P1370R1: Generic numerical algorithm development with(out) numeric_limits

The LEWG-I minutes were not posted, but the authors remember the following discussion points.

1. Suggested name for what Fortran calls TINY: min_normal_v (a term of art from IEEE 754).
2. Suggested name for what LAPACK calls sfmin: reciprocal_overflow_threshold_v (there was another, more concise name, but I don't recall it and don't have the minutes). The group strongly objected to names with the word "safe" in them.

In discussion, one LEWG-I participant expressed a preference that the new traits proposed by P0437 should come in has_${PROPERTY}_v<T>, ${PROPERTY}_v pairs. They prefer the current numeric_limits-like behavior, in which ${PROPERTY}_v<T> exists but has some arbitrary value if has_${PROPERTY}_v<T> is false, to the approach in which ${PROPERTY}_v<T> does not exist if T does not have the property in question. SG18 did not vote on this topic, but this feedback could be helpful for the next revision of P0437. In particular, the last revision of P0437 before the 2019 Kona meeting proposes that use of ${PROPERTY}_v<T> be ill-formed if if T does not have the property in question. Section 6 of P0437R1 suggests a general mechanism, value_exists<Trait, T>, that would replace has_${PROPERTY}_v<T>. We prefer P0437R1's approach, since it prevents bugs at compile time. However, our proposal should work with both approaches.

An example of existing traits needing revision is the current definition of numeric_limits<T>::min, which is inconsistent for integer versus floating-point types T. For integer types, this function behaves as expected; it returns the minimum finite value. For signed integer types, this is the negative value of largest magnitude, and for unsigned integer types, it is zero. However, for a floating-point type, it returns the type's smallest positive normalized value. This is confusing in two different ways:

1. It is positive, so it is not the least finite floating-point value, namely -numeric_limits<T>::max().
2. It is normalized, so if the implementation of floating-point arithmetic has subnormal numbers, smaller positive floating-point numbers of type T could exist.³

This surprising behavior could lead to bugs when writing algorithms meant for either integers or floating-point values. Nevertheless, the actual value is useful in practice, as we will show below.

An example of a new trait to consider is the smallest positive floating-point number such that one divided by that number does not overflow. We propose calling this trait reciprocal_overflow_threshold_v<T>. As we will show below, floating-point types exist for which this trait's value differs from that of min_normal_v<T>. This new trait is also useful in practice.

LAPACK is a Fortran library, but it takes a "generic" approach to algorithms for different data types. LAPACK implements algorithms for four different data types:

• Single-precision real (S)
• Double-precision real (D)
• Single-precision complex (C)
• Double-precision complex (Z)

LAPACK does not rely on Fortran generic procedures or parameterized derived types, the closest analogs in Fortran to C++ templates. However, most of LAPACK's routines are implemented in such a way that one could generate all four versions automatically from a single "template."⁵ As a result, we find LAPACK a useful analog to a C++ library of generic numerical algorithms, written using templates and traits classes. Numerical algorithm developers who are not C++ programmers have plenty of experience writing generic numerical algorithms. See, for example, "Templates for the Solution of Linear Systems,"⁶ where "templates" means "recipes," not C++ templates. Thus, it should not be surprising to find design patterns in common between generic numerical algorithms not specifically using C++, and generic C++ libraries. In fact, our motivating examples will come from LAPACK's "floating-point traits" routines.

LAPACK's "generic" approach means that algorithm developers need a way to access floating-point arithmetic properties as a function of data type, just as if a C++ developer were writing an algorithm templated on a floating-point type. Many linear algebra algorithms depend on those properties to avoid unwarranted overflow or underflow, and to get accurate results. As a result, LAPACK provides the SLAMCH and DLAMCH routines, that return machine parameters for the given real floating-point type (single-precision real resp. double-precision real). (One can derive from these the properties for corresponding complex numbers.)

LAPACK routines have a uniform naming convention, where the first letter indicates the data type. LAPACK developers refer to the "generic" algorithm by omitting the first letter. For example, _GEQRF represents the same QR factorization for all data types for which it is implemented, in this case, SGEQRF, DGEQRF, CGEQRF, and ZGEQRF. Hence, we refer to the "floating-point traits" routines SLAMCH and DLAMCH generically as _LAMCH.

_LAMCH was designed to work on computers that may have non-IEEE-754 floating-point arithmetic. Older versions of the routine would actually compute the machine parameters. This is what LAPACK 3.1.1 does.⁷ More recent versions of LAPACK, including the most recent version, 3.8.0, rely on Fortran intrinsics to get the values of most of the machine parameters.⁸

_LAMCH thus offers functionality analogous to numeric_traits, for different real floating-point types. LAPACK's authors chose this functionality specifically for the needs of linear algebra algorithm development. _LAMCH gives developers the following constants:

• eps: relative machine precision
• sfmin: safe minimum, such that 1/sfmin does not overflow
• base: base of the machine
• prec: eps*base
• t: number of (base) digits in the mantissa
• rnd: 1.0 when rounding occurs in addition, 0.0 otherwise⁹
• emin: minimum exponent before (gradual) underflow
• rmin: underflow threshold - base**(emin-1)
• emax: largest exponent before overflow
• rmax: overflow threshold - (base**emax)*(1-eps)

Our proposed min_normal_v<T> corresponds to the underflow threshold rmin, and our reciprocal_overflow_threshold_v<T> corresponds to the "safe minimum" sfmin. LAPACK uses the "safe minimum" more often than the underflow threshold, but it does use both values.

LAPACK uses the safe minimum in algorithms (e.g., equilibration and balancing) that scale the rows and/or columns of a matrix to improve accuracy of a subsequent factorization. In the process of improving accuracy, one would not want to divide by too small of a number, and thus cause underflow that is not warranted by the user's data. For example, see DRSCL, DLADIV, and ZDRSCL. (The documentation of LAPACK's _LABAD routine explains how LAPACK uses this routine to work around surprising behavior at the extreme exponent ranges of certain floating-point systems.)

The safe minimum also helps in LAPACK's LU factorization with complete pivoting (see e.g., ZGETC2). If LAPACK finds a pivot U(k, k) that is too small in magnitude, it replaces the pivot with a tiny real number. This helps the factorization finish, which is an important goal for LU with complete pivoting (unlike the usual LU factorization with partial pivoting _GETRF, which stops on encountering a zero pivot). LAPACK derives the value for this tiny real number from the safe minimum, since LU factorization must divide numbers in the matrix by the pivot.

LAPACK uses the underflow threshold quite a bit in its tests. In its actual code, it uses this value when computing the eigenvalues of a real symmetric tridiagonal matrix (the _LARRD auxiliary routine, called from _STEMR).

In the most recent version of LAPACK, 3.8.0, LAPACK uses the Fortran intrinsic function TINY for the underflow threshold, and derives the "safe minimum" sfmin from that value and the overflow threshold (Fortran intrinsic HUGE) as follows:

	 sfmin = tiny(zero)
	 small = one / huge(zero)
	 IF( small.GE.sfmin ) THEN
*
*           Use SMALL plus a bit, to avoid the possibility of rounding
*           causing overflow when computing  1/sfmin.
*
	    sfmin = small*( one+eps )
	 END IF
	 rmach = sfmin

Here is the C++ equivalent:

template<class T>
T safe_minimum (const T& /* ignored */) {
  constexpr T one (1.0);
  constexpr T eps = std::numeric_limits<T>::epsilon();
  constexpr T tiny = std::numeric_limits<T>::min();
  constexpr T huge = std::numeric_limits<T>::max();
  constexpr T small = one / huge;
  T sfmin = tiny;
  if (small >= tiny) {
    sfmin = small * (one + eps);
  }
  return sfmin;
}

For IEEE 754 float and double, the IF branch never gets taken. (LAPACK was originally written to work on computers that did not implement IEEE 754 arithmetic, so the extra branches may have made sense for earlier computer architectures. They are also useful as a conservative check of floating-point properties.) Thus, for T=float and T=double, sfmin always equals numeric_limits<T>::min(). However, several historical floating-point formats have the property that one divided by the underflow threshold would overflow. Table 1 in Prof. William Kahan's essay¹⁰ gives examples. In general, this is possible whenever the absolute value of the underflow threshold exponent is greater than the overflow threshold exponent. Whenever this property holds, LAPACK's sfmin is larger than the minimum normalized floating-point value.

WG21 has seen proposals both to standardize new number formats, and to standardize ways for users to add support for their own number formats. Furthermore, WG21 has seen proposals for generic linear algebra (e.g., P1385) and other algorithms useful for machine learning. This makes it critical for C++ developers, including Standard Library developers, to have Standard Library support for writing type-independent floating-point algorithms. Ignoring the two proposed traits puts these developers at risk, especially for algorithms that rescale data to improve accuracy (a common feature when solving linear systems, for example). Such algorithms could cause infinities and/or Not-a-Numbers that are not warranted by the data, unless they use the two proposed traits.