# Decimal

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 and (+-)Infinity. -0 and +0 are two distinct values. Some operations 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). The other kind of NaN is signalling which is the value that can be reached in `Error.result/1` when the result is NaN. Any operation given a signalling NaN return will signal `:invalid_operation`.

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

This implementation follows the above standards as closely as possible. But at some places the implementation diverges from the specification. The reasons are different for each case but may be that the specification doesn’t map to this environment, ease of implementation or that API will be nicer. Still, the implementation is close enough that the specifications can be seen as additional documentation that can be used when things are unclear.

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 “underflow” 0 is returned instead of Etiny. This may happen when dividing a number with infinity. Additionally, overflow, underflow and clamped may never be signalled.

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## Summary↑

 abs(num) The absolute value of given number. Sets the number’s sign to positive add(num1, decimal) Adds two numbers together compare(num1, num2) Compares two numbers numerically. If the first number is greater than the second `#Decimal<1>` is returned, if less than `Decimal<-1>` is returned. Otherwise, if both numbers are equal `Decimal<0>` is returned decimal?(decimal) Returns `true` if argument is a decimal number; otherwise `false` div(num1, decimal) Divides two numbers div_int(num1, num2) Divides two numbers and returns the integer part div_rem(num1, decimal) 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)}` equal?(num1, num2) Compares two numbers numerically and returns `true` if they are equal, otherwise `false` get_context() Gets the process’ context inf?(decimal) Returns `true` if number is (+-)Infinity; otherwise `false` max(decimal, num2) 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 min(decimal, num2) 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 minus(num) Negates the given number mult(num1, decimal) Multiplies two numbers nan?(decimal) Returns `true` if number is NaN; otherwise `false` new(num) Creates a new decimal number from a string representation, an integer or a floating point number. Floating point numbers will be converted to decimal numbers with `:io_lib_format.fwrite_g/1`, since this conversion is not exact it is recommended to give an integer or a string when possible new(sign, coefficient, exponent) Creates a new decimal number from the sign, coefficient and exponent such that the number will be: `sign * coefficient * 10^exponent` plus(num) Applies the context to the given number rounding it to specified precision reduce(num) Reduces the given number. Removes trailing zeros from coefficient while keeping the number numerically equivalent by increasing the exponent rem(num1, num2) Remainder of integer division of two numbers. The result will have the sign of the first number round(num, places \\ 0, mode \\ :half_up) 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 set_context(context) Set the process’ context sub(num1, num2) Subtracts second number from the first. Equivalent to `Decimal.add/2` when the second number’s sign is negated to_string(num, type \\ :scientific) Converts given number to its string representation update_context(fun) Update the process’ context with_context(context, fun) Runs function with given context

## Types ↑

signal :: :invalid_operation | :division_by_zero | :rounded | :inexact

rounding :: :down | :half_up | :half_even | :ceiling | :floor | :half_down | :up

t :: %Decimal{sign: 1 | -1, coef: non_neg_integer | :qNaN | :sNaN | :inf, exp: integer}

## Functions

abs(num)

Specs:

• abs(t) :: t

The absolute value of given number. Sets the number’s sign to positive.

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Specs:

## Exceptional conditions

• If one number is -Infinity and the other +Infinity `:invalid_operation` will be signalled.
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compare(num1, num2)

Specs:

• compare(t, t) :: t

Compares two numbers numerically. If the first number is greater than the second `#Decimal<1>` is returned, if less than `Decimal<-1>` is returned. Otherwise, if both numbers are equal `Decimal<0>` is returned.

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decimal?(decimal)

Specs:

• decimal?(any) :: boolean

Returns `true` if argument is a decimal number; otherwise `false`.

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div(num1, decimal)

Specs:

• div(t, t) :: t

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.
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div_int(num1, num2)

Specs:

• div_int(t, t) :: t

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.
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div_rem(num1, decimal)

Specs:

• div_rem(t, t) :: {t, t}

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.
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equal?(num1, num2)

Specs:

• equal?(t, t) :: boolean

Compares two numbers numerically and returns `true` if they are equal, otherwise `false`.

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get_context()

Specs:

Gets the process’ context.

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inf?(decimal)

Specs:

• inf?(t) :: boolean

Returns `true` if number is (+-)Infinity; otherwise `false`.

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max(decimal, num2)

Specs:

• max(t, t) :: t

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.

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min(decimal, num2)

Specs:

• min(t, t) :: t

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.

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minus(num)

Specs:

• minus(t) :: t

Negates the given number.

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mult(num1, decimal)

Specs:

• mult(t, t) :: t

Multiplies two numbers.

## Exceptional conditions

• If one number is (+-0) and the other is (+-)Infinity `:invalid_operation` is signalled.
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nan?(decimal)

Specs:

• nan?(t) :: boolean

Returns `true` if number is NaN; otherwise `false`.

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new(num)

Specs:

Creates a new decimal number from a string representation, an integer or a floating point number. Floating point numbers will be converted to decimal numbers with `:io_lib_format.fwrite_g/1`, since this conversion is not exact it is recommended to give an integer or a string when possible.

A decimal number will always be created exactly as specified with all digits kept - it will not be rounded with the context.

## BNFC

``````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] | ’sNaN’ [digits]
numeric-value  ::=  decimal-part [exponent-part] | infinity
numeric-string ::=  [sign] numeric-value | [sign] nan``````
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new(sign, coefficient, exponent)

Specs:

• new(1 | -1, non_neg_integer | :qNaN | :sNaN | :inf, integer) :: t

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.

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plus(num)

Specs:

• plus(t) :: t

Applies the context to the given number rounding it to specified precision.

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reduce(num)

Specs:

• reduce(t) :: t

Reduces the given number. Removes trailing zeros from coefficient while keeping the number numerically equivalent by increasing the exponent.

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rem(num1, num2)

Specs:

• rem(t, t) :: t

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.
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round(num, places \\ 0, mode \\ :half_up)

Specs:

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.

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set_context(context)

Specs:

Set the process’ context.

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sub(num1, num2)

Specs:

• sub(t, t) :: t

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.
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to_string(num, type \\ :scientific)

Specs:

• to_string(t, :scientific | :normal | :raw) :: String.t

Converts given number to its string representation.

## Options

• `:scientific` - Number converted to scientific notation.
• `:normal` - Number converted without a exponent.
• `:raw` - Number converted to it’s raw, internal format.
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update_context(fun)

Specs:

Update the process’ context.

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with_context(context, fun)

Specs:

Runs function with given context.

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