Number Representation

Intel® C++ Compiler Classic Developer Guide and Reference

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ID 767249

Date 12/16/2022

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Number Representation

When specifying extents, positions inside of, or bounds of a container, numeric values can be represented three different ways: fixed, aligned, and int. Fixed is most precise and int is least precise. It is advised to use as precise specification as possible. The compiler may optimize better with more information.

Fixed

Represent a numerical constant whose value specified at compile time.

template <int NumberT> class fixed;

If offsets applied to index values inside a SIMD loop are known at compile time, then the compiler can use that information. For example, to maintain aligned access, if boundary is fixed and known to be aligned when accessing underlying data layout. When multiple accesses are happening near each other, the compiler will have the opportunity to detect which accesses occur in the same cache lines and potentially avoid prefetching the same cache line repeatedly. Additionally, if the start of an iteration space is known at compile time, if it's a multiple of the SIMD lane count, the compiler could skip generating a peel loop. Whenever possible, fixed values should be used over aligned or arbitrary integer values.

Although std::integral_constant<int> provides the same functionality, the library defines own type to provide overloaded operators and avoid collisions with any other code's interactions with std::integral_constant<int>.

The following table provides information about the template arguments for fixed.

Template Argument	Description
int Number T	The numerical value the fixed will represent.

Template Argument

Description

int Number T

The numerical value the fixed will represent.

The following table provides information about the members of fixed.

Member	Description
static constexpr int value = NumberT	The numerical value known at compile-time.
constexpr operator value_type() const	Returns: The numerical value
constexpr value_type operator()() const;	Returns: The numerical value

Constant expression arithmetic operators +,- (both unary and binary), * and / are defined for type sdlt::fixed<> and will be evaluated at compile-time.

The suffix _fixed is a C++11 user-defined equivalent literal. For example, 1080_fixed is equivalent to fixed<1080>. Consider the readability of the two samples below.

foo3d(fixed<1080>(), fixed<1920>());

versus

foo3d(1080_fixed, 1920_fixed);

NOTE:

The sdlt::fixed<NumberT> type supersedes the deprecated sdlt::fixed_offset<OffsetT> type found in SDLT v1. It is strongly advised to use sdlt::fixed<NumberT>. However, in this release, a template alias is provided mapping sdlt::fixed_offset<OffsetT> onto sdlt::fixed<NumberT>.

Aligned

Represent integer value known at compile time to be a multiple of an IndexAlignment.

template <int IndexAlignmentT> class aligned;

If you can tell the compiler that you know that an integer will be a multiple of known value, then, when combined with a loop index inside a SIMD loop, the compiler can use that information to maintain aligned access when accessing underlying data layout.

Internally, the integer value is converted to a block count, where:

block_count = value/IndexAlignmentT;

Overloaded math operations can then use that aligned block count as needed. The value() is represented by AlignmentT*block_count allowing the compiler to prove that the value() is a multiple of AlignmentT, which can utilize alignment optimizations.

The following table provides information about the template arguments for aligned.

Template Argument	Description
int IndexAlignmentT	The alignment the user is stating that the number is a multiple of. `IndexAlignmentT` must be a power of two.

Template Argument

Description

int IndexAlignmentT

The alignment the user is stating that the number is a multiple of. IndexAlignmentT must be a power of two.

The following table provides information about the types defined as members of aligned.

Member Type	Description
typedef int value_type	The type of the numerical value.
typedef int block_type	The type of the `block_count`.

Member Type

Description

typedef int value_type

The type of the numerical value.

typedef int block_type

The type of the block_count.

The following table provides information about the members of aligned.

Member	Description
static const int index_alignment	The `IndexAlignmentT` value.
aligned()	Constructs empty (uninitialized) object
explicit aligned(value_type)	Constructs computing `block_count=a_value/IndexAlignmentT`.
aligned(const aligned& a_other)	Constructs copying `block_count` from `a_other`. `a_other` must have same `IndexAlignmentT.`
template<int OtherAlignment> explicit aligned(const aligned& other)	Constructs computing `block_count` optimized by avoiding computing other.value(). Must have `IndexAlignmentT` of a_other < `IndexAlignmentT` and other.value() be multiple of `IndexAlignmentT`.
template<int OtherAlignment> aligned(const aligned& other)	Constructs computing `block_count` with a multiply instead of divide. Must have `IndexAlignmentT` of a_other > `IndexAlignmentT`
static aligned from_block_count(block_type block_count)	Creates an instance of `aligned` avoiding any math by directly using supplied `block_count`
value_type value() const	Computes the value represented by the aligned. Returns: `aligned_block_count()*IndexAlignmentT`
operator value_type()	Conversion to `int`. Returns: `value()`
block_type aligned_block_count() const	Conversion to `int`. Returns: The block count

The following operations are supported for the aligned type.

Operation	Description
operator *(int), commutative	Scale value. Returns: `aligned<IndexAlignmentT >`
operator *(fixed<V>), commutative	Scales `IndexAlignment` by 2^M and value by K. Must have V=2^MK (V is a multiple of a power of 2). Returns:* `aligned<IndexAlignmentT*(2^M)>`
operator *(aligned<OtherAl>)	Scales `IndexAlignment` by `OtherAl` and `block_count` by argument. Returns: `aligned<IndexAlignmentT*OtherAl>`
int operator/(fixed<IndexAlignmentT>)	Returns: `aligned_block_count()`
int operator/(fixed<-IndexAlignmentT>)	Returns: `-aligned_block_count();`
int operator/(fixed<V>)	Must have `abs(V)>IndexAlignmentT && IndexAlignmentT%V==0`. Returns: `aligned_block_count()/(V/IndexAlignmentT)`
int operator/(fixed<V>)	Must have `abs(V) < IndexAlignmentT && V%IndexAlignmentT==0` Returns: `aligned_block_count()*(IndexAlignmentT/V)`
aligned operator -()	Returns: Same type aligned for negated value.
aligned operator -(const aligned &) const	Returns: Same type aligned for value of difference.
template<int OtherAl> aligned<?> operator -(const aligned<OtherAl>&) const	Difference with other alignment. Behavior and returned alignment type depend on relation between alignments of operands. Returns: Value for difference as lower of incoming alignments
template<int V> aligned<?> operator -(const fixed<V> &) const	Difference with fixed value. Behavior and returned alignment type depend on relation between alignments of aligned<> operand and the value of V. Returns: Adjusted aligned value of a difference
aligned operator +(const aligned &)const	Returns: Same type aligned for value of sum
template<int OtherAl> aligned<?> operator +(const aligned<OtherAl>&) const	Sum with other alignment. Behavior and returned alignment type depend on relation between alignments of operands. Returns: Value for sum as lower of incoming alignments
template<int V> aligned<?> operator +(const fixed<V> &) const	Sum with fixed value. Behavior and returned alignment type depend on relation between alignments of aligned<> operand and the value of V. Returns: Adjusted aligned value of a sum.
template<int OtherAl> aligned operator +=(const aligned<OtherAl> &) const	Increments value for the aligned object if `IndexAlignmentT` is compatible with `OtherAl` Returns: Aligned with incremented value.
template<int OtherAl> aligned operator -=(const aligned<OtherAl> &) const	Decrements value for the aligned object if `IndexAlignmentT` is compatible with `OtherAl` Returns: Same type aligned with decremented value.
template<int OtherAl> aligned operator *=(const aligned<OtherAl> &) const	Multiplies value for the aligned object if `IndexAlignmentT` is compatible with `OtherAl`. Returns: Same type aligned with multiplied value.
template<int OtherAl> aligned operator /=(const aligned<OtherAl> &) const	Divides value for the aligned object if `IndexAlignmentT` is compatible with `OtherAl` Returns: Same type aligned with divided value.

NOTE:

The sdlt::aligned<> type supersedes the deprecated sdlt::aligned_offset<> type found in SDLT v1. It is strongly advised to use sdlt::aligned<>, however in this release a template alias is provided mapping sdlt::aligned_offset<> onto sdlt::aligned<>.

int

Represents an arbitrary integer value. In interfaces where fixed<> and aligned<> values supported you may also use plain old integer value. It provides least information among these three and so least facilitates compiler optimizations.

Parent topic: User-Level Interface

aligned_offset
Represent an integer based offset whose value is a multiple of an IndexAlignment specified at compile time. #include <sdlt/aligned_offset.h>
fixed_offset
Represent an integer based offset whose value specified at compile time. #include <sdlt/fixed_offset.h>

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Intel® C++ Compiler Classic Developer Guide and Reference

Number Representation

Fixed

Aligned

int