### Description

The `UFix` and `SFix` types correspond to a vector of bits that can be used for fixed point arithmetic.

### Declaration

The syntax to declare a fixed point number is as follows:

#### Unsigned Fixed Point

Syntax bit width resolution max min
UFix(peak:ExpNumber, resolution:ExpNumber) peak-resolution 2^resolution 2^peak-2^resolution 0
UFix(peak:ExpNumber, width:BitCount) width 2^(peak-width) 2^peak-2^(peak-width) 0

#### Signed Fixed Point

Syntax bit width resolution max min
SFix(peak:ExpNumber, resolution:ExpNumber) peak-resolution 2^resolution 2^peak-2^resolution 0
SFix(peak:ExpNumber, width:BitCount) width 2^(peak-width) 2^peak-2^(peak-width) 0

#### Format

The chosen format follows the usual way of defining fixed point number format using Q notation. More information there.

For example Q8.2 will mean an fixed point of 8+2 bits, where 8 bit are used for the natural part and 2 bits for the fractional part. If the fixed point number is signed, one more bit is used for the sign.

The resolution is defined as being the smallest power of two that can be represented in this number.

#### Examples

``````// Unsigned Fixed Point
val UQ_8_2 = UFix(peak = 8 exp,resolution = -2 exp)
val UQ_8_2 = UFix(8 exp, -2 exp)

val UQ_8_2 = UFix(peak = 8 exp,width = 10 bits)
val UQ_8_2 = UFix(8 exp, 10 bits)

// Signed Fixed Point
val Q_8_2 = SFix(peak = 8 exp,resolution = -2 exp)
val Q_8_2 = SFix(8 exp, -2 exp)

val Q_8_2 = SFix(peak = 8 exp,width = 11 bits)
val Q_8_2 = SFix(8 exp, 11 bits)
``````

### Assignments

#### Valid Assignments

An assignment to a fixed point value is valid when there is no bit loss. A bit loss occurrence will result in an error.

If the source fixed point value is too big, the .truncated function will allow you to resize the source number to match the destination size.

##### Example
``````val i16_m2 = SFix(16 exp,-2 exp)
val i16_0  = SFix(16 exp, 0 exp)
val i8_m2  = SFix( 8 exp,-2 exp)
val o16_m2 = SFix(16 exp,-2 exp)
val o16_m0 = SFix(16 exp, 0 exp)
val o14_m2 = SFix(14 exp,-2 exp)

o16_m2 := i16_m2            // OK
o16_m0 := i16_m2            // Not OK, Bit loss
o14_m2 := i16_m2            // Not OK, Bit loss
o16_m0 := i16_m2.truncated  // OK, as it is resized
o14_m2 := i16_m2.truncated  // OK, as it is resized
``````

#### From a Scala constant

It is allowed to assign a Scala Double or a Scala BigInt into a fixed point.

##### Example
``````val i4_m2 = SFix(4 exp,-2 exp)
i4_m2 := 1.25    //Will load 5 in i4_m2.raw
i4_m2 := 4       //Will load 16 in i4_m2.raw
``````

### Raw value

The integer representation of the fixed point number can be read or written using the `raw` property.

#### Example

``````val UQ_8_2 = UFix(8 exp, 10 bits)
UQ_8_2.raw := 4        //Assign the value corresponding to 1.0
UQ_8_2.raw := U(17)    //Assign the value corresponding to 4.25
``````

### Operators

The following operators are available for the `UFix` type

#### Arithmetic

Operator Description Returned resolution Returned amplitude
x + y Addition Min(x.resolution, y.resolution) Max(x.amplitude, y.amplitude)
x - y Subtraction Min(x.resolution, y.resolution) Max(x.amplitude, y.amplitude)
x * y Multiplication x.resolution * y.resolution) x.amplitude * y.amplitude
x » y Arithmetic shift right, y : Int x.amplitude » y x.resolution » y
x « y Arithmetic shift left, y : Int x.amplitude « y x.resolution « y
x »| y Arithmetic shift right, y : Int x.amplitude » y x.resolution
x «| y Arithmetic shift left, y : Int x.amplitude « y x.resolution

#### Comparison

Operator Description Return type
x === y Equality Bool
x =/= y Inequality Bool
x > y Greater than Bool
x >= y Greater than or equal Bool
x > y Less than Bool
x >= y Less than or equal Bool

#### Type cast

Operator Description Return
x.asBits Binary cast in Bits Bits(w(x) bits)
x.asUInt Binary cast in UInt UInt(w(x) bits)
x.asSInt Binary cast in SInt SInt(w(x) bits)
x.asBools Cast into a array of Bool Vec(Bool,width(x))
x.toUInt Return the corresponding UInt (with truncation) UInt
x.toSInt Return the corresponding SInt (with truncation) SInt
x.toUFix Return the corresponding UFix UFix
x.toSFix Return the corresponding SFix SFix

#### Misc

Name Return Description
x.maxValue Return the maximum value storable Double
x.minValue Return the minimum value storable Double
x.resolution x.amplitude * y.amplitude Double
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