An exact simplification challenge - 101 (tan, arctan)
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From: Vladimir Bondarenko on 4 Aug 2010 11:07 Hello, Mathematica: Tan[1/4 ArcTan[(16 Sqrt[2] (11030013328417+6423142241817* Sqrt[3]))/(80380562320289 + 43647808257288 Sqrt[3])]] Maple: tan(1/4*arctan(16*(6423142241817*3^(1/2)+11030013328417)* 2^(1/2)/(43647808257288*3^(1/2)+80380562320289))) Can you get rid of these tan/arctan ? Cheers, Vladimir Bondarenko Co-founder, CEO, Mathematical Director http://www.cybertester.com/ Cyber Tester Ltd. ---------------------------------------------------------------- "We must understand that technologies like these are the way of the future." ---------------------------------------------------------------- ---------------------------------------------------------------- http://groups.google.com/group/sci.math/msg/9f429c3ea5649df5 "...... the challenges imply that a solution is built within the framework of the existent CAS functions & built-in definitions." ---------------------------------------------------------------- ----------------------------------------------------------------
From: Martin Brown on 4 Aug 2010 12:36 On 04/08/2010 16:07, Vladimir Bondarenko wrote: > Hello, > > Mathematica: > > Tan[1/4 ArcTan[(16 Sqrt[2] (11030013328417+6423142241817* > Sqrt[3]))/(80380562320289 + 43647808257288 Sqrt[3])]] > > Maple: > > tan(1/4*arctan(16*(6423142241817*3^(1/2)+11030013328417)* > 2^(1/2)/(43647808257288*3^(1/2)+80380562320289))) > > Can you get rid of these tan/arctan ? Presumably something along the lines of let t = tan(x/2) Then tan(x) = s = 2t/(1 - t^2) and solve the quadratic for t Applied twice to the argument of arctan( ) Regards, Martin Brown
From: Waldek Hebisch on 4 Aug 2010 13:08 In sci.math.symbolic Vladimir Bondarenko <vb(a)cybertester.com> wrote: > Hello, > > Mathematica: > > Tan[1/4 ArcTan[(16 Sqrt[2] (11030013328417+6423142241817* > Sqrt[3]))/(80380562320289 + 43647808257288 Sqrt[3])]] > > Maple: > > tan(1/4*arctan(16*(6423142241817*3^(1/2)+11030013328417)* > 2^(1/2)/(43647808257288*3^(1/2)+80380562320289))) > > Can you get rid of these tan/arctan ? > (-18468*sqrt(2)*sqrt(3) + 34300*sqrt(2))/9959 In FriCAS 'normalize' finds out that the result is a root of degree 4 polynomial. Factor the polynomial and use numeric evaluation to choose correct root. -- Waldek Hebisch hebisch(a)math.uni.wroc.pl
From: clicliclic on 4 Aug 2010 14:51 Vladimir Bondarenko schrieb: > > Mathematica: > > Tan[1/4 ArcTan[(16 Sqrt[2] (11030013328417+6423142241817* > Sqrt[3]))/(80380562320289 + 43647808257288 Sqrt[3])]] > > Maple: > > tan(1/4*arctan(16*(6423142241817*3^(1/2)+11030013328417)* > 2^(1/2)/(43647808257288*3^(1/2)+80380562320289))) > > Can you get rid of these tan/arctan ? > Derive 6.10 does it automatically (in 50 steps plus one final step): [PrecisionDigits:=20,NotationDigits:=20] TAN(1/4*ATAN(16*SQRT(2)*(11030013328417+6423142241817*SQRT(3))/(~ 80380562320289+43647808257288*SQRT(3)))) 0.32838122499020253603 34300*SQRT(2)/9959-18468*SQRT(6)/9959 0.32838122499020253603 Stepwise simplification appended below. Martin. - TAN(1/4*ATAN(16*SQRT(2)*(11030013328417+6423142241817*SQRT(3))/(~ 80380562320289+43647808257288*SQRT(3)))) " 1/(z+w) -> (z-w)/(z^2-w^2) " TAN(ATAN((80380562320289-43647808257288*SQRT(3))*(10277027586907~ 2*SQRT(6)+176480213254672*SQRT(2))/745641301930928200605698689)/~ 4) " If x>0, ATAN(x) -> pi/2-ATAN(1/x) " TAN((pi/2-ATAN(1/(19640584598955090192*SQRT(6)/26256601665255541~ 729+25652535759856250000*SQRT(2)/26256601665255541729)))/4) " 1/(z+w) -> (z-w)/(z^2-w^2) " TAN((pi/2-ATAN(26256601665255541729*(19640584598955090192*SQRT(6~ )/26256601665255541729-25652535759856250000*SQRT(2)/262566016652~ 55541729)/38025111217344848896))/4) " TAN(-z) -> -TAN(z) " -TAN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160328~ 3484991015625*SQRT(2)/2376569451084053056)/4-pi/8) " TAN(z) -> SIN(z)/COS(z) " -SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160328~ 3484991015625*SQRT(2)/2376569451084053056)/4-pi/8)/COS(ATAN(1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)/4-pi/8) " SIN(z) -> -SIN(z+pi) " SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)/4+7*pi/8)/COS(ATAN(122~ 7536537434693137*SQRT(6)/2376569451084053056-1603283484991015625~ *SQRT(2)/2376569451084053056)/4-pi/8) " SIN(z+n*pi) -> COS(z+(n-1/2)*pi) " COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)/4+3*pi/8)/COS(ATAN(122~ 7536537434693137*SQRT(6)/2376569451084053056-1603283484991015625~ *SQRT(2)/2376569451084053056)/4-pi/8) " COS(z) -> -COS(z+pi) " -COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160328~ 3484991015625*SQRT(2)/2376569451084053056)/4+3*pi/8)/COS(ATAN(12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)/4+7*pi/8) " COS(z+n*pi) -> -SIN(z+(n-1/2)*pi) " COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)/4+3*pi/8)/SIN(ATAN(122~ 7536537434693137*SQRT(6)/2376569451084053056-1603283484991015625~ *SQRT(2)/2376569451084053056)/4+3*pi/8) " TAN(u/2) -> SIN(u)/(1+COS(u)) " COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)/2+pi/4)/(1-COS(ATAN(12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)/2+3*pi/4)) " COS(z+w) -> COS(z)*COS(w)-SIN(z)*SIN(w) " (COS(pi/4)*COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053~ 056-1603283484991015625*SQRT(2)/2376569451084053056)/2)-SIN(pi/4~ )*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-16032~ 83484991015625*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)/2+3*pi/4)) " COS(pi/4) -> SQRT(2)/2 " (SQRT(2)*COS(ATAN(1227536537434693137*SQRT(6)/237656945108405305~ 6-1603283484991015625*SQRT(2)/2376569451084053056)/2)/2-SIN(pi/4~ )*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-16032~ 83484991015625*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)/2+3*pi/4)) " COS(ATAN(z)/2) -> (SQRT(SQRT(z^2+1)+z)+SQRT(SQRT(z^2+1)-z))/(2~ *(z^2+1)^(1/4)) " (SQRT(2)*(SQRT(SQRT((1227536537434693137*SQRT(6)/237656945108405~ 3056-1603283484991015625*SQRT(2)/2376569451084053056)^2+1)+12275~ 36537434693137*SQRT(6)/2376569451084053056-1603283484991015625*S~ QRT(2)/2376569451084053056)+SQRT(SQRT((1227536537434693137*SQRT(~ 6)/2376569451084053056-1603283484991015625*SQRT(2)/2376569451084~ 053056)^2+1)-1227536537434693137*SQRT(6)/2376569451084053056+160~ 3283484991015625*SQRT(2)/2376569451084053056))/(2*2*((1227536537~ 434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2~ )/2376569451084053056)^2+1)^(1/4))-SIN(pi/4)*SIN(ATAN(1227536537~ 434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2~ )/2376569451084053056)/2))/(1-COS(ATAN(1227536537434693137*SQRT(~ 6)/2376569451084053056-1603283484991015625*SQRT(2)/2376569451084~ 053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (SQRT(2)*(SQRT(9845600625*(171538593*SQRT(6)/2-118358233*SQRT(2)~ /2)/1188284725542026528+1227536537434693137*SQRT(6)/237656945108~ 4053056-1603283484991015625*SQRT(2)/2376569451084053056)+SQRT(SQ~ RT((1227536537434693137*SQRT(6)/2376569451084053056-160328348499~ 1015625*SQRT(2)/2376569451084053056)^2+1)-1227536537434693137*SQ~ RT(6)/2376569451084053056+1603283484991015625*SQRT(2)/2376569451~ 084053056))/(4*((1227536537434693137*SQRT(6)/2376569451084053056~ -1603283484991015625*SQRT(2)/2376569451084053056)^2+1)^(1/4))-SI~ N(pi/4)*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056~ -1603283484991015625*SQRT(2)/2376569451084053056)/2))/(1-COS(ATA~ N(1227536537434693137*SQRT(6)/2376569451084053056-16032834849910~ 15625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (SQRT(2)*(2^(3/4)*SQRT(901101447300249*SQRT(3)-855421078500625)/~ 38322232+SQRT(9845600625*(171538593*SQRT(6)/2-118358233*SQRT(2)/~ 2)/1188284725542026528-1227536537434693137*SQRT(6)/2376569451084~ 053056+1603283484991015625*SQRT(2)/2376569451084053056))/(4*((12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)^2+1)^(1/4))-SIN(pi/4)*SIN(ATAN(12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(12275365374346931~ 37*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/23765~ 69451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (74267795346376658*SQRT(62715457547050997826111171548885625*SQRT~ (6)-43272424048945478196765666173360625*SQRT(2)+8825128680978058~ 2051760147315983424)/(99225*148535590692753316*SQRT(636989655946~ 4699780681097*SQRT(6)-4395102513001382096662657*SQRT(2)))-SIN(pi~ /4)*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160~ 3283484991015625*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)/2+3*pi/4)) " 1/(z+w) -> (z-w)/(z^2-w^2) " (74267795346376658*SQRT(62715457547050997826111171548885625*SQRT~ (6)-43272424048945478196765666173360625*SQRT(2)+8825128680978058~ 2051760147315983424)*2^(3/4)*SQRT((171538593*SQRT(3)+118358233)/~ 2)/1094591741712559781760737577178556905800-SIN(pi/4)*SIN(ATAN(1~ 227536537434693137*SQRT(6)/2376569451084053056-16032834849910156~ 25*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(1227536537434693~ 137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/2376~ 569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " (148535590692753316*SQRT(2)*SQRT(37133897673188329)*SQRT(7426779~ 5346376658)*(SQRT((5754384384*SQRT(6)+18823840000)/2)+SQRT((-265~ 149408*SQRT(6)+867361250)/2))/2189183483425119563521475154357113~ 811600-SIN(pi/4)*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451~ 084053056-1603283484991015625*SQRT(2)/2376569451084053056)/2))/(~ 1-COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-16032~ 83484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " ((56*SQRT(2)*(456*SQRT(6)+503)+SQRT((-265149408*SQRT(6)+86736125~ 0)/2))/198450-SIN(pi/4)*SIN(ATAN(1227536537434693137*SQRT(6)/237~ 6569451084053056-1603283484991015625*SQRT(2)/2376569451084053056~ )/2))/(1-COS(ATAN(1227536537434693137*SQRT(6)/237656945108405305~ 6-1603283484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " ((51072*SQRT(3)+28168*SQRT(2)+17*(456*SQRT(6)-503))/198450-SIN(p~ i/4)*SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-16~ 03283484991015625*SQRT(2)/2376569451084053056)/2))/(1-COS(ATAN(1~ 227536537434693137*SQRT(6)/2376569451084053056-16032834849910156~ 25*SQRT(2)/2376569451084053056)/2+3*pi/4)) " SIN(pi/4) -> SQRT(2)/2 " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)~ *SIN(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160328~ 3484991015625*SQRT(2)/2376569451084053056)/2)/2-8551/198450)/(1-~ COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " SIN(ATAN(z)/2) -> (SQRT(SQRT(z^2+1)+z)-SQRT(SQRT(z^2+1)-z))/(2~ *(z^2+1)^(1/4)) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)~ *(SQRT(SQRT((1227536537434693137*SQRT(6)/2376569451084053056-160~ 3283484991015625*SQRT(2)/2376569451084053056)^2+1)+1227536537434~ 693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/2~ 376569451084053056)-SQRT(SQRT((1227536537434693137*SQRT(6)/23765~ 69451084053056-1603283484991015625*SQRT(2)/2376569451084053056)^~ 2+1)-1227536537434693137*SQRT(6)/2376569451084053056+16032834849~ 91015625*SQRT(2)/2376569451084053056))/(2*2*((122753653743469313~ 7*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/237656~ 9451084053056)^2+1)^(1/4))-8551/198450)/(1-COS(ATAN(122753653743~ 4693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/~ 2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)~ *(SQRT(9845600625*(171538593*SQRT(6)/2-118358233*SQRT(2)/2)/1188~ 284725542026528+1227536537434693137*SQRT(6)/2376569451084053056-~ 1603283484991015625*SQRT(2)/2376569451084053056)-SQRT(SQRT((1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)^2+1)-1227536537434693137*SQRT(6)/23~ 76569451084053056+1603283484991015625*SQRT(2)/237656945108405305~ 6))/(4*((1227536537434693137*SQRT(6)/2376569451084053056-1603283~ 484991015625*SQRT(2)/2376569451084053056)^2+1)^(1/4))-8551/19845~ 0)/(1-COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1~ 603283484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)~ *(2^(3/4)*SQRT(901101447300249*SQRT(3)-855421078500625)/38322232~ -SQRT(9845600625*(171538593*SQRT(6)/2-118358233*SQRT(2)/2)/11882~ 84725542026528-1227536537434693137*SQRT(6)/2376569451084053056+1~ 603283484991015625*SQRT(2)/2376569451084053056))/(4*((1227536537~ 434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2~ )/2376569451084053056)^2+1)^(1/4))-8551/198450)/(1-COS(ATAN(1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-7426779~ 5346376658*SQRT(62715457547050997826111171548885625*SQRT(6)-4327~ 2424048945478196765666173360625*SQRT(2)-882512868097805820517601~ 47315983424)/(99225*148535590692753316*SQRT(63698965594646997806~ 81097*SQRT(6)-4395102513001382096662657*SQRT(2)))-8551/198450)/(~ 1-COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-16032~ 83484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " 1/(z+w) -> (z-w)/(z^2-w^2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-7426779~ 5346376658*SQRT(62715457547050997826111171548885625*SQRT(6)-4327~ 2424048945478196765666173360625*SQRT(2)-882512868097805820517601~ 47315983424)*2^(3/4)*SQRT((171538593*SQRT(3)+118358233)/2)/10945~ 91741712559781760737577178556905800-8551/198450)/(1-COS(ATAN(122~ 7536537434693137*SQRT(6)/2376569451084053056-1603283484991015625~ *SQRT(2)/2376569451084053056)/2+3*pi/4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-1485355~ 90692753316*SQRT(2)*SQRT(37133897673188329)*SQRT(742677953463766~ 58)*(SQRT((-5754384384*SQRT(6)+18823840000)/2)-SQRT((265149408*S~ QRT(6)+867361250)/2))/2189183483425119563521475154357113811600-8~ 551/198450)/(1-COS(ATAN(1227536537434693137*SQRT(6)/237656945108~ 4053056-1603283484991015625*SQRT(2)/2376569451084053056)/2+3*pi/~ 4)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-(56*SQR~ T(2)*(456*SQRT(6)-503)-SQRT((265149408*SQRT(6)+867361250)/2))/19~ 8450-8551/198450)/(1-COS(ATAN(1227536537434693137*SQRT(6)/237656~ 9451084053056-1603283484991015625*SQRT(2)/2376569451084053056)/2~ +3*pi/4)) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " (1292*SQRT(6)/33075+1216*SQRT(3)/4725+2012*SQRT(2)/14175-(51072*~ SQRT(3)-28168*SQRT(2)-17*(456*SQRT(6)+503))/198450-8551/198450)/~ (1-COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-1603~ 283484991015625*SQRT(2)/2376569451084053056)/2+3*pi/4)) " COS(z+n*pi) -> -SIN(z+(n-1/2)*pi) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+SIN(ATAN(122753653743~ 4693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/~ 2376569451084053056)/2+pi/4)) " SIN(z+w) -> SIN(z)*COS(w)+COS(z)*SIN(w) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+COS(pi/4)*SIN(ATAN(12~ 27536537434693137*SQRT(6)/2376569451084053056-160328348499101562~ 5*SQRT(2)/2376569451084053056)/2)+SIN(pi/4)*COS(ATAN(12275365374~ 34693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)~ /2376569451084053056)/2)) " COS(pi/4) -> SQRT(2)/2 " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+SQRT(2)*SIN(ATAN(1227~ 536537434693137*SQRT(6)/2376569451084053056-1603283484991015625*~ SQRT(2)/2376569451084053056)/2)/2+SIN(pi/4)*COS(ATAN(12275365374~ 34693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)~ /2376569451084053056)/2)) " SIN(ATAN(z)/2) -> (SQRT(SQRT(z^2+1)+z)-SQRT(SQRT(z^2+1)-z))/(2~ *(z^2+1)^(1/4)) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+SQRT(2)*(SQRT(SQRT((1~ 227536537434693137*SQRT(6)/2376569451084053056-16032834849910156~ 25*SQRT(2)/2376569451084053056)^2+1)+1227536537434693137*SQRT(6)~ /2376569451084053056-1603283484991015625*SQRT(2)/237656945108405~ 3056)-SQRT(SQRT((1227536537434693137*SQRT(6)/2376569451084053056~ -1603283484991015625*SQRT(2)/2376569451084053056)^2+1)-122753653~ 7434693137*SQRT(6)/2376569451084053056+1603283484991015625*SQRT(~ 2)/2376569451084053056))/(2*2*((1227536537434693137*SQRT(6)/2376~ 569451084053056-1603283484991015625*SQRT(2)/2376569451084053056)~ ^2+1)^(1/4))+SIN(pi/4)*COS(ATAN(1227536537434693137*SQRT(6)/2376~ 569451084053056-1603283484991015625*SQRT(2)/2376569451084053056)~ /2)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+SQRT(2)*(SQRT(9845600~ 625*(171538593*SQRT(6)/2-118358233*SQRT(2)/2)/118828472554202652~ 8+1227536537434693137*SQRT(6)/2376569451084053056-16032834849910~ 15625*SQRT(2)/2376569451084053056)-SQRT(SQRT((122753653743469313~ 7*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/237656~ 9451084053056)^2+1)-1227536537434693137*SQRT(6)/2376569451084053~ 056+1603283484991015625*SQRT(2)/2376569451084053056))/(4*((12275~ 36537434693137*SQRT(6)/2376569451084053056-1603283484991015625*S~ QRT(2)/2376569451084053056)^2+1)^(1/4))+SIN(pi/4)*COS(ATAN(12275~ 36537434693137*SQRT(6)/2376569451084053056-1603283484991015625*S~ QRT(2)/2376569451084053056)/2)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+SQRT(2)*(2^(3/4)*SQRT~ (901101447300249*SQRT(3)-855421078500625)/38322232-SQRT(98456006~ 25*(171538593*SQRT(6)/2-118358233*SQRT(2)/2)/1188284725542026528~ -1227536537434693137*SQRT(6)/2376569451084053056+160328348499101~ 5625*SQRT(2)/2376569451084053056))/(4*((1227536537434693137*SQRT~ (6)/2376569451084053056-1603283484991015625*SQRT(2)/237656945108~ 4053056)^2+1)^(1/4))+SIN(pi/4)*COS(ATAN(1227536537434693137*SQRT~ (6)/2376569451084053056-1603283484991015625*SQRT(2)/237656945108~ 4053056)/2)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+74267795346376658*SQR~ T(62715457547050997826111171548885625*SQRT(6)-432724240489454781~ 96765666173360625*SQRT(2)-88251286809780582051760147315983424)/(~ 99225*148535590692753316*SQRT(6369896559464699780681097*SQRT(6)-~ 4395102513001382096662657*SQRT(2)))+SIN(pi/4)*COS(ATAN(122753653~ 7434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(~ 2)/2376569451084053056)/2)) " 1/(z+w) -> (z-w)/(z^2-w^2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+74267795346376658*SQR~ T(62715457547050997826111171548885625*SQRT(6)-432724240489454781~ 96765666173360625*SQRT(2)-88251286809780582051760147315983424)*2~ ^(3/4)*SQRT((171538593*SQRT(3)+118358233)/2)/1094591741712559781~ 760737577178556905800+SIN(pi/4)*COS(ATAN(1227536537434693137*SQR~ T(6)/2376569451084053056-1603283484991015625*SQRT(2)/23765694510~ 84053056)/2)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+148535590692753316*SQ~ RT(2)*SQRT(37133897673188329)*SQRT(74267795346376658)*(SQRT((-57~ 54384384*SQRT(6)+18823840000)/2)-SQRT((265149408*SQRT(6)+8673612~ 50)/2))/2189183483425119563521475154357113811600+SIN(pi/4)*COS(A~ TAN(1227536537434693137*SQRT(6)/2376569451084053056-160328348499~ 1015625*SQRT(2)/2376569451084053056)/2)) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+(56*SQRT(2)*(456*SQRT~ (6)-503)-SQRT((265149408*SQRT(6)+867361250)/2))/198450+SIN(pi/4)~ *COS(ATAN(1227536537434693137*SQRT(6)/2376569451084053056-160328~ 3484991015625*SQRT(2)/2376569451084053056)/2)) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " (2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1+(51072*SQRT(3)-28168*~ SQRT(2)-17*(456*SQRT(6)+503))/198450+SIN(pi/4)*COS(ATAN(12275365~ 37434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT~ (2)/2376569451084053056)/2)) " SIN(pi/4) -> SQRT(2)/2 " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)*COS(ATAN(1227536537434~ 693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/2~ 376569451084053056)/2)/2-189899/198450) " COS(ATAN(z)/2) -> (SQRT(SQRT(z^2+1)+z)+SQRT(SQRT(z^2+1)-z))/(2~ *(z^2+1)^(1/4)) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)*(SQRT(SQRT((1227536537~ 434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2~ )/2376569451084053056)^2+1)+1227536537434693137*SQRT(6)/23765694~ 51084053056-1603283484991015625*SQRT(2)/2376569451084053056)+SQR~ T(SQRT((1227536537434693137*SQRT(6)/2376569451084053056-16032834~ 84991015625*SQRT(2)/2376569451084053056)^2+1)-122753653743469313~ 7*SQRT(6)/2376569451084053056+1603283484991015625*SQRT(2)/237656~ 9451084053056))/(2*2*((1227536537434693137*SQRT(6)/2376569451084~ 053056-1603283484991015625*SQRT(2)/2376569451084053056)^2+1)^(1/~ 4))-189899/198450) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)*(SQRT(9845600625*(1715~ 38593*SQRT(6)/2-118358233*SQRT(2)/2)/1188284725542026528+1227536~ 537434693137*SQRT(6)/2376569451084053056-1603283484991015625*SQR~ T(2)/2376569451084053056)+SQRT(SQRT((1227536537434693137*SQRT(6)~ /2376569451084053056-1603283484991015625*SQRT(2)/237656945108405~ 3056)^2+1)-1227536537434693137*SQRT(6)/2376569451084053056+16032~ 83484991015625*SQRT(2)/2376569451084053056))/(4*((12275365374346~ 93137*SQRT(6)/2376569451084053056-1603283484991015625*SQRT(2)/23~ 76569451084053056)^2+1)^(1/4))-189899/198450) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-SQRT(2)*(2^(3/4)*SQRT(90110144~ 7300249*SQRT(3)-855421078500625)/38322232+SQRT(9845600625*(17153~ 8593*SQRT(6)/2-118358233*SQRT(2)/2)/1188284725542026528-12275365~ 37434693137*SQRT(6)/2376569451084053056+1603283484991015625*SQRT~ (2)/2376569451084053056))/(4*((1227536537434693137*SQRT(6)/23765~ 69451084053056-1603283484991015625*SQRT(2)/2376569451084053056)^~ 2+1)^(1/4))-189899/198450) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-74267795346376658*SQRT(6271545~ 7547050997826111171548885625*SQRT(6)-432724240489454781967656661~ 73360625*SQRT(2)+88251286809780582051760147315983424)/(99225*148~ 535590692753316*SQRT(6369896559464699780681097*SQRT(6)-439510251~ 3001382096662657*SQRT(2)))-189899/198450) " 1/(z+w) -> (z-w)/(z^2-w^2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-74267795346376658*SQRT(6271545~ 7547050997826111171548885625*SQRT(6)-432724240489454781967656661~ 73360625*SQRT(2)+88251286809780582051760147315983424)*2^(3/4)*SQ~ RT((171538593*SQRT(3)+118358233)/2)/1094591741712559781760737577~ 178556905800-189899/198450) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-148535590692753316*SQRT(2)*SQR~ T(37133897673188329)*SQRT(74267795346376658)*(SQRT((5754384384*S~ QRT(6)+18823840000)/2)+SQRT((-265149408*SQRT(6)+867361250)/2))/2~ 189183483425119563521475154357113811600-189899/198450) " If x>y>0, SQRT(x+y) -> SQRT((x+SQRT(x^2-y^2))/2)+SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-(56*SQRT(2)*(456*SQRT(6)+503)+~ SQRT((-265149408*SQRT(6)+867361250)/2))/198450-189899/198450) " If x>y>0, SQRT(x-y) -> SQRT((x+SQRT(x^2-y^2))/2)-SQRT((x-SQRT(~ x^2-y^2))/2) " -(2584*SQRT(6)/33075+4024*SQRT(2)/14175)/(1292*SQRT(6)/33075-121~ 6*SQRT(3)/4725+2012*SQRT(2)/14175-(51072*SQRT(3)+28168*SQRT(2)+1~ 7*(456*SQRT(6)-503))/198450-189899/198450) " 1/(z+w) -> (z-w)/(z^2-w^2) " 9845600625*(2584*SQRT(6)/33075+4024*SQRT(2)/14175)*(-2432*SQRT(3~ )/4725+90674/99225)/396726724 " one final step " 34300*SQRT(2)/9959-18468*SQRT(6)/9959
From: Peter Pein on 4 Aug 2010 23:06
Am Wed, 4 Aug 2010 08:07:33 -0700 (PDT) schrieb Vladimir Bondarenko <vb(a)cybertester.com>: > Hello, > > Mathematica: > > Tan[1/4 ArcTan[(16 Sqrt[2] (11030013328417+6423142241817* > Sqrt[3]))/(80380562320289 + 43647808257288 Sqrt[3])]] > > Maple: > > tan(1/4*arctan(16*(6423142241817*3^(1/2)+11030013328417)* > 2^(1/2)/(43647808257288*3^(1/2)+80380562320289))) > > Can you get rid of these tan/arctan ? > > Cheers, > > Vladimir Bondarenko > > Co-founder, CEO, Mathematical Director > > http://www.cybertester.com/ Cyber Tester Ltd. > > ---------------------------------------------------------------- > > "We must understand that technologies > like these are the way of the future." > > ---------------------------------------------------------------- > ---------------------------------------------------------------- > > http://groups.google.com/group/sci.math/msg/9f429c3ea5649df5 > > "...... the challenges imply that a solution is built within the > framework of the existent CAS functions & built-in definitions." > > ---------------------------------------------------------------- > ---------------------------------------------------------------- using Mathematica without really knowing what I do: In[1]:= f[x_] = Tan[ArcTan[(16 Sqrt[2] (11030013328417 + 6423142241817 x)) / (80380562320289 + 43647808257288 x)] / 4]; In[2]:= simp = ToRadicals[RootReduce[FullSimplify[ TrigToExp[ComplexExpand[f[x], TargetFunctions -> {Abs, Log}]] /. x -> Sqrt[3]]]] Out[2]= 3848 / Sqrt[68740346 + 39590775 Sqrt[3]] and tan/arctan have gone :-) test: In[3]:= SeriesCoefficient[simp-f[x],{x,Sqrt[3],0}] Out[3]= 0 |