***************************** ********************\\(o/e`uc(F_[n$\"&p.)Fdp7$$\"\\oLLLLLLLLLLLLLLLLLL LLLLLLLLek3GSd(F_[n$\"&l6)Fdp7$$\"\\o********************************* *****************\\P%)4xre(F_[n$\"&ZP)Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmm mmmmm;/6E.g(F_[n$\"&\"*e)Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLLLL$e@#Hl( F_[n$\"&!*R)Fdp7$$\"\\o*********************************************** *****\\i?=bq(F_[n$\"&'[\")Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmT5:LH7t( F_[n$\"&dP)Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLL$3xc/%pv(F_[n$\"&$[()Fd p7$$\"\\o*************************************************\\7.#e^Ey(F_ [n$\"&28)Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm\"H2FO3yF_[n$\"&(ewFdp7$ $\"\\o************************************************\\(=#*p#=7#yF_[n $\"&R2)Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLL3_D$Q2MyF_[n$\"&jr)Fdp7$$\" \\o***********************************************\\(=nQh,0%yF_[n$\"&$ o*)Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm;H#=&RHp%yF_[n$\"&34*Fdp7$$\"\\o LLLLLLLLLLLLLLLLLLLLLLLLeR(\\wrN`yF_[n$\"&[/*Fdp7$$\"\\o************** ************************************\\7y&\\yfyF_[n$\"&W#))Fdp7$$\"\\oL LLLLLLLLLLLLLLLLLLLLLLL$3FW?0ksyF_[n$\"&9-)Fdp7$$\"\\ommmmmmmmmmmmmmmm mmmmmmmmm\"H2$3'\\&)yF_[n$\"&.G(Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL3-) QkQ#>*yF_[n$\"&U:(Fdp7$$\"\\o***************************************** *******\\7.dk^$)*yF_[n$\"&4E(Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm\"H#=q UzZ!zF_[n$\"&Bf(Fdp7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLLL$3s?6zF_[n$\"&n3 )Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm\"zpe()=!)F_[n$\"&!\\!*Fdp7$$\" \\o****************************************************\\7`Wl7)F_[n$\" &M&**Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmm;/,R$Ry\")F_[n$\"&B4(Fdp7$$\" \\oKLLLLLLLLLLLLLLLLLLLLLLLLLe*[ACI#)F_[n$\"&:O'Fdp7$$\"\\olmmmmmmmmmm mmmmmmmmmmmmmT5:jNBnB)F_[n$\"&$3\")Fdp7$$\"\\o)*********************** ************************\\(=njW?V#)F_[n$\"&/W(Fdp7$$\"\\oJLLLLLLLLLLLL LLLLLLLLLLLekG5d&o\\#)F_[n$\"&IU&Fdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmT &Qym;c#)F_[n$\"&H4%Fdp7$$\"\\oJLLLLLLLLLLLLLLLLLLLLLLL$e*)4$*)G\"p#)F_ [n$\"&1,'Fdp7$$\"\\o)************************************************* \\7y5\"4#G)F_[n$\"&l\"zFdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmm;/EDK`]H)F_[ n$\"&@u&Fdp7$$\"\\oJLLLLLLLLLLLLLLLLLLLLLLLLeRs`:!3$)F_[n$\"&.F%Fdp7$$ \"\\o)***********************************************\\7`>vx4K)F_[n$\" &&3gFdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmmm'*RRL)F_[n$\"&Zi(Fdp7$$\"\\ okmmmmmmmmmmmmmmmmmmmmmmmmmTN\"H'pQ)F_[n$\"&M^'Fdp7$$\"\\okmmmmmmmmmmm mmmmmmmmmmmmmm;/'e)*R%)F_[n$\"&:,&Fdp7$$\"\\okmmmmmmmmmmmmmmmmmmmmmmmm m\"H2)3I\\)F_[n$\"&ir&Fdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmmmmTvJga)F_[n $\"&v^'Fdp7$$\"\\oKLLLLLLLLLLLLLLLLLLLLLLLLL$e9tOc()F_[n$\"&bo&Fdp7$$ \"\\ommmmmmmmmmmmmmmmmmmmmmmmT5:j?2Cw)F_[n$\"&+Z&Fdp7$$\"\\o********** **************************************\\(o/)4Z%o()F_[n$\"&[M&Fdp7$$\" \\oLLLLLLLLLLLLLLLLLLLLLLLLeky(*)p[u()F_[n$\"&GO&Fdp7$$\"\\ommmmmmmmmm mmmmmmmmmmmmmmmT5:)o_!y)F_[n$\"&d`&Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL $eR(\\m1Ez)F_[n$\"&0<'Fdp7$$\"\\o************************************* *************\\P%[ko/))F_[n$\"&Je'Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm; /,>Bmn\"))F_[n$\"&;?'Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLLek`,Y)G))F_[n $\"&#H`Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmTN'42f)[$))F_[n$\"&I.&Fdp7$$ \"\\o************************************************\\7G))zD4%))F_[n$ \"&c*\\Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLe*)f0plp%))F_[n$\"&eD&Fdp7$$ \"\\ommmmmmmmmmmmmmmmmmmmmmmmmm\"H#e0I&))F_[n$\"&4v&Fdp7$$\"\\o******* ****************************************\\PM-ua/f))F_[n$\"&\">jFdp7$$ \"\\oLLLLLLLLLLLLLLLLLLLLLLLL$3_vl`3l))F_[n$\"&mv'Fdp7$$\"\\ommmmmmmmm mmmmmmmmmmmmmmm\"zp[d_7r))F_[n$\"&y(oFdp7$$\"\\o********************** ***************************\\(=#\\^;x))F_[n$\"&xg'Fdp7$$\"\\ommmmmmmmm mmmmmmmmmmmmmmm;H#oK\\C*)))F_[n$\"&'4`Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLL LLLL$e9;ZK,*)F_[n$\"&bn%Fdp7$$\"\\o*********************************** *************\\P4'*\\/M\"*)F_[n$\"&v&eFdp7$$\"\\ommmmmmmmmmmmmmmmmmmmm mmmm\"H2$G%[D*)F_[n$\"&*HrFdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL$ek`mScP* )F_[n$\"&,<'Fdp7$$\"\\o*********************************************** ********\\Qk\\*)F_[n$\"&]%Fd p7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm;H#oK15)**)F_[n$\"&&GVFdp7$$\"\\oLLLL LLLLLLLLLLLLLLLLLLLLLL3-.B]+*F_[n$\"&1*fFdp7$$\"\\o******************* *****************************\\PMxUX>,*F_[n$\"&<^(Fdp7$$\"\\ommmmmmmmm mmmmmmmmmmmmmmmmTg_#y')=!*F_[n$\"&^_'Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLL LL$ekyA-zD!*F_[n$\"&3C%Fdp7$$\"\\o************************************ **************\\7.i7F.*F_[n$\"&c*[Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm; aQy,N'R!*F_[n$\"&Qf(Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLLLek`Tdl/*F_[n$ \"&**R&Fdp7$$\"\\o************************************************\\i! *G\")zM0*F_[n$\"&j4&Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmmmm;/@-/1*F_[n$ \"&)f]Fdp7$$\"\\oLLLLLLLLLLLLLLLLLLLLLLLL$3F%zgCt1*F_[n$\"'uw7Fdp7$$\" \\o*************************************************\\(oa+ZU2*F_[n$\"' Xw:Fdp7$$\"\\ommmmmmmmmmmmmmmmmmmmmmmm;z%*HSp63*F_[n$\"'.V8Fdp7$$\"\\o LLLLLLLLLLLLLLLLLLLLLLLLL$3_+=4)3*F_[n$\"'Ux7Fdp7$$\"\\o************** **********************************\\(o/)>9]4*F_[n$\"'sc:Fdp7$$\"\\ommm mmmmmmmmmmmmmmmmmmmmmm\"Hd&fO>5*F_[n$\"'5!\\\"Fdp7$$\"\\oLLLLLLLLLLLLL LLLLLLLLLLL$e*)4$**e)3\"*F_[n$\"''zD\"Fdp7$$\"\\o********************* ******************************\\i!R\"y:\"*F_[n$\"'*RF\"Fdp7$$\"\\ommmm mmmmmmmmmmmmmmmmmmmm;/^\")y.F7*F_[n$\"'*=Z\"Fdp7$$\"\\oLLLLLLLLLLLLLLL LLLLLLLLLL3xc=E'H\"*F_[n$\"'#3b\"Fdp7$$\"\\o************************** **********************\\7.Ke[l8*F_[n$\"'%\\U\"Fdp7$$\"\\ommmmmmmmmmmmm mmmmmmmmmmmm;H2)4ZV\"*F_[n$\"'nm7Fdp7$$\"\\o************************** ***********************\\7yvdJd\"*F_[n$\"'7u8Fdp7$$\"\\oKLLLLLLLLLLLLL LLLLLLLLLLLLL3dg6<*F_[n$\"'7.:Fdp7$$\"\\o)**************************** ***********************\\(oTAq#*F_[n$\"'039Fdp7$$\"\\olmmmmmmmmmmmmmmm mmmmmmmmmmmmw(Gp$*F_[n$\"'\"oK\"Fdp7$$\"\\o)************************** ***********************\\7`**)4A%*F_[n$\"'fy7Fdp7$$\"\\oJLLLLLLLLLLLLL LLLLLLLLLLLLeRA5\\Z*F_[n$\"'kK7Fdp7$$\"\\o)*************************** *******************\\i!Rvi6:[*F_[n$\"'Ek7Fdp7$$\"\\olmmmmmmmmmmmmmmmmm mmmmmm;z>6.8\")[*F_[n$\"')*38Fdp7$$\"\\oJLLLLLLLLLLLLLLLLLLLLLLL3_+ZV9 Z\\*F_[n$\"'bH8Fdp7$$\"\\o)******************************************* *****\\7GQeJ,&*F_[n$\"'R,8Fdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmm\"z>'=C\"Fdp7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmm;/EX@x_*F_[n$\"'yo 7Fdp7$$\"\\oJLLLLLLLLLLLLLLLLLLLLLLLe*[=cGKM&*F_[n$\"'PM8Fdp7$$\"\\o)* **********************************************\\il(fU#4a*F_[n$\"'bC8Fd p7$$\"\\olmmmmmmmmmmmmmmmmmmmmmmmTNYLmDva*F_[n$\"'oM7Fdp7$$\"\\oJLLLLL LLLLLLLLLLLLLLLLLLL3Fp1FTb*F_[n$\"'i]6Fdp7$$\"\\o)******************** **************************\\7y]q%G2c*F_[n$\"'uq6Fdp7$$\"\\olmmmmmmmmmm mmmmmmmmmmmmm;a)3u)Htc*F_[n$\"'*RG\"Fdp7$$\"\\oJLLLLLLLLLLLLLLLLLLLLLL L3FpwFJRd*F_[n$\"'w_8Fdp7$$\"\\o)************************************* **************\\7oK0e*F_[n$\"'Kp7Fdp7$$\"\\o)************************* **********************\\(ozm5&oe*F_[n$\"';L6Fdp7$$\"\\o)************** **********************************\\PM_%pJf*F_[n$\"'hL6Fdp7$$\"\\o)*** ********************************************\\i!*y$y[*f*F_[n$\"'#**G\" Fdp7$$\"\\o)*************************************************\\PMA1eg* F_[n$\"'ha8Fdp7$$\"\\o)*********************************************** \\P%)*3Y77'*F_[n$\"'T%=\"Fdp7$$\"\\o)********************************* ***************\\7`%*HW='*F_[n$\"'v&3\"Fdp7$$\"\\o)******************* ****************************\\7y+QhZi*F_[n$\"'B$H\"Fdp7$$\"\\o)******* *******************************************\\il(z5j*F_[n$\"'C;8Fdp7$$ \"\\o)************************************************\\(=n`;Pk*F_[n$ \"'lE8Fdp7$$\"\\o)*************************************************\\7 yI`jl*F_[n$\"'(*F_[n$\"'Tx7Fdp7$$\"\\o)************* **********************************\\7G)[b&es*F_[n$\"',d7Fdp7$$\"\\o)** ************************************************\\PMR " 0 "" {MPLTEXT 1 0 185 "Digits := 5 5:\nst := time():\nfor i to 1000 do arcsin55(rand()*Float(1,-12)) end \+ do:\ntime()-st;\nst := time():\nfor i to 1000 do arcsin(rand()*Float(1 ,-12)) end do:\ntime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%Sb!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%y`!\" $" }}}{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 a rcsine function: " }{TEXT 0 7 "arcsin_" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "arcsin_: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 20 "Calling Sequence: " }{TEXT -1 13 "ar csin_( x ) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 265 21 "x - a real constant" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 268 12 "Des cription:" }{TEXT -1 1 " " }{TEXT 267 14 "The procedure " }{TEXT 0 7 " arcsin_" }{TEXT 266 42 " calculates arcsin(x) for a real number x." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 16 "How 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 23 " arcsin_: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12195 "arcsin_ := proc(x::algebraic)\n \+ local u,t;\n\n if nargs <> 1 then\n error \"expecting 1 argum ent, got %1\", nargs;\n end if;\n if type(x,'float') then evalf('a rcsin_'(x))\n elif (type(x,'numeric') or type(x,'radnum')) \n \+ and signum(0,x,0)<0 then -arcsin_(-x)\n elif type (x,`*`) and type(op(1,x),'numeric') \n and signum(0 ,op(1,x),0)<0 then -arcsin_(-x)\n elif type(x,'function') and nops(x )=1 then\n u := op(1,x);\n if op(0,x)='sin' or op(0,x)='sin_ ' then\n if type(u,'realcons') then \n t := u - 2*P i*floor(u/(2*Pi)+1/4);\n if floor(2*t/Pi)<1 then t else Pi- t end if;\n end if;\n elif op(0,x)='cos' or op(0,x)='cos_ ' then\n if type(u,'realcons') then \n Pi/2-abs(u-2 *Pi*floor(u/(2*Pi)+1/2));\n end if;\n else 'arcsin_'(x)\n end if;\n elif type(x,'realcons') then\n if x=0 then 0\n \+ elif x=1 then Pi/2\n elif x=sqrt(3)/2 then Pi/3\n elif \+ x=sqrt(2)/2 then Pi/4\n elif x=1/2 then Pi/6\n elif x=sqrt(2 -sqrt(2))/2 then Pi/8\n elif x=sqrt(2+sqrt(2))/2 then 3*Pi/8\n \+ elif x=sqrt(2)*(sqrt(3)-1)/4 then Pi/12\n elif x=sqrt(2)*(sqrt (3)+1)/4 then 5*Pi/12\n else 'arcsin_'(x)\n end if\n else \+ 'arcsin_'(x)\n end if;\nend proc: # of arcsin_\n\n`evalf/arcsin_` := proc(xx)\n local x,z,s,t,saveDigits,eps,maxit,val,\n isneg, flag,k,sum,term,pow,fact,pi,hfDigits;\n\n if xx=0 then return xx end if;\n if not type(xx,realcons) then return 'arcsin_'(xx) end if;\n \n if abs(evalf(xx))>1 then\n error \"argument must be in the i nterval [-1,1]\"\n end if;\n\n # Use the fixed precision procedure for low precision values\n hfDigits := trunc(evalhf(Digits));\n i f Digits<=hfDigits then\n x := evalf(xx,hfDigits+2);\n retur n evalf(evalhf(arcsin16(x)));\n end if;\n\n\011\011 # Increase preci sion for the computation by a few digits\n\011\011 saveDigits := Digit s;\n Digits := Digits + length(Digits)+2;\n x := evalf(xx);\n\n \+ if abs(x)>1 then\n error \"argument must be in the interval [-1,1 ]\"\n end if;\n\n if saveDigits<28 then\n val := arcsin25(x); \n return evalf[saveDigits](val);\n elif saveDigits<58 then\n \+ val := arcsin55(x);\n return evalf[saveDigits](val);\n else \n if x<0 then\n isneg := true;\n x := -x;\n \+ else isneg := false end if;\n\n if x<.25881904510252076235 t hen\n flag := 1;\n else\n s := sqrt(1-x*x); # use Maple's square root\n if x>.96592582628906828675 then\n \+ flag := 4;\n x := s;\n else\n t := evalf(root3)*0.5; \n if x<.70710678118654752440 then\n \+ flag := 2;\n x := x*t-s*0.5;\n \+ else\n flag := 3;\n x := s*t-x*0.5;\n \+ end if;\n end if;\n end if;\n \n # Initiali sation for Maclaurin series loop.\n eps := Float(1,-saveDigits); \n maxit := Digits*3;\n\n z := x*x;\n term := x;\n \+ sum := term;\n fact := 1;\n pow := x; \n for k from 2 \+ to maxit by 2 do\n pow := pow*z;\n fact := fact*(k-1)/ k;\n term := fact*pow/(k+1);\n sum := sum+term;\n \+ if abs(term)<=eps*abs(sum) then break end if;\n end do;\n \n if flag=2 then sum := evalf(Pi)/6+sum\n elif flag=3 then su m := evalf(Pi)/3-sum\n elif flag=4 then sum := evalf(Pi)/2-sum en d if;\n if isneg then sum := -sum end if;\n\n # Return arcsi ne rounded to the original precision\n Digits := saveDigits;\n \+ return evalf(sum)\n end if;\nend proc: # of `evalf/arcsin_`\n\n`e valf/constant/root3` := proc()\nlocal d,r;\nglobal _root3;\n if Digi ts<=55 then evalf(1.73205080756887729352744634150587236694280525381038 0628)\n elif Digits<=length(op(1,_root3)) then evalf(_root3)\n els e\n d := length(op(1,_root3));\n r := _root3;\n while d 1 then\n error \"ar gument must be in the interval [-1,1]\"\n end if;\n\n # Handle the argument reduction\n if x<0 then\n isneg := true;\n x := \+ -x;\n else isneg := false end if;\n\n if x<.2588190451025208 then \n flag := 1;\n else\n s := sqrt(1-x*x); # use Maple's squ are root\n if x>.9659258262890683 then\n flag := 4;\n \+ x := s;\n else \n if x<.70710678118654752440 then \n flag := 2;\n x := x*sinPiBy3-s*0.5;\n \+ else\n flag := 3;\n x := s*sinPiBy3-x*0.5;\n \+ end if;\n end if;\n end if;\n \n # evaluate the ratio nal approximation\n z := x*x;\n num := (a1+(a2+(a3+a4*z)*z)*z)*z; \n den := 1+(b1+(b2+b3*z)*z)*z;\n val := (1+num/den)*x;\n\n if f lag=2 then val := val+.52359877559829887308;\n elif flag=3 then val \+ := 1.0471975511965977462-val\n elif flag=4 then val := 1.57079632679 48966192-val end if;\n if isneg then -val else val end if;\nend proc : # of arcsin16" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 7 "arcsin_" }{TEXT -1 10 ": exampl es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 21 "Testing the procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := evalf(sin(Pi/7) ,60);\nevalf(arcsin_(xx),55);\nevalf(evalf(arcsin(xx),60),55);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gnYXWw)fuyF2**4Yve$[GLodZ?\"e v6RP)QV!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Xfa'e`q0n\"G?T+&o/MjY \\0w#G^]*)z[%!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Xfa'e`q0n\"G?T+ &o/MjY\\0w#G^]*)z[%!#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Digits := 70:\nxx := 0.56789;\narcs in_(xx);\narcsin(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#xxG$\"&*yc!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"aoe`d8#4NN*y [#=SQ&ztj.oJl*4*R2XBY$)>e6SRg!#q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"aoe`d8#4NN*y[#=SQ&ztj.oJl*4*R2XBY$)>e6SRg!#q" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(arcsin_ (x),x=-1..1,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 287 258 258 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$!+Fjzq:!\"*7$$!+s%HaF1$!+ArURdF17$$!+]$*4)*\\F1$!+8KzL_F17$$!+ ]_&\\c%F1$!+RP_SZF17$$!+]1aZTF1$!+$3DnF%F17$$!+/#)[oPF1$!+,%>R'QF17$$! +$=exJ$F1$!+nR&=Q$F17$$!+L2$f$HF1$!+xR$)zHF17$$!+PYx\"\\#F1$!+j$3$=DF1 7$$!+L7i)4#F1$!+1%RV6#F17$$!+P'psm\"F1$!+>B4v;F17$$!+74_c7F1$!+Y5&)f7F 17$$!*3x%z#)F1$!+Yc'*)G)!#67$$!*?PQM%F1$!+?W?XVFgr7$$!(\"zr)*F1$!+.Ezr )*!#87$$\")!o2J%F-$\"+9U57VFgr7$$\")%Q#\\\")F-$\"+B`Ge\")Fgr7$$\"*;*[H 7F-$\"+0!3EB\"F17$$\"*qvxl\"F-$\"+#Rkam\"F17$$\"*`qn2#F-$\"+mj*>4#F17$ $\"*cp@[#F-$\"+=.wHF17$$\"*xanL$F-$\"+4$)*>S $F17$$\"*v+'oPF-$\"+04/kQF17$$\"*S<*fTF-$\"+C6L!H%F17$$\"*&)Hxe%F-$\"+ Ez8mZF17$$\"*.o-*\\F-$\"+')QvC_F17$$\"*TO5T&F-$\"+5#*[ " 0 "" {MPLTEXT 1 0 112 "Digits := 50:\nxx := sqrt(3 )/2;\narcsin_(xx);\nevalf(evalf(%,Digits+3));\narcsin_(evalf(xx,Digits +3));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\" #!\"\"\"\"$#\"\"\"F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\" \"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S]7LJsl!Gw;$4hW@ ahu(f'>^v>Z5!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S]7LJsl!Gw;$4hW @ahu(f'>^v>Z5!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 120 "Digits := 50:\nxx := sqrt(sqrt(2)+2)/2;\nar csin_(xx);\nevalf(evalf(%,Digits+3));\narcsin_(evalf(xx,Digits+3));\nD igits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"\"#!\"\",& F'\"\"\"*$F'#F*F'F*F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\" \"\"\"\")!\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SdwC&QRd \"e8)H(o7\\BWYsh4Xs4y6!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SdwC& QRd\"e8)H(o7\\BWYsh4Xs4y6!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 18 "Speed comparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "st := time():\nfor i to 500 0 do arcsin_(rand()*Float(1,-12)) end do:\ntime()-st;\nst := time():\n for i to 5000 do arcsin(rand()*Float(1,-12)) end do:\ntime()-st;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%tH!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%*R#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Digits := 25:\nst := time():\nfor \+ i to 5000 do arcsin_(rand()*Float(1,-12)) end do:\ntime()-st;\nst := t ime():\nfor i to 5000 do arcsin(rand()*Float(1,-12)) end do:\ntime()-s t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&#*G\"!\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"&M<#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "Digits := 55:\nst := time() :\nfor i to 5000 do arcsin_(rand()*Float(1,-12)) end do:\ntime()-st;\n st := time():\nfor i to 5000 do arcsin(rand()*Float(1,-12)) end do:\nt ime()-st;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&d0$!\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 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