{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 "Help Normal" -1 30 "Times" 1 12 0 0 0 0 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 "Grey Emphasis" -1 260 "Times " 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 " Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis " -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{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 1 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 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" 261 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 272 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text \+ Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 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 "Time s" 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 "Error" -1 8 1 {CSTYLE "" -1 -1 " Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Procedures for evaluating the exp onential function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana imo, 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 260 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 260 8 "numfcn.m" }{TEXT -1 72 " contains the code \+ for the alternative mathematical functions including " }{TEXT 0 4 "exp _" }{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 t he 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 260 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 260 8 "gkdata.m" }{TEXT -1 67 " contain the code and data for the nume rical integration procedure " }{TEXT 0 5 "GKint" }{TEXT -1 25 " used i n this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read i nto a Maple session by commands similar to those that follow, where th e 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 55 "Evaluating the exponential function using the func tion " }{XPPEDIT 18 0 "f(x) = x*(exp(x)+1)/(exp(x)-1);" "6#/-%\"fG6#% \"xG*(F'\"\"\",&-%$expG6#F'F)F)F)F),&-F,6#F'F)F)!\"\"F1" }{XPPEDIT 18 0 "``=x*coth(x/2)" "6#/%!G*&%\"xG\"\"\"-%%cothG6#*&F&F'\"\"#!\"\"F'" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 66 "We can evaluate the exponential function by mea ns of the function " }{XPPEDIT 18 0 "f(x) = x*(exp(x)+1)/(exp(x)-1)" " 6#/-%\"fG6#%\"xG*(F'\"\"\",&-%$expG6#F'F)F)F)F),&-F,6#F'F)F)!\"\"F1" } {XPPEDIT 18 0 "`` = x*coth(x/2)" "6#/%!G*&%\"xG\"\"\"-%%cothG6#*&F&F' \"\"#!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "The firs t few terms of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 5 " are " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "2+x^2/6-x^4/360+x^6/15120-x^8/604800+x^10/23950080-` . \+ . . `;" "6#,0\"\"#\"\"\"*&%\"xGF$\"\"'!\"\"F%*&F'\"\"%\"$g$F)F)*&F'F( \"&?^\"F)F%*&F'\"\")\"'+[gF)F)*&F'\"#5\")!3]R#F)F%%(~.~.~.~GF)" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "series(x*coth(x/2),x,12);\nconvert(%,polynom); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"#\"\"!#\"\"\"\"\"'F%#! \"\"\"$g$\"\"%#F(\"&?^\"F)#F+\"'+[g\"\")#F(\")!3]R#\"#5-%\"OG6#F(\"#7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"\"#\"\"\"*&#F%\"\"'F%)%\"xGF$F %F%*&#F%\"$g$F%*$)F*\"\"%F%F%!\"\"*&#F%\"&?^\"F%)F*F(F%F%*&#F%\"'+[gF% *$)F*\"\")F%F%F1*&#F%\")!3]R#F%)F*\"#5F%F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x*(exp(x)+1)/(exp(x)-1);" " 6#/-%\"fG6#%\"xG*(F'\"\"\",&-%$expG6#F'F)F)F)F),&-F,6#F'F)F)!\"\"F1" } {TEXT -1 7 ". Then " }{XPPEDIT 18 0 "v*exp(x)-v=x*exp(x)+x" "6#/,&*&% \"vG\"\"\"-%$expG6#%\"xGF'F'F&!\"\",&*&F+F'-F)6#F+F'F'F+F'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "v*exp(x)-x*exp(x)=v-x" "6#/,&*&%\"vG\" \"\"-%$expG6#%\"xGF'F'*&F+F'-F)6#F+F'!\"\",&F&F'F+F/" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "exp(x)=(v+x)/(v-x)" "6#/-%$expG6#%\"xG*&,&%\"vG\" \"\"F'F+F+,&F*F+F'!\"\"F-" }{XPPEDIT 18 0 "``= 1+2*x/(v-x)" "6#/%!G,& \"\"\"F&*(\"\"#F&%\"xGF&,&%\"vGF&F)!\"\"F,F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "exp(x)=1+2*x/(f(x)- x)" "6#/-%$expG6#%\"xG,&\"\"\"F)*(\"\"#F)F'F),&-%\"fG6#F'F)F'!\"\"F0F) " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "We can obtain better accuracy by using the function " }{XPPEDIT 18 0 "g(x)=x+2-x*coth(x/2) " "6#/-%\"gG6#%\"xG,(F'\"\"\"\"\"#F)*&F'F)-%%cothG6#*&F'F)F*!\"\"F)F0 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "The first few terms \+ of the Maclaurin series for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " are " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-x^2/6+x^4/360-x^6/15120+x^8/604800-x^10/23950080+` . . . `" "6#,0% \"xG\"\"\"*&F$\"\"#\"\"'!\"\"F)*&F$\"\"%\"$g$F)F%*&F$F(\"&?^\"F)F)*&F$ \"\")\"'+[gF)F%*&F$\"#5\")!3]R#F)F)%(~.~.~.~GF%" }{TEXT -1 1 "." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note t hat " }{XPPEDIT 18 0 "f(x)-x=2-g(x)" "6#/,&-%\"fG6#%\"xG\"\"\"F(!\"\", &\"\"#F)-%\"gG6#F(F*" }{TEXT -1 4 " so " }{XPPEDIT 18 0 "exp(x)=1+2*x/ (2-g(x)" "6#/-%$expG6#%\"xG,&\"\"\"F)*(\"\"#F)F'F),&F+F)-%\"gG6#F'!\" \"F0F)" }{XPPEDIT 18 0 "``=1+x+x*g(x)/(2-g(x))" "6#/%!G,(\"\"\"F&%\"xG F&*(F'F&-%\"gG6#F'F&,&\"\"#F&-F*6#F'!\"\"F0F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "g := x -> x+2-x*coth(x/2); \nE := x -> 1+x+x*g(x)/(2-g(x));\nseries(E(x),x,10):\nconvert(%,polyno m);\nsimplify(convert(E(x),exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$\"\"\"\"\"#F.*&F-F.-%%coth G6#,$F-#F.F/F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(,(\"\"\"F-9$F-*(F.F--%\"gG6#F.F-,&\"\" #F-F0!\"\"F5F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6\"\"\"F$%\"x GF$*&#F$\"\"#F$)F%F(F$F$*&#F$\"\"'F$)F%\"\"$F$F$*&#F$\"#CF$)F%\"\"%F$F $*&#F$\"$?\"F$)F%\"\"&F$F$*&#F$\"$?(F$)F%F,F$F$*&#F$\"%S]F$)F%\"\"(F$F $*&#F$\"&?.%F$)F%\"\")F$F$*&#F$\"'!)GOF$)F%\"\"*F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 64 "A fixed precision procedure to evaluate the exponential function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 97 "In this section we construct a fixed precision ver sion of the exponential function which can use " }{TEXT 259 34 "hardwa re floating point arithmetic" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 35 "Argument reduction to the interval " }{XPPEDIT 18 0 "[- ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F'6#F)F*F) F+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 44 "We can subtract a suitable integer multip le " }{XPPEDIT 18 0 "k*ln(2);" "6#*&%\"kG\"\"\"-%#lnG6#\"\"#F%" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT -1 6 " from " }{TEXT 274 1 "x" }{TEXT -1 37 ", or add a suitable integer \+ multiple " }{XPPEDIT 18 0 "k*ln(2);" "6#*&%\"kG\"\"\"-%#lnG6#\"\"#F%" }{TEXT -1 4 " to " }{TEXT 275 1 "x" }{TEXT -1 57 ", to ensure that the resulting number is in the interval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2 )/2];" "6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "ln(2)/2 < abs(x);" "6#2*&-%#lnG6#\"\"#\"\"\"F(!\"\"-%$absG6#%\"xG" }{TEXT -1 7 ", let " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k = floor (x/ln(2)+1/2);" "6#/%\"kG-%&floorG6#,&*&%\"xG\"\"\"-%#lnG6#\"\"#!\"\"F +*&F+F+F/F0F+" }{TEXT -1 14 " ------- (i), " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 271 11 "___________" }{TEXT -1 17 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "floor(x)" "6#-%&floorG6#%\"xG" }{TEXT 30 23 " is the largest integer" }{XPPEDIT 18 0 "``<= x" "6#1%!G%\"xG" }{TEXT 30 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Set " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z = x-k*`.`*ln(2);" "6#/%\"zG,&%\"xG\"\"\"*( %\"kGF'%\".GF'-%#lnG6#\"\"#F'!\"\"" }{TEXT -1 14 " ------- (ii)." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 8 "________" }{TEXT -1 17 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "Then z is in the in terval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"# \"\"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 4 " and" }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x) = exp(z)*`.`*2^k;" "6#/-%$expG 6#%\"xG*(-F%6#%\"zG\"\"\"%\".GF,)\"\"#%\"kGF," }{TEXT -1 16 " ------- \+ (iii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 272 6 "______" } {TEXT -1 20 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }{XPPEDIT 18 0 "k = \+ (x-z)/ln(2);" "6#/%\"kG*&,&%\"xG\"\"\"%\"zG!\"\"F(-%#lnG6#\"\"#F*" } {TEXT -1 10 ", so that " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "exp(z)*`.`*10^k = exp(z)*`.`*2^((x-z)/ln(2));" "6#/*(-% $expG6#%\"zG\"\"\"%\".GF))\"#5%\"kGF)*(-F&6#F(F)F*F))\"\"#*&,&%\"xGF)F (!\"\"F)-%#lnG6#F2F6F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z)*`.`*10^((x-z)*log[2](exp(1)));" "6#/ %!G*(-%$expG6#%\"zG\"\"\"%\".GF*)\"#5*&,&%\"xGF*F)!\"\"F*-&%$logG6#\" \"#6#-F'6#F*F*F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = exp(z)*`.`*2^log[2](exp(x-z));" "6#/%!G*(-%$expG6# %\"zG\"\"\"%\".GF*)\"\"#-&%$logG6#F-6#-F'6#,&%\"xGF*F)!\"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(z )*`.`*exp(x-z);" "6#/%!G*(-%$expG6#%\"zG\"\"\"%\".GF*-F'6#,&%\"xGF*F)! \"\"F*" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= exp(x)" "6#/%!G-%$expG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a function " }{TEXT 0 4 "exp1" }{TEXT -1 52 " using Maple's exponential function on the inte rval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\" \"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 58 ", but ensure that it gives no value outside this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "exp1 := x -> if abs(x)<=.34 65735903 then exp(x) else FAIL end if;\nplot('exp1(x)',x=-0.5..0.5,thi ckness=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp1Gf*6#%\"xG6\"6$% )operatorG%&arrowGF(@%1-%$absG6#9$$\"+.ftlM!#5-%$expGF0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 334 248 248 {PLOTDATA 2 "6&-%'CURVESG6$7P 7$$!3;+++CfFhM!#=$\"3E]3\\'>AU2(F*7$$!3/+++GjkaMF*$\"3Ciq'\\n8*yqF*7$$ !3#******>t;![MF*$\"37'=NZE3O3(F*7$$!3E+++NrQTMF*$\"3O%Rn&ofI)3(F*7$$! 3)*******o.N)Q$F*$\"3$4_v4#***f7(F*7$$!3;+++-OJNLF*$\"3i\">=RFj!HF*$\"3Eq#Gkh-zZ(F*7$$!3*******f4OZr#F*$\"3ro\\F%*RbAwF *7$$!3!******\\n\\!*\\#F*$\"3!4xO(zzu)y(F*7$$!33+++DwZ#G#F*$\"3oA0KX.F fzF*7$$!3-+++D.xt?F*$\"3Y?#f$H<8F\")F*7$$!37+++.TC%)=F*$\"3aNQK#4JEG)F *7$$!3!******>4z)e;F*$\"3Yu\"4g&=Tr%)F*7$$!33+++n`'zY\"F*$\"3sD2GRkpM' )F*7$$!3++++>t)eC\"F*$\"3,Eh_+#*fG))F*7$$!3%******ph5$\\5F*$\"3!fTxc\" f'Q+*F*7$$!3W*******=[jL)!#>$\"3A\"zq-xm,?*F*7$$!3'*******pXg#G'F_q$\" 3aK#4:@o5R*F*7$$!3;+++]&Q(RTF_q$\"3=Q'[a$yZ%f*F*7$$!3%*******4'=><#F_q $\"31lj!3x\\^y*F*7$$!3I++++q*e$\\!#@$\"3EiN&4Kl]***F*7$$\"33+++qRQb@F_ q$\"3zv^g,yy@5!#<7$$\"3!)******z\">Y2%F_q$\"3N%R#\\3xeT5F\\s7$$\"3++++ !yXu9'F_q$\"3Yh7_MLSj5F\\s7$$\"3%)*******[y))G)F_q$\"3M0 " 0 "" {MPLTEXT 1 0 272 "exp2 := proc(x)\n local xx,ln2,k;\n\n \+ ln2 := evalf(ln(2));\n xx := evalf(x);\n\n # Reduce the argument range to -ln(10)/2<=xx<=ln(10)/2\n k := 0;\n if abs(xx)>.34657359 03 then \n k := floor(xx/ln2+0.5);\n xx := xx-k*ln2;\n en d if;\n\n exp1(xx)*2^k;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%%exp2Gf*6#%\"xG6%%#xxG%$ln2G%\"kG6\"F,C'>8%-%&evalfG6#-%#lnG6#\"\" #>8$-F16#9$>8&\"\"!@$2$\"+.ftlM!#5-%$absG6#F8C$>F=-%&floorG6#,&*&F8\" \"\"F/!\"\"FN$\"\"&FOFN>F8,&F8FN*&F=FNF/FNFO*&-%%exp1GFFFN)F6F=FNF,F,F ," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exp2(-1.5);\nexp(-1.5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,;IJA!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,;IJA!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot('exp2(x)',x=-3..3,color =red,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%'CURVESG6#7V7$$!\"$\"\"!$\"3W%R'yOoqy\\!#>7$$!3!***** *\\2<#pG!#<$\"3%4+*)o\\LVn&F-7$$!3#)***\\7bBav#F1$\"3aWlPp$3#ejF-7$$!3 6++]K3XFEF1$\"3i1#R.nViA(F-7$$!3%)****\\F)H')\\#F1$\"3-JY*=ja(>#)F-7$$ !3#****\\i3@/P#F1$\"3[1Bh6r8W$*F-7$$!3;++Dr^b^AF1$\"3__04\"QaB0\"!#=7$ $!3$****\\7Sw%G@F1$\"3Mt^1A\\=!>\"FM7$$!3*****\\7;)=,?F1$\"3?tt(eyXFM7$$!3!******\\!)H%*\\\"F1$ \"3]/HHTUdKAFM7$$!3/+++vl[p8F1$\"38J?;zXPUDFM7$$!3\"******\\>iUC\"F1$ \"3IL)HwIM:)GFM7$$!3-++DhkaI6F1$\"3t8X)[zn&GKFM7$$!3s******\\XF`**FM$ \"3K!*[tlR-'p$FM7$$!3u*******>#z2))FM$\"3ENf7&HVk\"F17$$\"3A)****\\d6.B'FM$\"3yrX*RHrX'=F17$$\"3s*** *\\(o3lW(FM$\"32%Q9)Qhq0@F17$$\"35*****\\A))oz)FM$\"3t]rQ0'\\,T#F17$$ \"3e******Hk-,5F1$\"3p03!pQt5s#F17$$\"36+++D-eI6F1$\"39!y6')>`u4$F17$$ \"3u***\\(=_(zC\"F1$\"3SynSEHG$[$F17$$\"3M+++b*=jP\"F1$\"3m=?'Gt'HgRF1 7$$\"3g***\\(3/3(\\\"F1$\"3[;db_MioWF17$$\"33++vB4JB;F1$\"3!p=I/W[)p]F 17$$\"3u*****\\KCnu\"F1$\"3nM(zxI$yNdF17$$\"3s***\\(=n#f(=F1$\"3E$3/m) ['o_'F17$$\"3P+++!)RO+?F1$\"3gI`3dgu\"R(F17$$\"30++]_!>w7#F1$\"3y#f*)z Gb[R)F17$$\"3O++v)Q?QD#F1$\"3\")GhP-?0C&*F17$$\"3G+++5jypBF1$\"3o?wDg1 ^p5!#;7$$\"3<++]Ujp-DF1$\"3#Q=shkQ:A\"Fby7$$\"3++++gEd@EF1$\"3oRVI?Utv 8Fby7$$\"31+]PMh%\\o#F1$\"3m-\\$G=TdY\"Fby7$$\"39++v3'>$[FF1$\"33&GC:# ojh:Fby7$$\"39+++5h(*3GF1$\"3%*)3;$>?Hf;Fby7$$\"37++D6EjpGF1$\"3'4_*yu R0j " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "A special rational approximation for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 142 " In order to obtain a fixed precision procedure to evaluate the exponen tial function, we construct a polynomial approximation for the functio n " }{XPPEDIT 18 0 "f(x)=x*coth(x/2)" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%%co thG6#*&F'F)\"\"#!\"\"F)" }{TEXT -1 80 " on the interval [-0.34665, 0.3 4665], which is slightly wider than the interval " }{XPPEDIT 18 0 "[-l n(2)/2,ln(2)/2]" "6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F'6#F)F*F)F+ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 62 "We do this by constr ucting a minimax polynomial approximation " }{XPPEDIT 18 0 "q(x)" "6#- %\"qG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "h(x)=( x*coth(x/2)-2)/x^2" "6#/-%\"hG6#%\"xG*&,&*&F'\"\"\"-%%cothG6#*&F'F+\" \"#!\"\"F+F+F0F1F+*$F'F0F1" }{TEXT -1 9 ". Since " }{XPPEDIT 18 0 "f( x) = 2+h(x)*`.`*x^2;" "6#/-%\"fG6#%\"xG,&\"\"#\"\"\"*(-%\"hG6#F'F*%\". GF*F'F)F*" }{TEXT -1 17 ", the polynomial " }{XPPEDIT 18 0 "p(x)=2+q(x )*`.`*x^2" "6#/-%\"pG6#%\"xG,&\"\"#\"\"\"*(-%\"qG6#F'F*%\".GF*F'F)F*" }{TEXT -1 40 " will be a polynomial approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "In order to obtain desirable error characterstics for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 39 ", we use a weight function of the form " }{XPPEDIT 18 0 "w(x)=x^2+delta" "6#/-%\"wG6#%\"xG,&*$F' \"\"#\"\"\"%&deltaGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "delta" " 6#%&deltaG" }{TEXT -1 10 " is small." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The polynomial approximation " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 36 " is constructed u sing the procedure " }{TEXT 0 5 "remez" }{TEXT -1 51 ", which can be l oaded from the relevant worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "evalf(remez ((x*coth(x/2)-2)/x^2,x=-0.34665..0.34665,8,\n 'maxerr',errtype=abs olute,weight=x^2+1e-30,type=even,info=true),30):\nq := unapply(%,x);\n maxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calcula ting~minimax~error~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M--~minimising~the~weighted~ABSOLUTE~error~ --G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--- -----------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%Fprovisional~polynomial~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,$\"G&R&oh0X\"4gQh,mmmmmmm\"!#Q\"\"\"*&$\"G&z%Hg<(pA<+ EF%3qxxxxF!#SF')%\"xG\"\"#F'!\"\"*&$\"GQA1,7*o#3fJ7hHQ>jv8m!#UF')F-\" \"%F'F'*&$\"G,P:!>'p\"er37HYxc+!R`;!#VF')F-\"\"'F'F/*&$\"G?GN40h\"RRZO *))H%eQh!QT!#XF')F-\"\")F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%'*\\uxOlDUn)[^^y\"!#b" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%s\"o7!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D\"G9HxK bs\\sa1)G?lsk(!#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relati ve~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D_U^\\Divr/R`r5E lcI\"!#\\" }}{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 "61-%'CURVESG6% 7fs7$$\"\"!F)%%FAILG7$$\"D+++++++]P%[@0'31OO#!#Q$\"DGey`3#!#a7$$\"D+++++++++](oOv%RNc(FE$\"Dm*ejYm%etuA)G')*=hjp$Fg n7$$\"D+++++++]7GB6.r.a3\"!#O$\"D^\"zBC*Hzh&H5@]V&zvb(Fgn7$$\"D+++++++ +D\"yd3tMX99F`o$\"D>d7e%**\\iz+RU*=kP4F\"!#`7$$\"D++++++++]7[Y!=_ba@F` o$\"Dzw#H@%e())z<&=9R+bp&GFho7$$\"D++++++++](o.zPGc**GF`o$\"DkJUWy\\?+ L8fm#\\HsU\\Fho7$$\"D++++++++Dc\"[Om&H5k$F`o$\"D<\"o_8YnSS')Rd?U&f!RtF ho7$$\"D++++++++D\"G75QTYGVF`o$\"D.pXT3[T3b!3S+.v1v'*Fho7$$\"D++++++++ DJ!*p$>GES]F`o$\"DzCB\\0vP6Giyb!zHN-7!#_7$$\"D++++++++DJ]wP.1kx&F`o$\" DZ.)enaz:#e8)y'>(4Z<9Fbq7$$\"D++++++++D1WevU)=5lF`o$\"DG*[wQaX^0\\*4o& e'*)Qe\"Fbq7$$\"D+++++++]iSO@-7\"e()oF`o$\"Dzxyga0qX3HIQC\"3\"\\k\"Fbq 7$$\"D+++++++++vG%)G\"Q(\\E(F`o$\"DM\\&3:Wo)y'RsD5!*)oko\"Fbq7$$\"D+++ ++++]i:gUulx6V(F`o$\"D^8'QoNETwP#\\r,xO\")p\"Fbq7$$\"D++++++++Dc\"4+-: Q(f(F`o$\"D#=6C.))zs7`.J&fwfbq\"Fbq7$$\"D+++++++](oH#flM&ejxF`o$\"D#eU *yHzF`o$\"D$)on9-xVM)=R16v#Gsq \"Fbq7$$\"D+++++++]Pfo;[N&*o6)F`o$\"DoB\"pkEI0NO\\+O$z`#e\"[44.%[\"Fbq7$$\"D++ ++++++]()ebV*=R:5!#N$\"Diu')eITno\\xdL)QS%>C\"Fbq7$$\"D+++++++DJq?vlPu #[5F\\v$\"D+#\\g%pivqVf*eZI@t06Fbq7$$\"D+++++++]7`#[zeo:\"3\"F\\v$\"D5 ;r`03M$)p\\[l_')3ya*Fho7$$\"D+++++++DJXs:1Jd-7\"F\\v$\"D.I)o6FbLI3bjDa (4Be(Fho7$$\"D++++++++]PiOCwd$f6F\\v$\"D)QIY$pkcM?Jc:N,F,Y&Fho7$$\"D++ ++++++vVjVR(4[#>\"F\\v$\"D9&[kOQ1Z7&)*yikqW^c$Fho7$$\"D+++++++++]k]a=/ cA\"F\\v$\"DPrWU$oR,o,b;JY8J0;Fho7$$\"D+++++++D1kX$yaV8k7F\\v$!DY=s^K] )pT?1@`c>_JsFgn7$$\"D+++++++]7yE;T_kEI\"F\\v$!Dw,s1'\\=VPvWR7*)>ekIFho 7$$\"D+++++++Dc,vhEDqnL\"F\\v$!D$4)G;l8%)RjmM(pRTP8^Fho7$$\"D+++++++++ DB27)f(3P\"F\\v$!DU(=U1jXyb-#p8b]my5(Fho7$$\"D+++++++DcE\")))3P&H39F\\ v$!DfWlzqf]cbC_,p*z(p>*Fho7$$\"D+++++++]7GRq0w9dW\"F\\v$!D)Q9zsaegLhXp Gb4J96Fbq7$$\"D++++++++Dc3i%ooM\"[\"F\\v$!D(Gv\\1i*=up#p#3f@WFG\"Fbq7$ $\"D+++++++]P%yPNw*yp^\"F\\v$!DNMfrY+6CCK^JfKA9V\"Fbq7$$\"D++++++++D\" ylT4dL\"f\"F\\v$!D%Hn8ekjcK=Z`$GNJ_m\"Fbq7$$\"D+++++++v$fjBsbqZD;F\\v$ !D'G@S3W2AAk\"Rl(pW*Ht\"Fbq7$$\"D+++++++]i!\\\"G?S='f;F\\v$!DbV9HCW4-9 j*=M>kMtJ$)eH9.y;F\\v$!DF7GLr33u(fdK^.-E$y\"Fbq7$$\"D++++++vo /)**HxUzB(o\"F\\v$!DNy?#QI7;!3sep$)>N]y\"Fbq7$$\"D+++++++v=nhajXWkp\"F \\v$!DYg&[d1ROH(GGJN(ym%y\"Fbq7$$\"D++++++D\"GjLi$*p4l044l#*>S:&Q_;yfqy[\"Fb q7$$\"D+++++++](=A)33lVL7>F\\v$!DUq'eMqEEA=\\mw!*>,o6Fbq7$$\"D++++++++DJ\\w&fvH[>F\\v$! 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+++++](=-()=;3G$F1$\":Y_#)[2uteg9hA\"F>7$$\"?++++++D\"y+bLQ$)H$F1$\":Q 1Vz[!H%\\qP\"G9F>7$$\"?+++++++v$*H#[geJ$F1$\":q7M(omnI&=:[e\"F>7$$\"?+ ++++D1kc:_aFDLF1$\":w\"=5zLPqwR?X;F>7$$\"?+++++]7`>,A/pMLF1$\":lyl-B:! H#Gido\"F>7$$\"?++++]il(4Sp!zRRLF1$\":d1HXXx;nhqxp\"F>7$$\"?+++++v=U#o =R0TM$F1$\":K&)\\Wf;4en(z.7$$\"?++++](=nQ'zwG\")[LF1$\":-2;=Kjd2[yM q\"F>7$$\"?++++++DJXsh._`LF1$\":T.A*oF%z5?Okp\"F>7$$\"?+++++]P4rV,.NsL F1$\":7$$\"?++++++](o\\6C!=\"R$F1$\":w/,Zi:b#ois[8F>7$ $\"?+++++Dcwf+6_f+MF1$\":(pvf1&[RnbLV;\"F>7$$\"?+++++]ilA'3=5+T$F1$\"9 )3t5L%*oHUS,L*F>7$$\"?+++++voa&=2:D%>MF1$\"9ubCK6(QH6-Q]'F>7$$\"?+++++ +vV[d?,%)GMF1$\"9fFo*=%*)HFB!z6$F>7$$\"?++++]7G))H]0waLMF1$\"9C(o\"3CE ao,*)*>\"F>7$$\"?+++++D\"G8J/4b#QMF1$!8gy$*4Qu4tYiw)F>7$$\"?++++]PMx#f `diHW$F1$!9Ucj?**[N-n?=JF>7$$\"?+++++](=U(Gg+nZMF1$!9XIe\"z]*R&\\*fJbF >7$$\"?++++]iSmb@XvP_MF1$!9k!>fCz%3P#HP7)F>7$$\"?+++++v$4rV,.&3dMF1$!9 X]xX8?+J3 " 0 "" {MPLTEXT 1 0 71 "evalf[5](evalf[25] (numapprox[infnorm](f(x)-p(x),x=-ln(2)/2..ln(2)/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&Xq\"!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x) = x*coth(x/2);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%%c othG6#*&F'F)\"\"#!\"\"F)" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F' 6#F)F*F)F+" }{TEXT -1 11 " is about " }{XPPEDIT 18 0 "1.7*`. `*10^(-1 8);" "6#*(-%&FloatG6$\"# " 0 "" {MPLTEXT 1 0 93 "r := unapply(evalf[21](con vert(expand(x-q(x)*x^2),horner)),x);\ns := x -> 1+x+x*r(x)/(2-r(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arr owGF(*&,&$\"\"\"\"\"!F/*&,&$\"6,*>gmmmmmm;!#@!\"\"*&,&$\"6'e#*[9qxxxxF !#BF/*&,&$\"6'3?&3%=@jv8m!#DF6*&,&$\"6'\\#*pn\"\\>!R`;!#EF/*&$\"61'GlO D]u7QT!#GF/)9$\"\"#F/F6F/FJF/F/F/FJF/F/F/FJF/F/F/FKF/F/F/FKF/F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,(\"\"\"F-9$F-*(F.F--%\"rG6#F.F-,&\"\"#F-F0!\"\"F5F-F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "xx := evalf[20](ln(2)/2);\nevalf[18](evalf[20](s(xx)));\nevalf[18] (evalf[20](exp(xx)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5raE (*z-ftlM!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"30&4tBc8UT\"!#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"30&4tBc8UT\"!#<" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accurac y of the approximation " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 35 " with random numbers between 0 and " }{XPPEDIT 18 0 "ln(2)/2;" "6#*&-%#lnG6#\"\"#\"\"\"F'!\"\"" }{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 260 3 "eps" }{TEXT -1 3 " \".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "randomize():\neps \+ := Float(5,-16);\nfor i from 1 to 100 do\n xx := evalf[20](rand()/Fl oat(1,12)*.346573590279972654709);\n axx := evalf[20](exp(xx));\n \+ hxx := evalf[16](s(xx));\n e := evalf[20](abs((axx-hxx)/axx));\n \+ if e>=eps then \n printf(\" trial no. %d, x = %.16f,\\n\",i,xx) ;\n printf(\" exp(x) = %.16f, s(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. 1, \+ x = 0.1853883628823955," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1. 2036858162591512, s(x) = 1.2036858162591520, rel error = 6.53e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 8, x = 0.0882524669443684," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x ) = 1.0922638477370016, s(x) = 1.0922638477370010, rel error = 5.17e-1 6" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " tr ial no. 27, x = 0.1565680972337505," }}{PARA 6 "" 1 "" {TEXT -1 78 " \+ exp(x) = 1.1694903986685517, s(x) = 1.1694903986685510, rel error = 6 .40e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 31, x = 0.2332992081944465," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.2627592507237240, s(x) = 1.2627592507237230, rel e rror = 7.73e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 32, x = 0.0414409132200353," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.0423115731973736, s(x) = 1.04231157319737 30, rel error = 5.84e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 40 " trial no. 38, x = 0.3118451357327597," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.3659431408653652, s(x) = 1.3659431 408653660, rel error = 5.81e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 40 " trial no. 52, x = 0.2288162988871624, " }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.2571110850919509, s(x) = 1.2571110850919500, rel error = 7.23e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 54, x = 0.10338529807 01405," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.1089185909675860, \+ s(x) = 1.1089185909675850, rel error = 9.42e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 68, x = 0.15 36523281952356," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.166085401 2786142, s(x) = 1.1660854012786150, rel error = 7.23e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 75, x = 0.0640796293767516," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.06 61772944148273, s(x) = 1.0661772944148280, rel error = 6.27e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial n o. 79, x = 0.0515772721305326," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp( x) = 1.0529305453352404, s(x) = 1.0529305453352410, rel error = 5.36e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " t rial no. 84, x = 0.2894570637754566," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.3357020900709803, s(x) = 1.3357020900709810, rel error = \+ 5.61e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 86, x = 0.3294155068351275," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.3901553546536452, s(x) = 1.3901553546536460, rel e rror = 5.92e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 88, x = 0.0085110754828294," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.0085473976595678, s(x) = 1.00854739765956 70, rel error = 7.46e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 40 " trial no. 89, x = 0.0594859279287315," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.0612908263143114, s(x) = 1.0612908 263143120, rel error = 5.97e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 40 " trial no. 90, x = 0.1162573570384242, " }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.1232849202177537, s(x) = 1.1232849202177530, rel error = 6.01e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 93, x = 0.16382565609 08554," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.1780089185085608, \+ s(x) = 1.1780089185085600, rel error = 6.53e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial no. 94, x = 0.13 69029863013958," }}{PARA 6 "" 1 "" {TEXT -1 78 " exp(x) = 1.146716895 8766493, s(x) = 1.1467168958766500, rel error = 6.07e-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 5 "exp16" }}{PARA 0 "" 0 "" {TEXT -1 142 "Here is the code for the fixed precision version of the exponential function, which can be evaluated with hardware floating point arithmetic.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 700 "exp16 := proc(xx)\n local a1,a2,a3,a4,a5,x,z,t,v al,k,ln2,invln2;\n\n invln2 := 1.44269504088896338700; \n ln2 := . 693147180559945309417;\n\n # coeffs in polynomial approximation\n \+ a1 := -.166666666666666019901;\n a2 := .277777777770144892586e-2;\n \+ a3 := -.661375632118408520086e-4;\n a4 := .165339019491676992496e- 5;\n a5 := -.413812745025366528606e-7;\n x := xx;\n \n # Reduce \+ the argument range to -ln(2)/2<=x<=ln(2)/2\n k := 0;\n if abs(x)>. 3465735902799727 then\n k := floor(x*invln2+0.5);\n x := x-k *ln2;\n end if;\n \n # evaluate the rational approximation\n z := x*x;\n t := (1+(a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*x)*x;\n val := 1 -(x*t/(t-2)-x);\n val*2^k;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Testing the procedure exp16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot ('exp16'(x),x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"#\"\"!$\"3uEhOKGN`8!#=7$$!3MLLL$Q6G \">!#<$\"3QL4@0mkw9F-7$$!3bmm;M!\\p$=F1$\"3q(zuc7FIf\"F-7$$!3MLLL))Qj^ ;Te'f>CF-7$$!3SLL$3WDTL\"F1$\"3j%e(=NU)Qj#F-7$$!35++]d(Q &\\7F1$\"3![A)Hb(pj'GF-7$$!3gmmmc4`i6F1$\"3DqR_Tw$p7$F-7$$!3KLLLQW*e3 \"F1$\"3i#*H3()o*fP$F-7$$!3w++++()>'***F-$\"3'*[&ew4$>!o$F-7$$!3E++++0 \"*H\"*F-$\"3].%os_?K,%F-7$$!35++++83&H)F-$\"3Y\"z1k=QEO%F-7$$!3\\LLL3 k(p`(F-$\"3%*3#z8OJiq%F-7$$!3Anmmmj^NmF-$\"3'Hf#)HA*=]^F-7$$!3)zmmmYh= (eF-$\"3o![.*pF*)ebF-7$$!3+,++v#\\N)\\F-$\"33kY(yn#HvgF-7$$!3commmCC(> %F-$\"3+1'*[w-GslF-7$$!39*****\\FRXL$F-$\"3C8`(G1\\W;(F-7$$!3t*****\\# =/8DF-$\"3%y1$fnu&yx(F-7$$!3=mmm;a*el\"F-$\"3))R?QF)RRZ)F-7$$!3komm;Wn (o)!#>$\"3Kv$eR)3!z;*F-7$$!3IqLLL$eV(>!#?$\"3A!4+V*eF!)**F-7$$\"3)Qjmm \"f`@')Fjr$\"3I4?O`5/!4\"F17$$\"3%z****\\nZ)H;F-$\"3uVOhg(=q<\"F17$$\" 3ckmm;$y*eCF-$\"3e$Ryr\"*o(y7F17$$\"3f)******R^bJ$F-$\"3g:?^hH8$R\"F17 $$\"3'e*****\\5a`TF-$\"3iF\")*=32\\^\"F17$$\"3'o****\\7RV'\\F-$\"3@9IJ dA&Gk\"F17$$\"3Y'*****\\@fkeF-$\"3O(f&Qv@h(z\"F17$$\"3_ILLL&4Nn'F-$\"3 _MZg0t1\\>F17$$\"3A*******\\,s`(F-$\"3Rf[`T-*[7#F17$$\"3%[mm;zM)>$)F-$ \"31%[nV+syH#F17$$\"3M*******pfa<*F-$\"3B(Qnc1SJ]#F17$$\"39HLLeg`!)**F -$\"3V`MEEh*Hr#F17$$\"3w****\\#G2A3\"F1$\"3K'Go$pk=^HF17$$\"3;LLL$)G[k 6F1$\"3BQ8ivaE/KF17$$\"3#)****\\7yh]7F1$\"3GPfA,+]#\\$F17$$\"3xmmm')fd L8F1$\"3Y9#RDb)e%z$F17$$\"3bmmm,FT=9F1$\"3x+f)[!)e08%F17$$\"3FLL$e#pa- :F1$\"3MsqL[\"=J\\%F17$$\"3!*******Rv&)z:F1$\"3WGb\\BUEa[F17$$\"3ILLLG UYo;F1$\"3jp!4=y:SI&F17$$\"3_mmm1^rZF1$\"3%RGn\"Gt(Rx'F17$$\"\"#F*$\"3=_ 1$*)4c!*Q(F1-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fb[l-%+AXESLABELSG6$Q\"x6 \"Q!Fg[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "We can use Maple's exponential function t o check the accuracy of the procedure " }{TEXT 0 5 "exp16" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf(plot(1-'exp16'(x)/exp(x),x=0..3,color=blue),25) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6 #7gcm7$$\"\"!F)F(7$$\":+++++++D\"G$R<)!#F$!$w\"!#C7$$\":+++++++Dc'yM;! #E$!%o8F07$$\":++++++]P%)z@X#F4$!%QWF07$$\":+++++++]7t&pKF4$!%O**F07$$ \":+++++++vofV!\\F4$!&$GGF07$$\":++++++++DY\"RlF4$!&r<&F07$$\":+++++]P Mxb.D(F4$!&?4'F07$$\":++++++voHl:'zF4$!&:y'F07$$\":+++++v$fe+<<$)F4$!& d+(F07$$\":+++++]7.#[xs')F4$!&88(F07$$\":+++++DJ?ez$G!*F4$!&h9(F07$$\" :++++++]PM%)RQ*F4$!&#RqF07$$\":+++++]i!R.k!3\"!#D$!&*\\_F07$$\":++++++ ]PC#)GA\"Ffo$!&UD\"F07$$\":+++++voH>lGI\"Ffo$\"'aiFfo$\"'rU))Ffo7$$\":++++++]P4sP -#Ffo$\"'!f%RFfo7$$\":+++++]7ylGS1#Ffo$\"'4(3\"Ffo7$$\":++++++v=A&G/@F fo$!&P&>F07$$\":+++++]PfyTX9#Ffo$!&74&F07$$\":++++++++N)z%=#Ffo$!&%G#) F07$$\":+++++]iS\"\\0DAFfo$!''e7\"F07$$\":++++++D\"y9JlAFfo$!'#oS\"F07 $$\":+++++](=U!obI#Ffo$!'%Rl\"F07$$\":++++++]igCeM#Ffo$!'\\b=F07$$\":+ ++++]7.<\"3'Q#Ffo$!'<+?F07$$\":++++++vVtPjU#Ffo$!'rx?F07$$\":+++++D1k, mkW#Ffo$!'[)3#F07$$\":+++++]P%)H%fmCFfo$!'Nz?F07$$\":+++++vo/eAn[#Ffo$ !'`\\?F07$$\":+++++++D'3&o]#Ffo$!'Z)*>F07$$\":++++]iST4;pa#Ffo$!'EK=F0 7$$\":+++++D\"yD8)pe#Ffo$!'2!e\"F07$$\":++++](=Udl/FEFfo$!'TX7F07$$\": +++++]i!*y6rm#Ffo$!&9O)F07$$\":++++D\"G)[SWro#Ffo$!&K2'F07$$\":++++]7. 2-xrq#Ffo$!&0l$F07$$\":++++vVBlj4ss#Ffo$!&s6\"F07$$\":+++++vVBDUsu#Ffo $\"'V*\\\"Ffo7$$\":++++D1k\"o[FnFFfo$\"'%)oTFfo7$$\":++++]P%)R[2ty#Ffo $\"'YdoFfo7$$\":++++vo/)*4St!GFfo$\"']G&*Ffo7$$\":++++++Dcrst#GFfo$\"( aU@\"Ffo7$$\":++++]ils%zVnGFfo$\"(?Hq\"Ffo7$$\":+++++D1*yJ]2HFfo$\"('z :@Ffo7$$\":++++](oa5%ov%HFfo$\"(\"3:CFfo7$$\":+++++](=UOj()HFfo$\"(aLc #Ffo7$$\":++++]7GQ())pFIFfo$\"(,n_#Ffo7$$\":+++++voa5kx1$Ffo$\"(C(yAFf o7$$\":++++D1*G@nz(3$Ffo$\"(g02#Ffo7$$\":++++]P4rLHy5$Ffo$\"(*z0=Ffo7$ $\":++++voHH&>'y7$Ffo$\"(,e[\"Ffo7$$\":++++++](oX*y9$Ffo$\"(^O6\"Ffo7$ $\":+++++vozOnk;$Ffo$\"'CksFfo7$$\":+++++](=n,/&=$Ffo$\"'fXIFfo7$$\":+ ++++D1k'Hh.KFfo$!&:W\"F07$$\":++++++Dcw&=AKFfo$!&j4'F07$$\":+++++vV[ce 2C$Ffo$!'Kz5F07$$\":+++++]iSOJ$fKFfo$!'wP:F07$$\":+++++D\"GjT!zF$Ffo$! 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7$$\":++++]7y+7Qi'GF0$!&59(F07$$\":++++DcEA'*)>qGF0$!&f(oF07$$\":+++++ +vV!)fT(GF0$!&*RkF07$$\":+++++DJq'=u\")GF0$!&?6&F07$$\":+++++](oHRK*)G F0$!&'[JF07$$\":+++++vVB*f!p*GF0$!%HiF07$$\":+++++++]0)[/HF0$\"'h@BFfo 7$$\":+++++Dcw6q?\"HF0$\"'UuaFfo7$$\":+++++]7.=_'>HF0$\"'^o&)Ffo7$$\": +++++voHCMs#HF0$\"(.)H6Ffo7$$\":++++++DcI;[$HF0$\"(5VL\"Ffo7$$\":++++v VBlG!*)QHF0$\"(\"=09Ffo7$$\":++++](=UnU'H%HF0$\"(ALW\"Ffo7$$\":+++]P4r yD,]%HF0$\"(p*[9Ffo7$$\":++++DJ?$[#Qq%HF0$\"(\\^W\"Ffo7$$\":+++]7`p(Q_ 2\\HF0$\"(`:V\"Ffo7$$\":+++++v=#HA6^HF0$\"(0zS\"Ffo7$$\":++++]i:5>g#fH F0$\"(\\*47Ffo7$$\":+++++]7G:3u'HF0$\"'A([)Ffo7$$\":++++v$4rL@[rHF0$\" 'BChFfo7$$\":++++]P4Y6cb(HF0$\"'G[MFfo7$$\":++++D\"y]&4I'zHF0$\"&/A&Ff o7$$\":+++++D1k2/P)HF0$!&wd#F07$$\":++++vo/t0yx)HF0$!&+w&F07$$\":++++] 7.#Q?&=*HF0$!&H#*)F07$$\":++++Dc,\">g#f*HF0$!'S&>\"F07$$\"\"$F)$!'Vt9F 0-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fegw-%%V IEWG6$;F(Fffw%(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 63 "xx := -10.98;\nevalf(exp16(xx),17); \nevalf(evalf(exp(xx),20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$!%)4\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2\\X$p_(4Rq\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2[X$p_(4Rq\"!#@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 95 "xx := 100;\nevalf(evalhf(exp16(xx)),15);\neval f(evalhf(exp(xx)),15);\nevalf(evalf(exp(xx),18),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0 9;=9<\")o#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"08;=9<\")o#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"09;=9<\")o#\"#H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "We lose some accuracy when the argument is close to the maximum value which does not produc e overflow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 126 "xx := 705;\nevalf(evalf(exp16(xx),20),15);\nevalf( evalhf(exp16(xx)),15);\nevalf(evalhf(exp(xx)),15);\nevalf(evalf(exp(xx ),18),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"$0(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0>jI$QD0:\"$#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0;jI$QD0:\"$#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0>jI$QD0 :\"$#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0>jI$QD0:\"$#H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "xx := 710;\nevalf(evalf(exp16(xx), 20),15);\nevalf(evalhf(exp16(xx)),15);\nevalf(evalhf(exp(xx)),15);\nev alf(exp(xx),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"$5(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0rhhw%*RB#\"$%H" }}{PARA 8 "" 1 "" {TEXT -1 37 "Error, powering may produce overflow\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"\"\"%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"0rhhw%*RB#\"$%H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Translating from Maple code to C code " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "We can translate this program \+ into C using the Maple procedure C in the code generating package code gen. \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 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 30 ": The last line of code . . . " }} {PARA 6 "" 0 "" {TEXT -1 31 " return(val*pow(2.0,1.0*k));" }}{PARA 0 "" 0 "" {TEXT -1 35 " . . . should be replaced by . . . " }}{PARA 6 "" 0 "" {TEXT -1 26 " return(ldexp(val,k)); " }}{PARA 0 "" 0 "" {TEXT -1 117 "This computes val*2^k more efficiently by modifing the e xponent in the binary floating point representation of \"val\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(codegen,C):\nC(exp16,ansi);" }}{PARA 6 "" 1 "" {TEXT -1 17 "# include " }}{PARA 6 "" 1 "" {TEXT -1 23 "double exp16(double x x)" }}{PARA 6 "" 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 12 " \+ double a1;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a2;" }}{PARA 6 " " 1 "" {TEXT -1 12 " double a3;" }}{PARA 6 "" 1 "" {TEXT -1 12 " dou ble a4;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a5;" }}{PARA 6 "" 1 " " {TEXT -1 16 " double invln2;" }}{PARA 6 "" 1 "" {TEXT -1 8 " int k ;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double ln2;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double t;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double val; " }}{PARA 6 "" 1 "" {TEXT -1 11 " double x;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 34 " invln2 = 0.1442695040888963E1;" }}{PARA 6 "" 1 " " {TEXT -1 29 " ln2 = 0.6931471805599453;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a1 = -0.166666666666666;" }}{PARA 6 "" 1 "" {TEXT -1 31 " \+ a2 = 0.2777777777701449E-2;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a3 = -0.6613756321184085E-4;" }}{PARA 6 "" 1 "" {TEXT -1 30 " a4 = 0.16 5339019491677E-5;" }}{PARA 6 "" 1 "" {TEXT -1 32 " a5 = -0.41381274 50253665E-7;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }} {PARA 6 "" 1 "" {TEXT -1 10 " k = 0;" }}{PARA 6 "" 1 "" {TEXT -1 38 " if( 0.3465735902799727 < fabs(x) )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }}{PARA 6 "" 1 "" {TEXT -1 32 " k = floor(x*inv ln2+0.5);" }}{PARA 6 "" 1 "" {TEXT -1 20 " x += -k*ln2;" }} {PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 12 " \+ z = x*x;" }}{PARA 6 "" 1 "" {TEXT -1 49 " t = (1.0+(a1+(a2+(a3+( a4+a5*z)*z)*z)*z)*x)*x;" }}{PARA 6 "" 1 "" {TEXT -1 28 " val = 1.0- x*t/(t-2.0)+x;" }}{PARA 6 "" 1 "" {TEXT -1 31 " return(val*pow(2.0, 1.0*k));" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }}{PARA 6 "" 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 13 "Modified code" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 17 "#include \+ " }}{PARA 2 "" 1 "" {TEXT -1 23 "double exp16(double xx)" }} {PARA 2 "" 1 "" {TEXT -1 1 "\{" }}{PARA 2 "" 1 "" {TEXT -1 12 " doubl e a1;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a2;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a3;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a4; " }}{PARA 2 "" 1 "" {TEXT -1 12 " double a5;" }}{PARA 2 "" 1 "" {TEXT -1 16 " double invln2;" }}{PARA 2 "" 1 "" {TEXT -1 8 " int k; " }}{PARA 2 "" 1 "" {TEXT -1 13 " double ln2;" }}{PARA 2 "" 1 "" {TEXT -1 11 " double t;" }}{PARA 2 "" 1 "" {TEXT -1 13 " double val; " }}{PARA 2 "" 1 "" {TEXT -1 11 " double x;" }}{PARA 2 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 2 "" 1 "" {TEXT -1 3 " \{" }}{PARA 2 "" 1 "" {TEXT -1 34 " invln2 = 0.1442695040888963E1;" }}{PARA 2 "" 1 " " {TEXT -1 29 " ln2 = 0.6931471805599453;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a1 = -0.166666666666666;" }}{PARA 2 "" 1 "" {TEXT -1 31 " \+ a2 = 0.2777777777701449E-2;" }}{PARA 2 "" 1 "" {TEXT -1 32 " a3 = -0.6613756321184085E-4;" }}{PARA 2 "" 1 "" {TEXT -1 30 " a4 = 0.16 5339019491677E-5;" }}{PARA 2 "" 1 "" {TEXT -1 32 " a5 = -0.41381274 50253665E-7;" }}{PARA 2 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }} {PARA 2 "" 1 "" {TEXT -1 10 " k = 0;" }}{PARA 2 "" 1 "" {TEXT -1 38 " if( 0.3465735902799727 < fabs(x) )" }}{PARA 2 "" 1 "" {TEXT -1 7 " \{" }}{PARA 2 "" 1 "" {TEXT -1 32 " k = floor(x*inv ln2+0.5);" }}{PARA 2 "" 1 "" {TEXT -1 20 " x += -k*ln2;" }} {PARA 2 "" 1 "" {TEXT -1 7 " \}" }}{PARA 2 "" 1 "" {TEXT -1 12 " \+ z = x*x;" }}{PARA 2 "" 1 "" {TEXT -1 49 " t = (1.0+(a1+(a2+(a3+( a4+a5*z)*z)*z)*z)*x)*x;" }}{PARA 2 "" 1 "" {TEXT -1 28 " val = 1.0- x*t/(t-2.0)+x;" }}{PARA 2 "" 0 "" {TEXT -1 25 " return(ldexp(val,k) );" }}{PARA 2 "" 1 "" {TEXT -1 3 " \}" }}{PARA 2 "" 1 "" {TEXT -1 1 " \}" }}{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 ";" }}}{PARA 0 "" 0 "" {TEXT -1 113 "In this section we construct two fixed precision procedu res to evaluate the exponential function in the interval " }{XPPEDIT 18 0 "[-ln(10)/8,ln(10)/8]" "6#7$,$*&-%#lnG6#\"#5\"\"\"\"\")!\"\"F,*&- F'6#F)F*F+F," }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 25 digit rational approximation for " }{XPPEDIT 18 0 "exp(x);" "6#-%$ expG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 142 "In order to obtain a fixed prec ision procedure to evaluate the exponential function, we construct a p olynomial approximation for the function " }{XPPEDIT 18 0 "f(x)=x*coth (x/2)" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%%cothG6#*&F'F)\"\"#!\"\"F)" } {TEXT -1 78 " on the interval [-0.2879, 0.2879], which is slightly wid er than the interval " }{XPPEDIT 18 0 "[-ln(10)/8, ln(10)/8];" "6#7$,$ *&-%#lnG6#\"#5\"\"\"\"\")!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 62 "We do this by constructing a minimax poly nomial approximation " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 18 " for the function " }{XPPEDIT 18 0 "h(x)=(x*coth(x/2)-2)/x^2" " 6#/-%\"hG6#%\"xG*&,&*&F'\"\"\"-%%cothG6#*&F'F+\"\"#!\"\"F+F+F0F1F+*$F' F0F1" }{TEXT -1 9 ". Since " }{XPPEDIT 18 0 "f(x) = 2+h(x)*`.`*x^2;" "6#/-%\"fG6#%\"xG,&\"\"#\"\"\"*(-%\"hG6#F'F*%\".GF*F'F)F*" }{TEXT -1 17 ", the polynomial " }{XPPEDIT 18 0 "p(x)=2+q(x)*`.`*x^2" "6#/-%\"pG 6#%\"xG,&\"\"#\"\"\"*(-%\"qG6#F'F*%\".GF*F'F)F*" }{TEXT -1 40 " will b e a polynomial approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "In order to obtai n the desired error characterstics for " }{XPPEDIT 18 0 "f(x)" "6#-%\" fG6#%\"xG" }{TEXT -1 39 ", we use a weight function of the form " } {XPPEDIT 18 0 "w(x)=x^2+delta" "6#/-%\"wG6#%\"xG,&*$F'\"\"#\"\"\"%&del taGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" } {TEXT -1 11 " is small. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "evalf(remez((x*coth(x/2)-2)/x^2,x= -.2879..0.2879,[4,6],\n 'maxerr',errtype=absolute,weight=x^2+1e-50 ,type=even,info=true),40):\nq := unapply(%,x);\nmaxerr;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%-iteration~16G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,($\"R0L6qcqZrT5vsammmmmmmmmmm\"!#\\\"\"\"*&$\"R .dD6vxTl@-iO$4V,Q5?_\"e&*Rc#!#^F()%\"xG\"\"#F(F(*&$\"R;S[sA]hzVlE3'R\" [*f(ew%owcUi!#aF()F.\"\"%F(F(F(,*$F(\"\"!F(*&$\"RQF\"fA`NT'\\@ZBuVe%*G ()zb,k]?$!#]F(F-F(F(*&$\"R^i!Qcs\"[vH&>aY^PQZ;W:^M2[`rv[&*\\Ih\"f\"!#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5OV,#=[j Rc([&*\\Ih\"f\"!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\">NX$)yy#G4SG!3&[ov!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"OW3`9^(3yXkk@H\\xH1JXMH_v%!#i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"O->s6$f??-d%H'3]-c*GGt$y9o&!#i" }}{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)$\"Rikw2Bj3\\'>#[(*3s&QU2*yKjg#pb!#]$\"R+yrN.**pr>>'>O_VBy8\\(RL)39 7!#\\$\"Ru7M#QTl)o[oi_Fp&>YGb\"[n)ew_T!GGW4K'*oj@L&efJ9WTA F($\"Rd))*))G5JhuV[S^U[b7^'ewAu)*e#F($\"RnMhQ'yt$fk+yZ!HxFX/S`3R$e!GF( $\"O+++++++++++++++++++++zG!#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#*&,($\"I1x9aY^PQZ;W:^M2[%\"qGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(*&,($\"I1x9aY^PQZ;W:^M2[`rv[&*\\Ih\"f\"!#m" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "q:= x-> (.1666666666666666666666654727510417147706+.2563995581522010380143 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numapprox[infnorm](1-p(x)/f(x),x=-ln(10)/8..ln(10)/8),35),5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&g&z!#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in u sing " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to appro ximate " }{XPPEDIT 18 0 "f(x) = x*coth(x/2);" "6#/-%\"fG6#%\"xG*&F'\" \"\"-%%cothG6#*&F'F)\"\"#!\"\"F)" }{TEXT -1 17 " in the interval " } {XPPEDIT 18 0 "[-ln(10)/8, ln(10)/8];" "6#7$,$*&-%#lnG6#\"#5\"\"\"\"\" )!\"\"F,*&-F'6#F)F*F+F," }{TEXT -1 11 " is about " }{XPPEDIT 18 0 "8* `. `*10^(-26);" "6#*(\"\")\"\"\"%#.~GF%)\"#5,$\"#E!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "N ow construct the rational function " }{XPPEDIT 18 0 "r(x)=x-q(x)*x^2" "6#/-%\"rG6#%\"xG,&F'\"\"\"*&-%\"qG6#F'F)*$F'\"\"#F)!\"\"" }{TEXT -1 49 " and round the coefficients to 30 decimal digits." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "s(x)=1+x+x*r(x)/(2-r(x) )" 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" }}{PARA 0 "" 0 "" {TEXT -1 62 "We do \+ this by constructing a minimax polynomial approximation " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 18 " for the function " } {XPPEDIT 18 0 "h(x)=(x*coth(x/2)-2)/x^2" "6#/-%\"hG6#%\"xG*&,&*&F'\"\" \"-%%cothG6#*&F'F+\"\"#!\"\"F+F+F0F1F+*$F'F0F1" }{TEXT -1 9 ". Since \+ " }{XPPEDIT 18 0 "f(x) = 2+h(x)*`.`*x^2;" "6#/-%\"fG6#%\"xG,&\"\"#\"\" \"*(-%\"hG6#F'F*%\".GF*F'F)F*" }{TEXT -1 17 ", the polynomial " } {XPPEDIT 18 0 "p(x)=2+q(x)*`.`*x^2" "6#/-%\"pG6#%\"xG,&\"\"#\"\"\"*(-% \"qG6#F'F*%\".GF*F'F)F*" }{TEXT -1 40 " will be a polynomial approxima tion for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 56 "In order to obtain the desired error char acterstics for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 39 ", we use a weight function of the form " }{XPPEDIT 18 0 "w(x)=x^2+del ta" "6#/-%\"wG6#%\"xG,&*$F'\"\"#\"\"\"%&deltaGF+" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 11 " is small. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1219 "q:=x->(.1666666666666666666666666 666666666666666666666666666666665715648765513+.33950503303086046614094 57824891090536919169436836132754033801371907174e-2*x^2+.19841046835892 37072835798643959118061664937476717238467973020075906503e-4*x^4+.44729 54114528217502960803062487269016569971548004681905973430649017739e-7*x ^6+.403394031539763067033041400818864128288431392911408985198279372749 6674e-10*x^8+.12248225958360056892288894547647813924426330378633420850 63780676001315e-13*x^10+.571764694596822796978120873215841239524458473 9552715512575676590290543e-18*x^12)/(1.+.37036968648518294635123413616 01320988818168328768346264665910201936107e-1*x^2+.33950369499859564289 20460820790370897727655732392099909460314687904112e-3*x^4+.11501973029 91820973329442561251755141896766035562619712264393311807171e-5*x^6+.15 97463651815481147585981420148286834169550830522070934587149846665263e- 8*x^8+.849232569219197625336843378312648323130436490088335129238683052 6970317e-12*x^10+.1200762868256329698733370007130274996961630870570585 854103723077487183e-15*x^12):\np := unapply(evalf(convert(2+q(x)*x^2,h orner),63),x);\nf := x -> x*coth(x/2);\nevalf(plot(10^30*(1-p(x)/f(x)) ,x=-0.2879..0.2879,\n color=COLOR(RGB,.4,0,.9)),75); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,&$\"\"#\"\"!\"\"\"*(,0$\"jnl:dmmmmmmmmmmmmmmmmmmmmmmmmmmmm\"!# jF0*&$\"jnP,Q.aF8OoVp\">p`!4\"*[#yX49m/'3.L]]R$!#lF0)9$F.F0F0*&$\"jn3? 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%zGF*F+-%+AXESLABELSG6$Q\"x6\"Q!F`et-%&COLORG6&%$RGBG$\"\"%!\"\"\"\"!$ \"\"*Fhet-%%VIEWG6$;F(Fjdt%(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 77 "Digits := 60:\nxx := 0.00000 0001;\nevalf(f(xx),Digits+20);\np(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"hnMXE]XE]XE]&*)))))))))))))))))Qmmmmmmmmm;!#z" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"gngj-bk-bk-b*)))))))))))))))))Qmmmmmmmmm;!#y" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "exp55 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1318 "exp55 := proc(x)\n local a1,a2,a3,a4,a5,a6,a7,b1,b2,b3,b4,b5,b6,z,n,d,t;\n \+ \n a1 := \n .1666666666666666666666666666666666666666666666666666666 66571565;\n a2 := \n .3395050330308604661409457824891090536919169436 83613275403380137e-2;\n a3 := \n .1984104683589237072835798643959118 06166493747671723846797302008e-4;\n a4 := \n .4472954114528217502960 80306248726901656997154800468190597343065e-7;\n a5 := \n.40339403153 9763067033041400818864128288431392911408985198279373e-10;\n a6 := \n .122482259583600568922888945476478139244263303786334208506378068e-13; \n a7 := \n.57176469459682279697812087321584123952445847395527155125 7567659e-18;\n b1 := \n .3703696864851829463512341361601320988818168 32876834626466591020e-1;\n b2 := \n .3395036949985956428920460820790 37089772765573239209990946031469e-3;\n b3 := \n .1150197302991820973 32944256125175514189676603556261971226439331e-5;\n b4 := \n .1597463 65181548114758598142014828683416955083052207093458714985e-8;\n b5 := \n.849232569219197625336843378312648323130436490088335129238683053e-1 2;\n b6 := \n.120076286825632969873337000713027499696163087057058585 410372308e-15;\n \n # evaluate the approximation\n z := x*x;\n \+ n := (a1+(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z)*z)*z)*z)*z;\n d := 1+(b1+ (b2+(b3+(b4+(b5+b6*z)*z)*z)*z)*z)*z;\n t := x-n/d; \n 1-(x*t/(t- 2)-x);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 110 "Digits := 60:\nxx := evalf(ln(10)/8);\neval f(evalf(exp55(xx),65),60);\nevalf(evalf(exp(xx),65),60);\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gnmT+AmfygoP,&f_Xb$=$*[A ]5dDCm8ByG!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gnbHL*4PkZ'zm:C4J` H:<$fnDSKj@V@NL\"!#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gnbHL*4PkZ' zm:C4J`H:<$fnDSKj@V@NL\"!#f" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Digits := 60:\nxx := evalf( 10^(-3));\nevalf(evalf(exp55(xx),65),60);\nevalf(evalf(exp(xx),65),60) ;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gn+++++++ ++++++++++++++++++++++5!#i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gn^G- Y,2e+i2j:JeI*Rvb!o;M3n;+0+,5!#f" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" gn^G-Y,2e+i2j:JeI*Rvb!o;M3n;+0+,5!#f" }}}{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 exp55" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "xx := evalf( ln(10)/8,60);\nevalf(evalf(exp55(xx),60),57);\nevalf(evalf(exp(xx),60) ,57);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gnmT+AmfygoP,&f_Xb$= $*[A]5dDCm8ByG!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ZL$*4PkZ'zm:C4 J`H:<$fnDSKj@V@NL\"!#c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ZL$*4PkZ' zm:C4J`H:<$fnDSKj@V@NL\"!#c" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "evalf(plot(10^30*(1-'exp55'( x)/exp(x)),x=-ln(10)/8..ln(10)/8,color=COLOR(RGB,.4,0,.9)),67);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7ael 7$$!^o0i()f;/?i'fygoP,&f_Xb$=$*[A]5dDCm8ByG!#n$!%eO!#O7$$!^oT(*oS226^T n'fyq?r&e;d$Hz/IYJu9%zG5V(GF*$\"%96!#P7$$!^oyK]@)*4-S')[&e)Q+uak()e.a1 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"Curve 1" }}}}{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 53 "A procedur e for evaluating the exponential function: " }{TEXT 0 4 "exp_" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "exp_: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 264 20 "Calling Sequence: " }{TEXT -1 10 "exp_( x ) " }{TEXT 265 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 266 21 "x - a real constant" }} {PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 269 12 "Des cription:" }{TEXT -1 1 " " }{TEXT 268 14 "The procedure " }{TEXT 0 4 " exp_" }{TEXT 267 12 " calculates " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6 #%\"xG" }{TEXT 270 21 " 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 pro cedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "exp_: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 2 ": " }{TEXT 261 18 "evalhf(LNMAXFLOAT)" }{TEXT -1 302 " is \+ a number slightly smaller than the largest number x such that evalhf(e xp(x)) and evalhf(exp(-x)) both successfully compute, and such that th e results of those computations can continue to be used in evalhf. On \+ most machines, this value is set as evalhf(-ln(DBL_MIN)) truncated to \+ 6 decimal places. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12474 "exp_ := proc(x::algebraic)\n local u;\n \n if nargs <> 1 then\n error \"expecting 1 argument, got %1\", nargs;\n end if;\n if type(x,'float') then evalf('exp_'(x))\n e lif type(x,'function') and nops(x)=1 then\n u := op(1,x);\n \+ if op(0,x)='ln' or op(0,x)='ln_' then u\n else 'exp_'(x)\n e nd if;\n elif x=-'infinity' then 0\n elif x='infinity' then 'infin ity'\n else 'exp_'(x)\n end if;\nend proc: # of exp_\n\n`evalf/exp _` := proc(xx)\n local x,term,eps,k,i,maxit,saveDigits,\n extraDig its,sum,val,L,R,Q,r,m,hfDigits;\n\n if not type(xx,realcons) then re turn 'exp_'(xx) end if;\n\n # Use the fixed precision procedure for \+ low precision values\n hfDigits := trunc(evalhf(Digits));\n if Dig its<=hfDigits then\n x := evalf(xx,hfDigits+2);\n if abs(x)< evalhf(LNMAXFLOAT) then\n return evalf(evalhf(exp16(x)));\n \+ end if;\n end if;\n \n # increase precision \n saveDigits \+ := Digits;\n Digits := Digits+length(Digits)+1;\n x := evalf(xx); \n # argument reduction involves loss of precision\n # increase pr ecision further when x has large magnitude\n extraDigits := max(ilog 10(x),0);\n if extraDigits>0 then\n Digits := Digits+extraDigit s;\n x := evalf(xx)\n end if;\n \n # Reduce the range \+ to -ln(10)/8<=z<=ln(10)/8 approximately\n k := 0;\n if abs(x)>0.28 782313662425571050 then\n L := evalf(Ln10)*0.25;\n R := eval f(root4of10);\n # Determine the number of multiples of L=ln(10)/4 to add or subtract.\n k := floor(x/L+0.5);\n x := x - k*L; \n end if;\n\n Digits := Digits - extraDigits;\n\n if saveDigits <27 then \n val := exp25(x);\n elif saveDigits<60 then\n v al := exp55(x)\n else\n # use Maclaurin series\n eps := Fl oat(1,-saveDigits);\n maxit := Digits;\n term := 1.0;\n \+ sum := term;\n for i from 1 to maxit do\n term := term*x /i;\n sum := sum+term;\n if abs(term)<=eps*abs(sum) th en break end if;\n end do;\n val := sum;\n end if;\n\n m := iquo(k,4,`r`);\n if r<0 then\n r := r+4;\n m := m-1;\n end if;\n if r<2 then\n if r<1 then Q := 1 else Q := R end i f;\n else\n if r<3 then Q := R*R else Q := R*R*R end if;\n en d if;\n\n val := Scale10(val*Q,m); \n Digits := saveDigits;\n e valf(val);\nend proc: # of `evalf/exp_`\n\n`evalf/constant/root4of10` \+ := proc()\nlocal d,r,r2,r4;\nglobal _root4of10;\n if Digits<=55 then evalf(1.778279410038922801225421195192684844735790526402255358)\n e lif Digits<=length(op(1,_root4of10)) then evalf(_root4of10)\n else\n d := length(op(1,_root4of10));\n r := _root4of10;\n wh ile d.3465735902799727 then\n k := floor(x*invln2+0.5);\n \+ x := x-k*ln2;\n end if;\n \n # evaluate the rational approximat ion\n z := x*x;\n t := (1+(a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*x)*x;\n \+ val := 1-(x*t/(t-2)-x);\n val*2^k;\nend proc: # of exp16" }}}{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 "Examp les are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 4 "exp_" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Testing t he procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "xx := 123.4567;\nevalf(exp_(xx),20);\nevalf(evalf( exp(xx),25),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"(nXB\"!\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5-b7FjE%Qe8%\"#M" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"5-b7FjE%Qe8%\"#M" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xx := -706;\nevalf (exp_(xx));\nevalf(exp(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG !$1(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p%pRW#!$;$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p%pRW#!$;$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(exp_(x),x=-2..2);" }} {PARA 13 "" 1 "" {GLPLOT2D 345 242 242 {PLOTDATA 2 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\"+-FT=9F1$\"+1)e08%F17$$\"+Epa-:F1$\"+\\\"=J\\%F17$$\"+Sv&)z:F1$\"+BU Ea[F17$$\"+GUYo;F1$\"+!y:SI&F17$$\"+2^rZF1$\"+Dt(Rx'F17$$\"\"#F*$\"+*4c!*Q(F1-%'COLOURG6 &%$RGBG$\"#5!\"\"$F*F*F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;F(Fe z%(DEFAULTG" 1 2 0 1 10 0 2 9 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 49 "xx := 200;\nevalf(exp _(xx),65);\nevalf(exp(xx),65);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\\o7*ypKSG>t_GW(oN(p0$*= UqZx\"e#\\d7oP(fA(\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\\o7*ypKS G>t_GW(oN(p0$*=UqZx\"e#\\d7oP(fA(\"#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xx := 10^9;\nevalf(exp_(xx),55);\nevalf(exp(xx),55);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"+++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"XlXPwr*\\J#y)o+]Ou$4>/LD(4mq<)H+)\"*FWHM%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"XlXPwr*\\J#y)o+]Ou$4>/LD(4mq<)H+)\"*FWHM%" }}} {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 0 164 "st := time():\nDigits := 10:\nfor i to 5000 do exp_(rand()*Float( 1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 5000 do exp(rand( )*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"$o%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$.#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "st := time():\nDigits := 25:\nfor i to 5 000 do exp_(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nf or i to 5000 do exp(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits : = 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%\"G\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$s'!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "st := time():\nDigits := 55 :\nfor i to 500 do exp_(rand()*Float(1,-11)) end do:\ntime()-st;\nst : = time():\nfor i to 500 do exp(rand()*Float(1,-11)) end do:\ntime()-st ;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$>#!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#y!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot([exp_(x),4-x^2],x=- 2.2..2.2,-.2..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 428 228 228 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!#A!\"\"$\"+%eJ!36!#57$$!+AD4/@!\"*$\"+nHc>7F-7$ $!+QRk??F1$\"+22qD8F-7$$!+xszE>F1$\"+K\"QhX\"F-7$$!+S&GB$=F1$\"+HZS+;F -7$$!+(z3$QF-7$$!+%p#)3c\"F1$\"+s m]*4#F-7$$!+&)z`n9F1$\"+EE#\\I#F-7$$!+LE\\u8F1$\"+3\"z'HDF-7$$!+_Syy7F 1$\"+]ev$y#F-7$$!+#)Q[%>\"F1$\"+bAgGIF-7$$!+d=e*4\"F1$\"+0I5ILF-7$$!+b ,H/5F1$\"+?c/jOF-7$$!*V*eC\"*F1$\"+elN:SF-7$$!*0u1H)F1$\"+M8ckVF-7$$!* 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\"+*))H2>\"F27$$\"+.o-*\\\"Fcs$\"+nH2d5F27$$\"+TO5T:Fcs$\"+.\\W,$*F/7$ $\"+U9C#e\"Fcs$\"+X'=*z\")F/7$$\"+1*3`i\"Fcs$\"+$Q6X7(F/7$$\"+$*zym;Fc s$\"+n#Q^@'F/7$$\"+^j?4]#eMF/7$$\"+`v&Q(=Fcs$\"+%e \"o&)HF/7$$\"+Ol5;>Fcs$\"+:$>Ra#F/7$$\"+/Uac>Fcs$\"+ef;v@F/7$$\"\"#F)$ \"+*)QcJ=F/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fd [l-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "We calculate the integral " }{XPPEDIT 18 0 "Int(exp(-x^2),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\" \"/F+;\"\"!\"\"\"" }{TEXT -1 21 " using the procedure " }{TEXT 0 4 "ex p_" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "Int(exp_(-x^2),x=0..1);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%%exp_G6#,$*$)%\"xG\"\"#\"\"\"!\" \"/F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G8Cou!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "I'll use \+ one of the special numerical integration procedures to obtain a more a ccurate value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf[100](GKint(exp_(-x^2),x=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q(>>!H#)H%[y\\JGxhvK0'e*)\\aw-Hjg7oo*\\a `+`=8OuY*RDqU7G8Cou!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 106 "Compare with the value given by Maple's exponenti al function and standard numerical integration procedure." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(e xp(-x^2),x=0..1);\nevalf[100](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q(>>!H#)H%[y\\JGxhvK0'e*)\\aw-Hjg7oo*\\a`+`=8O uY*RDqU7G8Cou!$+\"" }}}{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 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