-- 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 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 SPECIAL_NUMBER = {
	-- x = ∞
	{ est   = function (a)   return math.huge                               end,
	  real  = function (a)   return math.huge                               end,
	  repr  = function (a)   return '1/0'                                   end,
	},
	-- x = a/b
	{ est   = calculate_fraction,
	  real  = function (a, b) return b ~= 1 and (a/b)                       end,
	  repr  = function (a, b) return a..'/'..b                              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,
    },
	-- 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,
    },
	-- x = 1/√a
	{ est   = function (n) return 1/(n^2)                                   end,
	  real  = function (a) return a >= 0 and 1/math.sqrt(a)                 end,
	  repr  = function (a) return ('1/math.sqrt(%.0f)'):format(a)             end,
    },
	-- x = lg a
	{ est   = function (n) return math.exp(n)                               end,
	  real  = function (a) return a >= 0 and math.log(a)                    end,
	  repr  = function (a) return ('math.log(%.0f)'):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(%.0f)'):format(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,
    },
	-- x = sqrt(a)
	{ est   = function (n) return n^2                                       end,
	  real  = function (a) return a >= 0 and math.sqrt(a)                   end,
	  repr  = function (a) return ('math.sqrt(%.0f)'):format(a)               end,
    },
}

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

local function format_soft_num (n)
	assert(type(n) == 'number')

	if     n ~= n   then  return '0/0'
	elseif n == 0   then  return '0'
	elseif n <  0   then  return '-' .. format_soft_num(-n)
	end

	-- Finding the shortest
	local shortest, length = nil, math.huge
	local function alternative_repr (repr)
		if #repr < length then  shortest, length  =  repr, #repr  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[1] then
			for i = 1, #a do  a[i] = math.floor(a[i] + 0.5)  end
			local num  = special_number_tests.real(unpack(a))
			if num == n then
				alternative_repr( special_number_tests.repr(unpack(a)) )
			elseif num then
				local repr = special_number_tests.repr(unpack(a))
				local native_precise = tonumber(('%'..#repr..'f'):format(n))
				if math.abs(num - n) <= math.abs( native_precise - n ) then
					alternative_repr( repr )
				end
			end
		end
	end

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

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

local function format_hard_num (n)
	assert(type(n) == 'number')

	-- All the fun special cases
	if     n ~=  n         then  return  '0/0'
	elseif n ==  0         then  return  '0'
	elseif n ==  math.huge then  return  '1/0'
	elseif n == -math.huge then  return '-1/0'
	end

	-- Now for the serious part.
	for i = 0, 99 do
		local repr = ('%.'..i..'f'):format(n)
		local num  = tonumber(repr)
		if num == n then  return  repr  end
	end

	assert(false)
end

return function (value, depth, l)
	-- Formats the number nicely. If depth is 0 and we have some space for extra
	-- info, we give some tidbits, to help investigation.

	assert(type(value) == 'number')
	assert(type(depth) == 'number' and type(l) == 'table')

	-- First format a "soft" version. This number is not guarenteed to accurate.
	-- It's purpose is to give a general idea of the value.
	l[#l+1] = format_soft_num(value)

	-- If we have space for it, format a "hard" value, also. This number is as
	-- short as possible, while evaluating precisely to the value of the number.
	if depth == 0 then
		local hard_repr = format_hard_num(value)
		if l[#l] ~= hard_repr then
			l[#l+1] = '  --  Approx: '
			l[#l+1] = hard_repr
		end
		-- TODO: Add extra information. I don't really know what is useful?
		-- Prime factorization is fun, but useless unless people are doing
		-- cryptography or general number theory.
		-- Bit pattern is maybe too lowlevel.
	end
end