We propose the addition of new functions to the C++0x standard library that provide floating-point operations (+, -, *, /, sqrt and fma) as well as conversion functions with directed rounding. This set of functions is necessary to provide efficient support for interval arithmetic and related computations, and they directly map to IEEE-754 specifications. These functions require special compiler support due to their "constexpr" nature. This document is submitted to the ISO/WG21 working group which standardizes the C++ language, under the document N2811 (http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2811.pdf)

Melquiond, Guillaume

Pion, Sylvain

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Directed Rounding Arithmetic Operations in C++

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HAL Id: inria-00345094https://hal.inria.fr/inria-00345094Submitted on 8 Dec 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Directed Rounding Arithmetic Operations in C++Guillaume Melquiond, Sylvain PionTo cite this version:Guillaume Melquiond, Sylvain Pion. Directed Rounding Arithmetic Operations in C++. [ResearchReport] RR-6757, INRIA. 2008, pp.11. ￿inria-00345094￿appor t  de  r ech er ch eISSN0249-6399ISRNINRIA/RR--6757--FR+ENGThème SYMINSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUEDirected Rounding Arithmetic Operations in C++Guillaume Melquiond — Sylvain PionN° 6757December 2008Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65Directed Rounding Arithmetic Operations in C++Guillaume Melquiond∗ , Sylvain Pion†Thème SYM — Systèmes symboliquesProjets Geometrica et ProvalRapport de recherche n° 6757 — December 2008 — 8 pagesAbstract: We propose the addition of new functions to the C++0x standard librarythat provide floating-point operations (+, -, *, /, sqrt and fma) as well as conversionfunctions with directed rounding. This set of functions is necessary to provide efficientsupport for interval arithmetic and related computations, and they directly map to IEEE-754 specifications. These functions require special compiler support due to their constexprnature. This document is submitted to the ISO/WG21 working group which standardizesthe C++ language, under the document N2811‡.Key-words: interval arithmetic, directed rounding, floating-point arithmetic, C++, stan-dardization, software design∗ INRIA Saclay, France. Email: guillaume.melquiond@inria.fr† INRIA Sophia-Antipolis, France. Email: Sylvain.Pion@sophia.inria.fr‡ http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2811.pdfOpérations arithmétiques avec arrondi dirigé en C++Résumé : Nous proposons l’ajout de nouvelles fonctions à la bibliothèque standard deC++0x, qui fournissent des opérations concernant l’arithmétique flottante (+, -, *, /, sqrtet fma), ainsi que des fonctions de conversions avec un mode d’arrondi dirigé. Cet ensemblede fonctions est nécessaire pour fournir un support efficace à l’arithmétique d’intervalles etautres types de calculs similaires. Ces fonctions nécessitent un support particulier de la partdu compilateur, à cause de leur nature constexpr. Ce document est soumis au groupe detravail ISO/WG21 qui standardise le language C++, sous la forme du document N2811§.Mots-clés : arithmétique d’intervalles, arrondis dirigés, arithmétique flottante, C++,standardisation, génie logiciel§ http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2811.pdfDirected Rounding Arithmetic Operations in C++ 3Contents1 Motivation and context 42 A different approach 42.1 Syntax and design choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Promotion rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Side effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Header file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Acknowledgements 64 Proposal 7RR n° 67574 Melquiond & Pion1 Motivation and contextWG21 has chosen to postpone the consideration of the proposal to add interval arithmeticto the standard library, N21371, after C++0x. While a new standard is being developedto provide a language-independent base for interval arithmetic (IEEE-1788), we believethat progress can be made to enhance low-level support for interval arithmetic in C++0x,and make it easier to build an interval arithmetic library on top of C++0x efficiently andportably.The current CD provides the fegetround and fesetround functions in the <cfenv>header to access and change the current rounding mode. Those functions were importedfrom C99, and they are at the root of every interval arithmetic implementation. There arehowever two problems which prevent efficient support for interval arithmetic:• the CD does not provide the accompanying FENV_ACCESS pragma, which instructs thecompiler that a block of code cares about the rounding mode. The consequence is thatone must use temporary volatile variables, or any other equivalent implementation-dependent method, which is inefficient. This can be considered as an incompleteaddition to C++0x compared to C++03.• building interval constants is very useful, but it is not possible to build constexprinterval operations since the restricted form of expressions allowed by constexpr func-tions forbids calling fesetround. Extending a general feature such as constexpr tosupport this, does not seem appropriate for this particular use.2 A different approachWe propose to go away from fesetround as the low-level primitive used to implement in-terval arithmetic. Instead of a global state rounding mode, we propose to add functionswith directed rounding modes that perform floating-point addition, subtraction, multipli-cation, division, square root and fma, as well as conversions between floating-point andintegral types, and between floating-point types themselves. These functions correspond tothe functionality mandated by IEEE-754-1985 (and by IEEE-754-2008 for fma).These functions, especially as they would be constexpr, require specific compiler sup-port.On hardwares that support a static rounding direction in the floating-point operationinstructions or that support several floating-point control registers, the new functions makeit easier for the compiler to generate those instructions than currently, since using thefesetround interface around a block of floating-point operations requires some (possiblyinter-procedural) flow analysis.Fundamentally, there is no reason why C++0x provides built-in operators +, -, *, /which allow to build constexpr functions with the default rounding mode, while it wouldnot be possible with its directed rounding cousins.1http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2006/n2137.pdfINRIADirected Rounding Arithmetic Operations in C++ 5Note on optimization: on hardware that needs to set a register holding the roundingmode, using fesetround in a block of code can be more efficient. In practice however, thiskind of computation is typically needed for adding vectors of intervals, and it is our beliefthat, in our approach, a quality compiler should be able to move the rounding mode changeinstructions outside of the loops.2.1 Syntax and design choicesThere are 4 rounding modes specified by IEEE-754-1985: to the nearest (default), towardszero, towards plus infinity and towards minus infinity.We propose that the new functions use a constant rounding mode. That is, it would notbe possible to pass it as an argument, and even less be a run time entity.Moreover, C++0x currently has 2 ways of refering to rounding modes:• fesetround takes an argument of type int with possible values FE_DOWNWARD,FE_TONEAREST, FE_TOWARDZERO and FE_UPWARD.• the floating_round_style enum which has 4 values plus a fifthround_indeterminate.We prefer the latter alternative for type checking and consistency reasons. Moreover,we suggest to pass the rounding mode as explicit template argument, to make sure it isa constant, while still not hardcoding it in the function name like add_up or mul_down.Passing round_indeterminate is not useful, and is not allowed.An example of use would then be:double d = add<round_toward_infinity>(a, b);float pi_up = rounded_cast<round_toward_infinity, float>(3.14159265358979323);int i = rounded_cast<round_toward_infinity, int>(134675./4247.);Side note: some languages which support more mathematical symbols for operators, likeFortress2, provide a nice syntax like a △+ b.2.2 Promotion rulesIdeally, the add function should behave just like the built-in floating-point operator+, con-cerning the types of the operands, following the promotion rules. And similarly for the otherfunctions. We propose to use constrained templates, auto and decltype to achieve this.2.3 Side effectsDivision by zero, or taking the square root of a negative number, can produce side effectslike raising floating-point exceptions and setting errno. Obviously, this cannot work for2http://research.sun.com/projects/plrg/Publications/fortress1.0beta.pdfRR n° 67576 Melquiond & Pionconstexpr functions at compile-time. We therefore propose that the evaluation of thesefunctions follow the IEEE-754 “default non-stop exception handling" mode (producing NaNsand Inf, but no floating-point exception nor errno change), and with the caveat that thecompile-time evaluation is done without raising the corresponding status flags.A compiler may choose to issue or not a warning in such cases.2.4 Header fileThe closest related headers are <cmath> and <numeric>. However, neither seem per-fectly appropriate to provide these functions. Moreover, since these functions are compiler-supported, it makes sense to put them in a separate header file, so as to ease the work oflibrary vendors. This header could then be provided by compiler vendors.We therefore propose a new header file <rounded_math>, and a new section to describeit.3 AcknowledgementsWe thank Marc Glisse for his useful comments.INRIADirected Rounding Arithmetic Operations in C++ 74 Proposal26.8 Directed rounding functions26.8.1 Header <rounded_math> synopsisnamespace std {template < float_round_style r, FloatingPointLike T, ArithmeticLike U >requires True<(r != round_indeterminate)>constexpr T rounded_cast(U u);template < float_round_style r, IntegralLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr T rounded_cast(U u);template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto add(T t, U u) -> decltype(t + u);template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto sub(T t, U u) -> decltype(t - u);template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto mul(T t, U u) -> decltype(t * u);template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto div(T t, U u) -> decltype(t / u);template < float_round_style r, FloatingPointLike T >requires True<(r != round_indeterminate)>constexpr T sqrt(T t);template < float_round_style r, FloatingPointLike T, FloatingPointLike U,FloatingPointLike V >requires True<(r != round_indeterminate)>constexpr auto fma(T t, U u, V v) -> decltype(fma(t, u, v));}1 The compile-time evaluation of these functions ignores any floating-point status flags raised,if any.RR n° 67578 Melquiond & Pion2 The run-time evaluation of these functions shall not raise any floating-point exception, norchange errno.template < float_round_style r, FloatingPointLike T, ArithmeticLike U >requires True<(r != round_indeterminate)>constexpr T rounded_cast(U u);template < float_round_style r, IntegralLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr T rounded_cast(U u);Returns: The conversion u to the type T rounded according to r.template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto add(T t, U u) -> decltype(t + u);Returns: The addition of t and u rounded according to r.template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto sub(T t, U u) -> decltype(t - u);Returns: The subtraction of t and u rounded according to r.template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto mul(T t, U u) -> decltype(t * u);Returns: The multiplication of t and u rounded according to r.template < float_round_style r, FloatingPointLike T, FloatingPointLike U >requires True<(r != round_indeterminate)>constexpr auto div(T t, U u) -> decltype(t / u);Returns: The division of t and u rounded according to r.template < float_round_style r, FloatingPointLike T >requires True<(r != round_indeterminate)>constexpr T sqrt(T t);Returns: The square root of t rounded according to r.template < float_round_style r, FloatingPointLike T, FloatingPointLike U,FloatingPointLike V >requires True<(r != round_indeterminate)>constexpr auto fma(T t, U u, V v) -> decltype(fma(t, u, v));Returns: The fused multiply-and-add of t, u, and v, rounded according to r.INRIAUnité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)http://www.inria.frISSN 0249-6399

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Directed Rounding Arithmetic Operations in C++

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