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last edited 9 years ago by Bill Page |
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Editor: Bill Page
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changed: - \begin{spad} )abbrev package EF ElementaryFunction ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 10 April 1995 ++ Keywords: elementary, function, logarithm, exponential. ++ Examples: )r EF INPUT ++ Description: Provides elementary functions over an integral domain. ElementaryFunction(R, F) : Exports == Implementation where R : Join(Comparable, IntegralDomain) F : Join(FunctionSpace R, RadicalCategory) B ==> Boolean L ==> List Z ==> Integer OP ==> BasicOperator K ==> Kernel F INV ==> error "Invalid argument" Exports ==> with exp : F -> F ++ exp(x) applies the exponential operator to x log : F -> F ++ log(x) applies the logarithm operator to x sin : F -> F ++ sin(x) applies the sine operator to x cos : F -> F ++ cos(x) applies the cosine operator to x tan : F -> F ++ tan(x) applies the tangent operator to x cot : F -> F ++ cot(x) applies the cotangent operator to x sec : F -> F ++ sec(x) applies the secant operator to x csc : F -> F ++ csc(x) applies the cosecant operator to x asin : F -> F ++ asin(x) applies the inverse sine operator to x acos : F -> F ++ acos(x) applies the inverse cosine operator to x atan : F -> F ++ atan(x) applies the inverse tangent operator to x acot : F -> F ++ acot(x) applies the inverse cotangent operator to x asec : F -> F ++ asec(x) applies the inverse secant operator to x acsc : F -> F ++ acsc(x) applies the inverse cosecant operator to x sinh : F -> F ++ sinh(x) applies the hyperbolic sine operator to x cosh : F -> F ++ cosh(x) applies the hyperbolic cosine operator to x tanh : F -> F ++ tanh(x) applies the hyperbolic tangent operator to x coth : F -> F ++ coth(x) applies the hyperbolic cotangent operator to x sech : F -> F ++ sech(x) applies the hyperbolic secant operator to x csch : F -> F ++ csch(x) applies the hyperbolic cosecant operator to x asinh : F -> F ++ asinh(x) applies the inverse hyperbolic sine operator to x acosh : F -> F ++ acosh(x) applies the inverse hyperbolic cosine operator to x atanh : F -> F ++ atanh(x) applies the inverse hyperbolic tangent operator to x acoth : F -> F ++ acoth(x) applies the inverse hyperbolic cotangent operator to x asech : F -> F ++ asech(x) applies the inverse hyperbolic secant operator to x acsch : F -> F ++ acsch(x) applies the inverse hyperbolic cosecant operator to x pi : () -> F ++ pi() returns the pi operator belong? : OP -> Boolean ++ belong?(p) returns true if operator p is elementary operator : OP -> OP ++ operator(p) returns an elementary operator with the same symbol as p -- the following should be local, but are conditional iisqrt2 : () -> F ++ iisqrt2() should be local but conditional iisqrt3 : () -> F ++ iisqrt3() should be local but conditional iiexp : F -> F ++ iiexp(x) should be local but conditional iilog : F -> F ++ iilog(x) should be local but conditional iisin : F -> F ++ iisin(x) should be local but conditional iicos : F -> F ++ iicos(x) should be local but conditional iitan : F -> F ++ iitan(x) should be local but conditional iicot : F -> F ++ iicot(x) should be local but conditional iisec : F -> F ++ iisec(x) should be local but conditional iicsc : F -> F ++ iicsc(x) should be local but conditional iiasin : F -> F ++ iiasin(x) should be local but conditional iiacos : F -> F ++ iiacos(x) should be local but conditional iiatan : F -> F ++ iiatan(x) should be local but conditional iiacot : F -> F ++ iiacot(x) should be local but conditional iiasec : F -> F ++ iiasec(x) should be local but conditional iiacsc : F -> F ++ iiacsc(x) should be local but conditional iisinh : F -> F ++ iisinh(x) should be local but conditional iicosh : F -> F ++ iicosh(x) should be local but conditional iitanh : F -> F ++ iitanh(x) should be local but conditional iicoth : F -> F ++ iicoth(x) should be local but conditional iisech : F -> F ++ iisech(x) should be local but conditional iicsch : F -> F ++ iicsch(x) should be local but conditional iiasinh : F -> F ++ iiasinh(x) should be local but conditional iiacosh : F -> F ++ iiacosh(x) should be local but conditional iiatanh : F -> F ++ iiatanh(x) should be local but conditional iiacoth : F -> F ++ iiacoth(x) should be local but conditional iiasech : F -> F ++ iiasech(x) should be local but conditional iiacsch : F -> F ++ iiacsch(x) should be local but conditional specialTrigs:(F, L Record(func:F,pole:B)) -> Union(F, "failed") ++ specialTrigs(x, l) should be local but conditional localReal? : F -> Boolean ++ localReal?(x) should be local but conditional Implementation ==> add ELEM := 'elem ipi : List F -> F iexp : F -> F ilog : F -> F iiilog : F -> F isin : F -> F icos : F -> F itan : F -> F icot : F -> F isec : F -> F icsc : F -> F iasin : F -> F iacos : F -> F iatan : F -> F iacot : F -> F iasec : F -> F iacsc : F -> F isinh : F -> F icosh : F -> F itanh : F -> F icoth : F -> F isech : F -> F icsch : F -> F iasinh : F -> F iacosh : F -> F iatanh : F -> F iacoth : F -> F iasech : F -> F iacsch : F -> F dropfun : F -> F kernel : F -> K posrem : (Z, Z) -> Z iisqrt1 : () -> F valueOrPole : Record(func : F, pole : B) -> F oppi := operator('pi)$CommonOperators oplog := operator('log)$CommonOperators opexp := operator('exp)$CommonOperators opsin := operator('sin)$CommonOperators opcos := operator('cos)$CommonOperators optan := operator('tan)$CommonOperators opcot := operator('cot)$CommonOperators opsec := operator('sec)$CommonOperators opcsc := operator('csc)$CommonOperators opasin := operator('asin)$CommonOperators opacos := operator('acos)$CommonOperators opatan := operator('atan)$CommonOperators opacot := operator('acot)$CommonOperators opasec := operator('asec)$CommonOperators opacsc := operator('acsc)$CommonOperators opsinh := operator('sinh)$CommonOperators opcosh := operator('cosh)$CommonOperators optanh := operator('tanh)$CommonOperators opcoth := operator('coth)$CommonOperators opsech := operator('sech)$CommonOperators opcsch := operator('csch)$CommonOperators opasinh := operator('asinh)$CommonOperators opacosh := operator('acosh)$CommonOperators opatanh := operator('atanh)$CommonOperators opacoth := operator('acoth)$CommonOperators opasech := operator('asech)$CommonOperators opacsch := operator('acsch)$CommonOperators -- Pi is a domain... Pie, isqrt1, isqrt2, isqrt3 : F -- following code is conditionalized on arbitraryPrecision to recompute in -- case user changes the precision if R has TranscendentalFunctionCategory then Pie := pi()$R :: F else Pie := kernel(oppi, nil()$List(F)) if R has TranscendentalFunctionCategory and R has arbitraryPrecision then pi() == pi()$R :: F else pi() == Pie if R has imaginary : () -> R then isqrt1 := imaginary()$R :: F else isqrt1 := sqrt(-1::F) if R has RadicalCategory then isqrt2 := sqrt(2::R)::F isqrt3 := sqrt(3::R)::F else isqrt2 := sqrt(2::F) isqrt3 := sqrt(3::F) iisqrt1() == isqrt1 if R has RadicalCategory and R has arbitraryPrecision then iisqrt2() == sqrt(2::R)::F iisqrt3() == sqrt(3::R)::F else iisqrt2() == isqrt2 iisqrt3() == isqrt3 ipi l == pi() log x == oplog x exp x == opexp x sin x == opsin x cos x == opcos x tan x == optan x cot x == opcot x sec x == opsec x csc x == opcsc x asin x == opasin x acos x == opacos x atan x == opatan x acot x == opacot x asec x == opasec x acsc x == opacsc x sinh x == opsinh x cosh x == opcosh x tanh x == optanh x coth x == opcoth x sech x == opsech x csch x == opcsch x asinh x == opasinh x acosh x == opacosh x atanh x == opatanh x acoth x == opacoth x asech x == opasech x acsch x == opacsch x kernel x == retract(x)@K posrem(n, m) == ((r := n rem m) < 0 => r + m; r) valueOrPole rec == (rec.pole => INV; rec.func) belong? op == has?(op, ELEM) operator op == is?(op, 'pi) => oppi is?(op, 'log) => oplog is?(op, 'exp) => opexp is?(op, 'sin) => opsin is?(op, 'cos) => opcos is?(op, 'tan) => optan is?(op, 'cot) => opcot is?(op, 'sec) => opsec is?(op, 'csc) => opcsc is?(op, 'asin) => opasin is?(op, 'acos) => opacos is?(op, 'atan) => opatan is?(op, 'acot) => opacot is?(op, 'asec) => opasec is?(op, 'acsc) => opacsc is?(op, 'sinh) => opsinh is?(op, 'cosh) => opcosh is?(op, 'tanh) => optanh is?(op, 'coth) => opcoth is?(op, 'sech) => opsech is?(op, 'csch) => opcsch is?(op, 'asinh) => opasinh is?(op, 'acosh) => opacosh is?(op, 'atanh) => opatanh is?(op, 'acoth) => opacoth is?(op, 'asech) => opasech is?(op, 'acsch) => opacsch error "Not an elementary operator" dropfun x == ((k := retractIfCan(x)@Union(K, "failed")) case "failed") or empty?(argument(k::K)) => 0 first argument(k::K) if R has RetractableTo Z then specialTrigs(x, values) == (r := retractIfCan(y := x/pi())@Union(Fraction Z, "failed")) case "failed" => "failed" q := r::Fraction(Integer) m := minIndex values (n := retractIfCan(q)@Union(Z, "failed")) case Z => even?(n::Z) => valueOrPole(values.m) valueOrPole(values.(m+1)) (n := retractIfCan(2*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 4)) => valueOrPole(values.(m+2)) (s := posrem(n::Z, 4)) = 1 => valueOrPole(values.(m+2)) valueOrPole(values.(m+3)) (n := retractIfCan(3*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 6)) => valueOrPole(values.(m+4)) (s := posrem(n::Z, 6)) = 1 => valueOrPole(values.(m+4)) s = 2 => valueOrPole(values.(m+5)) s = 4 => valueOrPole(values.(m+6)) valueOrPole(values.(m+7)) (n := retractIfCan(4*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 8)) => valueOrPole(values.(m+8)) (s := posrem(n::Z, 8)) = 1 => valueOrPole(values.(m+8)) s = 3 => valueOrPole(values.(m+9)) s = 5 => valueOrPole(values.(m+10)) valueOrPole(values.(m+11)) (n := retractIfCan(6*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 12)) => valueOrPole(values.(m+12)) (s := posrem(n::Z, 12)) = 1 => valueOrPole(values.(m+12)) s = 5 => valueOrPole(values.(m+13)) s = 7 => valueOrPole(values.(m+14)) valueOrPole(values.(m+15)) "failed" else specialTrigs(x, values) == "failed" isin x == zero? x => 0 y := dropfun x is?(x, opasin) => y is?(x, opacos) => sqrt(1 - y^2) is?(x, opatan) => y / sqrt(1 + y^2) is?(x, opacot) => inv sqrt(1 + y^2) is?(x, opasec) => sqrt(y^2 - 1) / y is?(x, opacsc) => inv y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[0, false], [0, false], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opsin, x) icos x == zero? x => 1 y := dropfun x is?(x, opasin) => sqrt(1 - y^2) is?(x, opacos) => y is?(x, opatan) => inv sqrt(1 + y^2) is?(x, opacot) => y / sqrt(1 + y^2) is?(x, opasec) => inv y is?(x, opacsc) => sqrt(y^2 - 1) / y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[1, false], [-1, false], [0, false], [0, false], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opcos, x) itan x == zero? x => 0 y := dropfun x is?(x, opasin) => y / sqrt(1 - y^2) is?(x, opacos) => sqrt(1 - y^2) / y is?(x, opatan) => y is?(x, opacot) => inv y is?(x, opasec) => sqrt(y^2 - 1) is?(x, opacsc) => inv sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, false], [0, false], [0, true], [0, true], [s3, false], [-s3, false], [s3, false], [-s3, false], [1, false], [-1, false], [1, false], [-1, false], [s33, false], [-s33, false], [s33, false], [-s33, false]]) u case F => u :: F kernel(optan, x) icot x == zero? x => INV y := dropfun x is?(x, opasin) => sqrt(1 - y^2) / y is?(x, opacos) => y / sqrt(1 - y^2) is?(x, opatan) => inv y is?(x, opacot) => y is?(x, opasec) => inv sqrt(y^2 - 1) is?(x, opacsc) => sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, true], [0, true], [0, false], [0, false], [s33, false], [-s33, false], [s33, false], [-s33, false], [1, false], [-1, false], [1, false], [-1, false], [s3, false], [-s3, false], [s3, false], [-s3, false]]) u case F => u :: F kernel(opcot, x) isec x == zero? x => 1 y := dropfun x is?(x, opasin) => inv sqrt(1 - y^2) is?(x, opacos) => inv y is?(x, opatan) => sqrt(1 + y^2) is?(x, opacot) => sqrt(1 + y^2) / y is?(x, opasec) => y is?(x, opacsc) => y / sqrt(y^2 - 1) s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[1, false], [-1, false], [0, true], [0, true], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opsec, x) icsc x == zero? x => INV y := dropfun x is?(x, opasin) => inv y is?(x, opacos) => inv sqrt(1 - y^2) is?(x, opatan) => sqrt(1 + y^2) / y is?(x, opacot) => sqrt(1 + y^2) is?(x, opasec) => y / sqrt(y^2 - 1) is?(x, opacsc) => y s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[0, true], [0, true], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opcsc, x) iasin x == zero? x => 0 -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x, opsin) => y -- is?(x, opcos) => pi() / (2::F) - y kernel(opasin, x) iacos x == zero? x => pi() / (2::F) -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x, opsin) => pi() / (2::F) - y -- is?(x, opcos) => y kernel(opacos, x) iatan x == zero? x => 0 -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => - pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (3::F) -- one?(x*r3) => pi() / (6::F) (x*r3) = 1 => pi() / (6::F) -- y := dropfun x -- is?(x, optan) => y -- is?(x, opcot) => pi() / (2::F) - y kernel(opatan, x) iacot x == zero? x => pi() / (2::F) -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => 3 * pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (6::F) x = -r3 => 5 * pi() / (6::F) -- one?(xx := x*r3) => pi() / (3::F) (xx := x*r3) = 1 => pi() / (3::F) xx = -1 => 2* pi() / (3::F) -- y := dropfun x -- is?(x, optan) => pi() / (2::F) - y -- is?(x, opcot) => y kernel(opacot, x) iasec x == zero? x => INV -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x, opsec) => y -- is?(x, opcsc) => pi() / (2::F) - y kernel(opasec, x) iacsc x == zero? x => INV -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x, opsec) => pi() / (2::F) - y -- is?(x, opcsc) => y kernel(opacsc, x) isinh x == zero? x => 0 y := dropfun x is?(x, opasinh) => y is?(x, opacosh) => sqrt(y^2 - 1) is?(x, opatanh) => y / sqrt(1 - y^2) is?(x, opacoth) => - inv sqrt(y^2 - 1) is?(x, opasech) => sqrt(1 - y^2) / y is?(x, opacsch) => inv y kernel(opsinh, x) icosh x == zero? x => 1 y := dropfun x is?(x, opasinh) => sqrt(y^2 + 1) is?(x, opacosh) => y is?(x, opatanh) => inv sqrt(1 - y^2) is?(x, opacoth) => y / sqrt(y^2 - 1) is?(x, opasech) => inv y is?(x, opacsch) => sqrt(y^2 + 1) / y kernel(opcosh, x) itanh x == zero? x => 0 y := dropfun x is?(x, opasinh) => y / sqrt(y^2 + 1) is?(x, opacosh) => sqrt(y^2 - 1) / y is?(x, opatanh) => y is?(x, opacoth) => inv y is?(x, opasech) => sqrt(1 - y^2) is?(x, opacsch) => inv sqrt(y^2 + 1) kernel(optanh, x) icoth x == zero? x => INV y := dropfun x is?(x, opasinh) => sqrt(y^2 + 1) / y is?(x, opacosh) => y / sqrt(y^2 - 1) is?(x, opatanh) => inv y is?(x, opacoth) => y is?(x, opasech) => inv sqrt(1 - y^2) is?(x, opacsch) => sqrt(y^2 + 1) kernel(opcoth, x) isech x == zero? x => 1 y := dropfun x is?(x, opasinh) => inv sqrt(y^2 + 1) is?(x, opacosh) => inv y is?(x, opatanh) => sqrt(1 - y^2) is?(x, opacoth) => sqrt(y^2 - 1) / y is?(x, opasech) => y is?(x, opacsch) => y / sqrt(y^2 + 1) kernel(opsech, x) icsch x == zero? x => INV y := dropfun x is?(x, opasinh) => inv y is?(x, opacosh) => inv sqrt(y^2 - 1) is?(x, opatanh) => sqrt(1 - y^2) / y is?(x, opacoth) => - sqrt(y^2 - 1) is?(x, opasech) => y / sqrt(1 - y^2) is?(x, opacsch) => y kernel(opcsch, x) iasinh x == -- is?(x, opsinh) => first argument kernel x kernel(opasinh, x) iacosh x == -- is?(x, opcosh) => first argument kernel x kernel(opacosh, x) iatanh x == -- is?(x, optanh) => first argument kernel x kernel(opatanh, x) iacoth x == zero? x => INV -- is?(x, opcoth) => first argument kernel x kernel(opacoth, x) iasech x == zero? x => INV -- is?(x, opsech) => first argument kernel x kernel(opasech, x) iacsch x == zero? x => INV -- is?(x, opcsch) => first argument kernel x kernel(opacsch, x) iexp x == zero? x => 1 is?(x, oplog) => first argument kernel x smaller?(x, 0) and empty? variables x => inv iexp(-x) R has RetractableTo Z => i := iisqrt1() xi := x / i y := xi / pi() -- this test saves us a lot of effort in the common case -- when no trigonometic simplifiaction is possible retractIfCan(y)@Union(Fraction Z, "failed") case "failed" => kernel(opexp, x) h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(xi, [[1, false], [-1, false], [i, false], [-i, false], [h + i * s3, false], [-h + i * s3, false], [-h - i * s3, false], [h - i * s3, false], [s2 + i * s2, false], [-s2 + i * s2, false], [-s2 - i * s2, false], [s2 - i * s2, false], [s3 + i * h, false], [-s3 + i * h, false], [-s3 - i * h, false], [s3 - i * h, false]]) u case F => u :: F kernel(opexp, x) kernel(opexp, x) -- THIS DETERMINES WHEN TO PERFORM THE log exp f -> f SIMPLIFICATION -- CURRENT BEHAVIOR: -- IF R IS COMPLEX(S) THEN ONLY ELEMENTS WHICH ARE RETRACTABLE TO R -- AND EQUAL TO THEIR CONJUGATES ARE DEEMED REAL (OVERRESTRICTIVE FOR NOW) -- OTHERWISE (e.g. R = INT OR FRAC INT), ALL THE ELEMENTS ARE DEEMED REAL if (R has imaginary : () -> R) and (R has conjugate : R -> R) then localReal? x == (u := retractIfCan(x)@Union(R, "failed")) case R and (u::R) = conjugate(u::R) else localReal? x == true iiilog x == zero? x => INV -- one? x => 0 (x = 1) => 0 (u := isExpt(x, opexp)) case Record(var : K, exponent : Integer) => rec := u::Record(var : K, exponent : Integer) arg := first argument(rec.var) localReal? arg => rec.exponent * first argument(rec.var) ilog x ilog x ilog x == -- ((num1 := one?(num := numer x)) or num = -1) and (den := denom x) ~= 1 ((num1 := ((num := numer x) = 1)) or num = -1) and (den := denom x) ~= 1 and empty? variables x => - kernel(oplog, (num1 => den; -den)::F) kernel(oplog, x) if R has ElementaryFunctionCategory then iilog x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iiilog x log(r::R)::F iiexp x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iexp x exp(r::R)::F else iilog x == iiilog x iiexp x == iexp x if R has TrigonometricFunctionCategory then iisin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isin x sin(r::R)::F iicos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icos x cos(r::R)::F iitan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itan x tan(r::R)::F iicot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icot x cot(r::R)::F iisec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isec x sec(r::R)::F iicsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsc x csc(r::R)::F else iisin x == isin x iicos x == icos x iitan x == itan x iicot x == icot x iisec x == isec x iicsc x == icsc x if R has ArcTrigonometricFunctionCategory then iiasin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasin x asin(r::R)::F iiacos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacos x acos(r::R)::F iiatan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatan x atan(r::R)::F iiacot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacot x acot(r::R)::F iiasec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasec x asec(r::R)::F iiacsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsc x acsc(r::R)::F else iiasin x == iasin x iiacos x == iacos x iiatan x == iatan x iiacot x == iacot x iiasec x == iasec x iiacsc x == iacsc x if R has HyperbolicFunctionCategory then iisinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isinh x sinh(r::R)::F iicosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icosh x cosh(r::R)::F iitanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itanh x tanh(r::R)::F iicoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icoth x coth(r::R)::F iisech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isech x sech(r::R)::F iicsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsch x csch(r::R)::F else iisinh x == isinh x iicosh x == icosh x iitanh x == itanh x iicoth x == icoth x iisech x == isech x iicsch x == icsch x if R has ArcHyperbolicFunctionCategory then iiasinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasinh x asinh(r::R)::F iiacosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacosh x acosh(r::R)::F iiatanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatanh x atanh(r::R)::F iiacoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacoth x acoth(r::R)::F iiasech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasech x asech(r::R)::F iiacsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsch x acsch(r::R)::F else iiasinh x == iasinh x iiacosh x == iacosh x iiatanh x == iatanh x iiacoth x == iacoth x iiasech x == iasech x iiacsch x == iacsch x import from BasicOperatorFunctions1(F) evaluate(oppi, ipi) evaluate(oplog, iilog) evaluate(opexp, iiexp) evaluate(opsin, iisin) evaluate(opcos, iicos) evaluate(optan, iitan) evaluate(opcot, iicot) evaluate(opsec, iisec) evaluate(opcsc, iicsc) evaluate(opasin, iiasin) evaluate(opacos, iiacos) evaluate(opatan, iiatan) evaluate(opacot, iiacot) evaluate(opasec, iiasec) evaluate(opacsc, iiacsc) evaluate(opsinh, iisinh) evaluate(opcosh, iicosh) evaluate(optanh, iitanh) evaluate(opcoth, iicoth) evaluate(opsech, iisech) evaluate(opcsch, iicsch) evaluate(opasinh, iiasinh) evaluate(opacosh, iiacosh) evaluate(opatanh, iiatanh) evaluate(opacoth, iiacoth) evaluate(opasech, iiasech) evaluate(opacsch, iiacsch) derivative(opexp, exp) derivative(oplog, inv) derivative(opsin, cos) derivative(opcos, (x : F) : F +-> - sin x) derivative(optan, (x : F) : F +-> 1 + tan(x)^2) derivative(opcot, (x : F) : F +-> - 1 - cot(x)^2) derivative(opsec, (x : F) : F +-> tan(x) * sec(x)) derivative(opcsc, (x : F) : F +-> - cot(x) * csc(x)) derivative(opasin, (x : F) : F +-> inv sqrt(1 - x^2)) derivative(opacos, (x : F) : F +-> - inv sqrt(1 - x^2)) derivative(opatan, (x : F) : F +-> inv(1 + x^2)) derivative(opacot, (x : F) : F +-> - inv(1 + x^2)) derivative(opasec, (x : F) : F +-> inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opacsc, (x : F) : F +-> - inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opsinh, cosh) derivative(opcosh, sinh) derivative(optanh, (x : F) : F +-> 1 - tanh(x)^2) derivative(opcoth, (x : F) : F +-> 1 - coth(x)^2) derivative(opsech, (x : F) : F +-> - tanh(x) * sech(x)) derivative(opcsch, (x : F) : F +-> - coth(x) * csch(x)) derivative(opasinh, (x : F) : F +-> inv sqrt(1 + x^2)) derivative(opacosh, (x : F) : F +-> inv sqrt(x^2 - 1)) derivative(opatanh, (x : F) : F +-> inv(1 - x^2)) derivative(opacoth, (x : F) : F +-> inv(1 - x^2)) derivative(opasech, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) - 1))) derivative(opacsch, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) + 1))) --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. -- SPAD files for the functional world should be compiled in the -- following order: -- -- op kl fspace algfunc ELEMNTRY expr \end{spad}
spad
)abbrev package EF ElementaryFunction ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 10 April 1995 ++ Keywords: elementary,function, logarithm, exponential. ++ Examples: )r EF INPUT ++ Description: Provides elementary functions over an integral domain. ElementaryFunction(R, F) : Exports == Implementation where R : Join(Comparable, IntegralDomain) F : Join(FunctionSpace R, RadicalCategory)
B ==> Boolean L ==> List Z ==> Integer OP ==> BasicOperator K ==> Kernel F INV ==> error "Invalid argument"
Exports ==> with exp : F -> F ++ exp(x) applies the exponential operator to x log : F -> F ++ log(x) applies the logarithm operator to x sin : F -> F ++ sin(x) applies the sine operator to x cos : F -> F ++ cos(x) applies the cosine operator to x tan : F -> F ++ tan(x) applies the tangent operator to x cot : F -> F ++ cot(x) applies the cotangent operator to x sec : F -> F ++ sec(x) applies the secant operator to x csc : F -> F ++ csc(x) applies the cosecant operator to x asin : F -> F ++ asin(x) applies the inverse sine operator to x acos : F -> F ++ acos(x) applies the inverse cosine operator to x atan : F -> F ++ atan(x) applies the inverse tangent operator to x acot : F -> F ++ acot(x) applies the inverse cotangent operator to x asec : F -> F ++ asec(x) applies the inverse secant operator to x acsc : F -> F ++ acsc(x) applies the inverse cosecant operator to x sinh : F -> F ++ sinh(x) applies the hyperbolic sine operator to x cosh : F -> F ++ cosh(x) applies the hyperbolic cosine operator to x tanh : F -> F ++ tanh(x) applies the hyperbolic tangent operator to x coth : F -> F ++ coth(x) applies the hyperbolic cotangent operator to x sech : F -> F ++ sech(x) applies the hyperbolic secant operator to x csch : F -> F ++ csch(x) applies the hyperbolic cosecant operator to x asinh : F -> F ++ asinh(x) applies the inverse hyperbolic sine operator to x acosh : F -> F ++ acosh(x) applies the inverse hyperbolic cosine operator to x atanh : F -> F ++ atanh(x) applies the inverse hyperbolic tangent operator to x acoth : F -> F ++ acoth(x) applies the inverse hyperbolic cotangent operator to x asech : F -> F ++ asech(x) applies the inverse hyperbolic secant operator to x acsch : F -> F ++ acsch(x) applies the inverse hyperbolic cosecant operator to x pi : () -> F ++ pi() returns the pi operator belong? : OP -> Boolean ++ belong?(p) returns true if operator p is elementary operator : OP -> OP ++ operator(p) returns an elementary operator with the same symbol as p -- the following should be local,but are conditional iisqrt2 : () -> F ++ iisqrt2() should be local but conditional iisqrt3 : () -> F ++ iisqrt3() should be local but conditional iiexp : F -> F ++ iiexp(x) should be local but conditional iilog : F -> F ++ iilog(x) should be local but conditional iisin : F -> F ++ iisin(x) should be local but conditional iicos : F -> F ++ iicos(x) should be local but conditional iitan : F -> F ++ iitan(x) should be local but conditional iicot : F -> F ++ iicot(x) should be local but conditional iisec : F -> F ++ iisec(x) should be local but conditional iicsc : F -> F ++ iicsc(x) should be local but conditional iiasin : F -> F ++ iiasin(x) should be local but conditional iiacos : F -> F ++ iiacos(x) should be local but conditional iiatan : F -> F ++ iiatan(x) should be local but conditional iiacot : F -> F ++ iiacot(x) should be local but conditional iiasec : F -> F ++ iiasec(x) should be local but conditional iiacsc : F -> F ++ iiacsc(x) should be local but conditional iisinh : F -> F ++ iisinh(x) should be local but conditional iicosh : F -> F ++ iicosh(x) should be local but conditional iitanh : F -> F ++ iitanh(x) should be local but conditional iicoth : F -> F ++ iicoth(x) should be local but conditional iisech : F -> F ++ iisech(x) should be local but conditional iicsch : F -> F ++ iicsch(x) should be local but conditional iiasinh : F -> F ++ iiasinh(x) should be local but conditional iiacosh : F -> F ++ iiacosh(x) should be local but conditional iiatanh : F -> F ++ iiatanh(x) should be local but conditional iiacoth : F -> F ++ iiacoth(x) should be local but conditional iiasech : F -> F ++ iiasech(x) should be local but conditional iiacsch : F -> F ++ iiacsch(x) should be local but conditional specialTrigs:(F, L Record(func:F, pole:B)) -> Union(F, "failed") ++ specialTrigs(x, l) should be local but conditional localReal? : F -> Boolean ++ localReal?(x) should be local but conditional
Implementation ==> add
ELEM := 'elem
ipi : List F -> F iexp : F -> F ilog : F -> F iiilog : F -> F isin : F -> F icos : F -> F itan : F -> F icot : F -> F isec : F -> F icsc : F -> F iasin : F -> F iacos : F -> F iatan : F -> F iacot : F -> F iasec : F -> F iacsc : F -> F isinh : F -> F icosh : F -> F itanh : F -> F icoth : F -> F isech : F -> F icsch : F -> F iasinh : F -> F iacosh : F -> F iatanh : F -> F iacoth : F -> F iasech : F -> F iacsch : F -> F dropfun : F -> F kernel : F -> K posrem : (Z,Z) -> Z iisqrt1 : () -> F valueOrPole : Record(func : F, pole : B) -> F
oppi := operator('pi)$CommonOperators oplog := operator('log)$CommonOperators opexp := operator('exp)$CommonOperators opsin := operator('sin)$CommonOperators opcos := operator('cos)$CommonOperators optan := operator('tan)$CommonOperators opcot := operator('cot)$CommonOperators opsec := operator('sec)$CommonOperators opcsc := operator('csc)$CommonOperators opasin := operator('asin)$CommonOperators opacos := operator('acos)$CommonOperators opatan := operator('atan)$CommonOperators opacot := operator('acot)$CommonOperators opasec := operator('asec)$CommonOperators opacsc := operator('acsc)$CommonOperators opsinh := operator('sinh)$CommonOperators opcosh := operator('cosh)$CommonOperators optanh := operator('tanh)$CommonOperators opcoth := operator('coth)$CommonOperators opsech := operator('sech)$CommonOperators opcsch := operator('csch)$CommonOperators opasinh := operator('asinh)$CommonOperators opacosh := operator('acosh)$CommonOperators opatanh := operator('atanh)$CommonOperators opacoth := operator('acoth)$CommonOperators opasech := operator('asech)$CommonOperators opacsch := operator('acsch)$CommonOperators
-- Pi is a domain... Pie,isqrt1, isqrt2, isqrt3 : F
-- following code is conditionalized on arbitraryPrecision to recompute in -- case user changes the precision
if R has TranscendentalFunctionCategory then Pie := pi()$R :: F else Pie := kernel(oppi,nil()$List(F))
if R has TranscendentalFunctionCategory and R has arbitraryPrecision then pi() == pi()$R :: F else pi() == Pie
if R has imaginary : () -> R then isqrt1 := imaginary()$R :: F else isqrt1 := sqrt(-1::F)
if R has RadicalCategory then isqrt2 := sqrt(2::R)::F isqrt3 := sqrt(3::R)::F else isqrt2 := sqrt(2::F) isqrt3 := sqrt(3::F)
iisqrt1() == isqrt1 if R has RadicalCategory and R has arbitraryPrecision then iisqrt2() == sqrt(2::R)::F iisqrt3() == sqrt(3::R)::F else iisqrt2() == isqrt2 iisqrt3() == isqrt3
ipi l == pi() log x == oplog x exp x == opexp x sin x == opsin x cos x == opcos x tan x == optan x cot x == opcot x sec x == opsec x csc x == opcsc x asin x == opasin x acos x == opacos x atan x == opatan x acot x == opacot x asec x == opasec x acsc x == opacsc x sinh x == opsinh x cosh x == opcosh x tanh x == optanh x coth x == opcoth x sech x == opsech x csch x == opcsch x asinh x == opasinh x acosh x == opacosh x atanh x == opatanh x acoth x == opacoth x asech x == opasech x acsch x == opacsch x kernel x == retract(x)@K
posrem(n,m) == ((r := n rem m) < 0 => r + m; r) valueOrPole rec == (rec.pole => INV; rec.func) belong? op == has?(op, ELEM)
operator op == is?(op,'pi) => oppi is?(op, 'log) => oplog is?(op, 'exp) => opexp is?(op, 'sin) => opsin is?(op, 'cos) => opcos is?(op, 'tan) => optan is?(op, 'cot) => opcot is?(op, 'sec) => opsec is?(op, 'csc) => opcsc is?(op, 'asin) => opasin is?(op, 'acos) => opacos is?(op, 'atan) => opatan is?(op, 'acot) => opacot is?(op, 'asec) => opasec is?(op, 'acsc) => opacsc is?(op, 'sinh) => opsinh is?(op, 'cosh) => opcosh is?(op, 'tanh) => optanh is?(op, 'coth) => opcoth is?(op, 'sech) => opsech is?(op, 'csch) => opcsch is?(op, 'asinh) => opasinh is?(op, 'acosh) => opacosh is?(op, 'atanh) => opatanh is?(op, 'acoth) => opacoth is?(op, 'asech) => opasech is?(op, 'acsch) => opacsch error "Not an elementary operator"
dropfun x == ((k := retractIfCan(x)@Union(K,"failed")) case "failed") or empty?(argument(k::K)) => 0 first argument(k::K)
if R has RetractableTo Z then specialTrigs(x,values) == (r := retractIfCan(y := x/pi())@Union(Fraction Z, "failed")) case "failed" => "failed" q := r::Fraction(Integer) m := minIndex values (n := retractIfCan(q)@Union(Z, "failed")) case Z => even?(n::Z) => valueOrPole(values.m) valueOrPole(values.(m+1)) (n := retractIfCan(2*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 4)) => valueOrPole(values.(m+2)) (s := posrem(n::Z, 4)) = 1 => valueOrPole(values.(m+2)) valueOrPole(values.(m+3)) (n := retractIfCan(3*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 6)) => valueOrPole(values.(m+4)) (s := posrem(n::Z, 6)) = 1 => valueOrPole(values.(m+4)) s = 2 => valueOrPole(values.(m+5)) s = 4 => valueOrPole(values.(m+6)) valueOrPole(values.(m+7)) (n := retractIfCan(4*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 8)) => valueOrPole(values.(m+8)) (s := posrem(n::Z, 8)) = 1 => valueOrPole(values.(m+8)) s = 3 => valueOrPole(values.(m+9)) s = 5 => valueOrPole(values.(m+10)) valueOrPole(values.(m+11)) (n := retractIfCan(6*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 12)) => valueOrPole(values.(m+12)) (s := posrem(n::Z, 12)) = 1 => valueOrPole(values.(m+12)) s = 5 => valueOrPole(values.(m+13)) s = 7 => valueOrPole(values.(m+14)) valueOrPole(values.(m+15)) "failed"
else specialTrigs(x,values) == "failed"
isin x == zero? x => 0 y := dropfun x is?(x,opasin) => y is?(x, opacos) => sqrt(1 - y^2) is?(x, opatan) => y / sqrt(1 + y^2) is?(x, opacot) => inv sqrt(1 + y^2) is?(x, opasec) => sqrt(y^2 - 1) / y is?(x, opacsc) => inv y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[0, false], [0, false], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opsin, x)
icos x == zero? x => 1 y := dropfun x is?(x,opasin) => sqrt(1 - y^2) is?(x, opacos) => y is?(x, opatan) => inv sqrt(1 + y^2) is?(x, opacot) => y / sqrt(1 + y^2) is?(x, opasec) => inv y is?(x, opacsc) => sqrt(y^2 - 1) / y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[1, false], [-1, false], [0, false], [0, false], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opcos, x)
itan x == zero? x => 0 y := dropfun x is?(x,opasin) => y / sqrt(1 - y^2) is?(x, opacos) => sqrt(1 - y^2) / y is?(x, opatan) => y is?(x, opacot) => inv y is?(x, opasec) => sqrt(y^2 - 1) is?(x, opacsc) => inv sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, false], [0, false], [0, true], [0, true], [s3, false], [-s3, false], [s3, false], [-s3, false], [1, false], [-1, false], [1, false], [-1, false], [s33, false], [-s33, false], [s33, false], [-s33, false]]) u case F => u :: F kernel(optan, x)
icot x == zero? x => INV y := dropfun x is?(x,opasin) => sqrt(1 - y^2) / y is?(x, opacos) => y / sqrt(1 - y^2) is?(x, opatan) => inv y is?(x, opacot) => y is?(x, opasec) => inv sqrt(y^2 - 1) is?(x, opacsc) => sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, true], [0, true], [0, false], [0, false], [s33, false], [-s33, false], [s33, false], [-s33, false], [1, false], [-1, false], [1, false], [-1, false], [s3, false], [-s3, false], [s3, false], [-s3, false]]) u case F => u :: F kernel(opcot, x)
isec x == zero? x => 1 y := dropfun x is?(x,opasin) => inv sqrt(1 - y^2) is?(x, opacos) => inv y is?(x, opatan) => sqrt(1 + y^2) is?(x, opacot) => sqrt(1 + y^2) / y is?(x, opasec) => y is?(x, opacsc) => y / sqrt(y^2 - 1) s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[1, false], [-1, false], [0, true], [0, true], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opsec, x)
icsc x == zero? x => INV y := dropfun x is?(x,opasin) => inv y is?(x, opacos) => inv sqrt(1 - y^2) is?(x, opatan) => sqrt(1 + y^2) / y is?(x, opacot) => sqrt(1 + y^2) is?(x, opasec) => y / sqrt(y^2 - 1) is?(x, opacsc) => y s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[0, true], [0, true], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opcsc, x)
iasin x == zero? x => 0 -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x,opsin) => y -- is?(x, opcos) => pi() / (2::F) - y kernel(opasin, x)
iacos x == zero? x => pi() / (2::F) -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x,opsin) => pi() / (2::F) - y -- is?(x, opcos) => y kernel(opacos, x)
iatan x == zero? x => 0 -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => - pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (3::F) -- one?(x*r3) => pi() / (6::F) (x*r3) = 1 => pi() / (6::F) -- y := dropfun x -- is?(x,optan) => y -- is?(x, opcot) => pi() / (2::F) - y kernel(opatan, x)
iacot x == zero? x => pi() / (2::F) -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => 3 * pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (6::F) x = -r3 => 5 * pi() / (6::F) -- one?(xx := x*r3) => pi() / (3::F) (xx := x*r3) = 1 => pi() / (3::F) xx = -1 => 2* pi() / (3::F) -- y := dropfun x -- is?(x,optan) => pi() / (2::F) - y -- is?(x, opcot) => y kernel(opacot, x)
iasec x == zero? x => INV -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x,opsec) => y -- is?(x, opcsc) => pi() / (2::F) - y kernel(opasec, x)
iacsc x == zero? x => INV -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x,opsec) => pi() / (2::F) - y -- is?(x, opcsc) => y kernel(opacsc, x)
isinh x == zero? x => 0 y := dropfun x is?(x,opasinh) => y is?(x, opacosh) => sqrt(y^2 - 1) is?(x, opatanh) => y / sqrt(1 - y^2) is?(x, opacoth) => - inv sqrt(y^2 - 1) is?(x, opasech) => sqrt(1 - y^2) / y is?(x, opacsch) => inv y kernel(opsinh, x)
icosh x == zero? x => 1 y := dropfun x is?(x,opasinh) => sqrt(y^2 + 1) is?(x, opacosh) => y is?(x, opatanh) => inv sqrt(1 - y^2) is?(x, opacoth) => y / sqrt(y^2 - 1) is?(x, opasech) => inv y is?(x, opacsch) => sqrt(y^2 + 1) / y kernel(opcosh, x)
itanh x == zero? x => 0 y := dropfun x is?(x,opasinh) => y / sqrt(y^2 + 1) is?(x, opacosh) => sqrt(y^2 - 1) / y is?(x, opatanh) => y is?(x, opacoth) => inv y is?(x, opasech) => sqrt(1 - y^2) is?(x, opacsch) => inv sqrt(y^2 + 1) kernel(optanh, x)
icoth x == zero? x => INV y := dropfun x is?(x,opasinh) => sqrt(y^2 + 1) / y is?(x, opacosh) => y / sqrt(y^2 - 1) is?(x, opatanh) => inv y is?(x, opacoth) => y is?(x, opasech) => inv sqrt(1 - y^2) is?(x, opacsch) => sqrt(y^2 + 1) kernel(opcoth, x)
isech x == zero? x => 1 y := dropfun x is?(x,opasinh) => inv sqrt(y^2 + 1) is?(x, opacosh) => inv y is?(x, opatanh) => sqrt(1 - y^2) is?(x, opacoth) => sqrt(y^2 - 1) / y is?(x, opasech) => y is?(x, opacsch) => y / sqrt(y^2 + 1) kernel(opsech, x)
icsch x == zero? x => INV y := dropfun x is?(x,opasinh) => inv y is?(x, opacosh) => inv sqrt(y^2 - 1) is?(x, opatanh) => sqrt(1 - y^2) / y is?(x, opacoth) => - sqrt(y^2 - 1) is?(x, opasech) => y / sqrt(1 - y^2) is?(x, opacsch) => y kernel(opcsch, x)
iasinh x == -- is?(x,opsinh) => first argument kernel x kernel(opasinh, x)
iacosh x == -- is?(x,opcosh) => first argument kernel x kernel(opacosh, x)
iatanh x == -- is?(x,optanh) => first argument kernel x kernel(opatanh, x)
iacoth x == zero? x => INV -- is?(x,opcoth) => first argument kernel x kernel(opacoth, x)
iasech x == zero? x => INV -- is?(x,opsech) => first argument kernel x kernel(opasech, x)
iacsch x == zero? x => INV -- is?(x,opcsch) => first argument kernel x kernel(opacsch, x)
iexp x == zero? x => 1 is?(x,oplog) => first argument kernel x smaller?(x, 0) and empty? variables x => inv iexp(-x) R has RetractableTo Z => i := iisqrt1() xi := x / i y := xi / pi() -- this test saves us a lot of effort in the common case -- when no trigonometic simplifiaction is possible retractIfCan(y)@Union(Fraction Z, "failed") case "failed" => kernel(opexp, x) h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(xi, [[1, false], [-1, false], [i, false], [-i, false], [h + i * s3, false], [-h + i * s3, false], [-h - i * s3, false], [h - i * s3, false], [s2 + i * s2, false], [-s2 + i * s2, false], [-s2 - i * s2, false], [s2 - i * s2, false], [s3 + i * h, false], [-s3 + i * h, false], [-s3 - i * h, false], [s3 - i * h, false]]) u case F => u :: F kernel(opexp, x) kernel(opexp, x)
-- THIS DETERMINES WHEN TO PERFORM THE log exp f -> f SIMPLIFICATION -- CURRENT BEHAVIOR: -- IF R IS COMPLEX(S) THEN ONLY ELEMENTS WHICH ARE RETRACTABLE TO R -- AND EQUAL TO THEIR CONJUGATES ARE DEEMED REAL (OVERRESTRICTIVE FOR NOW) -- OTHERWISE (e.g. R = INT OR FRAC INT),ALL THE ELEMENTS ARE DEEMED REAL
if (R has imaginary : () -> R) and (R has conjugate : R -> R) then localReal? x == (u := retractIfCan(x)@Union(R,"failed")) case R and (u::R) = conjugate(u::R)
else localReal? x == true
iiilog x == zero? x => INV -- one? x => 0 (x = 1) => 0 (u := isExpt(x,opexp)) case Record(var : K, exponent : Integer) => rec := u::Record(var : K, exponent : Integer) arg := first argument(rec.var) localReal? arg => rec.exponent * first argument(rec.var) ilog x ilog x
ilog x == -- ((num1 := one?(num := numer x)) or num = -1) and (den := denom x) ~= 1 ((num1 := ((num := numer x) = 1)) or num = -1) and (den := denom x) ~= 1 and empty? variables x => - kernel(oplog,(num1 => den; -den)::F) kernel(oplog, x)
if R has ElementaryFunctionCategory then iilog x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iiilog x log(r::R)::F
iiexp x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iexp x exp(r::R)::F
else iilog x == iiilog x iiexp x == iexp x
if R has TrigonometricFunctionCategory then iisin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isin x sin(r::R)::F
iicos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icos x cos(r::R)::F
iitan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itan x tan(r::R)::F
iicot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icot x cot(r::R)::F
iisec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isec x sec(r::R)::F
iicsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsc x csc(r::R)::F
else iisin x == isin x iicos x == icos x iitan x == itan x iicot x == icot x iisec x == isec x iicsc x == icsc x
if R has ArcTrigonometricFunctionCategory then iiasin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasin x asin(r::R)::F
iiacos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacos x acos(r::R)::F
iiatan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatan x atan(r::R)::F
iiacot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacot x acot(r::R)::F
iiasec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasec x asec(r::R)::F
iiacsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsc x acsc(r::R)::F
else iiasin x == iasin x iiacos x == iacos x iiatan x == iatan x iiacot x == iacot x iiasec x == iasec x iiacsc x == iacsc x
if R has HyperbolicFunctionCategory then iisinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isinh x sinh(r::R)::F
iicosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icosh x cosh(r::R)::F
iitanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itanh x tanh(r::R)::F
iicoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icoth x coth(r::R)::F
iisech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isech x sech(r::R)::F
iicsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsch x csch(r::R)::F
else iisinh x == isinh x iicosh x == icosh x iitanh x == itanh x iicoth x == icoth x iisech x == isech x iicsch x == icsch x
if R has ArcHyperbolicFunctionCategory then iiasinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasinh x asinh(r::R)::F
iiacosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacosh x acosh(r::R)::F
iiatanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatanh x atanh(r::R)::F
iiacoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacoth x acoth(r::R)::F
iiasech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasech x asech(r::R)::F
iiacsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsch x acsch(r::R)::F
else iiasinh x == iasinh x iiacosh x == iacosh x iiatanh x == iatanh x iiacoth x == iacoth x iiasech x == iasech x iiacsch x == iacsch x
import from BasicOperatorFunctions1(F)
evaluate(oppi,ipi) evaluate(oplog, iilog) evaluate(opexp, iiexp) evaluate(opsin, iisin) evaluate(opcos, iicos) evaluate(optan, iitan) evaluate(opcot, iicot) evaluate(opsec, iisec) evaluate(opcsc, iicsc) evaluate(opasin, iiasin) evaluate(opacos, iiacos) evaluate(opatan, iiatan) evaluate(opacot, iiacot) evaluate(opasec, iiasec) evaluate(opacsc, iiacsc) evaluate(opsinh, iisinh) evaluate(opcosh, iicosh) evaluate(optanh, iitanh) evaluate(opcoth, iicoth) evaluate(opsech, iisech) evaluate(opcsch, iicsch) evaluate(opasinh, iiasinh) evaluate(opacosh, iiacosh) evaluate(opatanh, iiatanh) evaluate(opacoth, iiacoth) evaluate(opasech, iiasech) evaluate(opacsch, iiacsch) derivative(opexp, exp) derivative(oplog, inv) derivative(opsin, cos) derivative(opcos, (x : F) : F +-> - sin x) derivative(optan, (x : F) : F +-> 1 + tan(x)^2) derivative(opcot, (x : F) : F +-> - 1 - cot(x)^2) derivative(opsec, (x : F) : F +-> tan(x) * sec(x)) derivative(opcsc, (x : F) : F +-> - cot(x) * csc(x)) derivative(opasin, (x : F) : F +-> inv sqrt(1 - x^2)) derivative(opacos, (x : F) : F +-> - inv sqrt(1 - x^2)) derivative(opatan, (x : F) : F +-> inv(1 + x^2)) derivative(opacot, (x : F) : F +-> - inv(1 + x^2)) derivative(opasec, (x : F) : F +-> inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opacsc, (x : F) : F +-> - inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opsinh, cosh) derivative(opcosh, sinh) derivative(optanh, (x : F) : F +-> 1 - tanh(x)^2) derivative(opcoth, (x : F) : F +-> 1 - coth(x)^2) derivative(opsech, (x : F) : F +-> - tanh(x) * sech(x)) derivative(opcsch, (x : F) : F +-> - coth(x) * csch(x)) derivative(opasinh, (x : F) : F +-> inv sqrt(1 + x^2)) derivative(opacosh, (x : F) : F +-> inv sqrt(x^2 - 1)) derivative(opatanh, (x : F) : F +-> inv(1 - x^2)) derivative(opacoth, (x : F) : F +-> inv(1 - x^2)) derivative(opasech, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) - 1))) derivative(opacsch, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) + 1)))
--Copyright (c) 1991-2002,The Numerical ALgorithms Group Ltd. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-- SPAD files for the functional world should be compiled in the -- following order: -- -- op kl fspace algfunc ELEMNTRY expr
spad
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6831497810020874664-25px001.spad
using old system compiler.
EF abbreviates package ElementaryFunction
------------------------------------------------------------------------
initializing NRLIB EF for ElementaryFunction
compiling into NRLIB EF
****** Domain: R already in scope
****** Domain: F already in scope
****** Domain: R already in scope
augmenting R: (TranscendentalFunctionCategory)
****** Domain: R already in scope
augmenting R: (TranscendentalFunctionCategory)
****** Domain: R already in scope
augmenting R: (arbitraryPrecision)
compiling exported pi : () -> F
Time: 0.05 SEC.
compiling exported pi : () -> F
Time: 0 SEC.
compiling exported pi : () -> F
Time: 0 SEC.
augmenting R: (SIGNATURE R imaginary (R))
****** Domain: R already in scope
augmenting R: (RadicalCategory)
compiling local iisqrt1 : () -> F
Time: 0.01 SEC.
****** Domain: R already in scope
augmenting R: (RadicalCategory)
****** Domain: R already in scope
augmenting R: (arbitraryPrecision)
compiling exported iisqrt2 : () -> F
Time: 0 SEC.
compiling exported iisqrt3 : () -> F
Time: 0 SEC.
compiling exported iisqrt2 : () -> F
Time: 0 SEC.
compiling exported iisqrt3 : () -> F
Time: 0 SEC.
compiling exported iisqrt2 : () -> F
Time: 0 SEC.
compiling exported iisqrt3 : () -> F
Time: 0 SEC.
compiling local ipi : List F -> F
Time: 0 SEC.
compiling exported log : F -> F
Time: 0 SEC.
compiling exported exp : F -> F
Time: 0 SEC.
compiling exported sin : F -> F
Time: 0 SEC.
compiling exported cos : F -> F
Time: 0 SEC.
compiling exported tan : F -> F
Time: 0 SEC.
compiling exported cot : F -> F
Time: 0 SEC.
compiling exported sec : F -> F
Time: 0 SEC.
compiling exported csc : F -> F
Time: 0 SEC.
compiling exported asin : F -> F
Time: 0 SEC.
compiling exported acos : F -> F
Time: 0 SEC.
compiling exported atan : F -> F
Time: 0 SEC.
compiling exported acot : F -> F
Time: 0 SEC.
compiling exported asec : F -> F
Time: 0 SEC.
compiling exported acsc : F -> F
Time: 0 SEC.
compiling exported sinh : F -> F
Time: 0 SEC.
compiling exported cosh : F -> F
Time: 0 SEC.
compiling exported tanh : F -> F
Time: 0 SEC.
compiling exported coth : F -> F
Time: 0 SEC.
compiling exported sech : F -> F
Time: 0 SEC.
compiling exported csch : F -> F
Time: 0 SEC.
compiling exported asinh : F -> F
Time: 0 SEC.
compiling exported acosh : F -> F
Time: 0 SEC.
compiling exported atanh : F -> F
Time: 0 SEC.
compiling exported acoth : F -> F
Time: 0 SEC.
compiling exported asech : F -> F
Time: 0 SEC.
compiling exported acsch : F -> F
Time: 0.01 SEC.
compiling local kernel : F -> Kernel F
Time: 0 SEC.
compiling local posrem : (Integer,
compiling local valueOrPole : Record(func: F,
compiling exported belong? : BasicOperator -> Boolean
Time: 0 SEC.
compiling exported operator : BasicOperator -> BasicOperator
Time: 0.01 SEC.
compiling local dropfun : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
compiling exported specialTrigs : (F,
compiling exported specialTrigs : (F,
compiling local isin : F -> F
Time: 0.03 SEC.
compiling local icos : F -> F
Time: 0.02 SEC.
compiling local itan : F -> F
Time: 0.02 SEC.
compiling local icot : F -> F
Time: 0.02 SEC.
compiling local isec : F -> F
Time: 0.03 SEC.
compiling local icsc : F -> F
Time: 0.02 SEC.
compiling local iasin : F -> F
Time: 0.01 SEC.
compiling local iacos : F -> F
Time: 0 SEC.
compiling local iatan : F -> F
Time: 0.01 SEC.
compiling local iacot : F -> F
Time: 0.03 SEC.
compiling local iasec : F -> F
Time: 0 SEC.
compiling local iacsc : F -> F
Time: 0 SEC.
compiling local isinh : F -> F
Time: 0.02 SEC.
compiling local icosh : F -> F
Time: 0.01 SEC.
compiling local itanh : F -> F
Time: 0.01 SEC.
compiling local icoth : F -> F
Time: 0.01 SEC.
compiling local isech : F -> F
Time: 0.01 SEC.
compiling local icsch : F -> F
Time: 0.02 SEC.
compiling local iasinh : F -> F
Time: 0 SEC.
compiling local iacosh : F -> F
Time: 0 SEC.
compiling local iatanh : F -> F
Time: 0 SEC.
compiling local iacoth : F -> F
Time: 0 SEC.
compiling local iasech : F -> F
Time: 0 SEC.
compiling local iacsch : F -> F
Time: 0 SEC.
compiling local iexp : F -> F
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
Time: 0.03 SEC.
augmenting R: (SIGNATURE R imaginary (R))
augmenting R: (SIGNATURE R conjugate (R R))
compiling exported localReal? : F -> Boolean
Time: 0 SEC.
compiling exported localReal? : F -> Boolean
EF;localReal?;FB;72 is replaced by QUOTET
Time: 0 SEC.
compiling exported localReal? : F -> Boolean
EF;localReal?;FB;73 is replaced by QUOTET
Time: 0 SEC.
compiling local iiilog : F -> F
Time: 0.01 SEC.
compiling local ilog : F -> F
Time: 0.02 SEC.
****** Domain: R already in scope
augmenting R: (ElementaryFunctionCategory)
compiling exported iilog : F -> F
Time: 0 SEC.
compiling exported iiexp : F -> F
Time: 0 SEC.
compiling exported iilog : F -> F
Time: 0 SEC.
compiling exported iiexp : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (TrigonometricFunctionCategory)
compiling exported iisin : F -> F
Time: 0 SEC.
compiling exported iicos : F -> F
Time: 0 SEC.
compiling exported iitan : F -> F
Time: 0 SEC.
compiling exported iicot : F -> F
Time: 0 SEC.
compiling exported iisec : F -> F
Time: 0.01 SEC.
compiling exported iicsc : F -> F
Time: 0 SEC.
compiling exported iisin : F -> F
Time: 0 SEC.
compiling exported iicos : F -> F
Time: 0 SEC.
compiling exported iitan : F -> F
Time: 0 SEC.
compiling exported iicot : F -> F
Time: 0 SEC.
compiling exported iisec : F -> F
Time: 0 SEC.
compiling exported iicsc : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (ArcTrigonometricFunctionCategory)
compiling exported iiasin : F -> F
Time: 0 SEC.
compiling exported iiacos : F -> F
Time: 0 SEC.
compiling exported iiatan : F -> F
Time: 0.01 SEC.
compiling exported iiacot : F -> F
Time: 0 SEC.
compiling exported iiasec : F -> F
Time: 0 SEC.
compiling exported iiacsc : F -> F
Time: 0 SEC.
compiling exported iiasin : F -> F
Time: 0 SEC.
compiling exported iiacos : F -> F
Time: 0 SEC.
compiling exported iiatan : F -> F
Time: 0 SEC.
compiling exported iiacot : F -> F
Time: 0 SEC.
compiling exported iiasec : F -> F
Time: 0 SEC.
compiling exported iiacsc : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (HyperbolicFunctionCategory)
compiling exported iisinh : F -> F
Time: 0 SEC.
compiling exported iicosh : F -> F
Time: 0 SEC.
compiling exported iitanh : F -> F
Time: 0 SEC.
compiling exported iicoth : F -> F
Time: 0.01 SEC.
compiling exported iisech : F -> F
Time: 0 SEC.
compiling exported iicsch : F -> F
Time: 0 SEC.
compiling exported iisinh : F -> F
Time: 0 SEC.
compiling exported iicosh : F -> F
Time: 0 SEC.
compiling exported iitanh : F -> F
Time: 0 SEC.
compiling exported iicoth : F -> F
Time: 0 SEC.
compiling exported iisech : F -> F
Time: 0 SEC.
compiling exported iicsch : F -> F
Time: 0 SEC.
****** Domain: R already in scope
augmenting R: (ArcHyperbolicFunctionCategory)
compiling exported iiasinh : F -> F
Time: 0 SEC.
compiling exported iiacosh : F -> F
Time: 0.01 SEC.
compiling exported iiatanh : F -> F
Time: 0 SEC.
compiling exported iiacoth : F -> F
Time: 0 SEC.
compiling exported iiasech : F -> F
Time: 0 SEC.
compiling exported iiacsch : F -> F
Time: 0 SEC.
compiling exported iiasinh : F -> F
Time: 0 SEC.
compiling exported iiacosh : F -> F
Time: 0 SEC.
compiling exported iiatanh : F -> F
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compiling exported iiacoth : F -> F
Time: 0 SEC.
compiling exported iiasech : F -> F
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compiling exported iiacsch : F -> F
Time: 0 SEC.
importing BasicOperatorFunctions1 F
(time taken in buildFunctor: 0)
;;; *** |ElementaryFunction| REDEFINED
;;; *** |ElementaryFunction| REDEFINED
Time: 0.37 SEC.
Warnings:
[1] pi: Pie has no value
[2] iisqrt1: isqrt1 has no value
[3] iisqrt2: isqrt2 has no value
[4] iisqrt3: isqrt3 has no value
[5] valueOrPole: pole has no value
[6] valueOrPole: func has no value
[7] iiilog: var has no value
[8] iiilog: exponent has no value
Cumulative Statistics for Constructor ElementaryFunction
Time: 0.87 seconds
finalizing NRLIB EF
Processing ElementaryFunction for Browser database:
--------constructor---------
--------(exp (F F))---------
--------(log (F F))---------
--------(sin (F F))---------
--------(cos (F F))---------
--------(tan (F F))---------
--------(cot (F F))---------
--------(sec (F F))---------
--------(csc (F F))---------
--------(asin (F F))---------
--------(acos (F F))---------
--------(atan (F F))---------
--------(acot (F F))---------
--------(asec (F F))---------
--------(acsc (F F))---------
--------(sinh (F F))---------
--------(cosh (F F))---------
--------(tanh (F F))---------
--------(coth (F F))---------
--------(sech (F F))---------
--------(csch (F F))---------
--------(asinh (F F))---------
--------(acosh (F F))---------
--------(atanh (F F))---------
--------(acoth (F F))---------
--------(asech (F F))---------
--------(acsch (F F))---------
--------(pi (F))---------
--------(belong? ((Boolean) (BasicOperator)))---------
--------(operator ((BasicOperator) (BasicOperator)))---------
--------(iisqrt2 (F))---------
--------(iisqrt3 (F))---------
--------(iiexp (F F))---------
--------(iilog (F F))---------
--------(iisin (F F))---------
--------(iicos (F F))---------
--------(iitan (F F))---------
--------(iicot (F F))---------
--------(iisec (F F))---------
--------(iicsc (F F))---------
--------(iiasin (F F))---------
--------(iiacos (F F))---------
--------(iiatan (F F))---------
--------(iiacot (F F))---------
--------(iiasec (F F))---------
--------(iiacsc (F F))---------
--------(iisinh (F F))---------
--------(iicosh (F F))---------
--------(iitanh (F F))---------
--------(iicoth (F F))---------
--------(iisech (F F))---------
--------(iicsch (F F))---------
--------(iiasinh (F F))---------
--------(iiacosh (F F))---------
--------(iiatanh (F F))---------
--------(iiacoth (F F))---------
--------(iiasech (F F))---------
--------(iiacsch (F F))---------
--------(specialTrigs ((Union F failed) F (List (Record (: func F) (: pole (Boolean))))))---------
--------(localReal? ((Boolean) F))---------
; compiling file "/var/aw/var/LatexWiki/EF.NRLIB/EF.lsp" (written 27 FEB 2015 01:03:16 PM):
; /var/aw/var/LatexWiki/EF.NRLIB/EF.fasl written
; compilation finished in 0:00:01.370
------------------------------------------------------------------------
ElementaryFunction is now explicitly exposed in frame initial
ElementaryFunction will be automatically loaded when needed from
/var/aw/var/LatexWiki/EF.NRLIB/EF