Edit detail for SandBox Manip revision 1 of 1
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changed: - Demonstration of modifying and testing a SPAD file I changed the abbreviation to TRMANIP2 and the name to !TranscendentalManipulations2 so I can call both the old and new package if necessary. \begin{spad} )abbrev package TRMANIP2 TranscendentalManipulations2 ++ Transformations on transcendental objects ++ Author: Bob Sutor, Manuel Bronstein ++ Date Created: Way back ++ Date Last Updated: 22 January 1996, added simplifyLog MCD. ++ Description: ++ TranscendentalManipulations provides functions to simplify and ++ expand expressions involving transcendental operators. ++ Keywords: transcendental, manipulation. TranscendentalManipulations2(R, F): Exports == Implementation where R : Join(OrderedSet, GcdDomain) F : Join(FunctionSpace R, TranscendentalFunctionCategory) Z ==> Integer K ==> Kernel F P ==> SparseMultivariatePolynomial(R, K) UP ==> SparseUnivariatePolynomial P POWER ==> "%power"::Symbol POW ==> Record(val: F,exponent: Z) PRODUCT ==> Record(coef : Z, var : K) FPR ==> Fraction Polynomial R Exports ==> with expand : F -> F ++ expand(f) performs the following expansions on f:\begin{items} ++ \item 1. logs of products are expanded into sums of logs, ++ \item 2. trigonometric and hyperbolic trigonometric functions ++ of sums are expanded into sums of products of trigonometric ++ and hyperbolic trigonometric functions. ++ \item 3. formal powers of the form \spad{(a/b)**c} are expanded into ++ \spad{a**c * b**(-c)}. ++ \end{items} simplify : F -> F ++ simplify(f) performs the following simplifications on f:\begin{items} ++ \item 1. rewrites trigs and hyperbolic trigs in terms ++ of \spad{sin} ,\spad{cos}, \spad{sinh}, \spad{cosh}. ++ \item 2. rewrites \spad{sin**2} and \spad{sinh**2} in terms ++ of \spad{cos} and \spad{cosh}, ++ \item 3. rewrites \spad{exp(a)*exp(b)} as \spad{exp(a+b)}. ++ \item 4. rewrites \spad{(a**(1/n))**m * (a**(1/s))**t} as a single ++ power of a single radical of \spad{a}. ++ \end{items} htrigs : F -> F ++ htrigs(f) converts all the exponentials in f into ++ hyperbolic sines and cosines. simplifyExp: F -> F ++ simplifyExp(f) converts every product \spad{exp(a)*exp(b)} ++ appearing in f into \spad{exp(a+b)}. simplifyLog : F -> F ++ simplifyLog(f) converts every \spad{log(a) - log(b)} appearing in f ++ into \spad{log(a/b)}, every \spad{log(a) + log(b)} into \spad{log(a*b)} ++ and every \spad{n*log(a)} into \spad{log(a^n)}. expandPower: F -> F ++ expandPower(f) converts every power \spad{(a/b)**c} appearing ++ in f into \spad{a**c * b**(-c)}. expandLog : F -> F ++ expandLog(f) converts every \spad{log(a/b)} appearing in f into ++ \spad{log(a) - log(b)}, and every \spad{log(a*b)} into ++ \spad{log(a) + log(b)}.. cos2sec : F -> F ++ cos2sec(f) converts every \spad{cos(u)} appearing in f into ++ \spad{1/sec(u)}. cosh2sech : F -> F ++ cosh2sech(f) converts every \spad{cosh(u)} appearing in f into ++ \spad{1/sech(u)}. cot2trig : F -> F ++ cot2trig(f) converts every \spad{cot(u)} appearing in f into ++ \spad{cos(u)/sin(u)}. coth2trigh : F -> F ++ coth2trigh(f) converts every \spad{coth(u)} appearing in f into ++ \spad{cosh(u)/sinh(u)}. csc2sin : F -> F ++ csc2sin(f) converts every \spad{csc(u)} appearing in f into ++ \spad{1/sin(u)}. csch2sinh : F -> F ++ csch2sinh(f) converts every \spad{csch(u)} appearing in f into ++ \spad{1/sinh(u)}. sec2cos : F -> F ++ sec2cos(f) converts every \spad{sec(u)} appearing in f into ++ \spad{1/cos(u)}. sech2cosh : F -> F ++ sech2cosh(f) converts every \spad{sech(u)} appearing in f into ++ \spad{1/cosh(u)}. sin2csc : F -> F ++ sin2csc(f) converts every \spad{sin(u)} appearing in f into ++ \spad{1/csc(u)}. sinh2csch : F -> F ++ sinh2csch(f) converts every \spad{sinh(u)} appearing in f into ++ \spad{1/csch(u)}. tan2trig : F -> F ++ tan2trig(f) converts every \spad{tan(u)} appearing in f into ++ \spad{sin(u)/cos(u)}. tanh2trigh : F -> F ++ tanh2trigh(f) converts every \spad{tanh(u)} appearing in f into ++ \spad{sinh(u)/cosh(u)}. tan2cot : F -> F ++ tan2cot(f) converts every \spad{tan(u)} appearing in f into ++ \spad{1/cot(u)}. tanh2coth : F -> F ++ tanh2coth(f) converts every \spad{tanh(u)} appearing in f into ++ \spad{1/coth(u)}. cot2tan : F -> F ++ cot2tan(f) converts every \spad{cot(u)} appearing in f into ++ \spad{1/tan(u)}. coth2tanh : F -> F ++ coth2tanh(f) converts every \spad{coth(u)} appearing in f into ++ \spad{1/tanh(u)}. removeCosSq: F -> F ++ removeCosSq(f) converts every \spad{cos(u)**2} appearing in f into ++ \spad{1 - sin(x)**2}, and also reduces higher ++ powers of \spad{cos(u)} with that formula. removeSinSq: F -> F ++ removeSinSq(f) converts every \spad{sin(u)**2} appearing in f into ++ \spad{1 - cos(x)**2}, and also reduces higher powers of ++ \spad{sin(u)} with that formula. removeCoshSq:F -> F ++ removeCoshSq(f) converts every \spad{cosh(u)**2} appearing in f into ++ \spad{1 - sinh(x)**2}, and also reduces higher powers of ++ \spad{cosh(u)} with that formula. removeSinhSq:F -> F ++ removeSinhSq(f) converts every \spad{sinh(u)**2} appearing in f into ++ \spad{1 - cosh(x)**2}, and also reduces higher powers ++ of \spad{sinh(u)} with that formula. if R has PatternMatchable(R) and R has ConvertibleTo(Pattern(R)) and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then expandTrigProducts : F -> F ++ expandTrigProducts(e) replaces \axiom{sin(x)*sin(y)} by ++ \spad{(cos(x-y)-cos(x+y))/2}, \axiom{cos(x)*cos(y)} by ++ \spad{(cos(x-y)+cos(x+y))/2}, and \axiom{sin(x)*cos(y)} by ++ \spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses ++ the pattern matcher and so is relatively expensive. To avoid ++ getting into an infinite loop the transformations are applied ++ at most ten times. Implementation ==> add -- for debugging only import OutputForm import OutputPackage -- end debugging import FactoredFunctions(P) import PolynomialCategoryLifting(IndexedExponents K, K, R, P, F) import PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F) smpexp : P -> F termexp : P -> F exlog : P -> F smplog : P -> F smpexpand : P -> F smp2htrigs: P -> F kerexpand : K -> F expandpow : K -> F logexpand : K -> F sup2htrigs: (UP, F) -> F supexp : (UP, F, F, Z) -> F ueval : (F, String, F -> F) -> F ueval2 : (F, String, F -> F) -> F powersimp : (P, List K) -> F t2t : F -> F c2t : F -> F c2s : F -> F s2c : F -> F s2c2 : F -> F th2th : F -> F ch2th : F -> F ch2sh : F -> F sh2ch : F -> F sh2ch2 : F -> F simplify0 : F -> F simplifyLog1 : F -> F logArgs : List F -> F import F import List F if R has PatternMatchable R and R has ConvertibleTo Pattern R and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then XX : F := coerce new()$Symbol YY : F := coerce new()$Symbol sinCosRule : RewriteRule(R,R,F) := rule(cos(XX)*sin(YY),(sin(XX+YY)-sin(XX-YY))/2::F) sinSinRule : RewriteRule(R,R,F) := rule(sin(XX)*sin(YY),(cos(XX-YY)-cos(XX+YY))/2::F) cosCosRule : RewriteRule(R,R,F) := rule(cos(XX)*cos(YY),(cos(XX-YY)+cos(XX+YY))/2::F) expandTrigProducts(e:F):F == applyRules([sinCosRule,sinSinRule,cosCosRule],e,10)$ApplyRules(R,R,F) logArgs(l:List F):F == -- This function will take a list of Expressions (implicitly a sum) and -- add them up, combining log terms. It also replaces n*log(x) by -- log(x^n). import K sum : F := 0 arg : F := 1 for term in l repeat is?(term,"log"::Symbol) => arg := arg * simplifyLog(first(argument(first(kernels(term))))) -- Now look for multiples, including negative ones. prod : Union(PRODUCT, "failed") := isMult(term) (prod case PRODUCT) and is?(prod.var,"log"::Symbol) => arg := arg * simplifyLog ((first argument(prod.var))**(prod.coef)) sum := sum+term sum+log(arg) simplifyLog(e:F):F == simplifyLog1(numerator e)/simplifyLog1(denominator e) simplifyLog1(e:F):F == freeOf?(e,"log"::Symbol) => e -- Check for n*log(u) prod : Union(PRODUCT, "failed") := isMult(e) (prod case PRODUCT) and is?(prod.var,"log"::Symbol) => log simplifyLog ((first argument(prod.var))**(prod.coef)) termList : Union(List(F),"failed") := isTimes(e) -- I'm using two variables, termList and terms, to work round a -- bug in the old compiler. not (termList case "failed") => -- We want to simplify each log term in the product and then multiply -- them together. However, if there is a constant or arithmetic -- expression (i.e. somwthing which looks like a Polynomial) we would -- like to combine it with a log term. terms :List F := [simplifyLog(term) for term in termList::List(F)] exprs :List F := [] for i in 1..#terms repeat if retractIfCan(terms.i)@Union(FPR,"failed") case FPR then exprs := cons(terms.i,exprs) terms := delete!(terms,i) if not empty? exprs then foundLog := false i : NonNegativeInteger := 0 while (not(foundLog) and (i < #terms)) repeat i := i+1 if is?(terms.i,"log"::Symbol) then args : List F := argument(retract(terms.i)@K) setelt(terms,i, log simplifyLog1(first(args)**(*/exprs))) foundLog := true -- The next line deals with a situation which shouldn't occur, -- since we have checked whether we are freeOf log already. if not foundLog then terms := append(exprs,terms) */terms terms : Union(List(F),"failed") := isPlus(e) not (terms case "failed") => logArgs(terms) expt : Union(POW, "failed") := isPower(e) -- (expt case POW) and not one? expt.exponent => (expt case POW) and not (expt.exponent = 1) => simplifyLog(expt.val)**(expt.exponent) kers : List K := kernels e -- not(one?(#kers)) => e -- Have a constant not(((#kers) = 1)) => e -- Have a constant kernel(operator first kers,[simplifyLog(u) for u in argument first kers]) if R has RetractableTo Integer then simplify x == rootProduct(simplify0 x)$AlgebraicManipulations(R,F) else simplify x == simplify0 x expandpow k == a := expandPower first(arg := argument k) b := expandPower second arg -- ne:F := (one? numer a => 1; numer(a)::F ** b) ne:F := (((numer a) = 1) => 1; numer(a)::F ** b) -- de:F := (one? denom a => 1; denom(a)::F ** (-b)) de:F := (((denom a) = 1) => 1; denom(a)::F ** (-b)) ne * de termexp p == exponent:F := 0 coef := (leadingCoefficient p)::P lpow := select(is?(#1, POWER)$K, lk := variables p)$List(K) for k in lk repeat d := degree(p, k) if is?(k, "exp"::Symbol) then exponent := exponent + d * first argument k else if not is?(k, POWER) then -- Expand arguments to functions as well ... MCD 23/1/97 --coef := coef * monomial(1, k, d) coef := coef * monomial(1, kernel(operator k,[simplifyExp u for u in argument k], height k), d) coef::F * exp exponent * powersimp(p, lpow) expandPower f == l := select(is?(#1, POWER)$K, kernels f)$List(K) eval(f, l, [expandpow k for k in l]) -- l is a list of pure powers appearing as kernels in p powersimp(p, l) == empty? l => 1 k := first l -- k = a**b a := first(arg := argument k) exponent := degree(p, k) * second arg empty?(lk := select(a = first argument #1, rest l)) => (a ** exponent) * powersimp(p, rest l) for k0 in lk repeat exponent := exponent + degree(p, k0) * second argument k0 (a ** exponent) * powersimp(p, setDifference(rest l, lk)) t2t x == sin(x) / cos(x) c2t x == cos(x) / sin(x) c2s x == inv sin x s2c x == inv cos x s2c2 x == 1 - cos(x)**2 th2th x == sinh(x) / cosh(x) ch2th x == cosh(x) / sinh(x) ch2sh x == inv sinh x sh2ch x == inv cosh x sh2ch2 x == cosh(x)**2 - 1 ueval(x, s,f) == eval(x, s::Symbol, f) ueval2(x,s,f) == eval(x, s::Symbol, 2, f) cos2sec x == ueval(x, "cos", inv sec #1) sin2csc x == ueval(x, "sin", inv csc #1) csc2sin x == ueval(x, "csc", c2s) sec2cos x == ueval(x, "sec", s2c) tan2cot x == ueval(x, "tan", inv cot #1) cot2tan x == ueval(x, "cot", inv tan #1) tan2trig x == ueval(x, "tan", t2t) cot2trig x == ueval(x, "cot", c2t) cosh2sech x == ueval(x, "cosh", inv sech #1) sinh2csch x == ueval(x, "sinh", inv csch #1) csch2sinh x == ueval(x, "csch", ch2sh) sech2cosh x == ueval(x, "sech", sh2ch) tanh2coth x == ueval(x, "tanh", inv coth #1) coth2tanh x == ueval(x, "coth", inv tanh #1) tanh2trigh x == ueval(x, "tanh", th2th) coth2trigh x == ueval(x, "coth", ch2th) removeCosSq x == ueval2(x, "cos", 1 - (sin #1)**2) removeSinSq x == ueval2(x, "sin", s2c2) removeCoshSq x== ueval2(x, "cosh", 1 + (sinh #1)**2) removeSinhSq x== ueval2(x, "sinh", sh2ch2) expandLog x == smplog(numer x) / smplog(denom x) simplifyExp x == (smpexp numer x) / (smpexp denom x) expand x == (smpexpand numer x) / (smpexpand denom x) smpexpand p == map(kerexpand, #1::F, p) smplog p == map(logexpand, #1::F, p) smp2htrigs p == map(htrigs(#1::F), #1::F, p) htrigs f == (m := mainKernel f) case "failed" => f op := operator(k := m::K) arg := [htrigs x for x in argument k]$List(F) num := univariate(numer f, k) den := univariate(denom f, k) is?(op, "exp"::Symbol) => g1 := cosh(a := first arg) + sinh(a) g2 := cosh(a) - sinh(a) supexp(num,g1,g2,b:= (degree num)::Z quo 2)/supexp(den,g1,g2,b) sup2htrigs(num, g1:= op arg) / sup2htrigs(den, g1) supexp(p, f1, f2, bse) == ans:F := 0 while p ^= 0 repeat g := htrigs(leadingCoefficient(p)::F) if ((d := degree(p)::Z - bse) >= 0) then ans := ans + g * f1 ** d else ans := ans + g * f2 ** (-d) p := reductum p ans sup2htrigs(p, f) == (map(smp2htrigs, p)$SparseUnivariatePolynomialFunctions2(P, F)) f exlog p == +/[r.coef * log(r.logand::F) for r in log squareFree p] logexpand k == nullary?(op := operator k) => k::F is?(op, "log"::Symbol) => exlog(numer(x := expandLog first argument k)) - exlog denom x op [expandLog x for x in argument k]$List(F) kerexpand k == nullary?(op := operator k) => k::F is?(op, POWER) => expandpow k arg := first argument k is?(op, "sec"::Symbol) => inv expand cos arg is?(op, "csc"::Symbol) => inv expand sin arg is?(op, "log"::Symbol) => exlog(numer(x := expand arg)) - exlog denom x num := numer arg den := denom arg num := numer arg den := denom arg -- for debugging output num := numer arg den := denom arg output(message "num:") output(num::OutputForm) output(message "den:") output(den::OutputForm) -- end debugging (b := (reductum num) / den) ^= 0 => a := (leadingMonomial num) / den is?(op, "exp"::Symbol) => exp(expand a) * expand(exp b) is?(op, "sin"::Symbol) => sin(expand a) * expand(cos b) + cos(expand a) * expand(sin b) is?(op, "cos"::Symbol) => cos(expand a) * expand(cos b) - sin(expand a) * expand(sin b) is?(op, "tan"::Symbol) => ta := tan expand a tb := expand tan b (ta + tb) / (1 - ta * tb) is?(op, "cot"::Symbol) => cta := cot expand a ctb := expand cot b (cta * ctb - 1) / (ctb + cta) op [expand x for x in argument k]$List(F) op [expand x for x in argument k]$List(F) smpexp p == ans:F := 0 while p ^= 0 repeat ans := ans + termexp leadingMonomial p p := reductum p ans -- this now works in 3 passes over the expression: -- pass1 rewrites trigs and htrigs in terms of sin,cos,sinh,cosh -- pass2 rewrites sin**2 and sinh**2 in terms of cos and cosh. -- pass3 groups exponentials together simplify0 x == simplifyExp eval(eval(x, ["tan"::Symbol,"cot"::Symbol,"sec"::Symbol,"csc"::Symbol, "tanh"::Symbol,"coth"::Symbol,"sech"::Symbol,"csch"::Symbol], [t2t,c2t,s2c,c2s,th2th,ch2th,sh2ch,ch2sh]), ["sin"::Symbol, "sinh"::Symbol], [2, 2], [s2c2, sh2ch2]) \end{spad} Testing the change \begin{axiom} -- old ex1:=expandTrigProducts(sin(x)*sin(y))$TRMANIP(INT,Expression Integer) ex2:=expand(ex1)$TRMANIP(INT,Expression Integer) --new ex3:=expand(ex1)$TRMANIP2(INT,Expression Integer) \end{axiom} From unknown Mon Sep 12 18:13:59 -0500 2005 From: unknown Date: Mon, 12 Sep 2005 18:13:59 -0500 Subject: Message-ID: <20050912181359-0500@page.axiom-developer.org> \begin{axiom} sinCosProducts := rule sin(x)*sin(y) == (cos(x-y) - cos(x+y))/2 cos(x)*cos(y) == (cos(x-y) + cos(x+y))/2 sin(x)*cos(y) == (sin(x-y) + sin(x+y))/2 sin(x)^2 == (1 - cos(2*x))/2 sin(x)^3 == sin(x)*(1 - cos(2*x))/2 \end{axiom}
Demonstration of modifying and testing a SPAD file
I changed the abbreviation to TRMANIP2 and the name to TranscendentalManipulations2 so I can call both the old and new package if necessary.
spad
)abbrev package TRMANIP2 TranscendentalManipulations2 ++ Transformations on transcendental objects ++ Author: Bob Sutor, Manuel Bronstein ++ Date Created: Way back ++ Date Last Updated: 22 January 1996, added simplifyLog MCD. ++ Description: ++ TranscendentalManipulations provides functions to simplify and ++ expand expressions involving transcendental operators. ++ Keywords: transcendental, manipulation. TranscendentalManipulations2(R, F): Exports == Implementation where R : Join(OrderedSet, GcdDomain) F : Join(FunctionSpace R, TranscendentalFunctionCategory)
Z ==> Integer K ==> Kernel F P ==> SparseMultivariatePolynomial(R, K) UP ==> SparseUnivariatePolynomial P POWER ==> "%power"::Symbol POW ==> Record(val: F,exponent: Z) PRODUCT ==> Record(coef : Z, var : K) FPR ==> Fraction Polynomial R
Exports ==> with expand : F -> F ++ expand(f) performs the following expansions on f:\begin{items} ++ \item 1. logs of products are expanded into sums of logs, ++ \item 2. trigonometric and hyperbolic trigonometric functions ++ of sums are expanded into sums of products of trigonometric ++ and hyperbolic trigonometric functions. ++ \item 3. formal powers of the form \spad{(a/b)**c} are expanded into ++ \spad{a**c * b**(-c)}. ++ \end{items} simplify : F -> F ++ simplify(f) performs the following simplifications on f:\begin{items} ++ \item 1. rewrites trigs and hyperbolic trigs in terms ++ of \spad{sin} ,\spad{cos}, \spad{sinh}, \spad{cosh}. ++ \item 2. rewrites \spad{sin**2} and \spad{sinh**2} in terms ++ of \spad{cos} and \spad{cosh}, ++ \item 3. rewrites \spad{exp(a)*exp(b)} as \spad{exp(a+b)}. ++ \item 4. rewrites \spad{(a**(1/n))**m * (a**(1/s))**t} as a single ++ power of a single radical of \spad{a}. ++ \end{items} htrigs : F -> F ++ htrigs(f) converts all the exponentials in f into ++ hyperbolic sines and cosines. simplifyExp: F -> F ++ simplifyExp(f) converts every product \spad{exp(a)*exp(b)} ++ appearing in f into \spad{exp(a+b)}. simplifyLog : F -> F ++ simplifyLog(f) converts every \spad{log(a) - log(b)} appearing in f ++ into \spad{log(a/b)}, every \spad{log(a) + log(b)} into \spad{log(a*b)} ++ and every \spad{n*log(a)} into \spad{log(a^n)}. expandPower: F -> F ++ expandPower(f) converts every power \spad{(a/b)**c} appearing ++ in f into \spad{a**c * b**(-c)}. expandLog : F -> F ++ expandLog(f) converts every \spad{log(a/b)} appearing in f into ++ \spad{log(a) - log(b)}, and every \spad{log(a*b)} into ++ \spad{log(a) + log(b)}.. cos2sec : F -> F ++ cos2sec(f) converts every \spad{cos(u)} appearing in f into ++ \spad{1/sec(u)}. cosh2sech : F -> F ++ cosh2sech(f) converts every \spad{cosh(u)} appearing in f into ++ \spad{1/sech(u)}. cot2trig : F -> F ++ cot2trig(f) converts every \spad{cot(u)} appearing in f into ++ \spad{cos(u)/sin(u)}. coth2trigh : F -> F ++ coth2trigh(f) converts every \spad{coth(u)} appearing in f into ++ \spad{cosh(u)/sinh(u)}. csc2sin : F -> F ++ csc2sin(f) converts every \spad{csc(u)} appearing in f into ++ \spad{1/sin(u)}. csch2sinh : F -> F ++ csch2sinh(f) converts every \spad{csch(u)} appearing in f into ++ \spad{1/sinh(u)}. sec2cos : F -> F ++ sec2cos(f) converts every \spad{sec(u)} appearing in f into ++ \spad{1/cos(u)}. sech2cosh : F -> F ++ sech2cosh(f) converts every \spad{sech(u)} appearing in f into ++ \spad{1/cosh(u)}. sin2csc : F -> F ++ sin2csc(f) converts every \spad{sin(u)} appearing in f into ++ \spad{1/csc(u)}. sinh2csch : F -> F ++ sinh2csch(f) converts every \spad{sinh(u)} appearing in f into ++ \spad{1/csch(u)}. tan2trig : F -> F ++ tan2trig(f) converts every \spad{tan(u)} appearing in f into ++ \spad{sin(u)/cos(u)}. tanh2trigh : F -> F ++ tanh2trigh(f) converts every \spad{tanh(u)} appearing in f into ++ \spad{sinh(u)/cosh(u)}. tan2cot : F -> F ++ tan2cot(f) converts every \spad{tan(u)} appearing in f into ++ \spad{1/cot(u)}. tanh2coth : F -> F ++ tanh2coth(f) converts every \spad{tanh(u)} appearing in f into ++ \spad{1/coth(u)}. cot2tan : F -> F ++ cot2tan(f) converts every \spad{cot(u)} appearing in f into ++ \spad{1/tan(u)}. coth2tanh : F -> F ++ coth2tanh(f) converts every \spad{coth(u)} appearing in f into ++ \spad{1/tanh(u)}. removeCosSq: F -> F ++ removeCosSq(f) converts every \spad{cos(u)**2} appearing in f into ++ \spad{1 - sin(x)**2}, and also reduces higher ++ powers of \spad{cos(u)} with that formula. removeSinSq: F -> F ++ removeSinSq(f) converts every \spad{sin(u)**2} appearing in f into ++ \spad{1 - cos(x)**2}, and also reduces higher powers of ++ \spad{sin(u)} with that formula. removeCoshSq:F -> F ++ removeCoshSq(f) converts every \spad{cosh(u)**2} appearing in f into ++ \spad{1 - sinh(x)**2}, and also reduces higher powers of ++ \spad{cosh(u)} with that formula. removeSinhSq:F -> F ++ removeSinhSq(f) converts every \spad{sinh(u)**2} appearing in f into ++ \spad{1 - cosh(x)**2}, and also reduces higher powers ++ of \spad{sinh(u)} with that formula. if R has PatternMatchable(R) and R has ConvertibleTo(Pattern(R)) and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then expandTrigProducts : F -> F ++ expandTrigProducts(e) replaces \axiom{sin(x)*sin(y)} by ++ \spad{(cos(x-y)-cos(x+y))/2}, \axiom{cos(x)*cos(y)} by ++ \spad{(cos(x-y)+cos(x+y))/2}, and \axiom{sin(x)*cos(y)} by ++ \spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses ++ the pattern matcher and so is relatively expensive. To avoid ++ getting into an infinite loop the transformations are applied ++ at most ten times.
Implementation ==> add -- for debugging only import OutputForm import OutputPackage -- end debugging import FactoredFunctions(P) import PolynomialCategoryLifting(IndexedExponents K, K, R, P, F) import PolynomialCategoryQuotientFunctions(IndexedExponents K,K,R,P,F)
smpexp : P -> F termexp : P -> F exlog : P -> F smplog : P -> F smpexpand : P -> F smp2htrigs: P -> F kerexpand : K -> F expandpow : K -> F logexpand : K -> F sup2htrigs: (UP, F) -> F supexp : (UP, F, F, Z) -> F ueval : (F, String, F -> F) -> F ueval2 : (F, String, F -> F) -> F powersimp : (P, List K) -> F t2t : F -> F c2t : F -> F c2s : F -> F s2c : F -> F s2c2 : F -> F th2th : F -> F ch2th : F -> F ch2sh : F -> F sh2ch : F -> F sh2ch2 : F -> F simplify0 : F -> F simplifyLog1 : F -> F logArgs : List F -> F
import F import List F
if R has PatternMatchable R and R has ConvertibleTo Pattern R and F has ConvertibleTo(Pattern(R)) and F has PatternMatchable R then XX : F := coerce new()$Symbol YY : F := coerce new()$Symbol sinCosRule : RewriteRule(R,R,F) := rule(cos(XX)*sin(YY),(sin(XX+YY)-sin(XX-YY))/2::F) sinSinRule : RewriteRule(R,R,F) := rule(sin(XX)*sin(YY),(cos(XX-YY)-cos(XX+YY))/2::F) cosCosRule : RewriteRule(R,R,F) := rule(cos(XX)*cos(YY),(cos(XX-YY)+cos(XX+YY))/2::F) expandTrigProducts(e:F):F == applyRules([sinCosRule,sinSinRule,cosCosRule],e,10)$ApplyRules(R,R,F)
logArgs(l:List F):F == -- This function will take a list of Expressions (implicitly a sum) and -- add them up, combining log terms. It also replaces n*log(x) by -- log(x^n). import K sum : F := 0 arg : F := 1 for term in l repeat is?(term,"log"::Symbol) => arg := arg * simplifyLog(first(argument(first(kernels(term))))) -- Now look for multiples, including negative ones. prod : Union(PRODUCT, "failed") := isMult(term) (prod case PRODUCT) and is?(prod.var,"log"::Symbol) => arg := arg * simplifyLog ((first argument(prod.var))**(prod.coef)) sum := sum+term sum+log(arg)
simplifyLog(e:F):F == simplifyLog1(numerator e)/simplifyLog1(denominator e)
simplifyLog1(e:F):F == freeOf?(e,"log"::Symbol) => e
-- Check for n*log(u) prod : Union(PRODUCT, "failed") := isMult(e) (prod case PRODUCT) and is?(prod.var,"log"::Symbol) => log simplifyLog ((first argument(prod.var))**(prod.coef))
termList : Union(List(F),"failed") := isTimes(e) -- I'm using two variables, termList and terms, to work round a -- bug in the old compiler. not (termList case "failed") => -- We want to simplify each log term in the product and then multiply -- them together. However, if there is a constant or arithmetic -- expression (i.e. somwthing which looks like a Polynomial) we would -- like to combine it with a log term. terms :List F := [simplifyLog(term) for term in termList::List(F)] exprs :List F := [] for i in 1..#terms repeat if retractIfCan(terms.i)@Union(FPR,"failed") case FPR then exprs := cons(terms.i,exprs) terms := delete!(terms,i) if not empty? exprs then foundLog := false i : NonNegativeInteger := 0 while (not(foundLog) and (i < #terms)) repeat i := i+1 if is?(terms.i,"log"::Symbol) then args : List F := argument(retract(terms.i)@K) setelt(terms,i, log simplifyLog1(first(args)**(*/exprs))) foundLog := true -- The next line deals with a situation which shouldn't occur, -- since we have checked whether we are freeOf log already. if not foundLog then terms := append(exprs,terms) */terms
terms : Union(List(F),"failed") := isPlus(e) not (terms case "failed") => logArgs(terms)
expt : Union(POW, "failed") := isPower(e) -- (expt case POW) and not one? expt.exponent => (expt case POW) and not (expt.exponent = 1) => simplifyLog(expt.val)**(expt.exponent)
kers : List K := kernels e -- not(one?(#kers)) => e -- Have a constant not(((#kers) = 1)) => e -- Have a constant kernel(operator first kers,[simplifyLog(u) for u in argument first kers])
if R has RetractableTo Integer then simplify x == rootProduct(simplify0 x)$AlgebraicManipulations(R,F)
else simplify x == simplify0 x
expandpow k == a := expandPower first(arg := argument k) b := expandPower second arg -- ne:F := (one? numer a => 1; numer(a)::F ** b) ne:F := (((numer a) = 1) => 1; numer(a)::F ** b) -- de:F := (one? denom a => 1; denom(a)::F ** (-b)) de:F := (((denom a) = 1) => 1; denom(a)::F ** (-b)) ne * de
termexp p == exponent:F := 0 coef := (leadingCoefficient p)::P lpow := select(is?(#1, POWER)$K, lk := variables p)$List(K) for k in lk repeat d := degree(p, k) if is?(k, "exp"::Symbol) then exponent := exponent + d * first argument k else if not is?(k, POWER) then -- Expand arguments to functions as well ... MCD 23/1/97 --coef := coef * monomial(1, k, d) coef := coef * monomial(1, kernel(operator k,[simplifyExp u for u in argument k], height k), d) coef::F * exp exponent * powersimp(p, lpow)
expandPower f == l := select(is?(#1, POWER)$K, kernels f)$List(K) eval(f, l, [expandpow k for k in l])
-- l is a list of pure powers appearing as kernels in p powersimp(p, l) == empty? l => 1 k := first l -- k = a**b a := first(arg := argument k) exponent := degree(p, k) * second arg empty?(lk := select(a = first argument #1, rest l)) => (a ** exponent) * powersimp(p, rest l) for k0 in lk repeat exponent := exponent + degree(p, k0) * second argument k0 (a ** exponent) * powersimp(p, setDifference(rest l, lk))
t2t x == sin(x) / cos(x) c2t x == cos(x) / sin(x) c2s x == inv sin x s2c x == inv cos x s2c2 x == 1 - cos(x)**2 th2th x == sinh(x) / cosh(x) ch2th x == cosh(x) / sinh(x) ch2sh x == inv sinh x sh2ch x == inv cosh x sh2ch2 x == cosh(x)**2 - 1 ueval(x, s,f) == eval(x, s::Symbol, f) ueval2(x,s,f) == eval(x, s::Symbol, 2, f) cos2sec x == ueval(x, "cos", inv sec #1) sin2csc x == ueval(x, "sin", inv csc #1) csc2sin x == ueval(x, "csc", c2s) sec2cos x == ueval(x, "sec", s2c) tan2cot x == ueval(x, "tan", inv cot #1) cot2tan x == ueval(x, "cot", inv tan #1) tan2trig x == ueval(x, "tan", t2t) cot2trig x == ueval(x, "cot", c2t) cosh2sech x == ueval(x, "cosh", inv sech #1) sinh2csch x == ueval(x, "sinh", inv csch #1) csch2sinh x == ueval(x, "csch", ch2sh) sech2cosh x == ueval(x, "sech", sh2ch) tanh2coth x == ueval(x, "tanh", inv coth #1) coth2tanh x == ueval(x, "coth", inv tanh #1) tanh2trigh x == ueval(x, "tanh", th2th) coth2trigh x == ueval(x, "coth", ch2th) removeCosSq x == ueval2(x, "cos", 1 - (sin #1)**2) removeSinSq x == ueval2(x, "sin", s2c2) removeCoshSq x== ueval2(x, "cosh", 1 + (sinh #1)**2) removeSinhSq x== ueval2(x, "sinh", sh2ch2) expandLog x == smplog(numer x) / smplog(denom x) simplifyExp x == (smpexp numer x) / (smpexp denom x) expand x == (smpexpand numer x) / (smpexpand denom x) smpexpand p == map(kerexpand, #1::F, p) smplog p == map(logexpand, #1::F, p) smp2htrigs p == map(htrigs(#1::F), #1::F, p)
htrigs f == (m := mainKernel f) case "failed" => f op := operator(k := m::K) arg := [htrigs x for x in argument k]$List(F) num := univariate(numer f, k) den := univariate(denom f, k) is?(op, "exp"::Symbol) => g1 := cosh(a := first arg) + sinh(a) g2 := cosh(a) - sinh(a) supexp(num,g1,g2,b:= (degree num)::Z quo 2)/supexp(den,g1,g2,b) sup2htrigs(num, g1:= op arg) / sup2htrigs(den, g1)
supexp(p, f1, f2, bse) == ans:F := 0 while p ^= 0 repeat g := htrigs(leadingCoefficient(p)::F) if ((d := degree(p)::Z - bse) >= 0) then ans := ans + g * f1 ** d else ans := ans + g * f2 ** (-d) p := reductum p ans
sup2htrigs(p, f) == (map(smp2htrigs, p)$SparseUnivariatePolynomialFunctions2(P, F)) f
exlog p == +/[r.coef * log(r.logand::F) for r in log squareFree p]
logexpand k == nullary?(op := operator k) => k::F is?(op, "log"::Symbol) => exlog(numer(x := expandLog first argument k)) - exlog denom x op [expandLog x for x in argument k]$List(F)
kerexpand k == nullary?(op := operator k) => k::F is?(op, POWER) => expandpow k arg := first argument k is?(op, "sec"::Symbol) => inv expand cos arg is?(op, "csc"::Symbol) => inv expand sin arg is?(op, "log"::Symbol) => exlog(numer(x := expand arg)) - exlog denom x num := numer arg den := denom arg num := numer arg den := denom arg
-- for debugging output num := numer arg den := denom arg output(message "num:") output(num::OutputForm) output(message "den:") output(den::OutputForm) -- end debugging
(b := (reductum num) / den) ^= 0 => a := (leadingMonomial num) / den is?(op, "exp"::Symbol) => exp(expand a) * expand(exp b) is?(op, "sin"::Symbol) => sin(expand a) * expand(cos b) + cos(expand a) * expand(sin b) is?(op, "cos"::Symbol) => cos(expand a) * expand(cos b) - sin(expand a) * expand(sin b) is?(op, "tan"::Symbol) => ta := tan expand a tb := expand tan b (ta + tb) / (1 - ta * tb) is?(op, "cot"::Symbol) => cta := cot expand a ctb := expand cot b (cta * ctb - 1) / (ctb + cta) op [expand x for x in argument k]$List(F) op [expand x for x in argument k]$List(F)
smpexp p == ans:F := 0 while p ^= 0 repeat ans := ans + termexp leadingMonomial p p := reductum p ans
-- this now works in 3 passes over the expression: -- pass1 rewrites trigs and htrigs in terms of sin,cos,sinh,cosh -- pass2 rewrites sin**2 and sinh**2 in terms of cos and cosh. -- pass3 groups exponentials together simplify0 x == simplifyExp eval(eval(x, ["tan"::Symbol,"cot"::Symbol,"sec"::Symbol,"csc"::Symbol, "tanh"::Symbol,"coth"::Symbol,"sech"::Symbol,"csch"::Symbol], [t2t,c2t,s2c,c2s,th2th,ch2th,sh2ch,ch2sh]), ["sin"::Symbol, "sinh"::Symbol], [2, 2], [s2c2, sh2ch2])
spad
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/6388374989627192641-25px001.spad using
old system compiler.
TRMANIP2 abbreviates package TranscendentalManipulations2
processing macro definition Z ==> Integer
processing macro definition K ==> Kernel F
processing macro definition P ==> SparseMultivariatePolynomial(R,Kernel F)
processing macro definition UP ==> SparseUnivariatePolynomial SparseMultivariatePolynomial(R,Kernel F)
processing macro definition POWER ==> ::(%power,Symbol)
processing macro definition POW ==> Record(val: F,exponent: Integer)
processing macro definition PRODUCT ==> Record(coef: Integer,var: Kernel F)
processing macro definition FPR ==> Fraction Polynomial R
processing macro definition Exports ==> -- the constructor category
processing macro definition Implementation ==> -- the constructor capsule
------------------------------------------------------------------------
initializing NRLIB TRMANIP2 for TranscendentalManipulations2
compiling into NRLIB TRMANIP2
****** Domain: R already in scope
****** Domain: F already in scope
importing OutputForm
importing OutputPackage
importing FactoredFunctions SparseMultivariatePolynomial(R,Kernel F)
importing PolynomialCategoryLifting(IndexedExponents Kernel F,Kernel F,R,SparseMultivariatePolynomial(R,Kernel F),F)
importing PolynomialCategoryQuotientFunctions(IndexedExponents Kernel F,Kernel F,R,SparseMultivariatePolynomial(R,Kernel F),F)
importing F
importing List F
****** Domain: R already in scope
augmenting R: (PatternMatchable R)
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern R))
****** Domain: F already in scope
augmenting F: (ConvertibleTo (Pattern R))
****** Domain: F already in scope
augmenting F: (PatternMatchable R)
augmenting $: (SIGNATURE $ expandTrigProducts (F F))
compiling exported expandTrigProducts : F -> F
Time: 0.87 SEC.
compiling local logArgs : List F -> F
Time: 0.23 SEC.
compiling exported simplifyLog : F -> F
Time: 0.02 SEC.
compiling local simplifyLog1 : F -> F
Time: 0.34 SEC.
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
compiling exported simplify : F -> F
Time: 0.01 SEC.
compiling exported simplify : F -> F
Time: 0 SEC.
compiling local expandpow : Kernel F -> F
Time: 0.14 SEC.
compiling local termexp : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.33 SEC.
compiling exported expandPower : F -> F
Time: 0.03 SEC.
compiling local powersimp : (SparseMultivariatePolynomial(R,Kernel F),List Kernel F) -> F
Time: 0.28 SEC.
compiling local t2t : F -> F
Time: 0.02 SEC.
compiling local c2t : F -> F
Time: 0.02 SEC.
compiling local c2s : F -> F
Time: 0 SEC.
compiling local s2c : F -> F
Time: 0 SEC.
compiling local s2c2 : F -> F
Time: 0.09 SEC.
compiling local th2th : F -> F
Time: 0.02 SEC.
compiling local ch2th : F -> F
Time: 0.02 SEC.
compiling local ch2sh : F -> F
Time: 0 SEC.
compiling local sh2ch : F -> F
Time: 0 SEC.
compiling local sh2ch2 : F -> F
Time: 0.02 SEC.
compiling local ueval : (F,String,F -> F) -> F
Time: 0.10 SEC.
compiling local ueval2 : (F,String,F -> F) -> F
Time: 0.13 SEC.
compiling exported cos2sec : F -> F
Time: 0.01 SEC.
compiling exported sin2csc : F -> F
Time: 0.02 SEC.
compiling exported csc2sin : F -> F
Time: 0.01 SEC.
compiling exported sec2cos : F -> F
Time: 0.01 SEC.
compiling exported tan2cot : F -> F
Time: 0.01 SEC.
compiling exported cot2tan : F -> F
Time: 0.01 SEC.
compiling exported tan2trig : F -> F
Time: 0.01 SEC.
compiling exported cot2trig : F -> F
Time: 0.01 SEC.
compiling exported cosh2sech : F -> F
Time: 0.02 SEC.
compiling exported sinh2csch : F -> F
Time: 0.01 SEC.
compiling exported csch2sinh : F -> F
Time: 0.01 SEC.
compiling exported sech2cosh : F -> F
Time: 0.01 SEC.
compiling exported tanh2coth : F -> F
Time: 0.01 SEC.
compiling exported coth2tanh : F -> F
Time: 0.02 SEC.
compiling exported tanh2trigh : F -> F
Time: 0.01 SEC.
compiling exported coth2trigh : F -> F
Time: 0.01 SEC.
compiling exported removeCosSq : F -> F
Time: 0.03 SEC.
compiling exported removeSinSq : F -> F
Time: 0.01 SEC.
compiling exported removeCoshSq : F -> F
Time: 0.03 SEC.
compiling exported removeSinhSq : F -> F
Time: 0.01 SEC.
compiling exported expandLog : F -> F
Time: 0.02 SEC.
compiling exported simplifyExp : F -> F
Time: 0.02 SEC.
compiling exported expand : F -> F
Time: 0.03 SEC.
compiling local smpexpand : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.10 SEC.
compiling local smplog : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.03 SEC.
compiling local smp2htrigs : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.02 SEC.
compiling exported htrigs : F -> F
Time: 0.18 SEC.
compiling local supexp : (SparseUnivariatePolynomial SparseMultivariatePolynomial(R,Kernel F),F,F,Integer) -> F
Time: 0.22 SEC.
compiling local sup2htrigs : (SparseUnivariatePolynomial SparseMultivariatePolynomial(R,Kernel F),F) -> F
Time: 0.25 SEC.
compiling local exlog : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.20 SEC.
compiling local logexpand : Kernel F -> F
Time: 0.05 SEC.
compiling local kerexpand : Kernel F -> F
Time: 0.42 SEC.
compiling local smpexp : SparseMultivariatePolynomial(R,Kernel F) -> F
Time: 0.03 SEC.
compiling local simplify0 : F -> F
Time: 1.30 SEC.
****** Domain: F already in scope
augmenting F: (ConvertibleTo (Pattern R))
****** Domain: F already in scope
augmenting F: (PatternMatchable R)
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern R))
****** Domain: R already in scope
augmenting R: (PatternMatchable R)
augmenting $: (SIGNATURE $ expandTrigProducts (F F))
(time taken in buildFunctor: 0)
;;; *** |TranscendentalManipulations2| REDEFINED
;;; *** |TranscendentalManipulations2| REDEFINED
Time: 0.21 SEC.
Warnings:
[1] logArgs: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch))
[2] logArgs: arg has no value
[3] simplifyLog1: exprs has no value
[4] simplifyLog1: terms has no value
[5] simplifyLog1: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch))
[6] expandpow: not known that (OrderedSet) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch))
[7] termexp: exponent has no value
[8] htrigs: not known that (Ring) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch))
[9] exlog: not known that (IntegralDomain) is of mode (CATEGORY package (SIGNATURE expand (F F)) (SIGNATURE simplify (F F)) (SIGNATURE htrigs (F F)) (SIGNATURE simplifyExp (F F)) (SIGNATURE simplifyLog (F F)) (SIGNATURE expandPower (F F)) (SIGNATURE expandLog (F F)) (SIGNATURE cos2sec (F F)) (SIGNATURE cosh2sech (F F)) (SIGNATURE cot2trig (F F)) (SIGNATURE coth2trigh (F F)) (SIGNATURE csc2sin (F F)) (SIGNATURE csch2sinh (F F)) (SIGNATURE sec2cos (F F)) (SIGNATURE sech2cosh (F F)) (SIGNATURE sin2csc (F F)) (SIGNATURE sinh2csch (F F)) (SIGNATURE tan2trig (F F)) (SIGNATURE tanh2trigh (F F)) (SIGNATURE tan2cot (F F)) (SIGNATURE tanh2coth (F F)) (SIGNATURE cot2tan (F F)) (SIGNATURE coth2tanh (F F)) (SIGNATURE removeCosSq (F F)) (SIGNATURE removeSinSq (F F)) (SIGNATURE removeCoshSq (F F)) (SIGNATURE removeSinhSq (F F)) (IF (has R (PatternMatchable R)) (IF (has R (ConvertibleTo (Pattern R))) (IF (has F (ConvertibleTo (Pattern R))) (IF (has F (PatternMatchable R)) (SIGNATURE expandTrigProducts (F F)) noBranch) noBranch) noBranch) noBranch))
[10] logexpand: IN has no value
[11] logexpand: x has no value
[12] kerexpand: IN has no value
[13] kerexpand: x has no value
Cumulative Statistics for Constructor TranscendentalManipulations2
Time: 6.02 seconds
finalizing NRLIB TRMANIP2
Processing TranscendentalManipulations2 for Browser database:
--------(expand (F F))---------
--------(simplify (F F))---------
--------(htrigs (F F))---------
--------(simplifyExp (F F))---------
--------(simplifyLog (F F))---------
--------(expandPower (F F))---------
--------(expandLog (F F))---------
--------(cos2sec (F F))---------
--------(cosh2sech (F F))---------
--------(cot2trig (F F))---------
--------(coth2trigh (F F))---------
--------(csc2sin (F F))---------
--------(csch2sinh (F F))---------
--------(sec2cos (F F))---------
--------(sech2cosh (F F))---------
--------(sin2csc (F F))---------
--------(sinh2csch (F F))---------
--------(tan2trig (F F))---------
--------(tanh2trigh (F F))---------
--------(tan2cot (F F))---------
--------(tanh2coth (F F))---------
--------(cot2tan (F F))---------
--------(coth2tanh (F F))---------
--------(removeCosSq (F F))---------
--------(removeSinSq (F F))---------
--------(removeCoshSq (F F))---------
--------(removeSinhSq (F F))---------
--------(expandTrigProducts (F F))---------
--------constructor---------
------------------------------------------------------------------------
TranscendentalManipulations2 is now explicitly exposed in frame
initial
TranscendentalManipulations2 will be automatically loaded when
needed from /var/zope2/var/LatexWiki/TRMANIP2.NRLIB/codeTesting the change
axiom
-- old ex1:=expandTrigProducts(sin(x)*sin(y))$TRMANIP(INT,Expression Integer)
 | (1) |
Type: Expression Integer
axiom
ex2:=expand(ex1)$TRMANIP(INT,Expression Integer)
 | (2) |
Type: Expression Integer
axiom
--new ex3:=expand(ex1)$TRMANIP2(INT,Expression Integer)
num: y + x den: 1 num: x den: 1 num: x den: 1 num: y - x den: 1 num: x den: 1 num: x den: 1
 | (3) |
Type: Expression Integer
... --unknown, Mon, 12 Sep 2005 18:13:59 -0500 reply
axiom
sinCosProducts := rule sin(x)*sin(y) == (cos(x-y) - cos(x+y))/2 cos(x)*cos(y) == (cos(x-y) + cos(x+y))/2 sin(x)*cos(y) == (sin(x-y) + sin(x+y))/2 sin(x)^2 == (1 - cos(2*x))/2 sin(x)^3 == sin(x)*(1 - cos(2*x))/2
 | (4) |
Type: Ruleset(Integer,Integer,Expression Integer)