Rounding and Overflow in C++

ISO/IEC JTC1 SC22 WG21 P0105R0 - 2015-09-27

Lawrence Crowl, Lawrence@Crowl.org

Introduction
Rounding
    Current Status
    Base Requirements
    Modes
    Functions
Overflow
    Current Status
    Base Requirements
    Modes
    Functions
Both Rounding and Overflow
General Conversion
Revisions

Introduction

C++ currently provides relatively poor facilities for controlling rounding. It has even fewer facilities for controlling overflow. The lack of such facilities often leads programmers to ignore the issue, making software less robust than it could be (and should be).

This paper presents the issues and provides some candidate enumerations and operations. The intent of the paper is to gather feedback on support for and direction of future work.

Rounding

Rounding is necessary whenever the resolution of a variable is coarser than the resolution of a value to be placed in that variable.

Current Status

The numeric_limits field round_style provides information on the style of rounding employed by a type.

namespace std {
  enum float_round_style {
    round_indeterminate = -1, // indeterminable
    round_toward_zero = 0, // toward zero
    round_to_nearest = 1, // to the nearest representable value
    round_toward_infinity = 2, // toward [positive] infinity
    round_toward_neg_infinity = 3 // toward negative infinity
  };
}

This specification is incomplete in that it fails to specify what happens when the value is equally far from the two nearest representable values.

The standard also says "Specializations for integer types shall return round_toward_zero." This requirement is somewhat misleading as a right-shift operation on a two's complement representation does not round toward zero.

Headers <cfenv> and <fenv.h> provide functions for setting and getting the floating-point rounding mode, fesetround and fegetround, respectively. The mode is specified via a macro constant:

Again, the specification is incomplete with respect to FE_TONEAREST

Base Requirements

The base requirements on a round function are:

• Given a value x and two adjacent representable values y < z such that y ≤ x ≤ z then

  • if x = y then round(x) = y,

  • if x = z then round(x) = z,

  • and otherwise round(x) = y or round(x) = z.

• Given an additional value w such that y ≤ w ≤ x ≤ z then

  • y ≤ round(w) ≤ round(x) ≤ z

Modes

The number of rounding modes is perhaps unlimited. However, we can explore the space of reasonably efficient rounding modes with two notions, its direction and its domain.

There are six precisely-defined rounding directions and at least three additional practical directions. They are:

Of these directions, only towards even and towards odd are unbiased.

Rounding towards odd has two desirable properties. First, the direction will not induce a carry out of the units position. This property avoids overflow and increased representation size. Second, because most operations tend to preserve zeros in the lowest bit, the towards-even direction carries less information than towards-odd. This effect increases as the number of bits decreases. However, rounding towards even produces numbers that are "nicer" than those produced by rounding towards odd. For example, you are more likely to get 10 than 9.9999999 with rounding towards even.

There are at least two direction domains:

• All values between two representable values move in the given direction.

• Only values midway between two representable values move in the given direction. Other values move to the nearest representable value. That is, the direction is a tie breaker.

Several of the precise rounding modes are in current use.

We represent the mode in C++ as an enumeration:

enum class rounding {
  all_to_neg_inf, all_to_pos_inf,
  all_to_zero, all_away_zero,
  all_to_even, all_to_odd,
  all_fastest, all_smallest,
  all_unspecified,
  tie_to_neg_inf, tie_to_pos_inf,
  tie_to_zero, tie_away_zero,
  tie_to_even, tie_to_odd,
  tie_fastest, tie_smallest,
  tie_unspecified
};

The unmotivated modes all_away_zero, all_to_even, all_to_odd, tie_to_neg_inf, tie_to_pos_inf, and tie_to_zero are conditionally supported.

Functions

Within the definition of the following functions, we use a defining function, which we do not expect will be directly represented in C++. It is T round(mode,U) where U either

• has a finer resolution than T or

• is evaluated as a real number expression.

We already have rounding functions for converting floating-point numbers to integers. However, we need a facility that extends to different sizes of floating-point and between other numeric types.

template<typename T, typename U> T convert(rounding mode, U value)

The result is round(mode, U).

template<rounding mode, typename T, typename U> T convert(U value)

The result is round<T>(mode, U).

A division function has obvious utility.

template<typename T> T divide(rounding mode, T dividend, T divisor)

The result is round(mode, dividend/divisor). Remember that division evaluates as a real number. Obviously, the implementation will use a different strategy, but it must yield the same result.

template<rounding mode, typename T> T divide(T dividend, T divisor)

The result is divide(mode, dividend, divisor).

Division by a power of two has substantial implementation efficiencies, and is used heavily in fixed-point arithmetic as a scaling mechanism. We represent the conjunction of these approaches with a rounding scale down (right shift).

template<typename T> T scale_down(rounding mode, T value, int bits)

The result is round(mode, dividend/2bits).

template<rounding mode, typename T> T scale_down(T value, int bits)

The result is scale_down(mode, dividend, bits).

We can add other functions as needed.

Overflow

Overflow is possible whenever the range of an expression exceeds the range of a variable.

Current Status

Signed integer overflow is undefined behavior. Programmers attempting to detect and handle overflow often get it wrong, in that they end up using overflow to detect overflow. Suffice it to say that present solutions are inadequate.

Unsigned integer overflow is defined to be mod 2^bits-in-type. While this definition is exactly right when coding in modular arithmetic, it is counter-productive when one is using unsigned arithmetic to state that the value is non-negative. In the latter environment, undefined behavior on overflow is better, as it enables tools to detect problems.

Floating-point overflow can be detected and altered via fegetexceptflag, fesetexceptflag, feclearexcept, and feraiseexcept with the value FE_OVERFLOW. However, such checking requires additional out-of-band effort. That is, any checking takes place in code separate from the operations themselves.

Base Requirements

The base requirements on a overflow function are:

• Given a value x and a representable range y ≤ z such that y ≤ x ≤ z then an overflow does not occur and

  • overflow(x) = x.

• Otherwise, an overflow has occured and the function may, for all overflow values, choose either:

  • Consider the expression an error, handling it or not as appropriate.

  • Return a normal value w = overflow(x) such that y ≤ w ≤ z.

  • Return a special value indicating overflow, e.g. IEEE infinities. This choice implies defining the result of operations given this special value as an argument.

Modes

Several overflow modes are possible. We categorize them based on the choices in the base requirements. Other modes may be possible or desirable as well.

Some error modes are as follows.

impossible

Mathematically, overflow cannot occur. This mode is useful when an overflow specification is necessary, but compiler-based range propogation is insufficient to eliminating a check. The mode is an assertion on the part of the programmer. It invites reviewers to examine the accompanying proof. Ignoring overflow and letting the program stray into undefined behavior is a suitable implementation.

undefined

The programmer states that overflow is sufficiently rare so that overflow is not a concern. Aborting on overflow is a suitable implementation. So is ignoring the issue and letting the program stray into undefined behavior.

abort

Abort the program on overflow. Detection is required.

quick_exit

Call quick_exit on overflow. Detection is required.

exception

Throw an exception on overflow. Detection is required.

A special substitution mode is as follows. Detection is required.

special

Return one of possibly several special values indicating overflow.

Some normal substitution modes are as follows. Detection is required.

saturate

Return the nearest value within the valid range.

modulo with shifted scale

For unsigned arguments and range from 0 to z, the result is simply x mod (z+1). Shifting the range such that 0 < y ≤ z requires a more complicated expression, y + ((x–y) mod (z–y+1)). We can also use this expression when y < 0. That is, it is a general purpose definition. However, it may not yield results consistent with division.

Various overflow modes are in current use.

We represent the mode in C++ as an enumeration:

enum class overflow {
  impossible, undefined, abort, exception,
  special,
  saturate, modulo_shifted
};

Functions

Within the definition of the following functions, we use a defining function, which we do not expect will be directly represented in C++. It is T overflow(mode,T lower,T upper,U value) where U either

• has a range that is not a subset of the range of T or

• is evaluated as a real number expression.

Many C++ conversions already reduce the range of a value, but they do not provide programmer control of that reduction. We can give programmers control.

template<typename T, typename U> T convert(overflow mode, U value)

The result is overflow(mode, numeric_limits<T>::min, numeric_limits<T>::max, value).

template<overflow mode, typename T, typename U> T convert(U value)

The result is convert(mode, value).

Being able to specify overflow from a range of values of the same type is also helpful.

template<typename T> T limit(overflow mode, T lower, T upper, T value)

The result is overflow(mode, lower, upper, value).

template<overflow mode, typename T> T limit(T lower, T upper, T value)

The result is limit(mode, lower, upper, value).

Common arguments can be elided with convenience functions.

template<typename T> T limit_nonnegative(overflow mode, T upper, T value)

The result is limit(mode, 0, upper, value).

template<overflow mode, typename T> T limit_nonnegative(T upper, T value)

The result is limit_nonnegative(mode, upper, value).

template<typename T> T limit_signed(overflow mode, T upper, T value)

The result is limit(mode, -upper, upper, value).

template<overflow mode, typename T> T limit_signed(T upper, T value)

The result is limit_signed(mode, upper, value).

Two's-complement numbers are a slight variant on the above.

template<typename T> T limit_twoscomp(overflow mode, T upper, T value)

The result is limit(mode, -upper-1, upper, value).

template<overflow mode, typename T> T limit_twoscomp(T upper, T value)

The result is limit_twoscomp(mode, upper, value).

For binary representations, we can also specify bits instead. While this specification may seem redundant, it enables faster implementations.

template<typename T> T limit_nonnegative_bits(overflow mode, T upper, T value)

The result is overflow(mode, 0, 2upper-1, value).

template<overflow mode, typename T> T limit_nonnegative_bits(T upper, T value)

The result is limit_nonnegative_bits(mode, upper, value).

template<typename T> T limit_signed_bits(overflow mode, T upper, T value)

The result is overflow(mode, -(2upper-1), 2upper-1, value).

template<overflow mode, typename T> T limit_signed_bits(T upper, T value)

The result is limit_signed_bits(mode, upper, value).

template<typename T> T limit_twoscomp_bits(overflow mode, T upper, T value)

The result is overflow(mode, -2upper, 2upper-1, value).

template<overflow mode, typename T> T limit_twoscomp_bits(T upper, T value)

The result is limit_twoscomp_bits(mode, upper, value).

Embedding overflow detection within regular operations can lead to enhanced performance. In particular, left shift is a important candidate operation within fixed-point arithmetic.

template<typename T> T scale_up(overflow mode, T value, int count)

The result is overflow(mode, numeric_limits<T>::min, numeric_limits<T>::max, value×2count).

template<overflow mode, typename T> T scale_up(T value, int count)

The result is scale_up(mode, value, count).

As before, finer specification of the limits is reasonable.

We can add other functions as needed.

Both Rounding and Overflow

Some operations may reasonably both require rounding and require overflow detection.

First and foremost, conversion from floating-point to integer may require handling a floating-point value that has both a finer resolution and a larger range than the integer can handle. The problem generalizes to arbitrary numeric types.

template<typename T, typename U> T convert(overflow omode, rounding rmode, U value)

The result is overflow(omode, numeric_limits<T>::min, numeric_limits<T>::max, round(rmode,value)).

template<overflow omode, rounding rmode, typename T, typename U> T convert(U value)

The result is convert(omode, rmode; value).

Consider shifting as multiplication by a power of two. It has an analogy in a bidirectional shift, where a positive power is a left shift and a negative power is a right shift.

template<typename T> T scale(overflow omode, rounding rmode, T value, int count)

The result is count < 0
? round(rmode,value×2count)
: overflow(omode, numeric_limits<T>::min, numeric_limits<T>::max, value×2count).

template<overflow omode, rounding rmode, typename T> T scale(T value, int count)

The result is scale(omode, rmode value count).

General Conversion

Conversion between arbitrary numeric types requires something more practical than implementing the full cross product of conversion possibilities.

To that end, we propose that each numeric type promotes to an unbound type in it same general category. For example, integers of fixed size would promote to an unbound integer. In this promotion, there can be no possibility of overflow or rounding. Each type also demotes from that type. The demotion may have both round and overflow.

The general template conversion algorithm from type S to type T is to:

• Promote S to its unbound type S'.

• Convert S' to unbound type T' of T.

• Demote T' to T.

We expect common conversions to have specialized implementations.

Revisions

This paper revises N4448 - 2015-04-12.

• Add contents entries.
• Add revisions section.
• Rename bshift to scale, lshift to scale_up and rshift to scale_down.
• Rename positive to nonnegative in function names.
• Make unmotivated rounding modes conditionally supported.
• Remove unmotivated modular overflow modes.
• Provide two sets of functions, one with modes as function parameters and one with modes as template parameters.
• Remove section Template Parameter versus Function Parameter.
• Add a General Conversion section with algorithm.