defmodule Decimal do
@moduledoc """
Decimal arithmetic on arbitrary precision floating-point numbers.
A number is represented by a signed coefficient and exponent such that: `sign
* coefficient * 10 ^ exponent`. All numbers are represented and calculated
exactly, but the result of an operation may be rounded depending on the
context the operation is performed with, see: `Decimal.Context`. Trailing
zeros in the coefficient are never truncated to preserve the number of
significant digits unless explicitly done so.
There are also special values such as NaN (not a number) and ±Infinity.
-0 and +0 are two distinct values.
Some operation results are not defined and will return NaN.
This kind of NaN is quiet, any operation returning a number will return
NaN when given a quiet NaN (the NaN value will flow through all operations).
Exceptional conditions are grouped into signals, each signal has a flag and a
trap enabler in the context. Whenever a signal is triggered it's flag is set
in the context and will be set until explicitly cleared. If the signal is trap
enabled `Decimal.Error` will be raised.
## Specifications
* [IBM's General Decimal Arithmetic Specification](http://speleotrove.com/decimal/decarith.html)
* [IEEE standard 854-1987](http://web.archive.org/web/20150908012941/http://754r.ucbtest.org/standards/854.pdf)
This library follows the above specifications for reference of arithmetic
operation implementations, but the public APIs may differ to provide a
more idiomatic Elixir interface.
The specification models the sign of the number as 1, for a negative number,
and 0 for a positive number. Internally this implementation models the sign as
1 or -1 such that the complete number will be `sign * coefficient *
10 ^ exponent` and will refer to the sign in documentation as either *positive*
or *negative*.
There is currently no maximum or minimum values for the exponent. Because of
that all numbers are "normal". This means that when an operation should,
according to the specification, return a number that "underflows" 0 is returned
instead of Etiny. This may happen when dividing a number with infinity.
Additionally, overflow, underflow and clamped may never be signalled.
"""
import Bitwise
import Kernel, except: [abs: 1, div: 2, max: 2, min: 2, rem: 2, round: 1]
import Decimal.Macros
alias Decimal.Context
alias Decimal.Error
@power_of_2_to_52 4_503_599_627_370_496
@typedoc """
The coefficient of the power of `10`. Non-negative because the sign is stored separately in `sign`.
* `non_neg_integer` - when the `t` represents a number, instead of one of the special values below.
* `:NaN` - Not a Number.
* `:inf` - Infinity.
"""
@type coefficient :: non_neg_integer | :NaN | :inf
@typedoc """
The exponent to which `10` is raised.
"""
@type exponent :: integer
@typedoc """
* `1` for positive
* `-1` for negative
"""
@type sign :: 1 | -1
@type signal ::
:invalid_operation
| :division_by_zero
| :rounded
| :inexact
@typedoc """
Rounding algorithm.
See `Decimal.Context` for more information.
"""
@type rounding ::
:down
| :half_up
| :half_even
| :ceiling
| :floor
| :half_down
| :up
@typedoc """
This implementation models the `sign` as `1` or `-1` such that the complete number will be: `sign * coef * 10 ^ exp`.
* `coef` - the coefficient of the power of `10`.
* `exp` - the exponent of the power of `10`.
* `sign` - `1` for positive, `-1` for negative.
"""
@type t :: %__MODULE__{
sign: sign,
coef: coefficient,
exp: exponent
}
@type decimal :: t | integer | String.t()
defstruct sign: 1, coef: 0, exp: 0
defmacrop error(flags, reason, result, context \\ nil) do
quote bind_quoted: binding() do
case handle_error(flags, reason, result, context) do
{:ok, result} -> result
{:error, error} -> raise Error, error
end
end
end
@doc """
Returns `true` if number is NaN, otherwise `false`.
## Examples
iex> Decimal.nan?(Decimal.new("NaN"))
true
iex> Decimal.nan?(Decimal.new(42))
false
"""
@spec nan?(t) :: boolean
def nan?(%Decimal{coef: :NaN}), do: true
def nan?(%Decimal{}), do: false
@doc """
Returns `true` if number is ±Infinity, otherwise `false`.
## Examples
iex> Decimal.inf?(Decimal.new("+Infinity"))
true
iex> Decimal.inf?(Decimal.new("-Infinity"))
true
iex> Decimal.inf?(Decimal.new("1.5"))
false
"""
@spec inf?(t) :: boolean
def inf?(%Decimal{coef: :inf}), do: true
def inf?(%Decimal{}), do: false
@doc """
Returns `true` if argument is a decimal number, otherwise `false`.
## Examples
iex> Decimal.is_decimal(Decimal.new(42))
true
iex> Decimal.is_decimal(42)
false
Allowed in guard tests on OTP 21+.
"""
doc_since("1.9.0")
defmacro is_decimal(term)
if function_exported?(:erlang, :is_map_key, 2) do
defmacro is_decimal(term) do
case __CALLER__.context do
nil ->
quote do
case unquote(term) do
%Decimal{} -> true
_ -> false
end
end
:match ->
raise ArgumentError,
"invalid expression in match, is_decimal is not allowed in patterns " <>
"such as function clauses, case clauses or on the left side of the = operator"
:guard ->
quote do
is_map(unquote(term)) and :erlang.is_map_key(:__struct__, unquote(term)) and
:erlang.map_get(:__struct__, unquote(term)) == Decimal
end
end
end
else
# TODO: remove when we require Elixir v1.10
defmacro is_decimal(term) do
quote do
case unquote(term) do
%Decimal{} -> true
_ -> false
end
end
end
end
@doc """
The absolute value of given number. Sets the number's sign to positive.
## Examples
iex> Decimal.abs(Decimal.new("1"))
Decimal.new("1")
iex> Decimal.abs(Decimal.new("-1"))
Decimal.new("1")
iex> Decimal.abs(Decimal.new("NaN"))
Decimal.new("NaN")
"""
@spec abs(t) :: t
def abs(%Decimal{coef: :NaN} = num), do: %{num | sign: 1}
def abs(%Decimal{} = num), do: context(%{num | sign: 1})
@doc """
Adds two numbers together.
## Exceptional conditions
* If one number is -Infinity and the other +Infinity, `:invalid_operation` will
be signalled.
## Examples
iex> Decimal.add(1, "1.1")
Decimal.new("2.1")
iex> Decimal.add(1, "Inf")
Decimal.new("Infinity")
"""
@spec add(decimal, decimal) :: t
def add(%Decimal{coef: :NaN} = num1, %Decimal{}), do: num1
def add(%Decimal{}, %Decimal{coef: :NaN} = num2), do: num2
def add(%Decimal{coef: :inf, sign: sign} = num1, %Decimal{coef: :inf, sign: sign} = num2) do
if num1.exp > num2.exp do
num1
else
num2
end
end
def add(%Decimal{coef: :inf}, %Decimal{coef: :inf}),
do: error(:invalid_operation, "adding +Infinity and -Infinity", %Decimal{coef: :NaN})
def add(%Decimal{coef: :inf} = num1, %Decimal{}), do: num1
def add(%Decimal{}, %Decimal{coef: :inf} = num2), do: num2
def add(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
{coef1, coef2} = add_align(coef1, exp1, coef2, exp2)
coef = sign1 * coef1 + sign2 * coef2
exp = Kernel.min(exp1, exp2)
sign = add_sign(sign1, sign2, coef)
context(%Decimal{sign: sign, coef: Kernel.abs(coef), exp: exp})
end
def add(num1, num2), do: add(decimal(num1), decimal(num2))
@doc """
Subtracts second number from the first. Equivalent to `Decimal.add/2` when the
second number's sign is negated.
## Exceptional conditions
* If one number is -Infinity and the other +Infinity `:invalid_operation` will
be signalled.
## Examples
iex> Decimal.sub(1, "0.1")
Decimal.new("0.9")
iex> Decimal.sub(1, "Inf")
Decimal.new("-Infinity")
"""
@spec sub(decimal, decimal) :: t
def sub(%Decimal{} = num1, %Decimal{sign: sign} = num2) do
add(num1, %{num2 | sign: -sign})
end
def sub(num1, num2) do
sub(decimal(num1), decimal(num2))
end
@doc """
Compares two numbers numerically. If the first number is greater than the second
`:gt` is returned, if less than `:lt` is returned, if both numbers are equal
`:eq` is returned.
Neither number can be a NaN.
## Examples
iex> Decimal.compare("1.0", 1)
:eq
iex> Decimal.compare("Inf", -1)
:gt
"""
@spec compare(decimal, decimal) :: :lt | :gt | :eq
def compare(%Decimal{coef: :inf, sign: sign}, %Decimal{coef: :inf, sign: sign}),
do: :eq
def compare(%Decimal{coef: :inf, sign: sign1}, %Decimal{coef: :inf, sign: sign2})
when sign1 < sign2,
do: :lt
def compare(%Decimal{coef: :inf, sign: sign1}, %Decimal{coef: :inf, sign: sign2})
when sign1 > sign2,
do: :gt
def compare(%Decimal{coef: :inf, sign: 1}, _num2), do: :gt
def compare(%Decimal{coef: :inf, sign: -1}, _num2), do: :lt
def compare(_num1, %Decimal{coef: :inf, sign: 1}), do: :lt
def compare(_num1, %Decimal{coef: :inf, sign: -1}), do: :gt
def compare(%Decimal{coef: :NaN} = num1, _num2),
do: error(:invalid_operation, "operation on NaN", num1)
def compare(_num1, %Decimal{coef: :NaN} = num2),
do: error(:invalid_operation, "operation on NaN", num2)
def compare(%Decimal{coef: 0}, %Decimal{coef: 0}), do: :eq
def compare(%Decimal{sign: 1}, %Decimal{coef: 0}), do: :gt
def compare(%Decimal{coef: 0}, %Decimal{sign: 1}), do: :lt
def compare(%Decimal{sign: -1}, %Decimal{coef: 0}), do: :lt
def compare(%Decimal{coef: 0}, %Decimal{sign: -1}), do: :gt
def compare(%Decimal{sign: 1}, %Decimal{sign: -1}), do: :gt
def compare(%Decimal{sign: -1}, %Decimal{sign: 1}), do: :lt
def compare(%Decimal{} = num1, %Decimal{} = num2) do
adjusted_exp1 = adjust_exp(num1)
adjusted_exp2 = adjust_exp(num2)
sign =
cond do
adjusted_exp1 == adjusted_exp2 ->
padded_num1 = pad_num(num1, num1.exp - num2.exp)
padded_num2 = pad_num(num2, num2.exp - num1.exp)
cond do
padded_num1 == padded_num2 -> 0
padded_num1 < padded_num2 -> -num1.sign
true -> num1.sign
end
adjusted_exp1 < adjusted_exp2 ->
-num1.sign
true ->
num1.sign
end
case sign do
0 -> :eq
1 -> :gt
-1 -> :lt
end
end
def compare(num1, num2) do
compare(decimal(num1), decimal(num2))
end
defp adjust_exp(%Decimal{coef: coef, exp: exp}) do
coef_adjustment = coef_length(coef)
exp + coef_adjustment - 1
end
def coef_length(0), do: 1
def coef_length(coef), do: coef_length(coef, 0)
def coef_length(0, length), do: length
def coef_length(coef, length), do: coef_length(Kernel.div(coef, 10), length + 1)
defp pad_num(%Decimal{coef: coef}, n) do
coef * pow10(Kernel.max(n, 0) + 1)
end
@deprecated "Use compare/2 instead"
@spec cmp(decimal, decimal) :: :lt | :eq | :gt
def cmp(num1, num2) do
compare(num1, num2)
end
@doc """
Compares two numbers numerically and returns `true` if they are equal,
otherwise `false`. If one of the operands is a quiet NaN this operation
will always return `false`.
## Examples
iex> Decimal.equal?("1.0", 1)
true
iex> Decimal.equal?(1, -1)
false
"""
@spec equal?(decimal, decimal) :: boolean
def equal?(num1, num2) do
eq?(num1, num2)
end
@doc """
Compares two numbers numerically and returns `true` if they are equal,
otherwise `false`. If one of the operands is a quiet NaN this operation
will always return `false`.
## Examples
iex> Decimal.eq?("1.0", 1)
true
iex> Decimal.eq?(1, -1)
false
"""
doc_since("1.8.0")
@spec eq?(decimal, decimal) :: boolean
def eq?(%Decimal{coef: :NaN}, _num2), do: false
def eq?(_num1, %Decimal{coef: :NaN}), do: false
def eq?(num1, num2), do: compare(num1, num2) == :eq
@doc """
Compares two numbers numerically and returns `true` if the the first argument
is greater than the second, otherwise `false`. If one the operands is a
quiet NaN this operation will always return `false`.
## Examples
iex> Decimal.gt?("1.3", "1.2")
true
iex> Decimal.gt?("1.2", "1.3")
false
"""
doc_since("1.8.0")
@spec gt?(decimal, decimal) :: boolean
def gt?(%Decimal{coef: :NaN}, _num2), do: false
def gt?(_num1, %Decimal{coef: :NaN}), do: false
def gt?(num1, num2), do: compare(num1, num2) == :gt
@doc """
Compares two numbers numerically and returns `true` if the the first number is
less than the second number, otherwise `false`. If one of the operands is a
quiet NaN this operation will always return `false`.
## Examples
iex> Decimal.lt?("1.1", "1.2")
true
iex> Decimal.lt?("1.4", "1.2")
false
"""
doc_since("1.8.0")
@spec lt?(decimal, decimal) :: boolean
def lt?(%Decimal{coef: :NaN}, _num2), do: false
def lt?(_num1, %Decimal{coef: :NaN}), do: false
def lt?(num1, num2), do: compare(num1, num2) == :lt
@doc """
Divides two numbers.
## Exceptional conditions
* If both numbers are ±Infinity `:invalid_operation` is signalled.
* If both numbers are ±0 `:invalid_operation` is signalled.
* If second number (denominator) is ±0 `:division_by_zero` is signalled.
## Examples
iex> Decimal.div(3, 4)
Decimal.new("0.75")
iex> Decimal.div("Inf", -1)
Decimal.new("-Infinity")
"""
@spec div(decimal, decimal) :: t
def div(%Decimal{coef: :NaN} = num1, %Decimal{}), do: num1
def div(%Decimal{}, %Decimal{coef: :NaN} = num2), do: num2
def div(%Decimal{coef: :inf}, %Decimal{coef: :inf}),
do: error(:invalid_operation, "±Infinity / ±Infinity", %Decimal{coef: :NaN})
def div(%Decimal{sign: sign1, coef: :inf} = num1, %Decimal{sign: sign2}) do
sign = if sign1 == sign2, do: 1, else: -1
%{num1 | sign: sign}
end
def div(%Decimal{sign: sign1, exp: exp1}, %Decimal{sign: sign2, coef: :inf, exp: exp2}) do
sign = if sign1 == sign2, do: 1, else: -1
# TODO: Subnormal
# exponent?
%Decimal{sign: sign, coef: 0, exp: exp1 - exp2}
end
def div(%Decimal{coef: 0}, %Decimal{coef: 0}),
do: error(:invalid_operation, "0 / 0", %Decimal{coef: :NaN})
def div(%Decimal{sign: sign1}, %Decimal{sign: sign2, coef: 0}) do
sign = if sign1 == sign2, do: 1, else: -1
error(:division_by_zero, nil, %Decimal{sign: sign, coef: :inf})
end
def div(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
sign = if sign1 == sign2, do: 1, else: -1
if coef1 == 0 do
context(%Decimal{sign: sign, coef: 0, exp: exp1 - exp2}, [])
else
prec10 = pow10(Context.get().precision)
{coef1, coef2, adjust} = div_adjust(coef1, coef2, 0)
{coef, adjust, _rem, signals} = div_calc(coef1, coef2, 0, adjust, prec10)
context(%Decimal{sign: sign, coef: coef, exp: exp1 - exp2 - adjust}, signals)
end
end
def div(num1, num2) do
div(decimal(num1), decimal(num2))
end
@doc """
Divides two numbers and returns the integer part.
## Exceptional conditions
* If both numbers are ±Infinity `:invalid_operation` is signalled.
* If both numbers are ±0 `:invalid_operation` is signalled.
* If second number (denominator) is ±0 `:division_by_zero` is signalled.
## Examples
iex> Decimal.div_int(5, 2)
Decimal.new("2")
iex> Decimal.div_int("Inf", -1)
Decimal.new("-Infinity")
"""
@spec div_int(decimal, decimal) :: t
def div_int(%Decimal{coef: :NaN} = num1, %Decimal{}), do: num1
def div_int(%Decimal{}, %Decimal{coef: :NaN} = num2), do: num2
def div_int(%Decimal{coef: :inf}, %Decimal{coef: :inf}),
do: error(:invalid_operation, "±Infinity / ±Infinity", %Decimal{coef: :NaN})
def div_int(%Decimal{sign: sign1, coef: :inf} = num1, %Decimal{sign: sign2}) do
sign = if sign1 == sign2, do: 1, else: -1
%{num1 | sign: sign}
end
def div_int(%Decimal{sign: sign1, exp: exp1}, %Decimal{sign: sign2, coef: :inf, exp: exp2}) do
sign = if sign1 == sign2, do: 1, else: -1
# TODO: Subnormal
# exponent?
%Decimal{sign: sign, coef: 0, exp: exp1 - exp2}
end
def div_int(%Decimal{coef: 0}, %Decimal{coef: 0}),
do: error(:invalid_operation, "0 / 0", %Decimal{coef: :NaN})
def div_int(%Decimal{sign: sign1}, %Decimal{sign: sign2, coef: 0}) do
div_sign = if sign1 == sign2, do: 1, else: -1
error(:division_by_zero, nil, %Decimal{sign: div_sign, coef: :inf})
end
def div_int(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
div_sign = if sign1 == sign2, do: 1, else: -1
cond do
compare(%{num1 | sign: 1}, %{num2 | sign: 1}) == :lt ->
%Decimal{sign: div_sign, coef: 0, exp: exp1 - exp2}
coef1 == 0 ->
context(%{num1 | sign: div_sign})
true ->
case integer_division(div_sign, coef1, exp1, coef2, exp2) do
{:ok, result} ->
result
{:error, error, reason, num} ->
error(error, reason, num)
end
end
end
def div_int(num1, num2) do
div_int(decimal(num1), decimal(num2))
end
@doc """
Remainder of integer division of two numbers. The result will have the sign of
the first number.
## Exceptional conditions
* If both numbers are ±Infinity `:invalid_operation` is signalled.
* If both numbers are ±0 `:invalid_operation` is signalled.
* If second number (denominator) is ±0 `:division_by_zero` is signalled.
## Examples
iex> Decimal.rem(5, 2)
Decimal.new("1")
"""
@spec rem(decimal, decimal) :: t
def rem(%Decimal{coef: :NaN} = num1, %Decimal{}), do: num1
def rem(%Decimal{}, %Decimal{coef: :NaN} = num2), do: num2
def rem(%Decimal{coef: :inf}, %Decimal{coef: :inf}),
do: error(:invalid_operation, "±Infinity / ±Infinity", %Decimal{coef: :NaN})
def rem(%Decimal{sign: sign1, coef: :inf}, %Decimal{}), do: %Decimal{sign: sign1, coef: 0}
def rem(%Decimal{sign: sign1}, %Decimal{coef: :inf} = num2) do
# TODO: Subnormal
# exponent?
%{num2 | sign: sign1}
end
def rem(%Decimal{coef: 0}, %Decimal{coef: 0}),
do: error(:invalid_operation, "0 / 0", %Decimal{coef: :NaN})
def rem(%Decimal{sign: sign1}, %Decimal{coef: 0}),
do: error(:division_by_zero, nil, %Decimal{sign: sign1, coef: 0})
def rem(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
cond do
compare(%{num1 | sign: 1}, %{num2 | sign: 1}) == :lt ->
%{num1 | sign: sign1}
coef1 == 0 ->
context(%{num2 | sign: sign1})
true ->
div_sign = if sign1 == sign2, do: 1, else: -1
case integer_division(div_sign, coef1, exp1, coef2, exp2) do
{:ok, result} ->
sub(num1, mult(num2, result))
{:error, error, reason, num} ->
error(error, reason, num)
end
end
end
def rem(num1, num2) do
rem(decimal(num1), decimal(num2))
end
@doc """
Integer division of two numbers and the remainder. Should be used when both
`div_int/2` and `rem/2` is needed. Equivalent to: `{Decimal.div_int(x, y),
Decimal.rem(x, y)}`.
## Exceptional conditions
* If both numbers are ±Infinity `:invalid_operation` is signalled.
* If both numbers are ±0 `:invalid_operation` is signalled.
* If second number (denominator) is ±0 `:division_by_zero` is signalled.
## Examples
iex> Decimal.div_rem(5, 2)
{Decimal.new(2), Decimal.new(1)}
"""
@spec div_rem(decimal, decimal) :: {t, t}
def div_rem(%Decimal{coef: :NaN} = num1, %Decimal{}), do: {num1, num1}
def div_rem(%Decimal{}, %Decimal{coef: :NaN} = num2), do: {num2, num2}
def div_rem(%Decimal{coef: :inf}, %Decimal{coef: :inf}) do
numbers = {%Decimal{coef: :NaN}, %Decimal{coef: :NaN}}
error(:invalid_operation, "±Infinity / ±Infinity", numbers)
end
def div_rem(%Decimal{sign: sign1, coef: :inf} = num1, %Decimal{sign: sign2}) do
sign = if sign1 == sign2, do: 1, else: -1
{%{num1 | sign: sign}, %Decimal{sign: sign1, coef: 0}}
end
def div_rem(%Decimal{} = num1, %Decimal{coef: :inf} = num2) do
%Decimal{sign: sign1, exp: exp1} = num1
%Decimal{sign: sign2, exp: exp2} = num2
sign = if sign1 == sign2, do: 1, else: -1
# TODO: Subnormal
# exponent?
{%Decimal{sign: sign, coef: 0, exp: exp1 - exp2}, %{num2 | sign: sign1}}
end
def div_rem(%Decimal{coef: 0}, %Decimal{coef: 0}) do
error = error(:invalid_operation, "0 / 0", %Decimal{coef: :NaN})
{error, error}
end
def div_rem(%Decimal{sign: sign1}, %Decimal{sign: sign2, coef: 0}) do
div_sign = if sign1 == sign2, do: 1, else: -1
div_error = error(:division_by_zero, nil, %Decimal{sign: div_sign, coef: :inf})
rem_error = error(:division_by_zero, nil, %Decimal{sign: sign1, coef: 0})
{div_error, rem_error}
end
def div_rem(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
div_sign = if sign1 == sign2, do: 1, else: -1
cond do
compare(%{num1 | sign: 1}, %{num2 | sign: 1}) == :lt ->
{%Decimal{sign: div_sign, coef: 0, exp: exp1 - exp2}, %{num1 | sign: sign1}}
coef1 == 0 ->
{context(%{num1 | sign: div_sign}), context(%{num2 | sign: sign1})}
true ->
case integer_division(div_sign, coef1, exp1, coef2, exp2) do
{:ok, result} ->
{result, sub(num1, mult(num2, result))}
{:error, error, reason, num} ->
error(error, reason, {num, num})
end
end
end
def div_rem(num1, num2) do
div_rem(decimal(num1), decimal(num2))
end
@doc """
Compares two values numerically and returns the maximum. Unlike most other
functions in `Decimal` if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.
## Examples
iex> Decimal.max(1, "2.0")
Decimal.new("2.0")
iex> Decimal.max(1, "NaN")
Decimal.new("1")
iex> Decimal.max("NaN", "NaN")
Decimal.new("NaN")
"""
@spec max(decimal, decimal) :: t
def max(%Decimal{coef: :NaN}, %Decimal{} = num2), do: num2
def max(%Decimal{} = num1, %Decimal{coef: :NaN}), do: num1
def max(%Decimal{sign: sign1, exp: exp1} = num1, %Decimal{sign: sign2, exp: exp2} = num2) do
case compare(num1, num2) do
:lt ->
num2
:gt ->
num1
:eq ->
cond do
sign1 != sign2 ->
if sign1 == 1, do: num1, else: num2
sign1 == 1 ->
if exp1 > exp2, do: num1, else: num2
sign1 == -1 ->
if exp1 < exp2, do: num1, else: num2
end
end
|> context()
end
def max(num1, num2) do
max(decimal(num1), decimal(num2))
end
@doc """
Compares two values numerically and returns the minimum. Unlike most other
functions in `Decimal` if a number is NaN the result will be the other number.
Only if both numbers are NaN will NaN be returned.
## Examples
iex> Decimal.min(1, "2.0")
Decimal.new("1")
iex> Decimal.min(1, "NaN")
Decimal.new("1")
iex> Decimal.min("NaN", "NaN")
Decimal.new("NaN")
"""
@spec min(decimal, decimal) :: t
def min(%Decimal{coef: :NaN}, %Decimal{} = num2), do: num2
def min(%Decimal{} = num1, %Decimal{coef: :NaN}), do: num1
def min(%Decimal{sign: sign1, exp: exp1} = num1, %Decimal{sign: sign2, exp: exp2} = num2) do
case compare(num1, num2) do
:lt ->
num1
:gt ->
num2
:eq ->
cond do
sign1 != sign2 ->
if sign1 == -1, do: num1, else: num2
sign1 == 1 ->
if exp1 < exp2, do: num1, else: num2
sign1 == -1 ->
if exp1 > exp2, do: num1, else: num2
end
end
|> context()
end
def min(num1, num2) do
min(decimal(num1), decimal(num2))
end
@doc """
Negates the given number.
## Examples
iex> Decimal.negate(1)
Decimal.new("-1")
iex> Decimal.negate("-Inf")
Decimal.new("Infinity")
"""
doc_since("1.9.0")
@spec negate(decimal) :: t
def negate(%Decimal{coef: :NaN} = num), do: num
def negate(%Decimal{sign: sign} = num), do: context(%{num | sign: -sign})
def negate(num), do: negate(decimal(num))
@doc """
Applies the context to the given number rounding it to specified precision.
"""
doc_since("1.9.0")
@spec apply_context(t) :: t
def apply_context(%Decimal{} = num), do: context(num)
@doc """
Returns `true` if given number is positive, otherwise `false`.
## Examples
iex> Decimal.positive?(Decimal.new("42"))
true
iex> Decimal.positive?(Decimal.new("-42"))
false
iex> Decimal.positive?(Decimal.new("0"))
false
iex> Decimal.positive?(Decimal.new("NaN"))
false
"""
doc_since("1.5.0")
@spec positive?(t) :: boolean
def positive?(%Decimal{coef: :NaN}), do: false
def positive?(%Decimal{coef: 0}), do: false
def positive?(%Decimal{sign: -1}), do: false
def positive?(%Decimal{sign: 1}), do: true
@doc """
Returns `true` if given number is negative, otherwise `false`.
## Examples
iex> Decimal.negative?(Decimal.new("-42"))
true
iex> Decimal.negative?(Decimal.new("42"))
false
iex> Decimal.negative?(Decimal.new("0"))
false
iex> Decimal.negative?(Decimal.new("NaN"))
false
"""
doc_since("1.5.0")
@spec negative?(t) :: boolean
def negative?(%Decimal{coef: :NaN}), do: false
def negative?(%Decimal{coef: 0}), do: false
def negative?(%Decimal{sign: 1}), do: false
def negative?(%Decimal{sign: -1}), do: true
@doc """
Multiplies two numbers.
## Exceptional conditions
* If one number is ±0 and the other is ±Infinity `:invalid_operation` is
signalled.
## Examples
iex> Decimal.mult("0.5", 3)
Decimal.new("1.5")
iex> Decimal.mult("Inf", -1)
Decimal.new("-Infinity")
"""
@spec mult(decimal, decimal) :: t
def mult(%Decimal{coef: :NaN} = num1, %Decimal{}), do: num1
def mult(%Decimal{}, %Decimal{coef: :NaN} = num2), do: num2
def mult(%Decimal{coef: 0}, %Decimal{coef: :inf}),
do: error(:invalid_operation, "0 * ±Infinity", %Decimal{coef: :NaN})
def mult(%Decimal{coef: :inf}, %Decimal{coef: 0}),
do: error(:invalid_operation, "0 * ±Infinity", %Decimal{coef: :NaN})
def mult(%Decimal{sign: sign1, coef: :inf, exp: exp1}, %Decimal{sign: sign2, exp: exp2}) do
sign = if sign1 == sign2, do: 1, else: -1
# exponent?
%Decimal{sign: sign, coef: :inf, exp: exp1 + exp2}
end
def mult(%Decimal{sign: sign1, exp: exp1}, %Decimal{sign: sign2, coef: :inf, exp: exp2}) do
sign = if sign1 == sign2, do: 1, else: -1
# exponent?
%Decimal{sign: sign, coef: :inf, exp: exp1 + exp2}
end
def mult(%Decimal{} = num1, %Decimal{} = num2) do
%Decimal{sign: sign1, coef: coef1, exp: exp1} = num1
%Decimal{sign: sign2, coef: coef2, exp: exp2} = num2
sign = if sign1 == sign2, do: 1, else: -1
%Decimal{sign: sign, coef: coef1 * coef2, exp: exp1 + exp2} |> context()
end
def mult(num1, num2) do
mult(decimal(num1), decimal(num2))
end
@doc """
Normalizes the given decimal: removes trailing zeros from coefficient while
keeping the number numerically equivalent by increasing the exponent.
## Examples
iex> Decimal.normalize(Decimal.new("1.00"))
Decimal.new("1")
iex> Decimal.normalize(Decimal.new("1.01"))
Decimal.new("1.01")
"""
doc_since("1.9.0")
@spec normalize(t) :: t
def normalize(%Decimal{coef: :NaN} = num), do: num
def normalize(%Decimal{coef: :inf} = num) do
# exponent?
%{num | exp: 0}
end
def normalize(%Decimal{sign: sign, coef: coef, exp: exp}) do
if coef == 0 do
%Decimal{sign: sign, coef: 0, exp: 0}
else
%{do_normalize(coef, exp) | sign: sign} |> context
end
end
@doc """
Rounds the given number to specified decimal places with the given strategy
(default is to round to nearest one). If places is negative, at least that
many digits to the left of the decimal point will be zero.
See `Decimal.Context` for more information about rounding algorithms.
## Examples
iex> Decimal.round("1.234")
Decimal.new("1")
iex> Decimal.round("1.234", 1)
Decimal.new("1.2")
"""
@spec round(decimal, integer, rounding) :: t
def round(num, places \\ 0, mode \\ :half_up)
def round(%Decimal{coef: :NaN} = num, _, _), do: num
def round(%Decimal{coef: :inf} = num, _, _), do: num
def round(%Decimal{} = num, n, mode) do
%Decimal{sign: sign, coef: coef, exp: exp} = normalize(num)
digits = :erlang.integer_to_list(coef)
target_exp = -n
value = do_round(sign, digits, exp, target_exp, mode)
context(value, [])
end
def round(num, n, mode) do
round(decimal(num), n, mode)
end
@doc """
Finds the square root.
## Examples
iex> Decimal.sqrt("100")
Decimal.new("10")
"""
doc_since("1.7.0")
@spec sqrt(decimal) :: t
def sqrt(%Decimal{coef: :NaN} = num),
do: error(:invalid_operation, "operation on NaN", num)
def sqrt(%Decimal{coef: 0, exp: exp} = num),
do: %{num | exp: exp >>> 1}
def sqrt(%Decimal{sign: -1} = num),
do: error(:invalid_operation, "less than zero", num)
def sqrt(%Decimal{sign: 1, coef: :inf} = num),
do: num
def sqrt(%Decimal{sign: 1, coef: coef, exp: exp}) do
precision = Context.get().precision + 1
digits = :erlang.integer_to_list(coef)
num_digits = length(digits)
# Since the root is calculated from integer operations only, it must be
# large enough to contain the desired precision. Calculate the amount of
# `shift` required (powers of 10).
case exp &&& 1 do
0 ->
# To get the desired `shift`, subtract the precision of `coef`'s square
# root from the desired precision.
#
# If `coef` is 10_000, the root is 100 (3 digits of precision).
# If `coef` is 100, the root is 10 (2 digits of precision).
shift = precision - ((num_digits + 1) >>> 1)
sqrt(coef, shift, exp)
_ ->
# If `exp` is odd, multiply `coef` by 10 and reduce shift by 1/2. `exp`
# must be even so the root's exponent is an integer.
shift = precision - ((num_digits >>> 1) + 1)
sqrt(coef * 10, shift, exp)
end
end
def sqrt(num) do
sqrt(decimal(num))
end
defp sqrt(coef, shift, exp) do
if shift >= 0 do
# shift `coef` up by `shift * 2` digits
sqrt(coef * pow10(shift <<< 1), shift, exp, true)
else
# shift `coef` down by `shift * 2` digits
operand = pow10(-shift <<< 1)
sqrt(Kernel.div(coef, operand), shift, exp, Kernel.rem(coef, operand) === 0)
end
end
defp sqrt(shifted_coef, shift, exp, exact) do
# the preferred exponent is `exp / 2` as per IEEE 754
exp = exp >>> 1
# guess a root 10x higher than desired precision
guess = pow10(Context.get().precision + 1)
root = sqrt_loop(shifted_coef, guess)
if exact and root * root === shifted_coef do
# if the root is exact, use preferred `exp` and shift `coef` to match
coef =
if shift >= 0,
do: Kernel.div(root, pow10(shift)),
else: root * pow10(-shift)
context(%Decimal{sign: 1, coef: coef, exp: exp})
else
# otherwise the calculated root is inexact (but still meets precision),
# so use the root as `coef` and get the final exponent by shifting `exp`
context(%Decimal{sign: 1, coef: root, exp: exp - shift})
end
end
# Babylonion method
defp sqrt_loop(coef, guess) do
quotient = Kernel.div(coef, guess)
if guess <= quotient do
guess
else
sqrt_loop(coef, (guess + quotient) >>> 1)
end
end
@doc """
Creates a new decimal number from an integer or a string representation.
A decimal number will always be created exactly as specified with all digits
kept - it will not be rounded with the context.
## Backus–Naur form
sign ::= "+" | "-"
digit ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
indicator ::= "e" | "E"
digits ::= digit [digit]...
decimal-part ::= digits "." [digits] | ["."] digits
exponent-part ::= indicator [sign] digits
infinity ::= "Infinity" | "Inf"
nan ::= "NaN" [digits]
numeric-value ::= decimal-part [exponent-part] | infinity
numeric-string ::= [sign] numeric-value | [sign] nan
## Floats
See also `from_float/1`.
## Examples
iex> Decimal.new(1)
Decimal.new("1")
iex> Decimal.new("3.14")
Decimal.new("3.14")
"""
@spec new(decimal) :: t
def new(%Decimal{sign: sign, coef: coef, exp: exp} = num)
when sign in [1, -1] and ((is_integer(coef) and coef >= 0) or coef in [:NaN, :inf]) and
is_integer(exp),
do: num
def new(int) when is_integer(int),
do: %Decimal{sign: if(int < 0, do: -1, else: 1), coef: Kernel.abs(int)}
def new(binary) when is_binary(binary) do
case parse(binary) do
{decimal, ""} -> decimal
_ -> raise Error, reason: "number parsing syntax: #{inspect(binary)}"
end
end
@doc """
Creates a new decimal number from the sign, coefficient and exponent such that
the number will be: `sign * coefficient * 10 ^ exponent`.
A decimal number will always be created exactly as specified with all digits
kept - it will not be rounded with the context.
## Examples
iex> Decimal.new(1, 42, 0)
Decimal.new("42")
"""
@spec new(sign :: 1 | -1, coef :: non_neg_integer | :NaN | :inf, exp :: integer) :: t
def new(sign, coef, exp)
when sign in [1, -1] and ((is_integer(coef) and coef >= 0) or coef in [:NaN, :inf]) and
is_integer(exp),
do: %Decimal{sign: sign, coef: coef, exp: exp}
@doc """
Creates a new decimal number from a floating point number.
Floating point numbers use a fixed number of binary digits to represent
a decimal number which has inherent inaccuracy as some decimal numbers cannot
be represented exactly in limited precision binary.
Floating point numbers will be converted to decimal numbers with
`:io_lib_format.fwrite_g/1`. Since this conversion is not exact and
because of inherent inaccuracy mentioned above, we may run into counter-intuitive results:
iex> Enum.reduce([0.1, 0.1, 0.1], &+/2)
0.30000000000000004
iex> Enum.reduce([Decimal.new("0.1"), Decimal.new("0.1"), Decimal.new("0.1")], &Decimal.add/2)
Decimal.new("0.3")
For this reason, it's recommended to build decimals with `new/1`, which is always precise, instead.
## Examples
iex> Decimal.from_float(3.14)
Decimal.new("3.14")
"""
doc_since("1.5.0")
@spec from_float(float) :: t
def from_float(float) when is_float(float) do
float
|> :io_lib_format.fwrite_g()
|> fix_float_exp()
|> IO.iodata_to_binary()
|> new()
end
@doc """
Creates a new decimal number from an integer, string, float, or existing decimal number.
Because conversion from a floating point number is not exact, it's recommended
to instead use `new/1` or `from_float/1` when the argument's type is certain.
See `from_float/1`.
## Examples
iex> {:ok, decimal} = Decimal.cast(3)
iex> decimal
Decimal.new("3")
iex> Decimal.cast("bad")
:error
"""
@spec cast(term) :: {:ok, t} | :error
def cast(integer) when is_integer(integer), do: {:ok, Decimal.new(integer)}
def cast(%Decimal{} = decimal), do: {:ok, decimal}
def cast(float) when is_float(float), do: {:ok, from_float(float)}
def cast(binary) when is_binary(binary) do
case parse(binary) do
{decimal, ""} -> {:ok, decimal}
_ -> :error
end
end
def cast(_), do: :error
@doc """
Parses a binary into a decimal.
If successful, returns a tuple in the form of `{decimal, remainder_of_binary}`,
otherwise `:error`.
## Examples
iex> Decimal.parse("3.14")
{%Decimal{coef: 314, exp: -2, sign: 1}, ""}
iex> Decimal.parse("3.14.15")
{%Decimal{coef: 314, exp: -2, sign: 1}, ".15"}
iex> Decimal.parse("-1.1e3")
{%Decimal{coef: 11, exp: 2, sign: -1}, ""}
iex> Decimal.parse("bad")
:error
"""
@spec parse(binary()) :: {t(), binary()} | :error
def parse("+" <> rest) do
parse_unsign(rest)
end
def parse("-" <> rest) do
case parse_unsign(rest) do
{%Decimal{} = num, rest} -> {%{num | sign: -1}, rest}
:error -> :error
end
end
def parse(binary) when is_binary(binary) do
parse_unsign(binary)
end
@doc """
Converts given number to its string representation.
## Options
* `:scientific` - number converted to scientific notation.
* `:normal` - number converted without a exponent.
* `:xsd` - number converted to the [canonical XSD representation](https://www.w3.org/TR/xmlschema-2/#decimal).
* `:raw` - number converted to its raw, internal format.
## Examples
iex> Decimal.to_string(Decimal.new("1.00"))
"1.00"
iex> Decimal.to_string(Decimal.new("123e1"), :scientific)
"1.23E+3"
iex> Decimal.to_string(Decimal.new("42.42"), :normal)
"42.42"
iex> Decimal.to_string(Decimal.new("1.00"), :xsd)
"1.0"
iex> Decimal.to_string(Decimal.new("4321.768"), :raw)
"4321768E-3"
"""
@spec to_string(t, :scientific | :normal | :xsd | :raw) :: String.t()
def to_string(num, type \\ :scientific)
def to_string(%Decimal{sign: sign, coef: :NaN}, _type) do
if sign == 1, do: "NaN", else: "-NaN"
end
def to_string(%Decimal{sign: sign, coef: :inf}, _type) do
if sign == 1, do: "Infinity", else: "-Infinity"
end
def to_string(%Decimal{sign: sign, coef: coef, exp: exp}, :normal) do
list = integer_to_charlist(coef)
list =
if exp >= 0 do
list ++ :lists.duplicate(exp, ?0)
else
diff = length(list) + exp
if diff > 0 do
List.insert_at(list, diff, ?.)
else
~c"0." ++ :lists.duplicate(-diff, ?0) ++ list
end
end
list = if sign == -1, do: [?- | list], else: list
IO.iodata_to_binary(list)
end
def to_string(%Decimal{sign: sign, coef: coef, exp: exp}, :scientific) do
list = integer_to_charlist(coef)
length = length(list)
adjusted = exp + length - 1
list =
cond do
exp == 0 ->
list
exp < 0 and adjusted >= -6 ->
abs_exp = Kernel.abs(exp)
diff = -length + abs_exp + 1
if diff > 0 do
list = :lists.duplicate(diff, ?0) ++ list
List.insert_at(list, 1, ?.)
else
List.insert_at(list, exp - 1, ?.)
end
true ->
list = if length > 1, do: List.insert_at(list, 1, ?.), else: list
list = list ++ ~c"E"
list = if exp >= 0, do: list ++ ~c"+", else: list
list ++ integer_to_charlist(adjusted)
end
list = if sign == -1, do: [?- | list], else: list
IO.iodata_to_binary(list)
end
def to_string(%Decimal{sign: sign, coef: coef, exp: exp}, :raw) do
str = Integer.to_string(coef)
str = if sign == -1, do: [?- | str], else: str
str = if exp != 0, do: [str, "E", Integer.to_string(exp)], else: str
IO.iodata_to_binary(str)
end
def to_string(%Decimal{} = decimal, :xsd) do
decimal |> canonical_xsd() |> to_string(:normal)
end
defp canonical_xsd(%Decimal{coef: 0} = decimal), do: %{decimal | exp: -1}
defp canonical_xsd(%Decimal{coef: coef, exp: 0} = decimal),
do: %{decimal | coef: coef * 10, exp: -1}
defp canonical_xsd(%Decimal{coef: coef, exp: exp} = decimal)
when exp > 0,
do: canonical_xsd(%{decimal | coef: coef * 10, exp: exp - 1})
defp canonical_xsd(%Decimal{coef: coef} = decimal)
when Kernel.rem(coef, 10) != 0,
do: decimal
defp canonical_xsd(%Decimal{coef: coef, exp: exp} = decimal),
do: canonical_xsd(%{decimal | coef: Kernel.div(coef, 10), exp: exp + 1})
@doc """
Returns the decimal represented as an integer.
Fails when loss of precision will occur.
## Examples
iex> Decimal.to_integer(Decimal.new("42"))
42
iex> Decimal.to_integer(Decimal.new("1.00"))
1
iex> Decimal.to_integer(Decimal.new("1.10"))
** (ArgumentError) cannot convert Decimal.new("1.1") without losing precision. Use Decimal.round/3 first.
"""
@spec to_integer(t) :: integer
def to_integer(%Decimal{sign: sign, coef: coef, exp: 0})
when is_integer(coef),
do: sign * coef
def to_integer(%Decimal{sign: sign, coef: coef, exp: exp})
when is_integer(coef) and exp > 0,
do: to_integer(%Decimal{sign: sign, coef: coef * 10, exp: exp - 1})
def to_integer(%Decimal{sign: sign, coef: coef, exp: exp})
when is_integer(coef) and exp < 0 and Kernel.rem(coef, 10) == 0,
do: to_integer(%Decimal{sign: sign, coef: Kernel.div(coef, 10), exp: exp + 1})
def to_integer(%Decimal{coef: coef} = decimal) when is_integer(coef) do
raise ArgumentError,
"cannot convert #{inspect(decimal)} without losing precision. Use Decimal.round/3 first."
end
@doc """
Returns the decimal converted to a float.
The returned float may have lower precision than the decimal. Fails if
the decimal cannot be converted to a float.
## Examples
iex> Decimal.to_float(Decimal.new("1.5"))
1.5
"""
@spec to_float(t) :: float
def to_float(%Decimal{sign: sign, coef: coef, exp: exp}) when is_integer(coef) do
# Convert back to float without loss
# http://www.exploringbinary.com/correct-decimal-to-floating-point-using-big-integers/
{num, den} = ratio(coef, exp)
boundary = den <<< 52
cond do
num == 0 ->
0.0
num >= boundary ->
{den, exp} = scale_down(num, boundary, 52)
decimal_to_float(sign, num, den, exp)
true ->
{num, exp} = scale_up(num, boundary, 52)
decimal_to_float(sign, num, den, exp)
end
end
@doc """
Returns the scale of the decimal.
A decimal's scale is the number of digits after the decimal point. This
includes trailing zeros; see `normalize/1` to remove them.
## Examples
iex> Decimal.scale(Decimal.new("42"))
0
iex> Decimal.scale(Decimal.new(1, 2, 26))
0
iex> Decimal.scale(Decimal.new("99.12345"))
5
iex> Decimal.scale(Decimal.new("1.50"))
2
"""
@spec scale(t) :: non_neg_integer()
def scale(%Decimal{exp: exp}), do: Kernel.max(0, -exp)
defp scale_up(num, den, exp) when num >= den, do: {num, exp}
defp scale_up(num, den, exp), do: scale_up(num <<< 1, den, exp - 1)
defp scale_down(num, den, exp) do
new_den = den <<< 1
if num < new_den do
{den >>> 52, exp}
else
scale_down(num, new_den, exp + 1)
end
end
defp decimal_to_float(sign, num, den, exp) do
quo = Kernel.div(num, den)
rem = num - quo * den
tmp =
case den >>> 1 do
den when rem > den -> quo + 1
den when rem < den -> quo
_ when (quo &&& 1) === 1 -> quo + 1
_ -> quo
end
sign = if sign == -1, do: 1, else: 0
tmp = tmp - @power_of_2_to_52
exp = if tmp < @power_of_2_to_52, do: exp, else: exp + 1
<<tmp::float>> = <<sign::size(1), exp + 1023::size(11), tmp::size(52)>>
tmp
end
@doc """
Returns `true` when the given `decimal` has no significant digits after the decimal point.
## Examples
iex> Decimal.integer?("1.00")
true
iex> Decimal.integer?("1.10")
false
"""
doc_since("2.0.0")
@spec integer?(decimal()) :: boolean
def integer?(%Decimal{coef: :NaN}), do: false
def integer?(%Decimal{coef: :inf}), do: false
def integer?(%Decimal{coef: coef, exp: exp}), do: exp >= 0 or zero_after_dot?(coef, exp)
def integer?(num), do: integer?(decimal(num))
defp zero_after_dot?(coef, exp) when coef >= 10 and exp < 0,
do: Kernel.rem(coef, 10) == 0 and zero_after_dot?(Kernel.div(coef, 10), exp + 1)
defp zero_after_dot?(coef, exp),
do: coef == 0 or exp == 0
## ARITHMETIC ##
defp add_align(coef1, exp1, coef2, exp2) when exp1 == exp2, do: {coef1, coef2}
defp add_align(coef1, exp1, coef2, exp2) when exp1 > exp2,
do: {coef1 * pow10(exp1 - exp2), coef2}
defp add_align(coef1, exp1, coef2, exp2) when exp1 < exp2,
do: {coef1, coef2 * pow10(exp2 - exp1)}
defp add_sign(sign1, sign2, coef) do
cond do
coef > 0 -> 1
coef < 0 -> -1
sign1 == -1 and sign2 == -1 -> -1
sign1 != sign2 and Context.get().rounding == :floor -> -1
true -> 1
end
end
defp div_adjust(coef1, coef2, adjust) when coef1 < coef2,
do: div_adjust(coef1 * 10, coef2, adjust + 1)
defp div_adjust(coef1, coef2, adjust) when coef1 >= coef2 * 10,
do: div_adjust(coef1, coef2 * 10, adjust - 1)
defp div_adjust(coef1, coef2, adjust), do: {coef1, coef2, adjust}
defp div_calc(coef1, coef2, coef, adjust, prec10) do
cond do
coef1 >= coef2 ->
div_calc(coef1 - coef2, coef2, coef + 1, adjust, prec10)
coef1 == 0 and adjust >= 0 ->
{coef, adjust, coef1, []}
coef >= prec10 ->
signals = [:rounded]
signals = if base10?(coef1), do: signals, else: [:inexact | signals]
{coef, adjust, coef1, signals}
true ->
div_calc(coef1 * 10, coef2, coef * 10, adjust + 1, prec10)
end
end
defp div_int_calc(coef1, coef2, coef, adjust, precision) do
cond do
coef1 >= coef2 ->
div_int_calc(coef1 - coef2, coef2, coef + 1, adjust, precision)
adjust != precision ->
div_int_calc(coef1 * 10, coef2, coef * 10, adjust + 1, precision)
true ->
{coef, coef1}
end
end
defp integer_division(div_sign, coef1, exp1, coef2, exp2) do
precision = exp1 - exp2
{coef1, coef2, adjust} = div_adjust(coef1, coef2, 0)
{coef, _rem} = div_int_calc(coef1, coef2, 0, adjust, precision)
prec10 = pow10(Context.get().precision)
if coef > prec10 do
{
:error,
:invalid_operation,
"integer division impossible, quotient too large",
%Decimal{coef: :NaN}
}
else
{:ok, %Decimal{sign: div_sign, coef: coef, exp: 0}}
end
end
defp do_normalize(coef, exp) do
if Kernel.rem(coef, 10) == 0 do
do_normalize(Kernel.div(coef, 10), exp + 1)
else
%Decimal{coef: coef, exp: exp}
end
end
defp ratio(coef, exp) when exp >= 0, do: {coef * pow10(exp), 1}
defp ratio(coef, exp) when exp < 0, do: {coef, pow10(-exp)}
pow10_max =
Enum.reduce(0..104, 1, fn int, acc ->
defp pow10(unquote(int)), do: unquote(acc)
defp base10?(unquote(acc)), do: true
acc * 10
end)
defp pow10(num) when num > 104, do: pow10(104) * pow10(num - 104)
defp base10?(num) when num >= unquote(pow10_max) do
if Kernel.rem(num, unquote(pow10_max)) == 0 do
base10?(Kernel.div(num, unquote(pow10_max)))
else
false
end
end
defp base10?(_num), do: false
## ROUNDING ##
defp do_round(sign, digits, exp, target_exp, rounding) do
num_digits = length(digits)
precision = num_digits - (target_exp - exp)
cond do
exp == target_exp ->
%Decimal{sign: sign, coef: digits_to_integer(digits), exp: exp}
exp < target_exp and precision < 0 ->
zeros = :lists.duplicate(target_exp - exp, ?0)
digits = zeros ++ digits
{signif, remain} = :lists.split(1, digits)
signif =
if increment?(rounding, sign, signif, remain),
do: digits_increment(signif),
else: signif
coef = digits_to_integer(signif)
%Decimal{sign: sign, coef: coef, exp: target_exp}
exp < target_exp and precision >= 0 ->
{signif, remain} = :lists.split(precision, digits)
signif =
if increment?(rounding, sign, signif, remain),
do: digits_increment(signif),
else: signif
coef = digits_to_integer(signif)
%Decimal{sign: sign, coef: coef, exp: target_exp}
exp > target_exp ->
digits = digits ++ Enum.map(1..(exp - target_exp), fn _ -> ?0 end)
coef = digits_to_integer(digits)
%Decimal{sign: sign, coef: coef, exp: target_exp}
end
end
defp digits_to_integer([]), do: 0
defp digits_to_integer(digits), do: :erlang.list_to_integer(digits)
defp precision(%Decimal{coef: :NaN} = num, _precision, _rounding) do
{num, []}
end
defp precision(%Decimal{coef: :inf} = num, _precision, _rounding) do
{num, []}
end
defp precision(%Decimal{sign: sign, coef: coef, exp: exp} = num, precision, rounding) do
digits = :erlang.integer_to_list(coef)
num_digits = length(digits)
if num_digits > precision do
do_precision(sign, digits, num_digits, exp, precision, rounding)
else
{num, []}
end
end
defp do_precision(sign, digits, num_digits, exp, precision, rounding) do
precision = Kernel.min(num_digits, precision)
{signif, remain} = :lists.split(precision, digits)
signif =
if increment?(rounding, sign, signif, remain), do: digits_increment(signif), else: signif
signals = if any_nonzero(remain), do: [:inexact, :rounded], else: [:rounded]
exp = exp + length(remain)
coef = digits_to_integer(signif)
dec = %Decimal{sign: sign, coef: coef, exp: exp}
{dec, signals}
end
defp increment?(_, _, _, []), do: false
defp increment?(:down, _, _, _), do: false
defp increment?(:up, _, _, _), do: true
defp increment?(:ceiling, sign, _, remain), do: sign == 1 and any_nonzero(remain)
defp increment?(:floor, sign, _, remain), do: sign == -1 and any_nonzero(remain)
defp increment?(:half_up, _, _, [digit | _]), do: digit >= ?5
defp increment?(:half_even, _, [], [?5 | rest]), do: any_nonzero(rest)
defp increment?(:half_even, _, signif, [?5 | rest]),
do: any_nonzero(rest) or Kernel.rem(:lists.last(signif), 2) == 1
defp increment?(:half_even, _, _, [digit | _]), do: digit > ?5
defp increment?(:half_down, _, _, [digit | rest]),
do: digit > ?5 or (digit == ?5 and any_nonzero(rest))
defp any_nonzero(digits), do: :lists.any(fn digit -> digit != ?0 end, digits)
defp digits_increment(digits), do: digits_increment(:lists.reverse(digits), [])
defp digits_increment([?9 | rest], acc), do: digits_increment(rest, [?0 | acc])
defp digits_increment([head | rest], acc), do: :lists.reverse(rest, [head + 1 | acc])
defp digits_increment([], acc), do: [?1 | acc]
## CONTEXT ##
defp context(num, signals \\ []) do
context = Context.get()
{result, prec_signals} = precision(num, context.precision, context.rounding)
error(put_uniq(signals, prec_signals), nil, result, context)
end
defp put_uniq(list, elems) when is_list(elems) do
Enum.reduce(elems, list, &put_uniq(&2, &1))
end
defp put_uniq(list, elem) do
if elem in list, do: list, else: [elem | list]
end
## PARSING ##
defp parse_unsign(<<first, remainder::size(7)-binary, rest::binary>>) when first in [?i, ?I] do
if String.downcase(remainder) == "nfinity" do
{%Decimal{coef: :inf}, rest}
else
:error
end
end
defp parse_unsign(<<first, remainder::size(2)-binary, rest::binary>>) when first in [?i, ?I] do
if String.downcase(remainder) == "nf" do
{%Decimal{coef: :inf}, rest}
else
:error
end
end
defp parse_unsign(<<first, remainder::size(2)-binary, rest::binary>>) when first in [?n, ?N] do
if String.downcase(remainder) == "an" do
{%Decimal{coef: :NaN}, rest}
else
:error
end
end
defp parse_unsign(bin) do
{int, rest} = parse_digits(bin)
{float, rest} = parse_float(rest)
{exp, rest} = parse_exp(rest)
if int == [] and float == [] do
:error
else
int = if int == [], do: ~c"0", else: int
exp = if exp == [], do: ~c"0", else: exp
number = %Decimal{
coef: List.to_integer(int ++ float),
exp: List.to_integer(exp) - length(float)
}
{number, rest}
end
end
defp parse_float("." <> rest), do: parse_digits(rest)
defp parse_float(bin), do: {[], bin}
defp parse_exp(<<e, rest::binary>>) when e in [?e, ?E] do
case rest do
<<sign, rest::binary>> when sign in [?+, ?-] ->
{digits, rest} = parse_digits(rest)
{[sign | digits], rest}
_ ->
parse_digits(rest)
end
end
defp parse_exp(bin) do
{[], bin}
end
defp parse_digits(bin), do: parse_digits(bin, [])
defp parse_digits(<<digit, rest::binary>>, acc) when digit in ?0..?9 do
parse_digits(rest, [digit | acc])
end
defp parse_digits(rest, acc) do
{:lists.reverse(acc), rest}
end
# Util
defp decimal(%Decimal{} = num), do: num
defp decimal(num) when is_integer(num), do: new(num)
defp decimal(num) when is_binary(num), do: new(num)
defp decimal(other) when is_float(other) do
raise ArgumentError,
"implicit conversion of #{inspect(other)} to Decimal is not allowed. Use Decimal.from_float/1"
end
defp handle_error(signals, reason, result, context) do
context = context || Context.get()
signals = List.wrap(signals)
flags = Enum.reduce(signals, context.flags, &put_uniq(&2, &1))
Context.set(%{context | flags: flags})
error_signal = Enum.find(signals, &(&1 in context.traps))
if error_signal do
error = [signal: error_signal, reason: reason]
{:error, error}
else
{:ok, result}
end
end
defp fix_float_exp(digits) do
fix_float_exp(digits, [])
end
defp fix_float_exp([?e | rest], [?0 | [?. | result]]) do
fix_float_exp(rest, [?e | result])
end
defp fix_float_exp([digit | rest], result) do
fix_float_exp(rest, [digit | result])
end
defp fix_float_exp([], result), do: :lists.reverse(result)
if Version.compare(System.version(), "1.3.0") == :lt do
defp integer_to_charlist(string), do: Integer.to_char_list(string)
else
defp integer_to_charlist(string), do: Integer.to_charlist(string)
end
end
defimpl Inspect, for: Decimal do
def inspect(dec, _opts) do
"Decimal.new(\"" <> Decimal.to_string(dec) <> "\")"
end
end
defimpl String.Chars, for: Decimal do
def to_string(dec) do
Decimal.to_string(dec)
end
end
