This library implements fixed-point number types. A fixed-point number represents a fractional, or non-integral, number. In contrast with the more widely known floating-point numbers, with fixed-point numbers the decimal point doesn't "float": fixed-point numbers are effectively integers that are interpreted as being scaled by a constant factor. Consequently, they have a fixed number of digits (bits) after the decimal (radix) point.
Fixed-point numbers can be used to perform arithmetic. Another practical application is to implicitly rescale integers without modifying the underlying representation.
This library exports two categories of fixed-point types. Fixed-point types are used like any other number: they can be added, multiplied, raised to a power, etc. In some cases these operations result in conversion to floating-point types.
This library defines an abstract type FixedPoint{T <: Integer, f}
as a
subtype of Real
. The parameter T
is the underlying machine representation and f
is the number of fraction bits.
For T<:Signed
(a signed integer), there is a fixed-point type
Fixed{T, f}
; for T<:Unsigned
(an unsigned integer), there is the
Normed{T, f}
type. However, there are slight differences in behavior
that go beyond signed/unsigned distinctions.
The Fixed{T,f}
types use 1 bit for sign, and f
bits to represent
the fraction. For example, Fixed{Int8,7}
uses 7 bits (all bits
except the sign bit) for the fractional part. The value of the number
is interpreted as if the integer representation has been divided by
2^f
. Consequently, Fixed{Int8,7}
numbers x
satisfy
-1.0 = -128/128 ≤ x ≤ 127/128 ≈ 0.992.
because the range of Int8
is from -128 to 127.
In contrast, the Normed{T,f}
, with f
fraction bits, map the closed
interval [0.0,1.0] to the span of numbers with f
bits. For example,
the N0f8
type (aliased to Normed{UInt8,8}
) is represented
internally by a UInt8
, and makes 0x00
equivalent to 0.0
and
0xff
to 1.0
. Consequently, Normed
numbers are scaled by 2^f-1
rather than 2^f
. The type aliases N6f10
, N4f12
,
N2f14
, and N0f16
are all based on UInt16
and reach the
value 1.0
at 10, 12, 14, and 16 bits, respectively (0x03ff
,
0x0fff
, 0x3fff
, and 0xffff
). The NXfY
notation is used for
compact printing and the fY
component informs about the number of
fractional bits and X+Y
equals the number of underlying bits used.
To construct such a number, use convert(N4f12, 1.3)
, N4f12(1.3)
,
Normed{UInt16,12}(1.3)
, or reinterpret(N4f12, 0x14cc)
.
The latter syntax means to construct a N4f12
(it ends in
uf12
) from the UInt16
value 0x14cc
.
More generally, an arbitrary number of bits from any of the standard unsigned
integer widths can be used for the fractional part. For example:
Normed{UInt32,16}
, Normed{UInt64,3}
, Normed{UInt128,7}
.