{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 "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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le 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 "Norma l" -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 57 "Procedures for evaluating the hyp erbolic tangent function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Ston e, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25. 3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "load interpolation and func tion approximation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 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 5 "tan h_" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read int o a Maple session by a command similar to the one that follows, where \+ 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 74 "A fixed precision procedure to evaluate the hyperb olic tangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 71 "In this section we construct a fixed precision version of the function " }{XPPEDIT 18 0 "tanh(x)" "6#-%%ta nhG6#%\"xG" }{TEXT -1 15 " which can use " }{TEXT 259 34 "hardware flo ating point arithmetic" }{TEXT -1 58 " and so run faster than the arbi trary precision procedure " }{TEXT 0 6 "tanhAP" }{TEXT -1 2 ". " }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A rational approximation for " } {XPPEDIT 18 0 "tanh(x)" "6#-%%tanhG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 30 "W e choose a rational function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "s(x) = (a[0]*x+a[1]*x^2+a[2]*x^3+` . . . `+a[m]*x^m)/(b [0]+b[1]*x+a[2]*x^2+` . . . `+b[n]*x^n);" "6#/-%\"sG6#%\"xG*&,,*&&%\"a G6#\"\"!\"\"\"F'F/F/*&&F,6#F/F/*$F'\"\"#F/F/*&&F,6#F4F/*$F'\"\"$F/F/%( ~.~.~.~GF/*&&F,6#%\"mGF/)F'F>F/F/F/,,&%\"bG6#F.F/*&&FB6#F/F/F'F/F/*&&F ,6#F4F/*$F'F4F/F/F:F/*&&FB6#%\"nGF/)F'FNF/F/!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "tanh(x) " "6#-%%tanhG6#%\"xG" }{TEXT -1 79 " on the interval [-0.34665, 0.3466 5] which is slightly wider than 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Such a rational approximation can be constructed using th e procedure " }{TEXT 0 5 "remez" }{TEXT -1 51 ", which can be loaded f rom the relevant worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 56 "First we \+ find a rational approximation for the function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([(x-tanh(x))/(x^3), x <> 0],[1/3, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&,&F'\"\"\"- %%tanhG6#F'!\"\"F.*$F'\"\"$F20F'\"\"!7$*&F.F.F4F2/F'F6" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Limit((x-tanh(x))/x^3 ,x=0);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&% \"xG\"\"\"-%%tanhG6#F(!\"\"F)F(!\"$/F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "evalf(remez((x-tanh(x))/x^3 ,x=-0.34665..0.34665,[4,4],\n 'maxerr',errtype=absolute,weight=x^2 +1e-30,type=even,info=true),35):\nr := unapply(%,x);\nmaxerr;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerror~g raph~drawn~for~the~right~half~of~the~intervalG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "61-%'CURVESG6%7gs7$$\"\"!F)%%FAILG7 $$\"I++++++++++vV[@0'31OO#!#V$\"IDWzu.s6<@a56X+D3.*GP%!#i7$$\"I+++++++ +++](oH/@<7ss%F.$\"IA(3dV.4!)4$oP99,#Gp?\"\\rzzg[LOa4&zpW%FX7$$\"I++++++++++D\"GB6.r.a3\"!#T$\"I-(> :L]QxF\"[)[l)>$3ic34*FX7$$\"I++++++++++]7yd3tMX99F`o$\"I]&)Q*R*yd>luAA ei[._XG:!#e7$$\"I+++++++++++D\"[Y!=_ba@F`o$\"I8\\2`TVX+4Z&3,FXu*oULMFh o7$$\"I+++++++++++vo.zPGc**GF`o$\"IEnvP%*p*[#fW@jbtGE;+MfFho7$$\"I++++ ++++++]i:[Om&H5k$F`o$\"IW8Lb.R2XJr!Q5d???.!*z)Fho7$$\"I++++++++++]7G75 QTYGVF`o$\"I-AclX?TDRZT'Qq?<&p7e6!#d7$$\"I++++++++++]7.*p$>GES]F`o$\"I U)H]jXT_?,&*o@Y8Y^>jV\"F]q7$$\"I++++++++++]7.lxLgSwdF`o$\"I,vN.^3YmWbN xF]q7$$ \"I+++++++++++](G%)G\"Q(\\E(F`o$\"IYvcf8\\(*\\zh.#>!>g.i)[*>F]q7$$\"I+ +++++++++Dc,EWdw*yHzF`o$\"Ia7E` RU0D9rXn1K6A$R.,#F]q7$$\"I++++++++++v$fo;[N&*o6)F`o$\"Izw*HLFC!GJbWMc/ i)\\M\"**>F]q7$$\"I++++++++++]7Ge^=:+/$)F`o$\"ILY@qlac@/0O+e196&o3)>F] q7$$\"I++++++++++DJq\\@#o26\\)F`o$\"Iak=w@UDm`g[OfX)yFoa&>F]q7$$\"I+++ ++++++++]7T\"f%Q@y')F`o$\"IWqs9Z]d@q(p47P?s*G*G#>F]q7$$\"I++++++++++++ vA3DF'R0*F`o$\"I))zIjhocv;C\"*p8&H)>J!e$=F]q7$$\"I+++++++++++]P/D/;rH% *F`o$\"If+,g$))Hg$4rC!p8or=A,s\"F]q7$$\"I+++++++++++v)ebV*=R:5!#S$\"IN auW_ZXqQg2g#)oR`LH?9F]q7$$\"I+++++++++]7.2_dwVF[5F\\v$\"IYAN()*>6]3:n& QOI/C)oLD\"F]q7$$\"I++++++++++DJD[zeo:\"3\"F\\v$\"I&4!oyRD27)zT#oRzJfQ Rp5F]q7$$\"I+++++++++]7`Cdh5tD?6F\\v$\"I[T_:ue?7+6J(frc(Q%)\\6$)Fho7$$ \"I+++++++++++vBmVixNf6F\\v$\"IOiq]?l35)R/IsPtVUuMv&Fho7$$\"I+++++++++ +]PMO%R(4[#>\"F\\v$\"I$y8,9YP#o5k28O+Z4y$3[$Fho7$$\"I++++++++++++X1X&= /cA\"F\\v$\"Isw)*4\\/Hkyw&\\u8I0$ePX*\\#3?/55 :6*Fho7$$\"I+++++++++]il7)))3P&H39F\\v$!Id\\mRX;LRudx#31c(f_Y_6F]q7$$ \"I++++++++++D\"GRq0w9dW\"F\\v$!ILl\\))yT78%3b.]8#QDB$\\P\"F]q7$$\"I++ ++++++++]i&3i%ooM\"[\"F\\v$!I6)f6:d%Q2\\!)=\"4&erF$3]c\"F]q7$$\"I+++++ +++++vVyPNw*yp^\"F\\v$!In\"ow.r'>u()*yC`'e8l5,IF]q7$$\"I+++++++++v$f3ZpDQ1%3;F\\v$!I 'e$HU]=w@lsa% *\\o4R?;U/#F]q7$$\"I+++++++++D\"GjDv)GxaU;F\\v$!IHg,B&e*z2ki'o.NR[6hW1 #F]q7$$\"I++++++++++D1\\\"G?S='f;F\\v$!I-JL30unJRWuU(>f6*H:w?F]q7$$\"I ++++++++]il(fw4c\"\\#)o;F\\v$!I-%e8N$4`8kUAiH9.y;F\\v$!I.Poj;lf&40!yL(H#yDj+z?F]q7$$\"I++++++++](o/)**Hx UzB(o\"F\\v$!I]%GG&[(4O?IFfG@F2.\"fw?F]q7$$\"I+++++++++](=nhajXWkp\"F \\v$!I&)y**4D)3t?\\:T)3=wSffr?F]q7$$\"I+++++++++voa]y^$[d[r\"F\\v$!Ik3 qz-!)3\\CQT#37/EYHQ0#F]q7$$\"I++++++++++]P%3\"o50FL,$[mD?F]q7$$\"I+++++++++]PMnANJC_r F]q7$$\"I++++++++++DJ]M-_Vx4=F\\v$!IdZ6n:5f(eb#GvYL$f!*R%)z\"F]q7$$\"I +++++++++++DOhThF2V=F\\v$!IW%pmO3&)H6uZ`z!evZ.2Y;F]q7$$\"I++++++++++v= A)33lVL7>F \\v$!IFEw)[(ft2n#R^[j:4;ceB\"F]q7$$\"I++++++++++]7$\\w&fvH[>F\\v$!I&z% yqmW_az.`1f+nZuK)z*Fho7$$\"I++++++++++D\"y4w$e5X&)>F\\v$!IgF3<\"yBkT_' ewE.%p_!32pFho7$$\"I+++++++++++]-d&)Fho7$$\"I+++++++++]Pf`\"Q%4 Oo/AF\\v$\"IRDm$)e#yz1Ua!)y4&HyE\"f;F]q7$$\"I+++++++++++]KC&Hh/RJ#F\\v$\"I/r%*GF-*3R= k<9cICv.%Q=F]q7$$\"I++++++++++]7)eN.*oO^BF\\v$\"I:Pk19cdeb([Lm&QJhJg!) >F]q7$$\"I+++++++++++vV(=x;H))Q#F\\v$\"I%GD:l()z=))ejGUbT;YJe1#F]q7$$ \"I++++++++]7`WJLaldJ(R#F\\v$\"IUe.W\\klR%p>.vj40gxm2#F]q7$$\"I+++++++ ++D19>zOjB!eS#F\\v$\"Io]L#G()[@93:P9UL9D)G%3#F]q7$$\"I++++++++]Pf$o]#> h*)G9CF\\v$\"IVE'>1W*RqXUFIQqY'o>')3#F]q7$$\"I+++++++++]7`%4/y?n?#eYhK'*3#F]q7$$\"I++++++++]ilA#oTo:i7V#F\\v$\"I\"Q \"[&e9$>%Rsb]`UQoM#H(3#F]q7$$\"I+++++++++v=#*pima([(RCF\\v$\"I(>cJ/dWP 2GQZM=='e*o:3#F]q7$$\"I++++++++](=scCF\\v$\"Ita>QiMVl**ei'\\zw.\"z()f?F]q7 $$\"I+++++++++]7Gt9UAV$Q\\#F\\v$\"I`T^HJn>)4Zf!\\qPbk$HW'>F]q7$$\"I+++ ++++++++D,v_%pY4`#F\\v$\"I_@d+jnUV]AhFA/Te!RL!=F]q7$$\"I+++++++++]7`%[ WWamec#F\\v$\"I@'3Wf=dC_^d=+V'Fho7$$\"I+++++++++]i!*Q^T,dZ4FF\\v$\"Ig9K3%e$z]8M'Gu(oPA> jhDFho7$$\"I+++++++++++v=vu)Ri^u#F\\v$!IoKd]l`7Mwrns4#*)RDo'p9Fho7$$\" I+++++++++]7`WNejI_#y#F\\v$!I?!3_oT]iUtn2K8:E**zAq&Fho7$$\"I++++++++++ DJq&>%GP))>GF\\v$!I$Hg$>IzEUM:Vbe(*Fho7$$\"I+++++++++]il2q2*[me& GF\\v$!I(pRq7jIZ([%4:1E+D8t/L\"F]q7$$\"I++++++++++++XWt\\#\\=*GF\\v$!I .h]e(\\3-#**[3#R#zULi4I;F]q7$$\"I++++++++++v$4&>h>okGHF\\v$!Is(*['*RtY 27e0[3&='ppPf=F]q7$$\"I++++++++++](oX*[*QWa'HF\\v$!I#H]Hs_g?7@8U2`CzV= 5*>F]q7$$\"I++++++++](=nQ^WjgnX(HF\\v$!Ill1&)fx=vNV_Fqh9mCX1?F]q7$$\"I +++++++++v$f3d*>B3p$)HF\\v$!I+$[4fA7>-vn!>s9&orbX,#F]q7$$\"I++++++++]i :&yia+/9G*HF\\v$!IYn1\\iZA.4*4\"QlhD>#4^,#F]q7$$\"I+++++++++]P%[o4pDP> +$F\\v$!IA$)[7\")\\&[([_CgM]$*\\A\"z+#F]q7$$\"I+++++++++D\"G))z>1p$=?I F\\v$!IHR(Q#H9QB(fRY'oKuKmcp>F]q7$$\"I++++++++++D\"G\"*HV7I%QIF\\v$!I8 76cFK`@:!)G:3;w<=N)*=F]q7$$\"I+++++++++]il2l[zK'>2$F\\v$!IOezf'3'*4D%o Hy8VE$)4iz;F]q7$$\"I+++++++++++]-JkMk\\0JF\\v$!I)3p;_BM&)zIoCTYS23AxM \"F]q7$$\"I+++++++++](o/9bY#GrCJF\\v$!I$o#oi8k0VXVwy\"Q)Fho7$$\"I+++++++++ ]iS;#zYgXJ;$F\\v$!IZ%*Hf4$Hk/?x(>p1LzEN!R&Fho7$$\"I++++++++++]Pa7p%*>O #=$F\\v$!I/Ifjf(RQ)e50\"zaW\\y!yn@Fho7$$\"I+++++++++]7`p#)*HO\\&*>$F\\ v$\"I#>oQ)fb*)y-;%4uTSV!)eWc)FX7$$\"I++++++++++vo%G08tOn@$F\\v$\"IR$)f nG*ePd\\zY66D$F\\v$\"Iyzu&fZGssNGN$ zs)HEtx+\"F]q7$$\"I+++++++++DJ?.=M(yO%pKF\\v$\"I6!yPtiD)G\"Ra'HB.ekhR5 8F]q7$$\"I+++++++++]iS\"Hkn5ixG$F\\v$\"IE\"\\6@g&ReI?CX$oKv9P%z:F]q7$$ \"I+++++++++v$4'zn=Eu31LF\\v$\"I.2%y:WR4X?x***=\\ZLT:*z\"F]q7$$\"I++++ ++++++D\"yE4cu7WK$F\\v$\"Ie(>puX#=sl)fC#3R,;3Y^>F]q7$$\"I+++++++++vV)R '>3!e#=LLF\\v$\"IcSJbu)\\(\\\")=l'eDy^?pX*>F]q7$$\"I+++++++++]i:gYb9C& >M$F\\v$\"IF_*pD6Qe=-4\")3@-g,p],#F]q7$$\"I++++++++](=U#35zJtLYLF\\v$ \"I&o)HLE;ju\"eop;;())pL-;?F]q7$$\"I+++++++++D\"GjNF!\\As]LF\\v$\"I_QI 9rm#H$ef@oL%HmED.,#F]q7$$\"I++++++++]iST/PEmr5bLF\\v$\"I:pNVoK\"=Z&flZ nf*fg?w*>F]q7$$\"I+++++++++++]_+]$3#\\fLF\\v$\"I+MT'y-Qxh%*o,P(p`C]ax> F]q7$$\"I+++++++++]P%[WXCvJqP$F\\v$\"IBf^@fB\\>/Zbb_OS,d(e\"=F]q7$$\"I ++++++++++v=P3R@9d%R$F\\v$\"Ij2Hhy,bhuuCc.%*e>`J/:F]q7$$\"I++++++++]i! RD)p@1e*RS$F\\v$\"I<=GCJmGwx\"zT@%oe3L5l7F]q7$$\"I+++++++++D1*y7V5>?MT $F\\v$\"I9Uj^\\^8!yA#om351Dpi&p*Fho7$$\"I++++++++](=UKFpedWGU$F\\v$\"I AS`(=;vo:55Hu#32uu%f7'Fho7$$\"I+++++++++]Pf=apg*oAV$F\\v$\"I,6owyA>fz% RN!*f'fRc9))=Fho7$$\"I++++++++DJ&p7\\3J:\")pV$F\\v$!Ih7eA-S5Be)y*4'>$o p>R*)\\FX7$$\"I++++++++]7`%Rc@bM$pTMF\\v$!ID#G!>tWn:#Hs1CX^D8-Y2$Fho7$ $\"I++++++++v$4@mjMz`0kW$F\\v$!I?.=/mDW(*4e9Iof@x)Hk%eFho7$$\"I+++++++ ++voH4xMIx6^MF\\v$!I_ER1\\y[OMAq\\'*fX1(4A#))Fho7$$\"I++++++++DcE(>ygF #*HeX$F\\v$!IkL&=+xnzI$e-dV*fjK$*4?\"F]q7$$\"I++++++++]P%[Y&Q<:@agMF\\ v$!Il`!RzGr_`X:!RF]q7$$\"I++++++++++++++++l'*pMF\\v$!ILhu=*R1h$>JQ))yG@gHy #H#F]q-%&COLORG6&%$RGBG$\"\"(!\"\"F)$\"\"*Fggm-%*THICKNESSG6#\"\"\"-F$ 6%7$7$$\"I0h0'=)y9Pv)f=IGo7FB9T(F`oF(7$Fbhm$\"Id+,mv#zuQ2_$Q-^*\\'4[0? F]q-Fbgm6&Fdgm$\"\"$Fggm$\"\")FggmFihmFjgm-F$6%7$7$$\"I3to')\\#>H*)pn0 *Q()p=$=Lg\"F\\vF(7$Faim$!I?2a$z#f^*R2_$Q-^*\\'4[0?F]qFghmFjgm-F$6%7$7 $$\"Idf&zJSE,f_>2q1a'oF**fBF\\vF(7$Fjim$\"IBtEZh7aHu?NQ-^*\\'4[0?F]qFg hmFjgm-F$6%7$7$$\"IiB)fVeW9\"\\<9POV^[^$Q(HF\\vF(7$Fcjm$!I&e8z$*e(40t? NQ-^*\\'4[0?F]qFghmFjgm-F$6%7$7$$\"I2@Pk+;Z&zf!>2@)HE(p*3#F]qFc\\nFg\\n-F $6%7$7$$\"I[e01?Jf:)=Pszxf(3:#*))HF\\vF(7$F`^n$!ISlE5[#>EA'oJU742\"z4e ,#F]qFc\\nFg\\n-F$6%7$7$$\"Ie')[!H]]N6Z&p%yj'z[zzWLF\\vF(7$Fi^n$\"I7,[ (4=T2i9L)Qx`&eCGk,#F]qFc\\nFg\\n-F$6%Fc[nFc\\nFg\\n-%+AXESLABELSG6$Q\" x6\"Q!Fd_n-%%VIEWG6$;F($\"0+++]m*pM!#:;$!06#)HE(p*3#!#K$\"06#)HE(p*3#F ``n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iterati on~15G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-------------------------- ------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rationa l~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,($\"LoE1QZ_#H: J&o!3YDLLLLLLL!#V\"\"\"*&$\"L*G\"Ggax\"zjK/s&=pS:X0Fz^B0iY[%F'F(F-F(F(*&$\"L6\"yE`Or'zb(3] $3sfql,;c9(Q?!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0B%z<7t10sfql,;c9(Q?!#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3# y/.OJT?D$!#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference: G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"IK@f8*4#y'Gvfji[U`-V^f\"!#h" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ix*e$>9C\\X0!3N'G5T$z`lJ%!#d" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Scritical~points~in~the~right~half~of~the~interval:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6($\"Lv$R=,_,uWB=tlEWY)*\\t&yu(!#W$\"L-w!>z5w=-R5(** faIlk/Ygn\"!#V$\"L+Al3Or?+dpl]%QC*\\$o*p(=CF($\"L%eXZj;V>:@t`3E^]Viac' )HF($\"LJ_TJ$)f^t%)3qvIn!pDjDDWLF($\"I+++++++++++++++++]mM!#S" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%7minimax~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,($\"D Z_#H:J&o!3YDLLLLLLL!#N\"\"\"*&$\"Dbx\"zjK/s&=pS:X0Fz^B0iY[%F'F(F-F(F(*&$\"Dl8nzb(3]%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,($\"DZ_#H:J &o!3YDLLLLLLL!#N\"\"\"*&$\"Dbx\"zjK/s&=pS:X0 Fz^B0iY[%F0F1F6F1F1*&$\"Dl8nzb(3]$3sfql,;c9(Q?!#_" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"x G" }{TEXT -1 12 " defined by " }{XPPEDIT 18 0 "s(x) = x-r(x)*`.`*x^3; " "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$!\"\"" } {TEXT -1 14 " approximates " }{XPPEDIT 18 0 "tanh(x)" "6#-%%tanhG6#%\" xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The relative error graph is drawn for " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 11 ", which is " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 72 " with the coefficients rounded to 20 digits and arranged in nested form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 384 "s:=x->x-(.33333333333333254608068531152925247+.16155 401745154069185720432637917755e-1*x^2+.6383900599517735025449664888863 1775e-5*x^4)/(1.+.44846620523517927051940716650320815*x^2+.17500871902 129655186559175579671365e-1*x^4)*x^3:\nh := unapply(evalf(convert(s(x) ,horner),20),x);\nevalf(plot((1-h(x)/tanh(x)),x=-ln(2)/2..ln(2)/2,\n \+ color=COLOR(RGB,.4,0,.9)),25);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\" \"\"**$F.\"\"!F.,($\"53YDLLLLLLL!#?F.*&$\"5'=pS:X(3])=%zK0IHcMF*$ \")7q38F*7$$!:DlO\\&\\?fc:d^MF*$\"(_#4(*F*7$$!:;Lz;s:cy5]oW$F*$\"(_bb' F*7$$!:2,A%)[E?\"f'G@W$F*$\"(Uuh$F*7$$!:)*ok^DP%Q5sSPMF*$\"'_m))F*7$$! :*ot!>-[[;w&oKMF*$!'&[k\"!#C7$$!:z/]')ye7HJkzU$F*$!'!\\)RFP7$$!:rs#Rb& pwT'GCBMF*$!'7ThFP7$$!:iSN@K!3W:9_=MF*$!'%47)FP7$$!:`3y))3\"\\qm**z8MF *$!'eJ**FP7$$!:Vw?c&=!pz^y!4MF*$!(3!e6FP7$$!:D71\"*QB(\\?cj*R$F*$!(!yT 9FP7$$!:1[\"fA\\a-BF>!R$F*$!(g'o;FP7$$!:p>i&*)z=3GpIrLF*$!(98(>FP7$$!: L\"H`c5$QJ8@CN$F*$!(6C5#FP7$$!:9F=+f_mcByHM$F*$!(*)Q6#FP7$$!:'HO]BTZ>Q ``LLF*$!(9V4#FP7$$!:y)*))pl&HsSC4CLF*$!(Xr/#FP7$$!:gMu/><^Ka\\YJ$F*$!( Hc(>FP7$$!:qI\"e0pD09r@)H$F*$!(e?!=FP7$$!:!o#)o?mR&[o%y\"G$F*$!($[y:FP 7$$!:!H_zNj`lbANlKF*$!(YyJ\"FP7$$!:,>-40wck#)>*[KF*$!(Z:.\"FP7$$!:5:4g w:esR([KKF*$!'e&H(FP7$$!:@6;6[bf!o\\0;KF*$!'S0UFP7$$!:J2Bi>&4')QDi*>$F *$!'_>6FP7$$!:U.I8\"\\Bm4,>$=$F*$\"(2&)*=F*7$$!:6#4!*\\+WTC,rkJF*$\"(? `9&F*7$$!:z!=Z)=Xm\"R,BYJF*$\"(&ew\")F*7$$!:[pUqK]=R:]x7$F*$\")_J%4\"F *7$$!::e8cYbq'o,F4JF*$\")7wS8F*7$$!:_NbFulu\")>5B2$F*$\")H;Lg(ywA]`.$F*$\")Ls$)>F*7$$!:a7*pt\"z#4Xxu;IF*$\")<*e0#F*7$$!:?7,vK# o]i_9)*HF*$\")(oH4#F*7$$!:.7-W!RQ@@S%)))HF*$\")tv)4#F*7$$!:'=JI\"[&3#* zFazHF*$\")gJ'4#F*7$$!:p6/#eqyiQ:CqHF*$\")K(e3#F*7$$!:^60^j)[L(HS4'HF* $\")xnn?F*7$$!:$3\"4F%\\H;K`tBHF*$\")h!H#>F*7$$!:958.D,\"*pOIl)GF*$\") '=xn\"F*7$$!:.0^&R0T_IA]\\GF*$\")N8b8F*7$$!:$***)yG)>dS4uC\"GF*$\"(%pg (*F*7$$!:P(zStWP#e-gRz#F*$\"(=9s(F*7$$!:#[p-=\"H!fdfWvFF*$\"(S7i&F*7$$ !:F#fkiPoN*)=$pv#F*$\"(m`[$F*7$$!:t*[E2%QB6#yTQFF*$\"(1zL\"F*7$$!:2#oh -kro)*z3/FF*$!'-)f#FP7$$!:UuozR%4Dw\"e(pEF*$!'SjjFP7$$!:xm?LRs9QNGaj#F *$!'hV)*FP7$$!:6fs')Q]y8`)4,EF*$!(NVH\"FP7$$!:-muh!=nq))>blDF*$!(irc\" FP7$$!:$HnnBK\\.Ya+IDF*$!(O\\y\"FP7$$!:$)zy6k9jL!*eW\\#F*$!(cO%>FP7$$! :u'3oeg8pgB\"*eCF*$!(p8/#FP7$$!:g#GYm16P*y@(\\CF*$!(fm0#FP7$$!:YyWUF&3 0=7`SCF*$!(xy1#FP7$$!:KuE?))fInkS8V#F*$!(Y]2#FP7$$!:=q3)*[M5a2]@U#F*$! (+#y?FP7$$!:/m!f(44!4/&fHT#F*$!(yt2#FP7$$!:!>EP0P)pF$*oPS#F*$!(GE2#FP7 $$!:wdaJJe\\9OyXR#F*$!(+S1#FP7$$!:i`O4#H$H,z(Q&Q#F*$!(a:0#FP7$$!:1Pk?N J[[]D'[BF*$!(f\\'>FP7$$!:]?#>$yHn&>K'=J#F*$!(wL#=FP7$$!:iZ1fAT,9$)=_F# F*$!(AOj\"FP7$$!:uu?'oEbBVWdQAF*$!('H-9FP7$$!:'=]L6T'p]0I>?#F*$!(:q8\" FP7$$!:)*G\\Sbv.pm&Gl@F*$!''zX)FP7$$!::$=!G\"*G2&4CfF@F*$!'8y_FP7$$!:J Pa:F#36_\"**)*3#F*$!'Z**>FP7$$!:[\"pIIcVr%*e?_?F*$\"(8;H\"F*7$$!:kXf!* )*)yJPE^9?F*$\"(.I^%F*7$$!:.yBTX@'o&R78)>F*$\"(`>B(F*7$$!:U5)=>RX0a@6[ >F*$\"(Z-z*F*7$$!:\"GCD%Q'GU7>\"\\\">F*$\")Ma97F*7$$!:?v;$\\)=\"zq;r\" )=F*$\")x/E9F*7$$!:7AMoIM[NXgp!=F*$\")$p!*z\"F*7$$!:/p^Vw\\0jB4At\"F*$ \")['3,#F*7$$!:eLd0#*zu!)pWMr\"F*$\"))*GP?F*7$$!:6)Hww+T%)f,o%p\"F*$\" )))*G0#F*7$$!:kioHBS8;i:fn\"F*$\")kud?F*7$$!:=Fu\"*Qq#Q$3^rl\"F*$\")`% >0#F*7$$!:Cc&e,28#p+A'>;F*$\")m44?F*7$$!:J&o*R,\"*f/$H4#e\"F*$\")6CE>F *7$$!:k/8.3F\"=#\\g(4:F*$\")/&Qm\"F*7$$!:'R#Hm9j-R0GuV\"F*$\")'oyG\"F* 7$$!:e*ewm:%Q@m&e/9F*$\")W&y3\"F*7$$!:?b-p)*>u.FV%FP7$$!:m!pi1$fy_OZ)\\6F*$!'@\\oFP7$$!:'yk+Q1^sPXw;6F*$!'J& )*)FP7$$!:10'Qp>;<5X@jkk]0;v,\"F*$!(o?X\"FP7$$!:!4UK`$*zv'o'\\0%*!#E$!(%om&Q_pG#G$ej)F*$!(=Z'>FP7$$!:`Sy$yOu$eN\">&H)F_[m$!()f8?FP7$$!:vLg:(\\ `!))))\\X&zF_[m$!(9)Q?FP7$$!:PI^\"=1$*Gb\"HUy(F_[m$!(1F/#FP7$$!:)pAuki Kx@%3Rh(F_[m$!(:4/#FP7$$!:fBL8\">sD)o(eVuF_[m$!(iN.#FP7$$!9-U#zbFP7$$!9Yf_JN;LUiJydF*$!(Kc q\"FP7$$!::tZ5&z2qSHam]F_[m$!(V%e9FP7$$!9<&p0P#*p!R'pZN%F*$!(ox<\"FP7$ $!:vv:#pc2uo)3@h$F_[m$!'`W()FP7$$!9)*>'y'*e6%)4[%pGF*$!'JieFP7$$!:l&H< G.xkxOX(=#F_[m$!'$[b$FP7$$!9:R[)o\"Q)oDfa]\"F*$!'FP7$$\":D\\ %)=OoF%\\\")RyMFe`m$!%6&*FP7$$\":(y\\U^a@;QBm)Q&Fe`m$!&!yAFP7$$\":]Yl4 ai'*o_E*)H(Fe`m$!&t;%FP7$$\":Qk/?nbO/\\X>6\"F_[m$!&-f*FP7$$\"9TF\"**3X $=G$)*R\\\"F*$!'b5(F*$!(2M,#FP7 $$\":I+B3Cwr-#R;ttF_[m$!(G*G?FP7$$\":?:TXxyu,n*y[vF_[m$!(`(Q?FP7$$\":5 If#38y2?aTCxF_[m$!(=F/#FP7$$\":+Xx>%Q3)*p6/+zF_[m$!(%oS?FP7$$\":!*f&pv jQ))>pmv!)F_[m$!(KD.#FP7$$\":![PT4*)oypEH^#)F_[m$!(o\"=?FP7$$\":q*=8V9 **o>%=pU)F_[m$!(@v*>FP7$$\"9Y+&o(RHfpTa-')F*$!(Y0(>FP7$$\":Ssz;(z&zr(R b#Q*F_[m$!(?Vx\"FP7$$\":-%4l'>iw%yjD;5F*$!(#4e9FP7$$\":ueTZiL*p`+I^5F* $!(![!G\"FP7$$\":YBKG00A*GPM'3\"F*$!(iM3\"FP7$$\":=)G#4[wWTS(Q@6F*$!'8 %p)FP7$$\":\"HN,4zuOz5Vc6F*$!'o5kFP7$$\":i+tJPV!f#oZQ>\"F*$!'S\\QFP7$$ \":M[Kt$)Q8eGk7B\"F*$!',,7FP7$$\":1'>\\,Vj.*)3oo7F*$\"(Q/\\\"F*7$$\":x V^cwHfA\\(418F*$\"(t$yTF*7$$\":%y0`O\"**zf\\-+M\"F*$\"(x1d'F*7$$\":#>( 4u]o+(*\\2RP\"F*$\"(&)R)))F*7$$\":*f))Gyy8U.D\"yS\"F*$\")]<36F*7$$\":1 +o\"\\s?92vrT9F*$\")iz78F*7$$\":J_V[keXiv^e^\"F*$\");s!p\"F*7$$\":c/>0 /5\\`+')**e\"F*$\")2(o%>F*7$$\":s6gW!RP%3Gj[i\"F*$\")e`zrn\"F*$\")kvd?F*7$$\":/EUBj,L=$ yh%p\"F*$\")B$H0#F*7$$\":iz7VcL!epk07r3C!=F*$\")Pu;=F*7$$\":0E8/\")y:mJA`(=F *$\")Msj9F*7$$\":C9O34N#e/b'4\">F*$\")%35C\"F*7$$\":U-f7P\"*[Dp3m%>F*$ \"(!f,**F*7$$\":h!>o^wa^!)=D#)>F*$\"(3q:(F*7$$\":zy/@$R?[o]*y,#F*$\"(, )GUF*7$$\":#Hv2#GHWI=5_0#F*$\"(13.\"F*7$$\":0F]?ja1wHDD4#F*$!'GHAFP7$$ \":<,B?)*zo@TS)H@F*$!'UraFP7$$\":Iv&*>L0Jn_br;#F*$!'b6')FP7$$\":d:_,&p 51aS\"FP7$$\":5'\\Y'=5@x >s\\F#F*$!(&>K;FP7$$\":OO@Y!=60)364J#F*$!(K!>=FP7$$\":rx5fS]#=%zjwM#F* $!(J>'>FP7$$\":1>+s+*QJ+lT%Q#F*$!(=+0#FP7$$\":RaAvltYon/OR#F*$!(hG1#FP 7$$\":t*[%yIezL&Gz-CF*$!(&*=2#FP7$$\":1Dn\"eHC\"*H5)>T#F*$!(fq2#FP7$$ \":Tg*[3w_W1#p6U#F*$!(-$y?FP7$$\":u&>\")eA\"yHQd.V#F*$!(ub2#FP7$$\":3J M\"4p4^fbaRCF*$!(R)o?FP7$$\":Umc%f:Q/OPt[CF*$!(g!e?FP7$$\":x,z(4imd7># zX#F*$!(9K/#FP7$$\":;\\lJ#R_u(GqV\\#F*$!(wR%>FP7$$\":b'>bO;Q\"Hm=3`#F* $!(.1y\"FP7$$\":&R%Q*\\$R#3QqEnDF*$!(B_b\"FP7$$\":M\"\\Kjq4D8ar.EF*$!( GAF\"FP7$$\":ye&='[3\\Yo2sj#F*$!'%=n*FP7$$\":AEY!4*>Zg&**pqEF*$!'VjiFP 7$$\":m$p!>LJXuA#>/FF*$!'F'e#FP7$$\":5hnZvUV))\\%oPFF*$\"(KHD\"F*7$$\" :BY-N7*\\:gu(ov#F*$\"(h!zMF*7$$\":PJPA\\bm9Uqgx#F*$\"((z#p&F*7$$\":];s 4'=\"yFQj_z#F*$\"(rs'yF*7$$\":l,2(H#o*3WjX9GF*$\"(VV(**F*7$$\":#>n8;5 cDHF*$\")+_K>F*7$$\":_;X=')*z#H%Q*)fHF*$\")q:l?F*7$$\":\"3]\\J1qi\\qZo HF*$\")&4I3#F*7$$\":5&[9,9gKc-1xHF*$\")g+_'\\f&GIF*$\")H?9?F*7$$\":!)Q&R=Eyv]a;lIF*$\")/z#z \"F*7$$\":v'ou<#f&*\\Sr<5$F*$\")u\\J9F*7$$\":ugAu^Z9@Qu+7$F*$\")G_+7F* 7$$\":rM)4$F*$\"'F+?F*7$$\":l()z z5=\"\\#))>+@$F*$!''z1$FP7$$\":9!\\u`47+n\"QvA$F*$!'jojFP7$$\":j#*4&*z B6:Xc]C$F*$!'<;'*FP7$$\":8&\\FXm7-OZdiKF*$!(m4F\"FP7$$\":i(*R5\\HJ0-$4 !G$F*$!(dJb\"FP7$$\":6+0oLKT]I6wH$F*$!(>Yz\"FP7$$\":f-qD=N^&*eH^J$F*$! (/)z>FP7$$\":yoyX_PHT[UXK$F*$!(q*\\?FP7$$\":'\\tem)R2(y`&RL$F*$!(&z&4# FP7$$\":9,'f3AaGt#oLM$F*$!(bS6#FP7$$\":Ln/1bWjy;\"y_LF*$!(585#FP7$$\": p*>iM#\\>q&pgrLF*$!(Qz'>FP7$$\":1KR'=RbxyPBMF*$!' &>3'FP7$$\":z'Rn'Gj([CV3GMF*$!'tFRFP7$$\":*)HywXkws\\PMF*$\"'Co$*F*7$$\":2'po*zmajm.AW$F*$\"(vCm$F*7$$\":;H \"pqzOk8,\"pW$F*$\"(mUf'F*7$$\":Di&pT\"pK4c;;X$F*$\"(!QS(*F*7$$\":M&** p7. " 0 "" {MPLTEXT 1 0 74 "evalf(evalf(numapprox[infnor m](1-h(x)/tanh(x),x=-ln(2)/2..ln(2)/2),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&U6#!#A" }}}{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 "tanh(x);" "6#-%%tanhG6#%\"xG" }{TEXT -1 17 " in the int erval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\" \"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 12 " is about " }{XPPEDIT 18 0 "2*`. `*10^(-18);" "6#*(\"\"#\"\"\"%#.~GF%)\"#5,$\"#=!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "Now round the coefficients to 16 decimal digits, which i s roughly equivalent to the precision of standard double precision bin ary hardware floating point numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "h := unapply(evalf(convert (s(x),horner),16),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\" xG6\"6$%)operatorG%&arrowGF(*(9$\"\"\",&$\"1+++++++5!#:F.*&,&$\"1cS]5b K^6!#;F.*&,&$\"1F=nfsWX8!#=F.*&$\"1$ecY)yR$Q'!#@F.)F-\"\"#F.!\"\"F.FAF .F.F.FAF.F.F.,&$F.\"\"!F.*&,&$\"1yt$Q%)eY[%F7F.*&$\"1ehV0U2] " 0 "" {MPLTEXT 1 0 81 "xx := eval f(ln(2)/2,20);\nevalf(evalf(h(xx),20),16);\nevalf(evalf(tanh(xx),20),1 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5raE(*z-ftlM!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1LLLLLLLL!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1LLLLLLLL!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approxima tion " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "tanh(x);" "6#-%%tanhG6#%\"xG" }{TEXT -1 35 " with rando m numbers between 0 and " }{XPPEDIT 18 0 "ln(2)/2;" "6#*&-%#lnG6#\"\"# \"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The prin tout occurs only when the relative error is greater than or equal to t he specified \"" }{TEXT 262 3 "eps" }{TEXT -1 3 "\".\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "random ize():\neps := Float(5,-16);\nfor i from 1 to 100 do\n xx := evalf(r and()/Float(1,12)*.34657359027997265471,20);\n axx := evalf(tanh(xx ),20);\n hxx := evalf(h(xx),16);\n e := evalf(abs((axx-hxx)/axx),2 0);\n if e>=eps then \n printf(\" trial no. %d, x = %.16f,\\n \",i,xx);\n printf(\" tanh(x) = %.16f, h(x) = %.16f, rel error = %.2e\\n\\n\",axx,hxx,e); \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"&!#;" }}{PARA 6 "" 1 "" {TEXT -1 39 " tria l no. 29, x = .1241919692473273," }}{PARA 6 "" 1 "" {TEXT -1 77 " ta nh(x) = .1235573863679499, h(x) = .1235573863679500, rel error = 6.06e -16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " \+ trial no. 43, x = .1701437365785448," }}{PARA 6 "" 1 "" {TEXT -1 77 " tanh(x) = .1685207038234639, h(x) = .1685207038234640, rel error = 5 .18e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 68, x = .0127726990436665," }}{PARA 6 "" 1 "" {TEXT -1 77 " tanh(x) = .0127720045017784, h(x) = .0127720045017784, rel er ror = 5.59e-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 6 "tanh16" }}{PARA 0 "" 0 "" {TEXT -1 158 " Here is the code for the fixed precision versions of the hyperbolic si ne and cosine functions, which can be evaluated with hardware floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 45 ": These procedures make use of the procedure " }{TEXT 0 5 "exp16" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 656 "tanh16 := proc(xx)\n local x,z,a1,a2,a3,b1,b2 ,num,den,val,k,t; \n \n x := evalf(xx); \n\n if abs(x)<.34657359 02799727 then\n # coefficients of numerator in rational approxima tion\n a1 := .33333333333333254608;\n a2 := .161554017451540 69186e-1;\n a3 := .63839005995177350254e-5;\n\n # coefficien ts of denominator in rational approximation\n b1 := .448466205235 17927052;\n b2 := .17500871902129655187e-1;\n\n # evaluate t he rational approximation\n z := x*x;\n num := (a1+(a2+a3*z) *z)*z;\n den := 1+(b1+b2*z)*z;\n (1-num/den)*x;\n else\n \+ t := exp16(-2*x);\n 1-2*t/(t+1);\n end if;\nend proc: # tan h16" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "Testing th e procedure tanh16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We can use Maple's built-in function " }{TEXT 0 4 "tanh" }{TEXT -1 41 " to check the accuracy of the procedures " }{TEXT 0 6 "t anh16" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf(plot(1-'tanh16'(x)/tanh(x),x=0..1,c olor=blue),25);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7][l7$$\":ILLLLL$3x1h6o!#G$!$l$!#C7$$\":mmmmmm;a8ABO\"! #F$!%h9F-7$$\":***********\\7.K[V?F1$!%'G$F-7$$\":KLLLLLL3FWYs#F1$!%Re F-7$$\":)***********\\iSmp3%F1$!&BJ\"F-7$$\":lmmmmmm;a)G\\aF1$!&%HBF-7 $$\":(************\\7G$R<)F1$!&z@&F-7$$\":LLLLLLL$3x&)*3\"!#E$!&%=#*F- 7$$\":+++++++Dc'yM;FP$!'BP?F-7$$\":mmmmmmmmT:(z@FP$!'>JNF-7$$\":****** ******\\7y%*z7$FP$!'&H$oF-7$$\":LLLLLLLe9ui2%FP$!(!*Q1\"F-7$$\":mmmmmm \"H2Q\\4YFP$!(.1G\"F-7$$\":************\\(oMrU^FP$!(-p[\"F-7$$\":LLLLL L3-8Lfn&FP$!(.In\"F-7$$\":mmmmmmm;z_\"4iFP$!(i&H=F-7$$\":lmmmmmm\"zp!f u'FP$!(s'[>F-7$$\":lmmmmmmm;hEG(FP$!(J;-#F-7$$\":kmmmmm;/EQ5b(FP$!(T)Q ?F-7$$\":kmmmmmmTN:%>yFP$!(rB/#F-7$$\":kmmmmmT5!Rg`zFP$!(^)Q?F-7$$\":k mmmmm;zW#z(3)FP$!(T<.#F-7$$\":kmmmmm\"z%*4)>A)FP$!(,5-#F-7$$\":lmmmmmm ;aphN)FP$!()f1?F-7$$\":ILLLLL3-js.*))FP$!(5E\">F-7$$\":(***********\\( =ddC%*FP$!(b-w\"F-7$$\":jmmmmm\"H2)y(e**FP$!([;b\"F-7$$\":LLLLLL$e*=)H \\5!#D$!(p6H\"F-7$$\":mmmmmmmTlD))4\"F_t$!(T!45F-7$$\":++++++](=JN[6F_ t$!'<[pF-7$$\":LLLLLLLLe!)y>\"F_t$!'QnNF-7$$\":mmmmmm;z/3uC\"F_t$!%OUF -7$$\":++++++](o3p)H\"F_t$\"(`)[OF_t7$$\":LLLLLL$e*ot*\\8F_t$\"([3E(F_ t7$$\":mmmmmm;/^c7S\"F_t$\")#\\n1\"F_t7$$\":++++++]7LRDX\"F_t$\")dMu8F _t7$$\":mmmmmm;z4wb]\"F_t$\")G#\\k\"F_t7$$\":LLLLLL$ekGhe:F_t$\")%*=b= F_t7$$\":++++++]7j\\;h\"F_t$\")Ww%*>F_t7$$\":mmmmmm;zR'ok;F_t$\")dab?F _t7$$\":+++++](=nI!zn\"F_t$\")*Rx0#F_t7$$\":LLLLLLekt>6p\"F_t$\")Mka?F _t7$$\":mmmmm;HdSOVq\"F_t$\")b@Y?F_t7$$\":+++++++]2`vr\"F_t$\")4VK?F_t 7$$\":mmmmmmTNT')Ru\"F_t$\")Fv))>F_t7$$\":LLLLLL$3_(>/x\"F_t$\")]nB>F_ t7$$\":mmmmmmm\"HkGB=F_t$\")auI))e9F_t7 $$\":LLLLL$3x\")H`I>F_t$\")u]16F_t7$$\":mmmmmm\"HdG\"\\)>F_t$\"(ZS%pF_ t7$$\":LLLLL3_]z-@,#F_t$\"(rZr%F_t7$$\":**********\\7Gt#HR?F_t$\"(PqS# F_t7$$\":mmmmm\"HdqE[m?F_t$\"&^*[F_t7$$\":LLLLLLL$3En$4#F_t$!'lHBF-7$$ \":mmmmmm\"HK/dT@F_t$!'ctkF-7$$\":************\\iDo%*=#F_t$!(![S5F-7$$ \":LLLLLL3-3mtB#F_t$!(YSR\"F-7$$\":mmmmmmmT!RE&G#F_t$!(#3!p\"F-7$$\":* **********\\P4b=RBF_t$!(BO$>F-7$$\":LLLLLL$e9r5$R#F_t$!(jA1#F-7$$\":mm mmmT&)e^(e1CF_t$!(+X2#F-7$$\":**********\\(=(GqW#F_t$!(n.1#F-7$$\":LLLLL$e RA&*)RZ#F_t$!(?u+#F-7$$\":+++++++]K]4]#F_t$!($>>>F-7$$\":+++++++vL$4bD F_t$!(trj\"F-7$$\":++++++++NO#4EF_t$!(IZA\"F-7$$\":++++++]i&yIOEF_t$!' %)e(*F-7$$\":+++++++DOzLm#F_t$!'uMqF-7$$\":++++++](o3X!p#F_t$!',@TF-7$ $\":+++++++]PAvr#F_t$!'8r5F-7$$\":+++++++vy5Ou#F_t$\"(l+%>F_t7$$\":+++ +++++?*ppFF_t$\"(P4'\\F_t7$$\":+++++++Dh(y&z#F_t$\"(\\e#zF_t7$$\":++++ +++]-w=#GF_t$\")TYw5F_t7$$\":+++++++vVkz%GF_t$\")POS8F_t7$$\":++++++++ &G0uGF_t$\")Wow:F_t7$$\":+++++++DET,!HF_t$\")!\\xx\"F_t7$$\":+++++++]n Hi#HF_t$\")/)f$>F_t7$$\":LLLLLekGX?*\\HF_t$\")FXO?F_t7$$\":mmmmm;H2B6O (HF_t$\")dc!4#F_t7$$\":LLLL$e9m>mX&)HF_t$\")(4))4#F_t7$$\":*********\\ Pf3?I(*HF_t$\")!QQ4#F_t7$$\":mmmm;/EvRZ\"4IF_t$\")A@v?F_t7$$\":LLLLLLe ky#*4-$F_t$\")QbU?F_t7$$\":mmmmm\"HKkNoWIF_t$\")#*)R$>F_t7$$\":******* ***\\(=UVPoIF_t$\")?nmF-7$$\":LLLLLLL $347TLF_t$!(!Q7@F-7$$\":KLLLLLeR7&)\\O$F_t$!(:E.#F-7$$\":KLLLLL$eR$\\) )Q$F_t$!(1mp\"F-7$$\":KLLLL$eRZ9y+MF_t$!(\"[59F-7$$\":KLLLLL3_b8FT$F_t $!(,D.\"F-7$$\":KLLLLe9\"4'z'=MF_t$!'Qd!)F-7$$\":KLLLL$3-jckCMF_t$!'B> bF-7$$\":KLLLL3Fpr61V$F_t$!'\"fp#F-7$$\":KLLLLLL3xxlV$F_t$\"'FxUF_t7$$ \":KLLLLeRZ#QaUMF_t$\"(ft'QF_t7$$\":KLLLL$eky)4&[MF_t$\"(+#RwF_t7$$\": KLLLL3_D$fZaMF_t$\")%**f<\"F_t7$$\":KLLLLLek)>WgMF_t$\")mqC;F_t7$$\":K LLLLek./3kY$F_t$!$4$F-7$$\":KLLLL$3F%4uBZ$F_t$!$5$F-7$$\":KLLLL3x\"[,M yMF_t$!$:$F-7$$\":KLLLLL$3-iI%[$F_t$!$I$F-7$$\":KLLLLL3xTq\"3NF_t$!$i& F-7$$\":LLLLLLLLjM?`$F_t$!%m7F-7$$\":**********\\7`9ave$F_t$!%xeF-7$$ \":mmmmmm\"HdO2VOF_t$!&ye\"F-7$$\":LLLLL$3FpJf)p$F_t$!&h6$F-7$$\":**** ********\\7o7Tv$F_t$!&L'\\F-7$$\":LLLLL$3xcoD.QF_t$!&Vc'F-7$$\":mmmmmm \"HK5S_QF_t$!&:z(F-7$$\":LLLLL3_+7tp(QF_t$!&)e\")F-7$$\":**********\\7 y?X:!RF_t$!&)4$)F-7$$\":mmmmm\"Hd&H)fYF_t$!'(=(=F-7$$\":KLLLLL3F9F-7$$\":KLLLL 3F>*[,4ZF_t$!'N()=F-7$$\":KLLLL$e9TEhLZF_t$!'DE#F_t7$$\":*********\\Pf3@`o\\F_t$\"(2W?#F_t7$$\":LLLL$3-Q 1\")z\")\\F_t$\"(?T9#F_t7$$\":mmmmmmmT5k]*\\F_t$\"(6U+#F_t7$$\":mmmmm; /wn#=]]F_t$\"'dLcF_t7$$\":mmmmmmT5D,`5&F_t$!'&=l\"F-7$$\":mmmm;/,W*33> ^F_t$!'6=Sm9&F_t$!'%yJ#F-7$$\":mmmm\"z%*z-+``^F_t$!'lx@F-7$$\" :mmmmm;zW#)>/;&F_t$!'oF>F-7$$\":mmmm;aQyY*>u^F_t$!''o-\"F-7$$\":mmmmm \"z>6\"zz=&F_t$\"'`]aF_t7$$\":mmmmTgxG$*o[>&F_t$\"(H4k\"F_t7$$\":mmmm; HdXveF_t7$$\":***********\\7y#> NE&F_t$\"(KnD#F_t7$$\":mmmmm;H#=(4vG&F_t$\"(\"R;Y2 aF_t$!'`k=F-7$$\":LLLLLLe*[K56bF_t$!'OO5F-7$$\":mmmmmmm\"zXu9cF_t$\"'h 'e)F_t7$$\":LLLLL$3-)[7:k&F_t$\"(y`<\"F_t7$$\":***********\\(oR!GocF_t $\"(G8Q\"F_t7$$\":mmmmm;HdI[]p&F_t$\"(a]Z\"F_t7$$\":LLLLLL$e9i\"=s&F_t $\"(LLY\"F_t7$$\":mmmmmm\"HK?NvdF_t$\"(e%z6F_t7$$\":***************\\y ))GeF_t$\"'$=s'F_t7$$\":++++++]ibOO$fF_t$!&>$RF-7$$\":*************\\i _QQgF_t$!&XK*F-7$$\":+++++vVBqAP1'F_t$!&(Q&*F-7$$\":+++++](=U,1*3'F_t$ !&PO*F-7$$\":+++++DJ?e(R9hF_t$!&Y%))F-7$$\":++++++v=-N(RhF_t$!&c.)F-7$ $\":+++++]i:!*4/>'F_t$!&Yz&F-7$$\":************\\7y%3TiF_t$!&R9$F-7$$ \":+++++]7`f]tH'F_t$!%%z#F-7$$\":++++++v$4kh`jF_t$\"'!4/#F_t7$$\":++++ +]PMA#))4kF_t$\"'OWNF_t7$$\":*************\\P![hY'F_t$\"'5qTF_t7$$\":m mmmmm;/risc'F_t$\"'7FNF_t7$$\":KLLLLLLLQx$omF_t$\"'')e=F_t7$$\":****** ********\\P+V)oF_t$!$1\"F-7$$\":mmmmmm;zpe*zqF_t$!%oTF-7$$\":LLLLLLe9 \"=\"p=(F_t$!&-f\"F-7$$\":**************\\#\\'QH(F_t$!&k,$F-7$$\":LLLL L$eR(>#=WtF_t$!&CT$F-7$$\":mmmmmm\"zp%*\\%R(F_t$!&6V$F-7$$\":********* *\\(=Un\"[W(F_t$!&t'HF-7$$\":KLLLLL$e9S8&\\(F_t$!&<)>F-7$$\":mmmmmm;/6 E.g(F_t$\"'uS9F_t7$$\":************\\i?=bq(F_t$\"'$e6&F_t7$$\":mmmmmT5 :LH7t(F_t$\"'*Gu&F_t7$$\":LLLLL$3xc/%pv(F_t$\"'dyhF_t7$$\":*********\\ 7.#e^Ey(F_t$\"'Z(Q'F_t7$$\":mmmmmm\"H2FO3yF_t$\"':TjF_t7$$\":LLLLL3_D$ Q2MyF_t$\"'??gF_t7$$\":**********\\7y&\\yfyF_t$\"'t;aF_t7$$\":mmmmm\"H 2$3'\\&)yF_t$\"'oNXF_t7$$\":KLLLLLL$3s?6zF_t$\"'I'R$F_t7$$\":mmmmmm\"z pe()=!)F_t$!&^9$F-7$$\":************\\7`Wl7)F_t$!&kD)F-7$$\":LLLLL3x1' *oC:)F_t$!&A_)F-7$$\":mmmmm;/,R$Ry\")F_t$!&lD)F-7$$\":*********\\7`>yJ /#)F_t$!&QU(F-7$$\":KLLLLLe*[ACI#)F_t$!&[-'F-7$$\":)*********\\7y5\"4# G)F_t$!&Pv\"F-7$$\":lmmmmmmm'*RRL)F_t$\"'?*e$F_t7$$\":kmmmm;/,W^/O)F_t $\"'-&='F_t7$$\":kmmmmmTN\"H'pQ)F_t$\"'[&H)F_t7$$\":kmmmm;zpQuMT)F_t$ \"'y$f*F_t7$$\":kmmmmm;/'e)*R%)F_t$\"'3p(*F_t7$$\":kmmmmm\"H2)3I\\)F_t $\"')[\"fF_t7$$\":lmmmmmmTvJga)F_t$!&k&HF-7$$\":)**********\\PM&*>^')F _t$\"&[$*)F_t7$$\":KLLLLL$e9tOc()F_t$\"':/tF_t7$$\":**********\\P%[ko/ ))F_t$!&L5\"F-7$$\":mmmmmm\"H#e0I&))F_t$!&!fsF-7$$\":*********\\(=#\\^ ;x))F_t$!&Gl)F-7$$\":LLLLL$e9;ZK,*)F_t$!&(Q*)F-7$$\":mmmmm\"H2$G%[D*)F _t$!&%y#)F-7$$\":***************\\Qk\\*)F_t$!&'*)oF-7$$\":LLLLLL3-.B]+ *F_t$!&\"=AF-7$$\":mmmmmm;/@-/1*F_t$\"'\")QDF_t7$$\":LLLLL$3_+=4)3*F_t $\"'!**R%F_t7$$\":***********\\i!R\"y:\"*F_t$\"'GbdF_t7$$\":mmmmm;H2)4 ZV\"*F_t$\"'xglF_t7$$\":KLLLLLL3dg6<*F_t$\"'NAoF_t7$$\":)***********\\ (oTAq#*F_t$\"'y\\UF_t7$$\":lmmmmmmmw(Gp$*F_t$!%@bF-7$$\":)*********\\7 `**)4A%*F_t$!&il#F-7$$\":JLLLLLeRA5\\Z*F_t$!&x)RF-7$$\":)********\\7GQ eJ,&*F_t$!&mK%F-7$$\":lmmmm;/EX@x_*F_t$!&=X%F-7$$\":JLLLL3Fp1FTb*F_t$! &*yVF-7$$\":)***********\\7oK0e*F_t$!&08%F-7$$\":)**************\\oi\" o*F_t$!&^5#F-7$$\":)***********\\(=5s#y*F_t$\"&#*z#F_t7$$\"\"\"\"\"!$ \"'(p%=F_t-%+AXESLABELSG6$Q\"x6\"Q!F`\\p-%'COLOURG6&%$RGBG$Fi[pFi[pFf \\p$\"*++++\"!\")-%%VIEWG6$;Ff\\pFg[p%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 60 "A procedure for evaluating the hyperbolic tangent funct ion: " }{TEXT 0 5 "tanh_" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "tanh_: usag e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 263 20 "Calling Sequence: " }{TEXT -1 11 "tanh_( x ) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 265 21 "x - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 268 12 "Description:" }{TEXT -1 1 " " }{TEXT 267 14 "The procedure " }{TEXT 0 5 "tanh_" }{TEXT 266 41 " calculates tanh(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 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 21 "tanh_: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 17 ": The procedures " }{TEXT 0 13 "`evalf/tanh_`" }{TEXT -1 33 " calls the subsidiary procedures " }{TEXT 0 6 "expm25" }{TEXT -1 5 " and " }{TEXT 0 6 "expm55" }{TEXT -1 62 " which use the same ration al approximations as the procedures " }{TEXT 0 5 "exp25" }{TEXT -1 5 " and " }{TEXT 0 5 "exp55" }{TEXT -1 35 " constructed for the evaluatio n of " }{TEXT 262 6 "exp(x)" }{TEXT -1 5 " via " }{TEXT 0 4 "exp_" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3178 "tanh_ := proc(x::algebraic)\n local n,t,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('tanh_'(x))\n \+ elif (type(x,'numeric') or type(x,'radnum')) \n \+ and signum(0,x,0)<0 then -tanh_(-x)\n elif type(x,`*`) and type(o p(1,x),'numeric') \n and signum(0,op(1,x),0)<0 then -tanh_(-x)\n elif type(x,'function') and nops(x)=1 then\n n := op(0,x);\n t := op(1,x);\n if n='arcsinh_' or n='arcsinh' t hen t/sqrt(1+t^2)\n elif n='arccosh_' or n='arccosh' then sqrt(t- 1)*sqrt(t+1)/t\n elif n='arctanh_' or n='arctanh' then t\n e lif n='ln_' or n='ln' then\n u := (t^2-1)/(1+t^2);\n i f type(u,radnum) then normal(u) else u end if; \n else 'tanh_'(x) \n end if\n elif type(x,'realcons') then\n if x=0 then 0\n elif x='infinity' then 'infinity'\n else 'tanh_'(x)\n \+ end if\n else 'tanh_'(x)\n end if;\nend proc: # of tanh_\n\n`evalf /tanh_` := proc(xx)\n local t,val,x,z,term,eps,k,maxit,saveDigits,hf Digits,sum,isneg;\n\n if not type(xx,realcons) then return 'tanh_'(x x) end if;\n\n # Use the fixed precision procedure for low precision values\n hfDigits := trunc(evalhf(Digits));\n if Digits<=hfDigits then\n x := evalf(xx,hfDigits+2);\n if abs(x)0.28782313662 425571050 then\n # Use formula tanh(x)=(exp(2*x)-1)/(exp(2*x)-1) \n t := `evalf/exp_`(-2*x);\n Digits := Digits+1;\n val := 1-2*t/(t+1);\n return evalf[saveDigits](val);\n else\n \+ if saveDigits<58 then\n # use the function exp(x)-1\n \+ if x<0 then\n x := -x;\n isneg := true;\n \+ else isneg := false end if;\n if saveDigits<26 then \+ \n t := expm25(-2*x);\n val := -t/(t+2);\n \+ val := evalf[saveDigits](val);\n if isneg then re turn -val else return val end if;\n elif saveDigits<57 then\n \+ t := expm55(2*x);\n val := -t/(t+2);\n \+ val := evalf[saveDigits](val);\n if isneg then return -va l else return val end if;\n end if;\n else\n t := `evalf/sinh_`(x);\n Digits := Digits+1;\n val := t/sq rt(1+t^2);\n return evalf[saveDigits](val);\n end if;\n \+ end if;\nend proc: # of `evalf/tanh_`\n\ntanh16 := proc(xx)\n local x,z,a1,a2,a3,b1,b2,num,den,val,k,t; \n \n x := evalf(xx); \n\n \+ if abs(x)<.3465735902799727 then\n # coefficients of numerator in rational approximation\n a1 := .33333333333333254608;\n a2 \+ := .16155401745154069186e-1;\n a3 := .63839005995177350254e-5;\n \n # coefficients of denominator in rational approximation\n \+ b1 := .44846620523517927052;\n b2 := .17500871902129655187e-1;\n \n # evaluate the rational approximation\n z := x*x;\n \+ num := (a1+(a2+a3*z)*z)*z;\n den := 1+(b1+b2*z)*z;\n (1-num/ den)*x;\n else\n t := exp16(-2*x);\n 1-2*t/(t+1);\n end \+ if;\nend proc: # tanh16" }}}{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 sectio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "tanh_" } {TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here are some numerical e xamples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Digits := 100:\nxx := 0.56789;\ntanh_(xx);\ntanh(xx); \nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"&*yc!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q]T!zlq!3qbc@s5b*)=)3AG\\Aq2b+3Z vK&4N'3%zCb^y?I%\\#zx#**z!Q^!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"_q]T!zlq!3qbc@s5b*)=)3AG\\Aq2b+3ZvK&4N'3%zCb^y?I%\\#zx#**z!Q^!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Digits := 100:\nxx := evalf(sqrt(2)/10,Digits+5):\ntanh_(xx); \ntanh(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q,g) G'G^yd[!=P4hi+4-**p\\.Vz&e\"\\s?I:w#**o)Rbq1[^L)>#35Hg[S\"!$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q,g)G'G^yd[!=P4hi+4-**p\\.Vz&e\"\\ s?I:w#**o)Rbq1[^L)>#35Hg[S\"!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "Some special cases can be handled.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Digits := 20:\nxx := ln(5); \ntanh_(xx);\nevalf(%);\ntanh_(evalf(xx,Digits+2));\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG-%#lnG6#\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"#7\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5#p2B p2Bp2B*!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5#p2Bp2Bp2B*!#?" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Digits := 20:\nxx := arcsinh(3/2);\ntanh_(xx);\nevalf(%);\ntanh_(e valf(xx,Digits+2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#xxG-%(arcsinhG6##\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$ -%%sqrtG6#\"#8\"\"\"#\"\"$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5-$ oVyL%H]?$)!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5.$oVyL%H]?$)!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We solve the eq uation " }{XPPEDIT 18 0 "tanh(x) = 1-x^2;" "6#/-%%tanhG6#%\"xG,&\"\" \"F)*$F'\"\"#!\"\"" }{TEXT -1 22 " using the functions " }{TEXT 0 5 " tanh_" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 " plot([tanh_(x),1-x^2,1,-1],x=-3..3,-1.5..1.1,color=[red,green,black$2] ,\n linestyle=[1$2,4$2],thickness=[2$2,1$2]);" }}{PARA 13 " " 1 "" {GLPLOT2D 503 183 183 {PLOTDATA 2 "6(-%'CURVESG6&7S7$$!\"$\"\"! $!+Pva]**!#57$$!+vq@pG!\"*$!+c0\"e$**F-7$$!+^NUbFF1$!+A>Z>**F-7$$!+K3X FEF1$!+G`5'*)*F-7$$!+G)H')\\#F1$!+H\"yd')*F-7$$!+'3@/P#F1$!+9d)o#)*F-7 $$!+r^b^AF1$!+ug$4y*F-7$$!+,kZG@F1$!+)3\\1s*F-7$$!+h\")=,?F1$!+&G96k*F -7$$!+O\"3V(=F1$!+94$)R&*F-7$$!+NkzViUC\"F1 $!+C#omY)F-7$$!+hkaI6F1$!+V?17\")F-7$$!*buK&**F1$!+K\"[if(F-7$$!*?#z2) )F1$!+GZ4oqF-7$$!*\"RKvuF1$!+S]uOjF-7$$!*qjeH'F1$!+MLnxbF-7$$!*\"*3=+& F1$!+nSfAYF-7$$!*ui&pPF1$!+4Dh+OF-7$$!*7VQ[#F1$!+=H*RV#F-7$$!*;^JI\"F1 $!+4U#eH\"F-7$$!(Q:'HF1$!+U$H:'H!#77$$\"*RIKH\"F1$\"+u(ogG\"F-7$$\"*^r ZW#F1$\"+M7?(R#F-7$$\"*[n%)o$F1$\"+Jc#)HNF-7$$\"*5FL(\\F1$\"+$*)o,g%F- 7$$\"*e6.B'F1$\"++(\\B`&F-7$$\"*p3lW(F1$\"+$))o%>jF-7$$\"*A))oz)F1$\"+ ZRjiqF-7$$\"+Ik-,5F1$\"+M*[-i(F-7$$\"+D-eI6F1$\"+%\\x@6)F-7$$\"+>_(zC \"F1$\"+_)[rZ)F-7$$\"+b*=jP\"F1$\"+EJC,))F-7$$\"+4/3(\\\"F1$\"+xE>Y!*F -7$$\"+C4JB;F1$\"+s1.^#*F-7$$\"+DVsYn#f(=F1$\"+Q SGT&*F-7$$\"+!)RO+?F1$\"+lG`S'*F-7$$\"+_!>w7#F1$\"+#Qw,s*F-7$$\"+*Q?QD #F1$\"+([:>y*F-7$$\"+5jypBF1$\"+.xmE)*F-7$$\"+Ujp-DF1$\"+&4eo')*F-7$$ \"+gEd@EF1$\"+cJ)[*)*F-7$$\"+4'>$[FF1$\"+GVK=**F-7$$\"+6EjpGF1$\"+4P'e $**F-7$$\"\"$F*$\"+Pva]**F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-% *THICKNESSG6#\"\"#-%*LINESTYLEG6#\"\"\"-F$6&7S7$F($F`[lF*7$$!3!****** \\2<#pG!#<$!3]a:ZBmSKsFb\\l7$$!3#)***\\7bBav#Fb\\l$!3CgJyY*eBf'Fb\\l7$ $!36++]K3XFEFb\\l$!3,W\\?xy\\.fFb\\l7$$!3%)****\\F)H')\\#Fb\\l$!3@zE([ ,^JC&Fb\\l7$$!3#****\\i3@/P#Fb\\l$!3mH'Qh7'*)=YFb\\l7$$!3;++Dr^b^AFb\\ l$!3KiUC\"Fb\\l$!3ax@1*4%)=[&Fa`l 7$$!3-++DhkaI6Fb\\l$!3)p(*[/,`8y#Fa`l7$$!3s******\\XF`**Fa`l$\"39LIABt DB$*!#?7$$!3u*******>#z2))Fa`l$\"3F;>;c'zAC#Fa`l7$$!3S++]7RKvuFa`l$\"3 do0KSK&>T%Fa`l7$$!3s,+++P'eH'Fa`l$\"3)*GA5F+@OgFa`l7$$!3q)***\\7*3=+&F a`l$\"3YeNGg2>)\\(Fa`l7$$!3[)***\\PFcpPFa`l$\"3M^^!onR!z&)Fa`l7$$!3;)* ***\\7VQ[#Fa`l$\"3uC!RIL_IQ*Fa`l7$$!32)***\\i6:.8Fa`l$\"3#**[nZqz,$)*F a`l7$$!3Wb+++v`hHFaal$\"33hj&HH7*****Fa`l7$$\"3]****\\(QIKH\"Fa`l$\"3H gY[;bvK)*Fa`l7$$\"38****\\7:xWCFa`l$\"3?Zo;D#4BS*Fa`l7$$\"3E,++vuY)o$F a`l$\"3[6neo2_R')Fa`l7$$\"3!z******4FL(\\Fa`l$\"3ng0kbDFb\\l7$$\"3P+++!)RO+?Fb\\l$!3aT9[_gX,IFb\\l7$$ \"30++]_!>w7#Fb\\l$!3l(*4cKGwENFb\\l7$$\"3O++v)Q?QD#Fb\\l$!3)G?XZM1(zS Fb\\l7$$\"3G+++5jypBFb\\l$!32$[FFb \\l$!3[`(R=ngKb'Fb\\l7$$\"37++D6EjpGFb\\l$!3G%\\\\NK\"zMsFb\\l7$FfzF^ \\l-F[[l6&F][lFa[lF^[lFa[lFb[lFf[l-F$6&7S7$F($Fi[lF*7$F`\\lFd[m7$Ff\\l Fd[m7$F[]lFd[m7$F`]lFd[m7$Fe]lFd[m7$Fj]lFd[m7$F_^lFd[m7$Fd^lFd[m7$Fi^l Fd[m7$F^_lFd[m7$Fc_lFd[m7$Fh_lFd[m7$F]`lFd[m7$Fc`lFd[m7$Fh`lFd[m7$F]al Fd[m7$FcalFd[m7$FhalFd[m7$F]blFd[m7$FbblFd[m7$FgblFd[m7$F\\clFd[m7$Fac lFd[m7$FfclFd[m7$F[dlFd[m7$F`dlFd[m7$FedlFd[m7$FjdlFd[m7$F_elFd[m7$Fde lFd[m7$FielFd[m7$F^flFd[m7$FcflFd[m7$FhflFd[m7$F]glFd[m7$FbglFd[m7$Fgg lFd[m7$F\\hlFd[m7$FahlFd[m7$FfhlFd[m7$F[ilFd[m7$F`ilFd[m7$FeilFd[m7$Fj ilFd[m7$F_jlFd[m7$FdjlFd[m7$FijlFd[m7$FfzFd[m-F[[l6&F][lF*F*F*-Fc[lFh[ l-Fg[l6#\"\"%-F$6&7S7$F($!\"\"F*7$F`\\lF__m7$Ff\\lF__m7$F[]lF__m7$F`]l F__m7$Fe]lF__m7$Fj]lF__m7$F_^lF__m7$Fd^lF__m7$Fi^lF__m7$F^_lF__m7$Fc_l F__m7$Fh_lF__m7$F]`lF__m7$Fc`lF__m7$Fh`lF__m7$F]alF__m7$FcalF__m7$Fhal F__m7$F]blF__m7$FbblF__m7$FgblF__m7$F\\clF__m7$FaclF__m7$FfclF__m7$F[d lF__m7$F`dlF__m7$FedlF__m7$FjdlF__m7$F_elF__m7$FdelF__m7$FielF__m7$F^f lF__m7$FcflF__m7$FhflF__m7$F]glF__m7$FbglF__m7$FgglF__m7$F\\hlF__m7$Fa hlF__m7$FfhlF__m7$F[ilF__m7$F`ilF__m7$FeilF__m7$FjilF__m7$F_jlF__m7$Fd jlF__m7$FijlF__m7$FfzF__mFe^mFg^mFh^m-%+AXESLABELSG6$Q\"x6\"Q!Febm-%%V IEWG6$;F(Ffz;$!#:F`_m$\"#6F`_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fsolve(tanh_(x)=1-x^2,x=0.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+?!>&Hl!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve(tanh_(x)=1-x^2,x=-1.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+\"=82P\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 21 "We find the va lue of " }{XPPEDIT 18 0 "Int(tanh(x^2),x = 0 .. 1);" "6#-%$IntG6$-%%ta nhG6#*$%\"xG\"\"#/F*;\"\"!\"\"\"" }{TEXT -1 20 " using the function " }{TEXT 0 5 "tanh_" }{TEXT -1 3 ".\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(tanh_(x^2),x=0..1.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"+D JdpK!#6$\"+N/,p5!#77$$\"+>7T9hF-$\"+Q]eQPF07$$\"+)=HPJ*F-$\"+lvLu')F07 $$\"+JaU`7!#5$\"+ig%4d\"F-7$$\"+%GZRd\"F>$\"+(R.oZ#F-7$$\"+s?6r=F>$\"+ @1j*\\$F-7$$\"+(**3)y@F>$\"+bdkVZF-7$$\"+(fHq\\#F>$\"+w\"*3FiF-7$$\"+f 'HU\"GF>$\"+X4P.zF-7$$\"+7*309$F>$\"+Q,%4$)*F-7$$\"+ce*yU$F>$\"+1'o'p6 F>7$$\"+)[D9v$F>$\"+h95)R\"F>7$$\"+iNGwSF>$\"+cP[Y;F>7$$\"+7XM*Q%F>$\" +\\[9.>F>7$$\"+ZQjtYF>$\"+ov>]@F>7$$\"+i8o6]F>$\"+^p7$$\"+]>0)H&F >$\"+.&oat#F>7$$\"+A!p6j&F>$\"+<'H)oIF>7$$\"+vS.EfF>$\"+)[0UP$F>7$$\"+ sxa\\iF>$\"+1?]=PF>7$$\"+;$4wb'F>$\"+!G+M0%F>7$$\"+>#R!zoF>$\"+zt!zS%F >7$$\"+4A@urF>$\"+^%Qft%F>7$$\"+chf#\\(F>$\"+k+3!4&F>7$$\"+(f2L#yF>$\" +(3DcX&F>7$$\"+yG>6\")F>$\"+p%Q(pdF>7$$\"+po6A%)F>$\"+ad]-hF>7$$\"+v$\"+%4cpV'F>7$$\"+%*ydd!*F>$\"+0*fGv'F>7$$\"+srih$*F>$\"+0b6YqF>7 $$\"+c?A*p*F>$\"+\"HicN(F>7$$\"+3mD+5!\"*$\"+qj4=wF>7$$\"+c]kK5Feu$\"+ WJ%3)yF>7$$\"+0Q*>1\"Feu$\"+bcv-\")F>7$$\"+R(zS4\"Feu$\"+mYSF$)F>7$$\" +-,FC6Feu$\"+)Q&e@&)F>7$$\"+Jx#e:\"Feu$\"+B3$pq)F>7$$\"+\"3\"o'=\"Feu$ \"+!QQ6())F>7$$\"+!o\")*=7Feu$\"+DbqD!*F>7$$\"+&*44]7Feu$\"+QAhe\"*F>7 $$\"+jZ!>G\"Feu$\"+lnHz#*F>7$$\"+(4bMJ\"Feu$\"+k(=[Q*F>7$$\"+ylWU8Feu$ \"+C`Hq%*F>7$$\"+'3ucP\"Feu$\"+1&*)eb*F>7$$\"+lJR09Feu$\"+)3xAi*F>7$$ \"+-*zqV\"Feu$\"+vub$o*F>7$$\"+`\"3uY\"Feu$\"+&[#*Rt*F>7$$\"#:!\"\"$\" +Z6E!y*F>-%'COLOURG6&%$RGBG$\"#5FfzF(F(-%+AXESLABELSG6$Q\"x6\"Q!Fc[l-% %VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(tanh_(x^2),x=0..1);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%&tanh_G6#*$)%\"xG\"\"# \"\"\"/F+;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gt,^H!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(tanh_(x^2),x=0..1);\nquad(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%&tanh_G6#*$)%\"xG\"\"#\"\"\"/F+;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gt,^H!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int(tanh(x^2),x=0..1);\neval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%%tanhG6#*$)%\"xG\" \"#\"\"\"/F+;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Gt,^H!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }