220 31536 <6bccd7e5-82ad-4c59-9f72-f7fb6639c0ca@isocpp.org> article Path: news.gmane.org!.POSTED!not-for-mail From: 3dw4rd@verizon.net Newsgroups: gmane.comp.lang.c++.isocpp.proposals Subject: Re: Math constants Date: Thu, 9 Mar 2017 14:44:32 -0800 (PST) Lines: 1184 Approved: news@gmane.org Message-ID: <6bccd7e5-82ad-4c59-9f72-f7fb6639c0ca@isocpp.org> References: Reply-To: std-proposals@isocpp.org NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_951_1972070828.1489099472968" X-Trace: blaine.gmane.org 1489099477 11565 195.159.176.226 (9 Mar 2017 22:44:37 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 9 Mar 2017 22:44:37 +0000 (UTC) To: ISO C++ Standard - Future Proposals Original-X-From: std-proposals+bncBCS2FYFYXYEBBUNVQ7DAKGQEF2ATQTY@isocpp.org Thu Mar 09 23:44:30 2017 Return-path: Envelope-to: gclcip-std-proposals@m.gmane.org Original-Received: from mail-io0-f198.google.com ([209.85.223.198]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1cm6nM-0002AO-VP for gclcip-std-proposals@m.gmane.org; Thu, 09 Mar 2017 23:44:29 +0100 Original-Received: by mail-io0-f198.google.com with SMTP id 68sf49467294ioh.4 for ; Thu, 09 Mar 2017 14:44:35 -0800 (PST) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=isocpp-org.20150623.gappssmtp.com; s=20150623; h=date:from:to:message-id:in-reply-to:references:subject:mime-version :x-original-sender:reply-to:precedence:mailing-list:list-id :x-spam-checked-in-group:list-post:list-help:list-archive :list-subscribe:list-unsubscribe; bh=s2exN+3OXxpyKywZmRxQyurTiQWEe0fuoRO+qhJMbX0=; b=XdBTA8q1KqTn+wWqZvXjBzFEuYSHA7erRvYuFAb6g5IEHhNxLHMBs3JcJhKS6s2h8Z AvrbaIxSNmnYmxIcLqC/XnaGLBcVNdhjeXqbMbKfVbxoqsr8F08fINnGPmLqBz+J2Z5i LSvszxjYcW+qka+XWCctvt8hqogeEYpMv2oCX/A+ZC41/T8eb8fmTXjCQJUBEvKhuw88 6wEtndOtPUj3Gq/vM52aehFakEtdlhfBjvkPdWautD0tAs/RgWPUhWm9XYxhn2kVsmzt ACerTqTOo4DEjgshDxzux5T34kITKzYMM8IdnO0St9LKVkBHukYlo+lW3YCHGPDkV49c 5V5g== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=x-gm-message-state:date:from:to:message-id:in-reply-to:references :subject:mime-version:x-original-sender:reply-to:precedence :mailing-list:list-id:x-spam-checked-in-group:list-post:list-help :list-archive:list-subscribe:list-unsubscribe; bh=s2exN+3OXxpyKywZmRxQyurTiQWEe0fuoRO+qhJMbX0=; b=pODKNV2HJIygFQoVD8ytgd7d96kw1AeE9IFr6z7xlCtOfM1bvMW7A/1569OzoyZcRK cFhjkINX1gSPtuzinmZ3BivYmPVSbu51CCUYx4mIJlhKt3FGbjaYPxWsbbS0LurD/b2v 2zXMcWExSpjnBd0pNiUHWSC5Fc1QSjwLxs0wgr/3terERXD39gxGtHRIq7qIbogURgEE l4S/0FghLOrFrMa2b3J/f0Fn3xCzCXFV2cThPN4c7kHIGRxjpU6NCgPyBYK9Jy4N+HTX rmuhplJXYBu6Xs/AnEmWTHL2UlaUDcCQFFdUqVYAmAfyyEtnF5aJvuXdx5u0R0zwDE6+ HxiQ== X-Gm-Message-State: AMke39k68xRY22nKEyz6DueaPZ4qpgDZRiT3mlRS/yzdHOC/+f3EneND2HsqQ1vlBdulRA== X-Received: by 10.107.175.217 with SMTP id p86mr5063440ioo.101.1489099474629; Thu, 09 Mar 2017 14:44:34 -0800 (PST) X-BeenThere: std-proposals@isocpp.org Original-Received: by 10.157.54.152 with SMTP id h24ls7519407otc.30.gmail; Thu, 09 Mar 2017 14:44:33 -0800 (PST) X-Received: by 10.157.13.82 with SMTP id 76mr1914288oti.6.1489099473672; Thu, 09 Mar 2017 14:44:33 -0800 (PST) In-Reply-To: X-Original-Sender: 3dw4rd@verizon.net Precedence: list Mailing-list: list std-proposals@isocpp.org; contact std-proposals+owners@isocpp.org List-ID: X-Google-Group-Id: 399137483710 List-Post: , List-Help: , List-Archive: List-Subscribe: , List-Unsubscribe: , Xref: news.gmane.org gmane.comp.lang.c++.isocpp.proposals:31536 Archived-At: ------=_Part_951_1972070828.1489099472968 Content-Type: multipart/alternative; boundary="----=_Part_952_1398975516.1489099472971" ------=_Part_952_1398975516.1489099472971 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable On Saturday, March 4, 2017 at 1:32:34 PM UTC-5, l...@compuserve.com wrote: > > > > > > > > *Hello everyone, Many areas of software development where C++ is= =20 > actively used involve mathematical calculations. The language inherited a= =20 > sizable math function library from C, and now has its own numerics librar= y=20 > that grows with every release. There is however a small but annoying=20 > omission: C++ standard headers do not include definitions of mathematical= =20 > constants. To this math major it appears to be a very strange gap: wherev= er=20 > common mathematical functions are used, the constants such as =CF=80 or = e=20 > appear quite frequently. Yes, it is not terribly difficult to define a=20 > constant, but from the C++ user's perspective this is as if they were=20 > served a dish at a restaurant but asked to bring their own salt. = =20 > True, many math.h headers include M_PI and other constant definitions. I= =20 > see a few problems with this. First,the predecessor defines are notorious= ly=20 > unsafe, and the C++ community has been trying to move away from them sinc= e=20 > its very first days. Second, relying on them is not helpful for code's=20 > portability, and sometimes requires an additional macro definition such= =20 > us _USE_MATH_DEFINES. Third, these definition are available only by the= =20 > grace of the specific vendor implementation. For example, the code=20 > fragment#include #define _USE_MATH_DEFINES#include const= =20 > double PI =3D M_PI;compiles in Microsoft Visual C++ 2015 but errors out i= n=20 > Visual C++ 2017 RC because the M_PI is no longer available. I think a=20 > solution would be very uncontroversial and easy to implement.The header= =20 > should include as part of the std namespace the definitions of=20 > common math constants specified at long double precision. For example,=20 > depending on the length of long double, that portion of the header may lo= ok=20 > as follows:namespace std { const long double PI =3D 3.1415926535897932384= 6;=20 > // pi const long double E =3D 2.71828182845904523536; // e const long dou= ble=20 > LOG2E =3D 1.44269504088896340736; // log2(e) const long double LOG10E =3D= =20 > 0.434294481903251827651; // log10(e) const long double LN2 =3D=20 > 0.693147180559945309417; // ln(2) const long double LN10 =3D=20 > 2.30258509299404568402; // ln(10) const long double SQRT2 =3D=20 > 1.41421356237309504880; // sqrt(2)}* > I had dabbled with this after variable templates were added. I took that= =20 route. I think that this could devolve into endless bikeshed discussions and what= =20 constants to include.=20 Design decisions: 1. variable templates. 2. In analogy with ud literals sub-directories place constants in constants= =20 sub-directories. 3. I prefixed the math constants with m_ but c_ for constant may be better. -------------------------------------------------------------------------- /* Variable template math constants for the standard library. Edward M. Smith-Rowland 2014-03-12 We have variable templates in C++14. In fact pi was offered as a use=20 case for that feature. We can do math constants right. The constants in math.h are double=20 precision and promote computations unless you cast or erode long double precision. These M_* constants are actually POSIX compatibility - std C++ does not= =20 mandate them. GCC has extension for long double versions suffixed by 'l' (but apparently no float versions with 'f' suffix). In any case these macros don't help with generic code. There are several bespoke versions of generic template math constants=20 around libstdc++ using all the pre-variable-template techniques and with all the=20 annoyances outlined by the variable template proposal. 'Grammar' + Bikeshedding I find 'm_' gives me flexibility in the 'grammar'. We could do just=20 '_'. Or 'c_'. Dividing numbers: 1_2 is 1/2, pi_2 is pi/2. Alt. _div_ seems wordy. Multiples of pi: Npi (2pi, 4pi). For some reason nobody cares about=20 multiples of e but 2e that would be my proposal for 2*e<> if someone were to want. Function args preceded by '_' (unlike math.h!) make things easier to=20 read. Alt. revert that to math.h. Lower case because these are not macros and because that's how we roll in= =20 the std library. What to put in? Help people with intensive calculations. Division and sqrt are are more expensive than multiplication and integral= =20 powers. Logs can take a while too. On the other hand, logs and square roots of integers appear in series=20 expansions of various functions. Rather than having just a few, perhaps a substantial table=20 would be in order. This is a superset of math.h, from libstdc++ (with different= =20 grammar). Prevent collision with an extra constants namespace after std. Add namespace versioning. Should I insert an experimental namespace between std and constants? Should we default to double? Doing so might play nice with auto. Although it spoils the genericity of the template variable constants we could offer typedefs to the different precisions. I'll hold off for= =20 now. I made the constants and math_constants name spaces inline for the same= =20 reason library literal operator namespaces are inline: using std to get the math= =20 functions should give you access to the constants as well. This would go in some or another. */ // Toy implementation: #if __cplusplus <=3D 201103L # include #else namespace std { namespace experimental { inline namespace maths_v1 { inline namespace constants { inline namespace math_constants { /// Constant: @f$ \pi @f$ (see POSIX @c M_PI). template constexpr _RealType m_pi =3D 3.1415'92653'58979'32384'62643'38327'95028'84195e+0= L; /// Constant: @f$ \pi / 2 @f$ (see POSIX @c M_PI_2). template constexpr _RealType m_pi_2 =3D 1.5707'96326'79489'66192'31321'69163'97514'42098e+0= L; /// Constant: @f$ \pi / 3 @f$. template constexpr _RealType m_pi_3 =3D 1.0471'97551'19659'77461'54214'46109'31676'28063e+0= L; /// Constant: @f$ \pi / 4 @f$ (see POSIX @c M_PI_4). template constexpr _RealType m_pi_4 =3D 7.8539'81633'97448'30961'56608'45819'87572'10488e-1= L; /// Constant: @f$ 4 \pi / 3 @f$. template constexpr _RealType m_4pi_3 =3D 4.1887'90204'78639'09846'16857'84437'26705'12253e+0= L; /// Constant: @f$ 2 \pi @f$. template constexpr _RealType m_2pi =3D 6.2831'85307'17958'64769'25286'76655'90057'68391e+0= L; /// Constant: @f$ 4 \pi @f$. template constexpr _RealType m_4pi =3D 1.2566'37061'43591'72953'85057'35331'18011'53678e+1= L; /// Constant: degrees per radian @f$ 180 / \pi @f$. template constexpr _RealType m_deg_rad =3D 5.7295'77951'30823'20876'79815'48141'05170'33237e+1= L; /// Constant: radians per degree @f$ \pi / 180 @f$. template constexpr _RealType m_rad_deg =3D 1.7453'29251'99432'95769'23690'76848'86127'13443e-2= L; /// Constant: @f$ \sqrt{\pi / 2} @f$. template constexpr _RealType m_sqrt_pi_2 =3D 1.2533'14137'31550'02512'07882'64240'55226'26505e+0= L; /// Constant: @f$ 1 / \pi @f$ (see POSIX @c M_1_PI). template constexpr _RealType m_1_pi =3D 3.1830'98861'83790'67153'77675'26745'02872'40691e-1= L; /// Constant: @f$ 2 / \pi @f$ (see POSIX @c M_2_PI). template constexpr _RealType m_2_pi =3D 6.3661'97723'67581'34307'55350'53490'05744'81383e-1= L; /// Constant: @f$ 1 / \sqrt{\pi} @f$. template constexpr _RealType m_1_sqrt_pi =3D 5.6418'95835'47756'28694'80794'51560'77258'58438e-1= L; /// Constant: @f$ \sqrt{\pi} @f$. template constexpr _RealType m_sqrt_pi =3D 1.7724'53850'90551'60272'98167'48334'11451'82797e+0= L; /// Constant: @f$ 2 / \sqrt{\pi} @f$ (see POSIX @c M_2_SQRTPI). template constexpr _RealType m_2_sqrt_pi =3D 1.1283'79167'09551'25738'96158'90312'15451'71688e+0= L; /// Constant: Euler's number @f$ e @f$ (see POSIX @c M_E). template constexpr _RealType m_e =3D 2.7182'81828'45904'52353'60287'47135'26624'97759e+0= L; /// Constant: @f$ 1 / e @f$. template constexpr _RealType m_1_e =3D 3.6787'94411'71442'32159'55237'70161'46086'74462e-1= L; /// Constant: @f$ \log_2(e) @f$ (see POSIX @c M_LOG2E). template constexpr _RealType m_log2_e =3D 1.4426'95040'88896'34073'59924'68100'18921'37427e+0= L; /// Constant: @f$ \log_2(10) @f$. template constexpr _RealType m_log2_10 =3D 3.3219'28094'88736'23478'70319'42948'93901'75867e+0= L; /// Constant: @f$ \log_10(2) @f$. template constexpr _RealType m_log10_2 =3D 3.0102'99956'63981'19521'37388'94724'49302'67680e-1= L; /// Constant: @f$ \log_10(e) @f$ (see POSIX @c M_LOG10E). template constexpr _RealType m_log10_e =3D 4.3429'44819'03251'82765'11289'18916'60508'22940e-1= L; /// Constant: @f$ \log_10(pi) @f$. template constexpr _RealType m_log10_pi =3D 4.9714'98726'94133'85435'12682'88290'89887'36507e-1= L; /// Constant: @f$ \ln(2) @f$ (see POSIX @c M_LN2). template constexpr _RealType m_ln_2 =3D 6.9314'71805'59945'30941'72321'21458'17656'80748e-1= L; /// Constant: @f$ \ln(3) @f$. template constexpr _RealType m_ln_3 =3D 1.0986'12288'66810'96913'95245'23692'25257'04648e+0= L; /// Constant: @f$ \ln(10)@f$ (see POSIX @c M_LN10). template constexpr _RealType m_ln_10 =3D 2.3025'85092'99404'56840'17991'45468'43642'07602e+0= L; /// Constant: Euler-Mascheroni @f$ \gamma_E @f$. template constexpr _RealType m_gamma_e =3D 5.7721'56649'01532'86060'65120'90082'40243'10432e-1= L; /// Constant: Golden Ratio @f$ \phi =3D (1 + \sqrt{5})/2 @f$. template constexpr _RealType m_phi =3D 1.6180'33988'74989'48482'04586'83436'56381'17720e+0= L; /// Constant: @f$ \sqrt{2}@f$ (see POSIX @c M_SQRT2). template constexpr _RealType m_sqrt_2 =3D 1.4142'13562'37309'50488'01688'72420'96980'78569e+0= L; /// Constant: @f$ \sqrt{3} @f$. template constexpr _RealType m_sqrt_3 =3D 1.7320'50807'56887'72935'27446'34150'58723'66945e+0= L; /// Constant: @f$ \sqrt{5} @f$. template constexpr _RealType m_sqrt_5 =3D 2.2360'67977'49978'96964'09173'66873'12762'35440e+0= L; /// Constant: @f$ \sqrt{7} @f$. template constexpr _RealType m_sqrt_7 =3D 2.6457'51311'06459'05905'01615'75363'92604'25706e+0= L; /// Constant: @f$ 1 / \sqrt{2}@f$ (see POSIX @c M_SQRT1_2). template constexpr _RealType m_1_sqrt_2 =3D 7.0710'67811'86547'52440'08443'62104'84903'92845e-1= L; /// Constant: Catalan's @f$ G =3D 1 - 1/9 + 1/25 - 1/49 + 1/81 - ... @f$. template constexpr _RealType m_catalan =3D 9.1596'55941'77219'01505'46035'14932'38411'07741e-1= L; /// Constant: @f$ \pi^2/6 @f$. template constexpr _RealType m_pi2_6 =3D 1.6449'34066'84822'64364'72415'16664'60251'89218e+0= L; } // inline namespace math_constants } // inline namespace constants } // inline namespace maths_v1 } // namespace experimental } // namespace std #endif // C++14 -------------------------------------------------------------------------- --=20 You received this message because you are subscribed to the Google Groups "= ISO C++ Standard - Future Proposals" group. To unsubscribe from this group and stop receiving emails from it, send an e= mail to std-proposals+unsubscribe@isocpp.org. To post to this group, send email to std-proposals@isocpp.org. To view this discussion on the web visit https://groups.google.com/a/isocpp= ..org/d/msgid/std-proposals/6bccd7e5-82ad-4c59-9f72-f7fb6639c0ca%40isocpp.or= g. ------=_Part_952_1398975516.1489099472971 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable


On Saturday, March 4, 2017 at 1:32:34 PM UTC-5, l.= ...@compuserve.com wrote:
Hello everyone,

=C2=A0 =C2= =A0 =C2=A0 =C2=A0 Many areas of software development where C++ is actively = used involve mathematical calculations. The language inherited a sizable ma= th function library from C, and now has its own numerics library that grows= with every release. There is however a small but annoying omission: C++ st= andard headers do not include definitions of mathematical constants. To thi= s math major it appears to be a very strange gap: wherever common mathemati= cal functions are used, the constants such as =C2=A0=CF=80 or e appear quit= e frequently. Yes, it is not terribly difficult to define a constant, but f= rom the C++ user's perspective this is as if they were served a dish at= a restaurant but asked to bring their own salt.=C2=A0

=
=C2=A0 =C2=A0 =C2=A0 =C2=A0 True, many math.h headers include M_PI and= other constant definitions. I see a few problems with this. First,
the predecessor defines are notoriously unsafe, and the C++ community ha= s been trying to move away from them since its very first days. Second, rel= ying on them is not helpful for code's portability, and sometimes requi= res an additional macro definition such us=C2=A0
_USE_MATH_DEFINE= S. Third, these definition are available only by the grace of the specific = vendor implementation. For example, the code fragment

<= div>#include <cstdlib>
#define _USE_MATH_DEFINES
= #include <cmath>
const double PI =3D M_PI;

compiles in Microsoft Visual C++ 2015 but errors out in Visual C++= 2017 RC because the M_PI is no longer available.

= I think a solution would be very un= controversial and easy to implement.The header <cmath> should include= as part of the std namespace the definitions of common math constants spec= ified at long double precision. For example, depending on the length of lon= g double, that portion of the header may look as follows:

namespace std {
const long double PI =3D 3.14159265358979323846; // pi
c= onst long double E =3D 2.71828182845904523536; // e
const lo= ng double LOG2E =3D 1.44269504088896340736;= // log2(e)
const = long double LOG10E =3D 0.434294481903251827651; // log10(e)
const long double LN2 =3D 0.693147180559= 945309417; // ln(2)
const long double LN10 =3D 2.302585092994= 04568402; // ln(10)
const long double SQRT2 =3D 1.41421356237= 309504880; // sqrt(2)
}


I had dabbled with this = after variable templates were added.=C2=A0 I took that route.
I think th= at this could devolve into endless bikeshed discussions and what constants = to include.

Design decisions:
1. variable templates.
2. In an= alogy with ud literals sub-directories place constants in constants sub-dir= ectories.
3. I prefixed the math constants with m_ but c_ for constant m= ay be better.

------------------------------------------------------= --------------------
/*
=C2=A0 Variable template math constants for t= he standard library.
=C2=A0 Edward M. Smith-Rowland <esmith-rowland a= t alionscience dot com>
=C2=A0 2014-03-12

=C2=A0 We have varia= ble templates in C++14.=C2=A0 In fact pi was offered as a use case for that= feature.
=C2=A0 We can do math constants right.=C2=A0 The constants in = math.h are double precision
=C2=A0 and promote computations unless you c= ast or erode long double precision.
=C2=A0 These M_* constants are actua= lly POSIX compatibility - std C++ does not mandate them.
=C2=A0 GCC has = extension for long double versions suffixed by 'l'
=C2=A0 (but a= pparently no float versions with 'f' suffix).
=C2=A0 In any case= these macros don't help with generic code.
=C2=A0 There are several= bespoke versions of generic template math constants around libstdc++
= =C2=A0 using all the pre-variable-template techniques and with all the anno= yances outlined
=C2=A0 by the variable template proposal.

=C2=A0 = 'Grammar' + Bikeshedding
=C2=A0 I find 'm_' gives me fle= xibility in the 'grammar'.=C2=A0 We could do just '_'.=C2= =A0 Or 'c_'.
=C2=A0 Dividing numbers: 1_2 is 1/2, pi_2 is pi/2. = Alt. _div_ seems wordy.
=C2=A0 Multiples of pi: Npi (2pi, 4pi).=C2=A0=C2= =A0 For some reason nobody cares about multiples of e
=C2=A0 but 2e that= would be my proposal for 2*e<> if someone were to want.
=C2=A0 Fu= nction args preceded by '_' (unlike math.h!) make things easier to = read.=C2=A0 Alt. revert that to math.h.
=C2=A0 Lower case because these = are not macros and because that's how we roll in the std library.
=C2=A0 What to put in?
=C2=A0 Help people with intensive calculations.=
=C2=A0 Division and sqrt are are more expensive than multiplication and= integral powers.
=C2=A0 Logs can take a while too.
=C2=A0 On the oth= er hand, logs and square roots of integers appear in series expansions of v= arious
=C2=A0 functions.=C2=A0 Rather than having just a few, perhaps a = substantial table would be in order.

=C2=A0 This is a superset of ma= th.h, <ext/cmath> from libstdc++ (with different grammar).

=C2= =A0 Prevent collision with an extra constants namespace after std.
=C2= =A0 Add namespace versioning.
=C2=A0 Should I insert an experimental nam= espace between std and constants?

=C2=A0 Should we default to double= ?=C2=A0 Doing so might play nice with auto.

=C2=A0 Although it spoil= s the genericity of the template variable constants
=C2=A0 we could offe= r typedefs to the different precisions.=C2=A0 I'll hold off for now.
=C2=A0 I made the constants and math_constants name spaces inline for = the same reason
=C2=A0 library literal operator namespaces are inline: u= sing std to get the math functions
=C2=A0 should give you access to the = constants as well.

=C2=A0 This would go in some <cmath> or ano= ther.
=C2=A0*/

// Toy implementation:

#if __cplusplus <=3D 201103L
# include <bits/c++14_warning.= h>
<= /span>#else

<= span style=3D"color: #008;" class=3D"styled-by-prettify">namespace std
{
namespace experimental
{
inline namespace maths_v1
{
inline namespa= ce constan= ts
{
inline namespace math_constants
{
=C2=A0 /// Constant: @f$ \pi @f$ (see P= OSIX @c M_PI).
=C2=A0 template<typ= ename _Rea= lType><= /span>
=C2=A0 = =C2=A0 con= stexpr _Re= alType
=C2=A0 =C2=A0 m_pi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 3.1415'92653'58979'32384'62643'38327'95028'84195e+0L;

=C2=A0 ///= Constant: @f$ \pi / 2 @f$ (see POSIX @c M_PI_2).
=C2=A0 template<typ= ename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m= _pi_2 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 1.5707'96326'79489'66192'31321'69163'97514'42098e+0L;

=C2=A0 /// Constant: @f$ \pi / 3 @f$.
=C2=A0 <= /span>template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0= =C2=A0 m_pi_3 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 1.0471'97551'19659'77461'54214'46109'31676'28063e+0L;

=C2=A0 /// Constant: @f$ \pi / 4= @f$ (see POSIX @c M_PI_4).
=C2=A0 template<typename _RealType>=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_pi_4 =C2=A0 =C2=A0 = =C2=A0 =C2=A0 =C2=A0=3D 7.8539'81633'97448'30961'56608'45819'87572'10488e-1L;

=C2=A0 /// Constant: @f$ 4 \pi / 3 @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_4pi_3 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =3D 4= ..1887'= 90204'= 78639'= 09846'= 16857'= 84437'= 26705'= 12253e+0L;

=C2=A0 /// Constant: @f$ 2 \pi @f$.
=C2=A0 template<= ;typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2= =A0 m_2pi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =3D 6.2831'85307'17958'64769'25286'76655'90057'68391e+0L;

=C2=A0 /// Constant: @f$ 4 \pi @f$.=
=C2=A0= template<= /span><= typename _RealType>
=C2=A0 =C2=A0 = constexpr<= span style=3D"color: #000;" class=3D"styled-by-prettify"> _RealType
=C2= =A0 =C2=A0 m_4pi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =3D 1.2566'37061'43591'72953'85057'35331'18011'53678e+1L;

=C2=A0 /// Constant: degrees= per radian @f$ 180 / \pi @f$.
=C2=A0 template<typename _RealType>=
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_deg_rad =C2=A0 =C2= =A0 =C2=A0 =3D 5.7295'77951'30823'20876'79815'48141'05170'33237e+1L;

=C2=A0 /// Constant: radians per degree @f$ \pi / 180 @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0= m_rad_deg =C2=A0 =C2=A0 =C2=A0 =3D 1.7453'29251'99432'95769'23690'76848'86127'13443e-2L;

=C2=A0 /// Constant: @f$ \sqrt{\pi / 2} @f$.
= =C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealT= ype
=C2=A0 =C2=A0 m_sqrt_pi_2 =C2=A0 =C2=A0 =3D 1.2533'14137'31550'02512'07882'64240'55226'26505e+0L;

=C2=A0 /// Constant: @f$ 1 / \pi @f= $ (see POSIX @c M_1_PI).
=C2=A0 template<typename _RealType><= br>=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_1_pi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0= =3D= 3.1830'98861'83790'67153'77675'26745'02872'40691e-1L;

=C2= =A0 /// Constant: @f$ 2 / \pi @f$ (see POSIX @c M_2_PI).
=C2=A0 template= <typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 = =C2=A0 m_2_pi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 6.3661'97723'67581'34307'55350'53490'05744'81383e-1L;

=C2=A0 /// Constant: @f$ 1 / \sqrt= {\pi} @f$.=
=C2=A0 template&= lt;typenam= e _RealTyp= e>
=C2=A0 =C2= =A0 conste= xpr _RealT= ype
=C2=A0 =C2=A0 m_1_sqrt_pi =C2=A0 =C2=A0 =3D 5.6418'95835'47756'28694'80794'51560'77258'58438e-1L;

=C2=A0 /// Constant: @f$ \sqrt{\p= i} @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 const= expr _RealType
=C2=A0 =C2=A0 m_sqrt_pi =C2=A0 =C2=A0 =C2=A0 =3D 1.7724&#= 39;53850'90551&#= 39;60272'98167&#= 39;48334'11451&#= 39;82797e+= 0L;=

=C2=A0 /// Constant= : @f$ 2 / \sqrt{\pi} @f$ (see POSIX @c M_2_SQRTPI).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_2_sqrt_pi = =C2=A0 =C2=A0 =3D 1.1283'79167'<= /span>09551'25738'<= /span>96158'90312'<= /span>15451'71688e+0L;<= br>
=C2=A0 /// Constant: Euler's number @f$ e @f$ (see POSIX @c M_E).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_e =C2=A0 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =3D 2.7182'81828'45904'52353'60287'47135'26624'97759e+0L;

=C2=A0 /// Constant: @f$ 1 / e @f$.
=C2=A0 te= mplate<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
= =C2=A0 =C2=A0 m_1_e =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =3D 3.6787'94411'71442'32159'55237'70161'46086'74462e-1L;

=C2=A0 /// Constant: @f$ \l= og_2(e) @f$ (see POSIX @c M_LOG2E).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_log2_e =C2=A0 =C2=A0 =C2=A0 = =C2=A0=3D<= /span> 1.4426'95040'88896'34073'59924'68100'18921'37427e+0L;

= =C2=A0 /// Constant: @f$ \log_2(10) @f$.
=C2=A0 template<typename _Re= alType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_log2_10 = =C2=A0 =C2=A0 =C2=A0 =3D 3.3219'28094'88736'23478'70319'42948'93901'75867e+0L;

=C2=A0 /// Constant: @f$ \log_10(2) @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_log10= _2 =C2=A0 =C2=A0 =C2=A0 =3D 3.= 0102'9= 9956'6= 3981'1= 9521'3= 7388'9= 4724'4= 9302'6= 7680e-1L;

=C2=A0 /// Constant: @f$ \log_10(e) @f$ (see POSIX @c M_LO= G10E).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 conste= xpr _RealType
=C2=A0 =C2=A0 m_log10_e =C2=A0 =C2=A0 =C2=A0 =3D 4.3429= 9;44819'03251= 9;82765'11289= 9;18916'60508= 9;22940e-1= L;<= span style=3D"color: #000;" class=3D"styled-by-prettify">

=C2=A0 /// Constant:= @f$ \log_10(pi) @f$.
=C2=A0 template<typename _RealType>
= =C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_log10_pi =C2=A0 =C2=A0 =C2=A0=3D 4.9714'98726'94133'85435'12682'88290'89887'36507e-1L;

=C2=A0 /// Constan= t: @f$ \ln(2) @f$ (see POSIX @c M_LN2).
=C2=A0 template<typename _Rea= lType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_ln_2 =C2= =A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 6.9314'71805'59945'30941'72321'21458'17656'80748e-1L;

=C2=A0 /// Constant: @f$ \ln(3) @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_ln_= 3 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 1.0986'12288'66810'96913'95245'23692'25257'04648e+0L;

=C2=A0 /// Constant: @f$ \ln(10)@f$ (see POSIX= @c M_LN10).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 = constexpr _RealType
=C2=A0 =C2=A0 m_ln_10 =C2=A0 =C2=A0 =C2=A0 =C2=A0 = =3D 2.3025'85092= '99404'56840= '17991'45468= '43642'07602e+0L;
<= br>=C2=A0 = /// Constant: Euler-Mascheroni @f$ \gamma_E @f$.
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_gamma_e =C2=A0= =C2=A0 =C2=A0 =3D <= /span>5.7721'56649'= 01532'86060'= 65120'90082'= 40243'10432e-1L;=

=C2=A0 /// Constant: Golden Ratio @f$ \phi =3D (1 + \sqrt{5})/2 @f$= ..
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _= RealType
=C2=A0 =C2=A0 m_phi =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =3D 1.61= 80'339= 88'749= 89'484= 82'045= 86'834= 36'563= 81'177= 20e+0L;

=C2= =A0 /// Co= nstant: @f$ \sqrt{2}@f$ (see POSIX @c M_SQRT2).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_sqrt_2 =C2=A0 = =C2=A0 =C2=A0 =C2=A0=3D 1.4142= '13562= '37309= '50488= '01688= '72420= '96980= '78569= e+0L;

=C2=A0 /// Constant: @f$ \sqrt{3} @f$.
=C2=A0 template<t= ypename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0= m_sqrt_3 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 1.7320'50807'56887'72935'27446'34150'58723'66945e+0L;

=C2=A0 /// Constant: @f$ \sqrt{5} @f$.=
=C2=A0 template<= span style=3D"color: #660;" class=3D"styled-by-prettify"><typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2= =A0 m_sqrt_5 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 2.2360'67977'49978'96964'09173'66873'12762'35440e+0L;

=C2=A0 /// Constant: @f$ \sqrt{7} @f$.=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _Real= Type
=C2=A0 =C2=A0 m_sqrt_7 =C2=A0 =C2=A0 =C2=A0 =C2=A0=3D 2.6457'51311= '06459'05905= '01615'75363= '92604'25706e+0L;

=C2=A0 = /// Constant: @f$= 1 / \sqrt{2}@f$ (see POSIX @c M_SQRT1_2).
=C2=A0 template<typename _RealType>
=C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_1_sqrt_2 =C2=A0 =C2= =A0 =C2=A0= =3D 7.0710'67811'86547'52440'08443'62104'84903'92845e-1L;
<= br>=C2=A0 /// Constant: Catalan's @f$ G =3D 1 - 1/9 + 1/25 - 1/49 + 1/81 - ... @f$.
=C2=A0 template<typename _RealType>
= =C2=A0 =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_catalan =C2=A0 =C2=A0 =C2=A0 =3D 9.1596'55941'77219'01505'46035'14932'38411'07741e-1L;

=C2=A0 /// Constan= t: @f$ \pi^2/6 @f$.
=C2=A0 template<typename _RealType>
=C2=A0 = =C2=A0 constexpr _RealType
=C2=A0 =C2=A0 m_pi2_6 =C2=A0 =C2=A0 =C2=A0 = =C2=A0 =3D 1.6449'34066'84822'64364'72415'16664'60251'89218e+0L;

= } <= span style=3D"color: #800;" class=3D"styled-by-prettify">// inline namespac= e math_constants
= } <= span style=3D"color: #800;" class=3D"styled-by-prettify">// inline namespac= e constants
} // inline namespace mat= hs_v1
<= /span>} // namespace experimental
} // namespace std

#endif // C++14



----------------= ----------------------------------------------------------

--
You received this message because you are subscribed to the Google Groups &= quot;ISO C++ Standard - Future Proposals" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to std-proposa= ls+unsubscribe@isocpp.org.
To post to this group, send email to std-proposals@isocpp.org.
To view this discussion on the web visit https://groups.google.= com/a/isocpp.org/d/msgid/std-proposals/6bccd7e5-82ad-4c59-9f72-f7fb6639c0ca= %40isocpp.org.
------=_Part_952_1398975516.1489099472971-- ------=_Part_951_1972070828.1489099472968-- .