-- Constants

local MAXIMUM_INT  = 2^53    -- The maximum double for where all integers can be represented exactly.
local MAXIMUM_ZERO = 10^-7   -- Used when attempting to determine fraction. Anything below is counted as 0.

--------------------------------------------------------------------------------
-- Util

local function is_integer (n)
	-- Predicate. Neither +inf, -inf nor nan are integer.
	return  n == math.floor(n)  and n ~= 1/0 and n ~= -1/0
end

local function factorial (n)
	-- Returns the factorial of n.
	-- That is n! = n * (n-1) * (n-2) * ... * 1
	local a = 1
	for i = 2, n do  a = a * i  end
	return a
end

local function inverse_factorial (n)
	-- Returns the inverse factorial of n.
	-- That is some m, where n = m!
	local a, i = 1, 1
	while a < n do
		i = i + 1
		a = a * i
	end
	return (a == n) and i or nil
end

local function calculate_fraction (n)
	-- Returns x and y such that x/y = n. If none could be found, returns nil.
	local a, b = 1, n % 1
	while MAXIMUM_ZERO < b and a <= MAXIMUM_INT do
		local r = math.pow(b, -1)
		a, b = a * r, r % 1
		-- Invariant:  n * a / a = n
	end
	-- Check the values make sense.
	local numerator, denumberator = math.floor(n * a), math.floor(a)
	if numerator / denumberator == n then
		return numerator, denumberator
	end
end

local UNICODE_SUPERSCRIPT = {
	['0'] = '⁰', ['1'] = '¹', ['2'] = '²', ['3'] = '³', ['4'] = '⁴',
	['5'] = '⁵', ['6'] = '⁶', ['7'] = '⁷', ['8'] = '⁸', ['9'] = '⁹',
	['-'] = '⁻', ['+'] = '⁺', ['='] = '⁼', ['('] = '⁽', [')'] = '⁾',
}
local function to_superscript (n)
	return tostring(n):gsub('.', function (a) return UNICODE_SUPERSCRIPT[a] end)
end

local UNICODE_CHAR_PATTERN = '[\01-\127\192-\255][\128-\191]*'
local function unicode_len (str)
	local len = 0
	for char in str:gmatch(UNICODE_CHAR_PATTERN) do  len = len + 1  end
	return len
end

--------------------------------------------------------------------------------

local SPECIAL_NUMBER = {
	-- x = ∞
	{ est   = function (...) return ...                                     end,
	  real  = function (a) return math.huge                                 end,
	  repr  = function (a) return '1/0'                                     end,
	  short = function (a) return '∞'                                       end,
	},
	-- Factorial
	{ est   = function (n) return inverse_factorial(n)                      end,
	  real  = function (a) return factorial(a)                              end,
	  repr  = function (a) return ('fac(%i)'):format(a)                     end,
	  short = function (a) return a..'!'                                    end,
    },
	-- x = 2^a
	{ est   = function (n) return math.log(n)/math.log(2)                   end,
	  real  = function (a) return 2^a                                       end,
	  repr  = function (a) return '2^'..a                                   end,
	  short = function (a) return '2'..to_superscript(a)                    end,
    },
	-- x = 10^a
	{ est   = function (n) return math.log(n)/math.log(10)                  end,
	  real  = function (a) return 10^a                                      end,
	  repr  = function (a) return '10^'..a                                  end,
	  short = function (a) return '10'..to_superscript(a)                   end,
    },
	-- x = 1/√a
	{ est   = function (n) return 1/(n^2)                                   end,
	  real  = function (a) return 1/math.sqrt(a)                            end,
	  repr  = function (a) return ('1/math.sqrt(%i)'):format(a)             end,
	  short = function (a) return '1/√'..a                                  end,
    },
	-- x = lg a
	{ est   = function (n) return math.exp(n)                               end,
	  real  = function (a) return math.log(a)                               end,
	  repr  = function (a) return ('math.log(%i)'):format(a)                end,
	  short = function (a) return ('lg(%i)'):format(a)                      end,
    },
	-- x = ℯ^a
	{ est   = function (n) return math.log(n)                               end,
	  real  = function (a) return math.exp(a)                               end,
	  repr  = function (a) return ('math.exp(%i)'):format(a)                end,
	  short = function (a) return a == 1 and 'ℯ' or 'ℯ'..to_superscript(a)  end,
    },
	-- x = aπ
	{ est   = function (n) return n/math.pi                                 end,
	  real  = function (a) return a*math.pi                                 end,
	  repr  = function (a) return a == 1 and 'math.pi' or a..'*math.pi'     end,
	  short = function (a) return (a == 1 and '' or a)..'π'                 end,
    },
	-- x = sqrt(a)
	{ est   = function (n) return n^2                                       end,
	  real  = function (a) return math.sqrt(a)                              end,
	  repr  = function (a) return ('math.sqrt(%i)'):format(a)               end,
	  short = function (a) return '√'..a                                    end,
    },
	-- a = x
	{ est   = function (n) return n                                         end,
	  real  = function (a) return a                                         end,
	  repr  = function (a) return ('%i'):format(a)                          end,
	  short = function (a) return ('%i'):format(a)                          end,
    },
}

--------------------------------------------------------------------------------

local LONG_ASS_CONSTANT = '___________________________________________________'

local function format_number (n, shorthand)
	if     n ~= n   then  return shorthand and 'NaN' or '0/0'
	elseif n <  0   then  return '-' .. format_number(-n, shorthand)
	end

	-- Finding the shortest
	local shortest, length = LONG_ASS_CONSTANT, unicode_len(LONG_ASS_CONSTANT)
	local function alternative_repr (repr)
		local repr_len = unicode_len(repr)
		if repr_len < length then  shortest, length  =  repr, repr_len  end
	end

	-- Maybe it's a "special" number?
	for _, special_number_tests in pairs(SPECIAL_NUMBER) do
		local a = special_number_tests.est(n)
		if a then
			local a_near = math.floor(a + 0.5)
			if n == special_number_tests.real(a_near) then
				alternative_repr( special_number_tests[shorthand and 'short' or 'repr'](a_near) )
			end
		end
	end

	-- Maybe it's a fractional number?
	do
		local numerator, denumberator = calculate_fraction(n)
		if numerator and denumberator ~= 1 then
			alternative_repr( format_number(numerator)..'/'..format_number(denumberator) )
		end
	end

	-- Maybe it's a decimal number?
	alternative_repr( tostring(n):gsub('([^e]+)e%+?(%-?)0+', function(a, b) return (a == '1' and '' or a..'*')..'10^'..b end))

	-- Well, this is not a pretty number!
	return shortest
end

return function (value, options, depth, l)
	l[#l+1] = format_number(value, options.math_shorthand)
end