Fixed{T <: Signed, f} <: FixedPoint{T, f}

Fixed{T,f} maps Signed integers from -2^f to 2^f to the range [-1.0, 1.0]. For example, Fixed{Int8,7} maps -128 to -1.0 and 127 to 127/128 ≈ 0.992.

There are the typealiases for Fixed in the QXfY notation, where Y is the number of fractional bits (i.e. f), and X+Y+1 equals the number of underlying bits used (+1 means the sign bit). For example, Q0f7 is aliased to Fixed{Int8,7} and Q3f12 is aliased to Fixed{Int16,12}.

FixedPoint{T <: Integer, f} <: Real

Supertype of the two fixed-point number types: Fixed{T, f} and Normed{T, f}.

The parameter T is the underlying machine representation and f is the number of fraction bits.

Normed{T <: Unsigned, f} <: FixedPoint{T, f}

Normed{T,f} maps Unsigned integers from 0 to 2^f-1 to the range [0.0, 1.0]. For example, Normed{UInt8,8} maps 0x00 to 0.0 and 0xff to 1.0.

There are the typealiases for Normed in the NXfY notation, where Y is the number of fractional bits (i.e. f), and X+Y equals the number of underlying bits used. For example, N0f8 is aliased to Normed{UInt8,8} and N4f12 is aliased to Normed{UInt16,12}.

isapprox(x::FixedPoint, y::FixedPoint; rtol=0, atol=max(eps(x), eps(y)))

For FixedPoint numbers, the default criterion is that x and y differ by no more than eps, the separation between adjacent fixed-point numbers.

floattype(::Type{T})::Type{<:AbstractFloat}

Return a minimal type suitable for performing computations with instances of type T without integer overflow.

The fallback definition of floattype(T) applies only to T<:AbstractFloat. However, it is permissible to extend floattype to return types that are not subtypes of AbstractFloat; the key characteristic is that the return type should support computation without integer overflow.

In general the returned type should have the minimum bitwidth needed to encode the full precision of the input type. however, a priority should be placed on computational efficiency; consequently, types like Float16 should be avoided except in scenarios where they are guaranteed to have hardware support.

Examples

A classic usage is to avoid overflow behavior by promoting FixedPoint to AbstractFloat

julia> x = N0f8(1.0)
1.0N0f8

julia> x + x # overflow
0.996N0f8

julia> T = floattype(x)
Float32

julia> T(x) + T(x)
2.0f0

The following represents a valid extension of floattype to non-AbstractFloats:

julia> using FixedPointNumbers, ColorTypes

julia> floattype(RGB{N0f8})
RGB{Float32}

RGB itself is not a subtype of AbstractFloat, but unlike RGB{N0f8} operations with RGB{Float32} are not subject to integer overflow.

sd, ad = scaledual(s::Number, a)

Return sd and ad such that sd * ad == s * a. When a is an array of FixedPoint numbers, sd*ad might be faster to compute than s*a.