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--- @class TrigExpression
--- Represents a trigonometric function from one expression to another.
--- @field name string
--- @field expression Expression
TrigExpression = {}
__TrigExpression = {}
----------------------------
-- Instance functionality --
----------------------------
--- Creates a new trig expression with the given name and expression.
--- @param name string|SymbolExpression
--- @param expression Expression
--- @return TrigExpression
function TrigExpression:new(name, expression)
local o = {}
local __o = Copy(__ExpressionOperations)
if not TrigExpression.NAMES[name] then
error("Argument error: " .. name .. " is not the name of a trigonometric function.")
end
o.name = name
o.expression = expression
o.expressions = {expression}
if expression:isatomic() then
o.variables = {expression}
else
o.variables = {SymbolExpression('x')}
end
o.derivatives = {Integer.zero()}
__o.__index = TrigExpression
__o.__tostring = function(a)
return tostring(a.name) .. '(' .. tostring(a.expression) .. ')'
end
__o.__eq = function(a, b)
-- if b:type() == FunctionExpression then
-- return a:tofunction() == b
-- end
-- This shouldn't be needed, since __eq should only fire if both metamethods have the same function, but for some reason Lua always runs this anyway
if not b:type() == TrigExpression then
return false
end
return a.name == b.name and a.expression == b.expression
end
o = setmetatable(o, __o)
return o
end
--- @return TrigExpression
function TrigExpression:evaluate()
local expression = self.expression:autosimplify()
if expression == Integer.zero() then
if self.name == "cos" or self.name == "sec" then
return Integer.one()
end
if self.name == "sin" or self.name == "tan" then
return Integer.zero()
end
if self.name == "arctan" or self.name == "arcsin" then
return Integer.zero()
end
if self.name == "arccos" or self.name == "arccot" then
return PI / Integer(2)
end
end
if expression == PI then
if self.name == "cos" or self.name == "sec" then
return Integer(-1)
end
if self.name == "sin" or self.name == "tan" then
return Integer.zero()
end
end
if expression:ismulratlPI() then
local coeff = expression.expressions[1]
if TrigExpression.COSVALUES[tostring(coeff)] ~= nil then
if self.name == "cos" then
return TrigExpression.COSVALUES[tostring(coeff)]:autosimplify()
end
if self.name == "sin" then
local sign = Integer.one()
if coeff > Integer.one() then
sign = Integer(-1)
end
return (sign*sqrt(Integer.one()-cos(expression)^Integer(2))):autosimplify()
end
if self.name == "tan" then
return (sin(expression) / cos(expression)):autosimplify()
end
if self.name == "sec" then
return (Integer.one() / cos(expression)):autosimplify()
end
if self.name == "csc" then
return (Integer.one() / sin(expression)):autosimplify()
end
if self.name == "cot" then
return (cos(expression) / sin(expression)):autosimplify()
end
end
end
if TrigExpression.ACOSVALUES[tostring(expression)] ~= nil then
if self.name == "arccos" then
return TrigExpression.ACOSVALUES[tostring(expression)]:autosimplify()
end
if self.name == "arcsin" then
if expression == Integer(-1) then
return TrigExpression.ACOSVALUES["-1"]:autosimplify()
elseif expression.expressions and expression.expressions[1] == Integer(-1) then
local expr = (Integer(-1)*sqrt(Integer.one() - expression ^ Integer(2))):autosimplify()
return TrigExpression.ACOSVALUES[tostring(expr)]:autosimplify()
else
local expr = (sqrt(Integer.one() - expression ^ Integer(2))):autosimplify()
return TrigExpression.ACOSVALUES[tostring(expr)]:autosimplify()
end
end
end
if self.name == "arctan" and TrigExpression.ATANVALUES[tostring(expression)] ~= nil then
return TrigExpression.ATANVALUES[tostring(expression)]:autosimplify()
end
return self
end
--- checks if expression is a rational multiple of pi
--- @return boolean
function Expression:ismulratlPI()
if self.operation == BinaryOperation.MUL and #self.expressions == 2 and (self.expressions[1]:type() == Integer or self.expressions[1]:type() == Rational) and self.expressions[2] == PI then
return true
end
return false
end
--- @return TrigExpression
function TrigExpression:autosimplify()
local expression = self.expression:autosimplify()
-- even and odd properties of trig functions
if (self.name == "sin" or self.name == "tan" or self.name == "csc" or self.name == "cot") and
expression.operation == BinaryOperation.MUL and expression.expressions[1]:isconstant() and expression.expressions[1] < Integer(0) then
return (-Integer.one() * TrigExpression(self.name, -expression)):autosimplify()
end
if (self.name == "cos" or self.name == "sec") and
expression.operation == BinaryOperation.MUL and expression.expressions[1]:isconstant() and expression.expressions[1] < Integer(0) then
expression = (-expression):autosimplify()
end
-- uses periodicity of sin and cos and friends
if self.name == "sin" or self.name == "cos" or self.name == "csc" or self.name == "sec" then
if expression == Integer.zero() or expression == PI then
goto skip
end
if expression.operation ~= BinaryOperation.ADD then
expression = BinaryOperation(BinaryOperation.ADD,{expression})
end
for index,component in ipairs(expression.expressions) do
if component:ismulratlPI() then
local coeff = component.expressions[1]
if coeff:type() == Integer then
coeff = coeff % Integer(2)
coeff = coeff:autosimplify()
end
if coeff:type() == Rational then
local n = coeff.numerator
local d = coeff.denominator
local m = {n:divremainder(d)}
coeff = (m[1] % Integer(2)) + m[2]/d
coeff = coeff:autosimplify()
end
expression.expressions[index].expressions[1] = coeff
end
expression = expression:autosimplify()
end
::skip::
end
-- uses periodicity of tan and cot
if self.name == "tan" or self.name == "cot" then
if expression == Integer.zero() or expression == PI then
goto skip
end
if expression.operation ~= BinaryOperation.ADD then
expression = BinaryOperation(BinaryOperation.ADD,{expression})
end
for index,component in ipairs(expression.expressions) do
if component:ismulratlPI() then
local coeff = component.expressions[1]
if coeff:type() == Integer then
coeff = Integer.zero()
end
if coeff:type() == Rational then
local n = coeff.numerator
local d = coeff.denominator
local m = {n:divremainder(d)}
coeff = m[2]/d
coeff = coeff:autosimplify()
end
expression.expressions[index].expressions[1] = coeff
end
if component == PI then
expression.expressions[index] = Integer.zero()
end
end
expression = expression:autosimplify()
::skip::
end
return TrigExpression(self.name, expression):evaluate()
end
--- @return table<number, Expression>
function TrigExpression:subexpressions()
return {self.expression}
end
--- @param subexpressions table<number, Expression>
--- @return TrigExpression
function TrigExpression:setsubexpressions(subexpressions)
return TrigExpression(self.name, subexpressions[1])
end
-- function TrigExpression:freeof(symbol)
-- return self.expression:freeof(symbol)
-- end
-- function TrigExpression:substitute(map)
-- for expression, replacement in pairs(map) do
-- if self == expression then
-- return replacement
-- end
-- end
-- return TrigExpression(self.name, self.expression:substitute(map))
-- end
-- function TrigExpression:order(other)
-- return self:tofunction():order(other)
-- end
-- function TrigExpression:tofunction()
-- return FunctionExpression(self.name, {self.expression}, true)
-- end
-----------------
-- Inheritance --
-----------------
__TrigExpression.__index = FunctionExpression
__TrigExpression.__call = TrigExpression.new
TrigExpression = setmetatable(TrigExpression, __TrigExpression)
----------------------
-- Static constants --
----------------------
TrigExpression.NAMES = {sin=1, cos=2, tan=3, csc=4, sec=5, cot=6,
arcsin=7, arccos=8, arctan=9, arccsc=10, arcsec=11, arccot=12}
TrigExpression.INVERSES = {sin="arcsin", cos="arccos", tan="arctan", csc="arccsc", sec="arcsec", cot="arccot",
arcsin="sin", arccos="cos", arctan="tan", arccsc="csc", arcsec="sec", arccot="cot"}
TrigExpression.COSVALUES = {
["0"] = Integer.one(),
["1/6"] = sqrt(Integer(3))/Integer(2),
["1/4"] = sqrt(Integer(2))/Integer(2),
["1/3"] = Integer.one()/Integer(2),
["1/2"] = Integer.zero(),
["2/3"] = -Integer.one()/Integer(2),
["3/4"] = -sqrt(Integer(2))/Integer(2),
["5/6"] = -sqrt(Integer(3))/Integer(2),
["1"] = -Integer.one(),
["7/6"] = -sqrt(Integer(3))/Integer(2),
["5/4"] = -sqrt(Integer(2))/Integer(2),
["4/3"] = -Integer.one()/Integer(2),
["3/2"] = Integer.zero(),
["5/3"] = Integer.one()/Integer(2),
["7/4"] = sqrt(Integer(2))/Integer(2),
["11/6"] = sqrt(Integer(3))/Integer(2),
}
TrigExpression.ACOSVALUES = {
["1"] = Integer.zero(),
["(1/2 * sqrt(3,2))"] = PI * Integer(6) ^ Integer(-1),
["(1/2 * sqrt(2,2))"] = PI * Integer(4) ^ Integer(-1),
["1/2"] = PI * Integer(3) ^ Integer(-1),
["0"] = PI * Integer(2) ^ Integer(-1),
["-1/2"] = PI * Integer(2) * Integer(3) ^ Integer(-1),
["(-1/2 * sqrt(2,2))"]= PI * Integer(3) * Integer(4) ^ Integer(-1),
["(-1/2 * sqrt(3,2))"]= PI * Integer(5) * Integer(6) ^ Integer(-1),
["-1"] = Integer(-1)*PI,
}
TrigExpression.ATANVALUES = {
["(-1 * sqrt(3,2))"] = Integer(-1) * PI * Integer(3) ^ Integer(-1),
["-1"] = Integer(-1) * PI * Integer(4) ^ Integer(-1),
["(-1/3 * sqrt(3,2))"] = Integer(-1) * Integer(6) ^ Integer(-1),
["0"] = Integer.zero(),
["(1/3 * sqrt(3,2))"] = PI * Integer(6) ^ Integer(-1),
["1"] = PI * Integer(4) ^ Integer(-1),
["sqrt(3,2)"] = PI * Integer(3) ^ Integer(-1)
}
SIN = function (a)
return TrigExpression("sin", a)
end
COS = function (a)
return TrigExpression("cos", a)
end
TAN = function (a)
return TrigExpression("tan", a)
end
CSC = function (a)
return TrigExpression("csc", a)
end
SEC = function (a)
return TrigExpression("sec", a)
end
COT = function (a)
return TrigExpression("cot", a)
end
ARCSIN = function (a)
return TrigExpression("arcsin", a)
end
ARCCOS = function (a)
return TrigExpression("arccos", a)
end
ARCTAN = function (a)
return TrigExpression("arctan", a)
end
ARCCSC = function (a)
return TrigExpression("arccsc", a)
end
ARCSEC = function (a)
return TrigExpression("arcsec", a)
end
ARCCOT = function (a)
return TrigExpression("arccot", a)
end
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