Document: | ISO/IEC JTC1 SC22 WG21 N3407=12–0097 | |||

Date: | 2012–09–14 | |||

Revises: | ||||

Project: | Programming Language C++ | |||

Addresses: | Library and Evolution Working Groups | |||

Reply To: | Dietmar Kühl, dkuhl@bloomberg.net | |||

Bloomberg L.P. 39–45 Finsbury Square London, EC2A 1PQ, United Kingdom |

This document proposes to add decimal floating point support to the C++ standard. The current version doesn’t spell out the details but instead refers to the Decimal TR (ISO/IEC TR 24733) as a basis and describes changes to be applied to this interface to bring the proposal up to date with C++ 2011 enhancements.

C++ provides built-in data types for the processing of numerical
values: `float`

, `double`

, and `long double`

. The constraints for these
types imply that a *floating point* representation is used, i.e., the
values are represented using a fixed size as
^{sign} * significand * base^{exponent}

In many areas, especially in finance, exact values need to be processed
and the inputs are commonly decimal. Unfortunately, decimal values
cannot, in general, be represented accurately using binary floating
points even when the decimal values only uses a few digits. Instead,
the values become an approximation. As long as the values are carefully
processed the original decimal value can be restored from a binary
floating point (assuming reasonable restrictions on the number of
decimal digits). However, computations and certain conversions
introduce subtle errors (e.g. `double`

to `float`

and back to `double`

,
even if `float`

is big enough to restore the original decimal value).
As a result, the processing of exact decimal values using binary
floating points is very error prone.

The use of decimal floating points avoids many of the problems caused by binary floating points. In particular, computations which need to accurately process decimal numbers can use decimal floating points. Decimal floating points provide a useful and sufficient compromise for these domains. Since they use a fixed size representation computations which are normally exact can introduce inaccuracies when the number of necessary digits becomes too big but for actual applications this is rarely a problem. Also, decimal floating points cannot represent the result of all operations exactly. For example, the result of a division with a prime other than 2 and 5 will, in general, be rounded. In the contexts where exact results are needed the corresponding operations aren’t needed.

The need for support of exact decimal computations is recognized in many communities and supported in several systems, although different alternatives for the support are chosen. Below is a list of example programming languages with decimal support:

- The C committee is working on a Decimal TR as
TR 24732.
The decimal support in C uses built-in types
`_Decimal32`

,`_Decimal64`

, and`_Decimal128`

. - Java provides decimal arithmetic by
`java.math.BigDecimal`

, an arbitrary sized integer with an integer scale for the decimal places. - Python provides
`decimal.Decimal`

which is a fixed point decimal representation. The number of decimal digits can be set globally. - .Net provides
`System.Decimal`

which is a 128 bit decimal floating point. The details of this represntation are slightly different from the 128 bit decimal floating point in IEEE 754–2008.`System.Decimal`

is accessible in C# as`decimal`

. - SQL provides a fixed point decimal representation where the number of digits and the number of fractional digits can be chosen for each context.
- Ruby provides
`BigDecimal`

, an arbitrary sized integer with an integer scale for the decimal places.

Since C++ is used in many places where accurate decimal arithmetic is required it seems reasonable to add similar support to the standard C++ library.

This document proposes to add the interfaces described by the Decimal TR, augmented to take advantage of C++ 2011 features as outlined below, as a mandatory part of the next revision of C++.

The Decimal TR was issued in 2009 and, thus, in 2014 a statement needs to be made whether it is to be affirmed, revised, or withdrawn. Since the next revision of C++ is scheduled to be released in 2016 it is also proposed that the Decimal TR is revised to reflect the changes outlined below for the 2014 systematic review. Assuming decimal floating point support is added to C++ 2016 the technical report can be withdrawn for the 2019 systematic review.

Adding anything to the standard C++ library isn’t free and any component may depend on specific infrastructure to be present to be implementable. This section discusses the involved costs and requirements.

The support for decimal floating point numbers described does not require specific hardware support. There are several software implementations of decimal floating points (Intel, IBM, HP) with suitable performance. It isn’t expected that decimal floating points are used for heavy number crunching because in these contexts the corresponding results will not be exact decimal values in the first place. Thus, the performance expectations for decimal floating points are different than those for floating points used for for number crunching.

When processing decimal values using binary floating points it is necessary to convert between fractional decimal values to the closest fractional binary value. These conversions are relatively expensive and are avoided when the processing is done with decimal floating points. Since the operations on binary floating points in general yield inaccurate decimal values the hardware support for binary floating point isn’t of much help when trying to process decimal values. Thus, performance comparisons between decimal floating points and binary floating points are misguided because they address different problems.

That said, dedicated hardware can improve the performance of operations using decimal floating points and the specification is written such that potentially available hardware support can be used for an implementation. For example, IBM provides a library detecting the presence of hardware support and which only uses a software implementation for decimal floating points where no hardware support is available (see the section on DFPAL; the decimal support in gcc is based on libdfp which also chooses between a hardware and a software implementation depending on availability of the hardware but which implements the interface of TR 24732).

The operations on decimal floating points are relatively complex. To yield predictable results for portable programs it is necessary to specify the details of rounding, retained precision, dealing with boundary conditions, etc. However, all of these details are already addressed by IEEE 754–2008. The specification in the C++ standard will have exactly the same semantics by referencing IEEE 754–2008 for the semantics. What needs to be specified are the interfaces to access the various features of IEEE 754–2008 in a natural way from C++.

The Decimal TR already spells out most aspects of a C++ binding. With the added C++ 2011 features it is possible to create a better user experience. There are some design areas open with respect to adding C++ 2011 support (see the section on Changes to the Decimal TR below). Thus, the overall cost of specification should be acceptable.

The implementation of decimal floating point support is certainly not trivial. However, it is also not as complex as, e.g., the implementation of the special math functions. Several independent implementations are available for different platforms, including open source versions. The libraries mentioned below are all using a C interface which can be used to implement the C++ support.

- Intel’s library is distributed as source.
- IBM’s implementation is distributed as source with multiple open source projects (gcc and ICU).
- HP provides support for decimal floating points with their C and C++ compilers.

Implementing the interfaces specified by the Decimal TR in terms of the C implementations is relatively straight forward. It also seems reasonable that a native C++ implementation can be provided with a reasonable amount of work.

IBM provides a set of language independent test cases for the decimal floating point semantics on the General Decimal Arithmetic page. These can be processed by a C++ program to yield a reasonable basis for testing. A comprehensive testsuite for the decimal floating point semantics is probably more involved but such testsuites can be shared with other languages also requiring support for decimal floating point support, e.g., C, ECMA Script, etc. Testing the various C++ interfaces, i.e., the languge specific parts which can’t be shared, shouldn’t be more involved than other C++ libraries.

The semantics of decimal floating points is very similar in spirit
to the semantics of binary floating points. The primary difference
is that the base is decimal rather than binary. The major difference
between binary and decimal floating points is that the latter are
not normalized, i.e., individual decimal values may have multiple
representations (a group of different representations for the same
value is refered to as *cohort* by
IEEE 754–2008). The freedom can
be used to keep track of the precision of values and needs to be
maintained during rounding. However, the overall complexity of the
decimal floating point semantics are on a similar level as those of
binary floating points. They are not dramatically more complex as is
the case, e.g., with the
special math functions.
Staff capable of providing support for use of binary floating points
will be able to also provide support for decimal floating points. To
some extent, using and providing support for decimal floating points is
easier than for binary floating points because all issues relating to
base conversions disappear.

The Decimal TR was targetting C++ 2003 and, thus, didn’t use any of the new C++ 2011 features. Several of the new features help in creating a better user experience and the specification in the Decimal TR needs to be updated to take these into account. This section describes the changes proposed to the Decimal TR.

C++ 2003 didn’t have any concept of standard-layout types and it was
impossible to make declared default constructors
trivial to take advantage of POD
types. In C++ 2011 the restrictions on types which can be treated
special are relaxed and *standard-layout types* are defined which
support types with private non-static data members. Standard-layout
types are, e.g., needed when communicating with other language. Thus,
all decimal types will be required to be standard-layout types.

To make a decimal type a POD type it needs to be a standard-layout type and a trivial class. Since there are several non-trivial constructors in each of the decimal types it is necessary to declare the default constructor. To keep the class trivial the default constructors need to be defaulted on the first declaration. The corresponding declarations will be changed to become

```
decimal32() = default;
decimal64() = default;
decimal128() = default;
```

The Decimal TR. couldn’t make the decimal types trivial because there was no way for C++ 2003 to make an user-declared default constructor trivial. Instead, the Decimal TR defined the default constructor to initialize the decimal floating point with a zero value. For decimal floating points there is a large cohort of zero values and whichever zero is chosen is unlikely to be the right one in practice. Thus, using an explicitly defaulted default constructor is a semantic change possibly resulting in non-initialized decimal floating points but the advantages of this change seem to outweight the disadvantages.

The decimal floating-point types all have a conversion operator to
`long long`

obtaining the value truncated towards zero. This conversion
yields an unspecified result when the integral part cannot be
represented by `long long`

or if the decimal type represents one of the
special values. Note, that the
Decimal TR
used the type `long long`

although it was introduced only with C++ 2011.
This was done to avoid compatibility issues between an implementation based on the TR
and an implementation augmented to use new C++ 2011 features.

Although the conversion is sometimes useful it shouldn’t be implicit,
i.e., these conversion operators will be made `explicit`

:

```
explicit operator long long() const;
```

The behavior of these conversion operators will remain unchanged.
Making the conversion explicit introduces an inconsistency with the
existing floating point types `float`

, `double`

, and `long double`

:
These can be converted implicitly to integer types. Since the implicit
conversions from floating point types to integers frequently introduce
surprises it seems to be reasonable to make the conversion explicit for
newly introduced types.

Section 4.2 (Conversions) of the Decimal TR describes how decimal floating-point types can be converted to basic floating types using a cast in C. Since implicit conversion between decimal floating-point types and basic floating types can easily create problems corresponding conversions are not available in the Decimal TR. With the possibility of disabling implicit conversions corresponding explicit conversions should be added:

```
explicit operator float() const;
```

**Returns**: If `std::numeric_limits<float>::is_iec559 == true`

, returns
the result of the conversion of `*this`

to `float`

, performed as in
IEEE 754–2008. Otherwise, the
returned value is implementation-defined.

```
explicit operator double() const;
```

**Returns**: If `std::numeric_limits<double>::is_iec559 == true`

, returns
the result of the conversion of `*this`

to `double`

, performed as in IEEE
754–2008. Otherwise, the returned
value is implementation-defined.

```
explicit operator long double() const;
```

**Returns**: If `std::numeric_limits<long double>::is_iec559 == true`

,
returns the result of the conversion of `*this`

to `long double`

,
performed as in IEEE 754–2008.
Otherwise, the returned value is implementation-defined.

Whether the various `decimal*_to_*()`

conversion functions used by the
current
Decimal TR.
are retained needs to be decided. In some contexts it may be
preferrable to use named functions. For example, the conversion
operators are not necessarily suitable to be used as function objects.
On the other hand, it is easy to create corresponding function objects
using the explicit conversions.

`final`

Although the decimal floating-point types are described as a library
feature, some restrictions are imposed on them to allow implementing
these types as built-in types. In particular, Section 2 (Conventions)
states that the result of deriving from the decimal floating-point
types is undefined. Instead of making this behavior undefined, all of
the decimal floating-point types should be made `final`

to prevent
deriving:

```
class decimal32 final { ... };
class decimal64 final { ... };
class decimal128 final { ... };
```

Use of other operations capable of detecting if the type is implemented as a class or is a built-in type will remain undefined.

Many of the operations on decimal floating-point types have wide
contracts and, thus, cannot throw any exception. Where appropriate the
corresponding operations should be declared to be `noexcept(true)`

.

Note that the
Decimal TR
refers to “raising floating-point exceptions”. This doesn’t
necessarily throw a C++ exception but may just setup an indication that
a specific condition occurred. However, an implementation may choose
to implement a mode of operation where C++ exceptions are thrown as a
result of raising certain floating-point exceptions. Thus, the use of
`noexcept(true)`

probably won’t apply to many operations.

C++ 2011 added the ability to create `constexpr`

functions. It may be
desirable to turn certain operations into `constexpr`

and it should be
explicitly permitted to do so. In general, the operations shouldn’t be
mandated to be `constexpr`

because the semantics of many operations
depend on run-time setting, e.g., because they use the rounding mode.
On the other hand, the use of `constexpr`

operations is especially
desirable, e.g., because constant initialization is preformed prior to
any dynamic initialization (3.6.2, [basic.start.init], paragraph 2),
thereby avoiding any issues relating to the order of initialization.

To really support the use of constant expressions for decimal
floating-point types it is necessary to restrict the semantics of the
operations. In particular, the operations need to be independent of
the floating point environment (however, it seems ISO/IEC 60559
requires a way to specify the rouding mode to be used when computing
constants). The conditional availability of the floating point
environment would raise the requirement that functions can be
overloaded on `constexpr`

arguments, for example (in C++ 2011 it is
**not** possible to overload these two functions):

```
constexpr decimal64 operator+ (constexpr decimal64 d1,
constexpr decimal64 d2);
decimal64 operator+ (decimal64 d1,
decimal64 d2);
```

The first function would be used if the arguments `d1`

and `d2`

are
constant expressions, otherwise the other function would be used. The
implementations of the version using constant expressions wouldn’t
raise any floating point exception and wouldn’t depend on the
dynamically specified floating point context. The implementations of
both functions would do similar operations but possibly in vastly
different ways. For example, the constant expression version would use
a software implementation while the other version could be implemented
to take advantage of hardware support for decimal floating points.
However, corresponding support isn’t available in C++ 2011.

Not having `constexpr`

support yields a viable library. If it is
controversial to add `constexpr`

it is probably safest to allow the
use of `constexpr`

but not to mandate it.

Section 4.1 (Literals) of the Decimal TR mentions that C uses literal suffixes for easy creation of decimal floating-point types. With the ability to define user-define literals a similar mechanism can be provided in C++. That is, the following operators should be added:

```
template <char... C> constexpr decimal32 operator "" DF();
template <char... C> constexpr decimal64 operator "" DD();
template <char... C> constexpr decimal128 operator "" DL();
template <char... C> constexpr decimal32 operator "" df();
template <char... C> constexpr decimal64 operator "" dd();
template <char... C> constexpr decimal128 operator "" dl();
```

It may be desirable to mandate that the return types of these operators
are `constexpr`

. However, the implementations aren’t necessarily trivial.
If mandated use of `constexpr`

is controversial the support should only
be allowed and not mandated.

Decimal floating points support a feature not available for binary
floating points: They can represent the precision of the original
number, i.e., they can keep track of trailing zeros after the decimal
point (unless the number of digits would exceed the number of decimals
for the decimalfloating point). To support a choice of formatting the
number using its own precision the
C Decimal TR
uses the `%a`

and `%A`

format specifiers which use the optionally
present precision to restrict the formatted to number to a maximum
number of digits.

Table 88 in 22.4.2.2.2 [facet.num.put.virtuals] paragraph 5 already
specifies that the format specifiers `%a`

and `%A`

are used for
floating point conversions when the `floatfield`

is set to
`std::ios_base::fixed | std::ios_base::scientific`

(this is used to
format binary floating point numbers using a hexadecimal format). The
Decimal TR
additionally expands the table for length modifiers to support the
modifiers `H`

, `D`

, and `DD`

for `decimal32`

, `decimal64`

, and
`decimal128`

, respectively. Thus, using
`std::ios_base::fixed | std::ios_base::scientific`

results in
formatting decimal floating points taking their own precision into
account when being formatted.

Unfortunately, this paragraph specifies that the precision
(`str.precision()`

) is only specified for floating pont types if
`floatflied != std::ios_base::fixed | std::ios_base::scientific`

.
However, it is desirable optionally impose an upper bound on the used
precision. One way to address this problem is to change the
corresponding paragraph to become

For conversion from a floating type, if

`floatfield != (ios_base::fixed | ios_base::scientific)`

or if a decimal floating-point type is formatted and`0 < str.presision()`

,`str.precision()`

is specified in the conversion specification. Otherwise, no precision is specified.

With this change the currently set precision would be taken into
account when formatting decimal floating points. When
`str.precision() == 0`

and `floatfield`

is set to
`std::ios_base::fixed | std::ios_base::scientific`

no precision would
be specified with the `%a`

or `%A`

specifiers when formatting a decimal
floating point, i.e., its own precision is used. Note, that using a
non-zero precision with the `%a`

or `%A`

format specifier affects all
digits, not just the fractional digits (this is consistent with the
way the `%g`

and `%G`

format specifiers work).

When setting `floatfield`

to
`std::ios_base::fixed | std::ios_base::scientific`

it would be desirabe
that the default precision used is the decimal floating point’s own
precision. This would imply that `str.precision() == 0`

and it seems
unlikely that the default for `str.precision()`

is changed. An
alternative approach could be to use a new attribute on
`std::ios_base`

, e.g., `decimal_precision()`

, which is used with
formatting of decimal floating points and whose initial value is `0`

.
To avoid any additional memory overhead this attribute could be
accessed using the `std::ios_base::iword()`

.

Independent on how the precision is set, it may also be worth to add an
alias for `std::hexfloat`

which gives the operation a name meaningful
in the context of decimal floating points. For example,
`std::decimal::ownprecision`

may be added which also sets the
`floatfield`

to `std::ios_bse::fixed | std::ios_based::scientific`

.