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Authors of MPFR (in chronological order of initial contribution):
Guillaume Hanrot Main author
Fabrice Rouillier Original version of mul_ui.c, gmp_op.c
Paul Zimmermann Main author
Sylvie Boldo Original version of agm.c and log.c
Jean-Luc Rémy Original version of zeta.c
Emmanuel Jeandel Original version of exp3.c, const_pi.c, sincos.c
Mathieu Dutour acos.c, asin.c, atan.c and early gamma.c
Vincent Lefèvre Main author
David Daney Hyperbolic and inverse hyperbolic functions, base-2
and base-10 exponential and logarithm, factorial
Alain Delplanque Rewritten get_str.c
Ludovic Meunier Error function (erf.c)
Patrick Pélissier Main author
Laurent Fousse Original version of sum.c
Damien Stehlé Function mpfr_get_ld_2exp
Philippe Théveny Main author
Sylvain Chevillard Original version of ai.c
Charles Karney mpfr_nrandom and mpfr_erandom functions
Fredrik Johansson New version of mpfr_const_euler
Mickaël Gastineau MPFRbench program
The main authors are included in the MPFR mailing-list <mpfr@inria.fr>.
This is the preferred way to contact us. For further information, please
look at the MPFR web page <https://www.mpfr.org/>.

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Copyright 1999, 2001-2019 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
##############################################################################
Known bugs:
* The overflow/underflow exceptions may be badly handled in some functions;
specially when the intermediary internal results have exponent which
exceeds the hardware limit (2^30 for a 32 bits CPU, and 2^62 for a 64 bits
CPU) or the exact result is close to an overflow/underflow threshold.
* Under Linux/x86 with the traditional FPU, some functions do not work
if the FPU rounding precision has been changed to single (this is a
bad practice and should be useless, but one never knows what other
software will do).
* Some functions do not use MPFR_SAVE_EXPO_* macros, thus do not behave
correctly in a reduced exponent range.
* Function hypot gives incorrect result when on the one hand the difference
between parameters' exponents is near 2*MPFR_EMAX_MAX and on the other hand
the output precision or the precision of the parameter with greatest
absolute value is greater than 2*MPFR_EMAX_MAX-4.
Potential bugs:
* Possible incorrect results due to internal underflow, which can lead to
a huge loss of accuracy while the error analysis doesn't take that into
account. If the underflow occurs at the last function call (just before
the MPFR_CAN_ROUND), the result should be correct (or MPFR gets into an
infinite loop). TODO: check the code and the error analysis.
* Possible bugs with huge precisions (> 2^30) and a 32-bit ABI, in particular
undetected integer overflows. TODO: use the MPFR_ADD_PREC macro.
* Possible bugs if the chosen exponent range does not allow to represent
the range [1/16, 16].
* Possible infinite loop in some functions for particular cases: when
the exact result is an exactly representable number or the middle of
consecutive two such numbers. However for non-algebraic functions, it is
believed that no such case exists, except the well-known cases like cos(0)=1,
exp(0)=1, and so on, and the x^y function when y is an integer or y=1/2^k.
* The mpfr_set_ld function may be quite slow if the long double type has an
exponent of more than 15 bits.
* mpfr_set_d may give wrong results on some non-IEEE architectures.
* Error analysis for some functions may be incorrect (out-of-date due
to modifications in the code?).
* Possible use of non-portable feature (pre-C99) of the integer division
with negative result.

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<h1>Frequently Asked Questions about <cite><acronym>GNU</acronym> <acronym>MPFR</acronym></cite></h1>
<p><strong>Important notice: Problems with a particular version of
<cite><acronym>MPFR</acronym></cite> are discussed in the corresponding
bugs page.</strong></p>
<p>The latest version of this <acronym>FAQ</acronym> is available at
<a href="https://www.mpfr.org/faq.html">https://www.mpfr.org/faq.html</a>.
Please look at this version if possible.</p>
<ol>
<li><a href="#mpfr_vs_mpf">What are the differences between
<cite><acronym>MPF</acronym></cite> from <cite><acronym>GMP</acronym></cite>
and <cite><acronym>MPFR</acronym></cite>?</a></li>
<li><a href="#mpf2mpfr">How to convert my program written using
<cite><acronym>MPF</acronym></cite> to
<cite><acronym>MPFR</acronym></cite>?</a></li>
<li><a href="#no_libgmp">At configure time, I get the error: <q>libgmp not found or uses a different ABI.</q></a></li>
<li><a href="#undef_ref1">I get undefined reference to <code>__gmp_get_memory_functions</code>.</a></li>
<li><a href="#undef_ref2">When I link my program with
<cite><acronym>MPFR</acronym></cite>, I get undefined reference
to <code>__gmpXXXX</code>.</a></li>
<li><a href="#crash_high_prec">My program crashes with high precisions.</a></li>
<li><a href="#accuracy">Though I have increased the precision, the results
are not more accurate.</a></li>
<li><a href="#detect_mpfr">How can I detect <cite><acronym>MPFR</acronym></cite>
installation using <cite>autoconf</cite> or <cite>pkg-config</cite>?</a></li>
<li><a href="#cite">How to cite <cite><acronym>MPFR</acronym></cite> in a
scientific publication?</a></li>
<li><a href="#fpic">When I build <cite><acronym>MPFR</acronym></cite>, I get
an error asking me to recompile with <samp>-fPIC</samp>.</a></li>
</ol>
<dl class="faq">
<dt id="mpfr_vs_mpf">1. What are the differences between
<cite><acronym>MPF</acronym></cite> from <cite><acronym>GMP</acronym></cite>
and <cite><acronym>MPFR</acronym></cite>?</dt>
<dd><p>The main differences are:</p>
<ul>
<li><p>The precision of a <cite><acronym>MPFR</acronym></cite> variable
is the <em>exact</em> number of bits used for its mantissa, whereas in
<cite><acronym>MPF</acronym></cite>, the precision requested by the user
is a minimum value (<cite><acronym>MPF</acronym></cite> generally uses a
higher precision). With the additional difference below, this implies that
the <cite><acronym>MPFR</acronym></cite> results do not depend on the
number of bits (16, 32, 64 or more) of the underlying architecture.</p></li>
<li><p>As a consequence, <cite><acronym>MPFR</acronym></cite> uses a
base-2 exponent, whereas in <cite><acronym>MPF</acronym></cite>, this
is a base-2<sup>32</sup> or base-2<sup>64</sup> exponent, depending on
the limb size. For this reason (and other internal ones), the maximum
exponent range in <cite><acronym>MPFR</acronym></cite> is different
(and smaller, if the exponent is represented by the same type as in
<cite><acronym>MPF</acronym></cite>).</p></li>
<li><p><cite><acronym>MPFR</acronym></cite> provides an additional rounding
mode argument to its functions; furthermore, it is guaranteed that the
result of any operation is the nearest possible floating-point value from
the exact result (considering the input variables as exact values), taking
into account the precision of the destination variable and the rounding
mode. <cite><acronym>MPFR</acronym></cite> also says whether the rounded
result is above or below the exact result.</p></li>
<li><p><cite><acronym>MPFR</acronym></cite> supports much more functions
(in particular transcendental functions such as exponentials, logarithms,
trigonometric functions and so on) and special values: signed zeros,
infinities, not-a-number (NaN).</p></li>
</ul></dd>
<dt id="mpf2mpfr">2. How to convert my program written using
<cite><acronym>MPF</acronym></cite> to
<cite><acronym>MPFR</acronym></cite>?</dt>
<dd><p>You need to add <q><code>r</code></q> to the function names, and to
specify the rounding mode (<code>MPFR_RNDN</code> for rounding to nearest,
<code>MPFR_RNDZ</code> for rounding toward zero, <code>MPFR_RNDU</code>
for rounding toward plus infinity, <code>MPFR_RNDD</code> for rounding
toward minus infinity). You can also define macros as follows:
<code class="block-code">#define mpf_add(a, b, c) mpfr_add(a, b, c, MPFR_RNDN)</code></p>
<p>The header file <samp>mpf2mpfr.h</samp> from the
<cite><acronym>MPFR</acronym></cite> distribution automatically
redefines all <cite><acronym>MPF</acronym></cite> functions in this
way, using the default <cite><acronym>MPFR</acronym></cite> rounding
mode. Thus you simply need to add the following line in all your files
using <cite><acronym>MPF</acronym></cite> functions:
<code class="block-code">#include &lt;mpf2mpfr.h&gt;</code>
just after the <samp>gmp.h</samp> and <samp>mpfr.h</samp>
header files. If the program uses <cite><acronym>MPF</acronym></cite>
internals (such as direct access to <code>__mpf_struct</code> members),
additional changes will be needed.</p></dd>
<dt id="no_libgmp">3. At configure time, I get the error: <q>libgmp not found or uses a different ABI.</q></dt>
<dd><p>This test (<samp>checking for __gmpz_init in -lgmp</samp>) comes
after the <samp>gmp.h</samp> detection. The failure occurs either because
the <cite><acronym>GMP</acronym></cite> library could not be found
(as it is not in the provided library search paths) or because the
<cite><acronym>GMP</acronym></cite> library that was found does not have
the expected <acronym title="Application Binary Interface">ABI</acronym>
(<abbr>e.g.</abbr> 32-bit <abbr>vs</abbr> 64-bit). The former problem can be
due to the fact that a static build of <cite><acronym>MPFR</acronym></cite>
was requested while only a shared <cite><acronym>GMP</acronym></cite> library
is installed (or the opposite, but another error can also show up in this
case, see the <a href="#fpic">question about <samp>-fPIC</samp></a>). The
latter problem can have several causes:</p>
<ul>
<li>A wrong libgmp library has been picked up. This can occur if you have
several <cite><acronym>GMP</acronym></cite> versions installed on the
machine and something is wrong with the provided library search paths.</li>
<li>Wrong compiler options (<samp>CFLAGS</samp>) were given. In general, the
presence or absence of the <samp>-m64</samp> compiler option must match the
library <acronym title="Application Binary Interface">ABI</acronym>.</li>
<li>A wrong <samp>gmp.h</samp> file has been picked up (if you have several
<cite><acronym>GMP</acronym></cite> versions installed). Indeed, by default,
<cite><acronym>MPFR</acronym></cite> gets the compiler options from the
<samp>gmp.h</samp> file (with <cite><acronym>GMP</acronym></cite> 4.2.3
or later); this is needed because <cite><acronym>GMP</acronym></cite> does
not necessarily use the default <acronym>ABI</acronym>. The consequence is
that if the <samp>gmp.h</samp> file is associated with a library using a
different <acronym>ABI</acronym>, the <acronym>ABI</acronym>-related options
will be incorrect. Hence the failure.</li>
</ul>
<p>Note: The <samp>config.log</samp> output gives more information
than the error message. In particular, see the output of the test:
<samp>checking for CC and CFLAGS in gmp.h</samp>; it should give you
the default compiler options (from <samp>gmp.h</samp>).</p>
<p>See also the answer to the <a href="#undef_ref1">next question</a>.</p></dd>
<dt id="undef_ref1">4. I get undefined reference to <code>__gmp_get_memory_functions</code>.</dt>
<dd><p>Note: this was mainly a problem when upgrading from
<cite><acronym>GMP</acronym></cite> 4.1.4 to a later version,
but information given below may still be useful in other cases,
when several <cite><acronym>GMP</acronym></cite> libraries are
installed on the same machine.</p>
<p>If you get such an error, in particular when running
<samp>make check</samp>, then this probably means that you are using
the header file from <cite><acronym>GMP</acronym></cite> 4.2.x but the
<cite><acronym>GMP</acronym></cite> 4.1.4 library. This can happen if
several <cite><acronym>GMP</acronym></cite> versions are installed on
your machine (<abbr>e.g.</abbr>, one provided by the system in
<samp>/usr/{include,lib}</samp> and a new one installed by the owner or
administrator of the machine in <samp>/usr/local/{include,lib}</samp>)
and your include and library search paths are inconsistent. On various
<acronym>GNU</acronym>/Linux machines, this is unfortunately the case
by default (<samp>/usr/local/include</samp> is in the default include
search path, but <samp>/usr/local/lib</samp> is <em>not</em> in the
default library search path). Typical errors are:
<samp class="block-code">undefined reference to `__gmp_get_memory_functions'</samp>
in <samp>make check</samp>. The best solution is to add
<samp>/usr/local/include</samp> to your <var class="env">C_INCLUDE_PATH</var>
environment variable and to add <samp>/usr/local/lib</samp> to your
<var class="env">LIBRARY_PATH</var> and <var class="env">LD_LIBRARY_PATH</var>
environment variables (and/or <var class="env">LD_RUN_PATH</var>).
Alternatively, you can use <samp>--with-gmp*</samp> configure options,
<abbr>e.g.</abbr> <samp>--with-gmp=/usr/local</samp>, but <strong>this is
not guaranteed to work</strong> (in particular with <samp>gcc</samp> and
system directories such as <samp>/usr</samp> or <samp>/usr/local</samp>),
and other software that uses <cite><acronym>GMP</acronym></cite> and/or
<cite><acronym>MPFR</acronym></cite> will need correct paths too;
environment variables allow you to set them in a global way.</p>
<p>Other information can be given in the <samp>INSTALL</samp> file and
<samp>ld</samp> manual. Please look at them for more details. See also
the <a href="#undef_ref2">next question</a>.</p></dd>
<dt id="undef_ref2">5. When I link my program with
<cite><acronym>MPFR</acronym></cite>, I get undefined reference
to <code>__gmpXXXX</code>.</dt>
<dd><p>Link your program with <cite><acronym>GMP</acronym></cite>. Assuming
that your program is <samp>foo.c</samp>, you should link it using:
<samp class="block-code">cc link.c -lmpfr -lgmp</samp>
<cite><acronym>MPFR</acronym></cite> library reference (<samp>-lmpfr</samp>)
should be before <cite><acronym>GMP</acronym></cite>'s one
(<samp>-lgmp</samp>). Another solution is, with <acronym>GNU</acronym>
<samp>ld</samp>, to give all the libraries inside a group:
<samp class="block-code">gcc link.c -Wl,--start-group libgmp.a libmpfr.a -Wl,--end-group</samp>
See <samp>INSTALL</samp> file and <samp>ld</samp> manual for more
details.</p>
<p>If you used correct link options, but still get an error, this may mean
that your include and library search paths are inconsistent. Please see the
<a href="#undef_ref1">previous question</a>.</p></dd>
<dt id="crash_high_prec">6. My program crashes with high precisions.</dt>
<dd><p>Your stack size limit may be too small; indeed, by default,
<cite><acronym>GMP</acronym></cite> 4.1.4 and below allocates all
temporary results on the stack, and in very high precisions, this
limit may be reached. You can solve this problem in different ways:</p>
<ul>
<li><p>You can upgrade to <cite><acronym>GMP</acronym></cite> 4.2 (or above),
which now makes temporary allocations on the stack only when they are
small.</p></li>
<li><p>You can increase the stack size limit with the <samp>limit</samp>,
<samp>unlimit</samp> or <samp>ulimit</samp> command, depending on your
shell. This may fail on some systems, where the maximum stack size cannot
be increased above some value.</p></li>
<li><p>You can rebuild both <cite><acronym>GMP</acronym></cite> and
<cite><acronym>MPFR</acronym></cite> to use another allocation method.</p></li>
</ul></dd>
<dt id="accuracy">7. Though I have increased the precision, the results
are not more accurate.</dt>
<dd><p>The reason may be the use of C floating-point numbers. If you want
to store a floating-point constant to a <code>mpfr_t</code>, you should use
<code>mpfr_set_str</code> (or one of the <cite><acronym>MPFR</acronym></cite>
constant functions, such as <code>mpfr_const_pi</code> for &#960;) instead
of <code>mpfr_set_d</code> or <code>mpfr_set_ld</code>. Otherwise the
floating-point constant will be first converted into a reduced-precision
(<abbr>e.g.</abbr>, 53-bit) binary number before
<cite><acronym>MPFR</acronym></cite> can work with it. This is the case
in particular for most exact decimal numbers, such as 0.17, which are
not exactly representable in binary.</p>
<p>Also remember that <cite><acronym>MPFR</acronym></cite> does not track
the accuracy of the results: copying a value <var>x</var> to <var>y</var>
with <code>mpfr_set (y, x, MPFR_RNDN)</code> where the variable <var>y</var>
is more precise than the variable <var>x</var> will not make it more
accurate; the (binary) value will remain unchanged.</p></dd>
<dt id="detect_mpfr">8. How can I detect <cite><acronym>MPFR</acronym></cite>
installation using <cite>autoconf</cite> or <cite>pkg-config</cite>?</dt>
<dd><p>The <cite><acronym>MPFR</acronym></cite> team does not currently
recommend any <cite>autoconf</cite> code, but a section will later
be added to the <cite><acronym>MPFR</acronym></cite> manual.
Limited <cite>pkg-config</cite> support has been added for
<cite><acronym>MPFR</acronym></cite><EFBFBD>4.0.0; example:</p>
<pre style="margin-left: 2em">cc myprogram.c $(pkg-config --cflags --libs mpfr)</pre></dd>
<dt id="cite">9. How to cite <cite><acronym>MPFR</acronym></cite> in a
scientific publication?</dt>
<dd><p>To properly cite <cite><acronym>MPFR</acronym></cite> in a scientific
publication, please cite the
<a href="https://doi.org/10.1145/1236463.1236468"><acronym title="Association for Computing Machinery">ACM</acronym>
<acronym title="Transactions on Mathematical Software">TOMS</acronym>
paper</a>
and/or the library web page
<a href="https://www.mpfr.org/">https://www.mpfr.org</a>. If your publication
is related to a particular release of <cite><acronym>MPFR</acronym></cite>,
for example if you report timings, please also indicate the release number
for future reference.</p></dd>
<dt id="fpic">10. When I build <cite><acronym>MPFR</acronym></cite>, I get
an error asking me to recompile with <samp>-fPIC</samp>.</dt>
<dd><p>A typical error looks like:</p>
<p><tt>/usr/bin/ld: <em>/path/to/</em>libgmp.a(realloc.o): relocation
R_X86_64_32 against `.rodata.str1.1' can not be used when making a
shared object; recompile with -fPIC<br />
<em>/path/to/</em>libgmp.a: could not read symbols: Bad value<br />
collect2: ld returned 1 exit status</tt></p>
<p>The probable reason is that you tried to build
<cite><acronym>MPFR</acronym></cite> with the shared library enabled (this
is the default), while only a static <cite><acronym>GMP</acronym></cite>
library could be found. To solve this problem, either rebuild and reinstall
<cite><acronym>GMP</acronym></cite> without the <samp>--disable-shared</samp>
configure option, or configure <cite><acronym>MPFR</acronym></cite> with
<samp>--disable-shared</samp>. If you did this and still get the above
error, the cause may be conflicting <cite><acronym>GMP</acronym></cite>
versions installed on your system; please check that your search path
settings are correct.</p>
<p>Additional note about the last sentence: Under <acronym>GNU</acronym>/Linux
(for instance), the linker takes the first library found in the library search
path, whether it is dynamic or static. The default behavior under darwin is
different, but <cite><acronym>MPFR</acronym></cite> will change it.</p></dd>
<!-- Reference concerning darwin: see MPFR_LD_SEARCH_PATHS_FIRST
in MPFR's configure.{ac,in} and acinclude.m4 -->
</dl>
</body>
</html>

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Copyright 2000-2019 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
##############################################################################
Changes from version 4.0.1 to version 4.0.2:
- Corrected minimal GMP version in the INSTALL file and the MPFR manual.
- Option -pedantic is now always removed from __GMP_CFLAGS (see INSTALL).
- Shared caches: cleanup; really detect lock failures (abort in this case).
- Improved MPFR manual. In particular, corrected/completed the
mpfr_get_str description in order to follow the historical behavior
and GMP's mpf_get_str function.
- Bug fixes (see ChangeLog file).
Changes from version 4.0.0 to version 4.0.1:
- Improved MPFR manual.
- Improved __GMP_CC and __GMP_CFLAGS retrieval (in particular for MS Windows).
- Fixed a build failure on some platforms when --with-gmp-build is used.
- Bug fixes (see ChangeLog file), in particular in mpfr_div_ui, which
could yield an incorrectly rounded result to nearest when using
different precisions; this bug had been present since the introduction
of mpfr_div_ui, and in MPFR 4.0.0, it was affecting mpfr_div too.
- New: optional "make check-exported-symbols", mainly for the MPFR developers
and binary distributions, to check that MPFR does not define symbols with a
GMP reserved prefix (experimental).
Changes from versions 3.1.* to version 4.0.0:
- The "dinde aux marrons" release.
- MPFR now depends on GMP 5.0+ instead of 4.1+.
- API change:
Applications that call GMP's mp_set_memory_functions function to change
the allocators must first call the new function mpfr_mp_memory_cleanup
in all threads where MPFR is potentially used; this new function is
currently equivalent to mpfr_free_cache.
The reason is that the way memory allocation is done by MPFR has changed
(again), so that the current GMP allocators are used (since for some
applications, the old allocators may become invalid).
Note: Freeing the caches like this might have a performance impact on some
particular applications; if this is an issue, this could be handled for a
future MPFR version.
- Mini-gmp support via the --enable-mini-gmp configure option (experimental).
- The minimum precision MPFR_PREC_MIN is now 1, with rounding defined as
in the errata of IEEE 754-2008 and in the following IEEE 754 revision
(ties rounded away from zero).
- Shared caches for multithreaded applications.
New function mpfr_free_cache2.
- Partial support of MPFR_RNDF (faithful rounding).
- New functions: mpfr_fpif_export and mpfr_fpif_import to export and import
numbers in a floating-point interchange format, independent both on the
number of bits per word and on the endianness.
- New function mpfr_fmodquo to return the low bits of the quotient
corresponding to mpfr_fmod.
- New functions mpfr_flags_clear, mpfr_flags_set, mpfr_flags_test,
mpfr_flags_save and mpfr_flags_restore to operate on groups of flags.
- New functions mpfr_set_float128 and mpfr_get_float128 to convert from/to
the __float128 type (requires --enable-float128 and compiler support).
- New functions mpfr_buildopt_float128_p and mpfr_buildopt_sharedcache_p.
- New functions mpfr_rint_roundeven and mpfr_roundeven, completing the
other similar round-to-integer functions for rounding to nearest with
the even-rounding rule.
- New macro mpfr_round_nearest_away to add partial emulation of the
rounding to nearest-away (as defined in IEEE 754-2008).
- New functions mpfr_nrandom and mpfr_erandom to generate random numbers
following normal and exponential distributions respectively.
- New functions mpfr_fmma and mpfr_fmms to compute a*b+c*d and a*b-c*d.
- New function mpfr_rootn_ui, similar to mpfr_root, but agreeing with the
rootn function of the IEEE 754-2008 standard.
- New functions mpfr_log_ui to compute the logarithm of an integer,
mpfr_gamma_inc for the incomplete Gamma function.
- New function mpfr_beta for the Beta function (incomplete, experimental).
- New function mpfr_get_q to convert a floating-point number into rational.
- The mpfr_dump function is now described in the manual; its output format
has slightly changed.
- The mpfr_eint function now returns the value of the E1/eint1 function
for negative argument.
- The behavior of the mpfr_set_exp function changed, as it could easily
yield undefined behavior in some cases (this modifies both the API and
the ABI).
- In function mpfr_urandom, the next random state no longer depends on the
current exponent range and the rounding mode. The exceptions due to the
rounding of the random number are now correctly generated, following the
uniform distribution.
- Functions mpfr_grandom and mpfr_root are deprecated and will be removed
in a future release.
- Complete rewrite of function mpfr_sum, which now works in all cases (the
old one could take all the memory and/or crash with inputs of different
magnitudes in case of huge cancellation or table maker's dilemma). The
sign of an exact zero result is now specified, and the return value is
now the usual ternary value. Note that the position of "const" in the
mpfr_sum prototype has been fixed (the manual was correct); user code
should not be affected.
- Old, deprecated macros mpfr_add_one_ulp and mpfr_sub_one_ulp removed.
The mpfr_next* functions should be used instead.
- Internally, improved caching: a minimum of 10% increase of the precision
is guaranteed to avoid too many recomputations.
- Added internal small-precision mpz_t pool, which aims to avoid the
overhead of memory allocation, in particular.
New function mpfr_free_pool.
- Added configure option --enable-assert=none to avoid checking any assertion.
- The --enable-decimal-float configure option no longer requires
--with-gmp-build, and support for decimal floats is now automatically
detected by default (similarly for support for __float128).
- Updated tuning parameters.
- Better support for Automake 1.13+ (now used to generate the tarball).
- Dropped K&R C compatibility.
- Improved MPFR manual.
- New MPFRbench program (see the tools/bench directory).
- Major speedup in mpfr_add, mpfr_sub, mpfr_mul, mpfr_div and mpfr_sqrt when
all operands have the same precision and this precision is less than twice
the number of bits per word, e.g., less than 128 on a 64-bit computer.
- Speedup by a factor of almost 2 in the double <--> mpfr conversions
(mpfr_set_d and mpfr_get_d).
- Speedup in mpfr_log1p and mpfr_atanh for small arguments.
- Speedup in the mpfr_const_euler function (contributed by Fredrik Johansson),
in the computation of Bernoulli numbers (used in mpfr_gamma, mpfr_li2,
mpfr_digamma, mpfr_lngamma and mpfr_lgamma), in mpfr_div, in mpfr_fma
and mpfr_fms.
- Test coverage: 96.3% lines of code.
- Bug fixes. In particular: a speed improvement when the --enable-assert
or --enable-assert=full configure option is used with GCC; mpfr_get_str
now sets the NaN flag on NaN input and the inexact flag when the conversion
is inexact. For a full list, see http://www.mpfr.org/mpfr-3.1.6/#fixed
and the same section for any previous 3.1.x version (follow the links
in the "Changes..." sections).
- Microsoft Windows: Added support for thread-safe DLL (shared library).
Tested with MinGW, ICC and MSVC.
- Limited pkg-config support.
- Autotools: Under Linux, make sure that the old dtags (when supported)
are used if LD_LIBRARY_PATH is defined; otherwise "make check" would
check an installed, compatible MPFR library found in LD_LIBRARY_PATH
instead of the one that has been built with "make".
- New: optional "make check-gmp-symbols", mainly for binary distributions,
to check that MPFR does not use GMP internal symbols (experimental).
Changes from versions 3.0.* to version 3.1.0:
- The "canard à l'orange" release.
- The MPFR source has been reorganized.
- Dropped ansi2knr support.
- TLS support is now detected automatically. If TLS is supported, MPFR is
built as thread safe by default. To disable TLS explicitly, configure
MPFR with --disable-thread-safe.
- New --enable-gmp-internals configure option to use GMP's undocumented
functions (not from the public API). Note that library versioning is
not guaranteed to work if this option is used.
- The mpfr_urandom and mpfr_urandomb functions now return identical values
on processors with different word size (assuming the same random seed, and
since the GMP random generator does not depend itself on the word size,
cf https://gmplib.org/list-archives/gmp-devel/2010-September/001642.html).
- The mpfr_add_one_ulp and mpfr_sub_one_ulp macros (which are obsolete and
no more documented) will be removed in a future release.
- Speed improvement for the mpfr_sqr and mpfr_div functions using Mulders'
algorithm. As a consequence, other functions using those routines are
also faster.
- Much faster formatted output (mpfr_printf, etc.) with %Rg and similar.
- The --with-gmp-build configure option can now be used when the GMP
source directory and the GMP build directory are different (without
having to copy header files manually as before).
- New functions mpfr_buildopt_gmpinternals_p, mpfr_buildopt_tune_case,
mpfr_frexp, mpfr_grandom and mpfr_z_sub.
- New divide-by-zero exception (flag) and associated functions.
- The mpfr.h header can be included several times, while still supporting
optional functions (see Section "Headers and Libraries" in the manual).
- Updated tuning parameters.
- Improved MPFR manual.
- MPFR tests: libtool no longer generates wrapper scripts with "make check"
(so that running the tests under valgrind or gdb is easier).
- Internal change: the logging mechanism has been improved.
- Test coverage: 95.2% lines of code.
- Bug fixes, in particular a huge inefficiency in mpfr_exp (when the
target precision is less than MPFR_EXP_THRESHOLD) on hard-to-round
cases, which can take several minutes.
Note: The mpfr_subnormalize implementation up to MPFR 3.0.0 did not change
the flags. In particular, it did not follow the generic rule concerning
the inexact flag (and no special behavior was specified). The case of the
underflow flag was more a lack of specification.
Changes from versions 2.4.* to version 3.0.0:
- The "boudin aux pommes" release.
- MPFR 3.0.0 is binary incompatible with previous versions but (almost)
API compatible. More precisely the obsolete functions mpfr_random
and mpfr_random2 have been removed, the meaning of the return type
of the function mpfr_get_f has changed, and the return type of the
function mpfr_get_z is now int instead of void. In practice, this
should not break any existing code.
- MPFR is now distributed under the GNU Lesser General Public License
version 3 or later (LGPL v3+).
- Rounding modes GMP_RNDx are now MPFR_RNDx (GMP_RNDx kept for
compatibility).
- A new rounding mode (MPFR_RNDA) is available to round away from zero.
- The rounding mode type is now mpfr_rnd_t (as in previous versions,
both mpfr_rnd_t and mp_rnd_t are accepted, but mp_rnd_t may be
removed in the future).
- The precision type is now mpfr_prec_t (as in previous versions, both
mpfr_prec_t and mp_prec_t are accepted, but mp_prec_t may be removed
in the future) and it is now signed (it was unsigned in MPFR 2.*, but
this was not documented). In practice, this change should not affect
existing code that assumed nothing on the precision type.
- MPFR now has its own exponent type mpfr_exp_t, which is currently
the same as GMP's mp_exp_t.
- Functions mpfr_random and mpfr_random2 have been removed.
- mpfr_get_f and mpfr_get_z now return a ternary value.
- mpfr_strtofr now accepts bases from 37 to 62.
- mpfr_custom_get_mantissa was renamed to mpfr_custom_get_significand
(mpfr_custom_get_mantissa is still available via a #define).
- Functions mpfr_get_si, mpfr_get_ui, mpfr_get_sj, mpfr_get_uj,
mpfr_get_z and mpfr_get_z_2exp no longer have cases with undefined
behavior; in these cases, the behavior is now specified, and in
particular, the erange flag is set.
- New functions mpfr_buildopt_tls_p and mpfr_buildopt_decimal_p giving
information about options used at MPFR build time.
- New function mpfr_regular_p.
- New function mpfr_set_zero.
- New function mpfr_digamma.
- New function mpfr_ai (incomplete, experimental).
- New functions mpfr_set_flt and mpfr_get_flt to convert from/to the
float type.
- New function mpfr_urandom.
- New function mpfr_set_z_2exp (companion to mpfr_get_z_2exp, which
was renamed from mpfr_get_z_exp in previous versions).
- New function mpfr_min_prec.
- Speed improvement for large precisions in the trigonometric functions
(mpfr_sin, mpfr_cos, mpfr_tan, mpfr_sin_cos): speedup of about 2.5
for 10^5 digits, of about 5 for 10^6 digits.
- Speed improvement for large precisions of the inverse trigonometric
functions (arcsin, arccos, arctan): about 2 for 10^3 digits, up to
2.7 for 10^6 digits.
- Some documentation files are installed in $docdir.
- The detection of a GMP build directory (more precisely, the internal
header files of GMP) was previously done separately from the use of
the --with-gmp-build configure option. This was not consistent with
the documentation and with other parts of the configure script. So,
as of MPFR 3.0.0, the internal header files of GMP are now used if
and only if the --with-gmp-build configure option is given.
- The configure script recognizes some extra "long double" formats
(double big endian, double little endian, double-double big endian).
- MPFR manual: added "API Compatibility" section.
- Test coverage: 97.1% lines of code.
- Bug fixes.
Changes from versions 2.3.* to version 2.4.0:
- The "andouillette sauce moutarde" release.
- MPFR is now a GNU package.
- Changes in the behavior of mpfr_strtofr and in its documentation
concerning particular cases where the code and the documentation
did not match; this change is also present in MPFR 2.3.1.
- Behavior of mpfr_check_range changed: if the value is an inexact
infinity, the overflow flag is set (in case it was lost); this
change is also present in MPFR 2.3.2.
- Function mpfr_init_gmp_rand (only defined when building MPFR without
the --with-gmp-build configure option) is no longer defined at all.
This function was private and not documented, and was used only in
the MPFR test suite. User code that calls it is regarded as broken
and may fail as a consequence. Running the old test suite against
MPFR 2.4.0 may also fail.
- New functions:
* between a MPFR number and a double: mpfr_add_d, mpfr_sub_d,
mpfr_d_sub, mpfr_mul_d, mpfr_div_d, mpfr_d_div,
* formatted input/output:
mpfr_printf, mpfr_fprintf, mpfr_vprintf, mpfr_vfprintf,
mpfr_sprintf, mpfr_snprintf, mpfr_vsprintf, mpfr_vsnprintf,
mpfr_asprintf, mpfr_vasprintf.
* mpfr_sinh_cosh, mpfr_li2, mpfr_modf, mpfr_fmod, mpfr_rec_sqrt.
- Configure test for TLS support.
- Get default $CC and $CFLAGS from gmp.h (__GMP_CC / __GMP_CFLAGS,
which are available as of GMP 4.2.3).
- Documented the fact that mpfr_random and mpfr_random2 will be
suppressed in the next release, and that the specification of
mpfr_eq may change in the next release (for compatibility with
the mpf layer of GMP).
- Test coverage: 96.7% lines of code.
- Bug fixes.
Changes from versions 2.2.* to version 2.3.0:
- The mpfr.info file is now installed in the share subdirectory
(as required by the Filesystem Hierarchy Standard); see output
of "./configure --help".
- The shared library is now enabled by default. If the MPFR build
fails on your platform, try the --disable-shared configure option
to disable the shared library.
- Thread-safe support with Microsoft Visual compiler.
- New functions mpfr_j0, mpfr_j1, mpfr_jn, mpfr_y0, mpfr_y1, mpfr_yn,
mpfr_lgamma, mpfr_remainder, mpfr_remquo, mpfr_fms, mpfr_signbit,
mpfr_setsign, mpfr_copysign, mpfr_get_patches.
- Functions mpfr_sin, mpfr_cos and mpfr_sin_cos improved (argument
reduction).
- More detailed MPFR manual.
- Improved tests (make check).
- Bug fixes.
Changes from versions 2.1.* to version 2.2.0:
- Bug fixes.
- new functions mpfr_set_overflow, mpfr_set_underflow, mpfr_set_inexflag,
mpfr_set_erangeflag, mpfr_set_nanflag, mpfr_erfc, mpfr_atan2, mpfr_pow_z,
mpfr_subnormalize, mpfr_const_catalan, mpfr_sec, mpfr_csc, mpfr_cot,
mpfr_root, mpfr_eint, mpfr_get_f, mpfr_sech, mpfr_csch, mpfr_coth,
mpfr_lngamma.
- new macro: MPFR_VERSION_STRING
- Remove the exported MPFR variables from mpfr.h to mpfr-impl.h.
(They were undocumented, so programs which respect the API still work).
- Grep CC and CFLAGS from GMP Makefile if possible.
- Math functions are faster (both average and worst cases).
- Better support for long double.
- Shared library of MPFR.
- Binary compatible with previous versions if you do not use undocumented
features.
- Thread safe (if built with --enable-thread-safe).
- Logging facility.
- Change in the semantics of mpfr_out_str/mpfr_get_str when n_digits=0.
- Better locale support.
Changes from version 2.1.0 to version 2.1.1:
- Better way to detect the GMP library.
- Bug fixes.
Changes from version 2.0.3 to version 2.1.0:
- Bug fixes.
- new functions mpfr_strtofr, mpfr_set_uj, mpfr_set_sj, mpfr_set_ui_2exp,
mpfr_set_si_2exp, mpfr_set_sj_2exp, mpfr_set_uj_2exp, mpfr_get_uj,
mpfr_get_sj, mpfr_get_z, mpfr_free_str, mpfr_si_sub, mpfr_sub_si,
mpfr_mul_si, mpfr_si_div, mpfr_div_si, mpfr_sqr, mpfr_cmp_z, mpfr_cmp_q,
mpfr_zero_p, mpfr_free_cache, mpfr_sum, mpfr_get_version,
mpfr_get_default_rounding_mode, mpfr_get_emin_min, mpfr_get_emin_max,
mpfr_get_emax_min, mpfr_get_emax_max, mpfr_inits, mpfr_inits2, mpfr_clears,
mpfr_fits_intmax_p, mpfr_fits_uintmax_p, mpfr_clear_erangeflag,
mpfr_erangeflag_p, mpfr_rint_round, mpfr_rint_trunc, mpfr_rint_ceil,
mpfr_rint_floor.
- new macros MPFR_DECL_INIT, MPFR_VERSION, MPFR_VERSION_NUM,
MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL.
- improved documentation.
- improved configure.
- improved portability (library and test suite).
- It handles correctly non IEEE-754 double.
- GMP internal files are not needed to install MPFR.
- It is faster with low-precision floating point.
- New global flag: ERANGE_FLAG.
- Binary incompatible with previous versions, but API compatible.
- mpfr_set_str doesn't allow anymore "@NAN@garbagechar" and "@INF@garbagechar",
allows base 0 (detection of the base), prefix (0x, 0b), leading whitespace.
Changes from version 2.0.2 to version 2.0.3:
- Bug fixes.
- Support GMP as a shared library (not fully tested).
Changes from version 2.0.1 to version 2.0.2:
- many bug fixes and other improvements.
- new functions mpfr_prec_round (replaces mpfr_round_prec), mpfr_get_exp,
mpfr_set_exp, mpfr_get_ld, mpfr_set_ld, mpfr_get_d_2exp, mpfr_get_si,
mpfr_get_ui, mpfr_nextabove, mpfr_nextbelow, mpfr_nexttoward, mpfr_frac,
mpfr_fits_*, mpfr_cmp_d, mpfr_cmpabs, mpfr_erf, mpfr_gamma, mpfr_zeta,
mpfr_greater_p, mpfr_greaterequal_p, mpfr_less_p, mpfr_lessequal_p,
mpfr_lessgreater_p, mpfr_equal_p, mpfr_unordered_p.
- removed functions: mpfr_print_binary, mpfr_round_prec (replaced by
mpfr_prec_round), mpfr_set_str_raw, mpfr_set_machine_rnd_mode.
- function mpfr_isinteger renamed mpfr_integer_p.
- return type of some functions changed from void to int, for consistency.
- return type of mpfr_set_prec changed from int to void.
- new values for exponent range.
- rename internal variables.
Changes from version 2001 to version 2.0.1:
- new mathematical functions: acos, acosh, asin, asinh, atan, atanh, cosh,
base-2 exponential and logarithm, base-10 logarithm, expm1, factorial,
pow, pow_si, pow_ui, sinh, tan, tanh, ui_pow, ui_pow_ui
- other new functions: mpfr_const_euler, mpfr_dim, mpfr_fma, mpfr_hypot,
mpfr_min, mpfr_max, mpfr_rint, mpfr_set_inf, mpfr_set_nan
- new operations with MPZ or MPQ: mpfr_{add,sub,mul,div}_[zq]
- new predicates: mpfr_inf_p, mpfr_nan_p, mpfr_number_p, mpfr_isinteger,
- add mechanism to set/check exponent range (overflow, underflow), partially
implemented in the mpfr functions.
- efficiency: mpfr_div is now faster when the divisor has a few limbs
- rounding: now mpfr_pow implements exact rounding, and most functions return a
ternary value indicating the position of the returned value wrt the exact one
(thus the return value is now 'int' instead of 'void')
- complete rewrite of the configuration files
- mpfr_get_d, mpfr_{add,sub}_one_ulp now get a rounding mode as 2nd argument
- some function names did change: mpz_set_fr is now mpfr_get_z_exp,
mpfr_print_raw is now mpfr_print_binary.
Changes from version 1.0 to version 2001:
- the default installation does not provide any more access to machine
rounding mode, and as a consequence does not compare MPFR results with
precision=53 to machine results. Add option -DTEST if you want to have
access to machine rounding mode, and to check MPFR results against.
- the MPFR files do not need <math.h> any more
- the header file <mpfr.h> was split into <mpfr.h> for exported functions
and <mpfr-impl.h> for internal functions. The user should not use functions
or macros from <mpfr-impl.h>, since those may change in further releases.
- <mpfr.h> was modified in order to make easy a C++ interface
- MPFR now deals with infinities (+infinity and -infinity) and NaN
- the missing function mpfr_swap is now available
- mpfr_zeta was removed (was incomplete)
- mpfr_init and mpfr_init2 now initialize the corresponding variable to 0
(like in other initialization functions from GNU MP)
- in case memory allocation fails, an error message is output
- several bugs of version 1.0 were fixed
Changes from version 0.4 to version 1.0:
- Version 1.0 now uses a standard configure/make installation.
- Version 1.0 implements all functions that are available in the MPF class
from GMP 3.1 (except mpf_swap) and a header file mpf2mpfr.h is included in
the distribution for easy change from MPF to MPFR.
- Version 1.0 implements new elementary functions: mpfr_sincos
- Some functions and macros have been renamed: mpfr_log2 is now
mpfr_const_log2, mpfr_pi is now mpfr_const_pi, SIGN is now MPFR_SIGN.
- Version 1.0 uses faster algorithms for mpfr_exp, mpfr_const_pi,
mpfr_const_log2. Compare the timings from version 1.0 and version 0.4.
- Version 1.0 corrects some bugs of version 0.4.
- The precision of MPFR variables is now named mpfr_prec, which makes it
easier to change it, to say unsigned long long. Same for the rounding mode
which is called mp_rnd_t.
You'll find other news concerning the GNU MPFR library on the web
page <http://www.mpfr.org/>.

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Copyright 1999-2019 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
Table of contents:
1. Documentation
2. Compiler/library detection
3. Changes in existing functions
4. New functions to implement
5. Efficiency
6. Miscellaneous
7. Portability
##############################################################################
1. Documentation
##############################################################################
- add a description of the algorithms used and a proof of correctness
##############################################################################
2. Compiler/library detection
##############################################################################
- if we want to distinguish GMP and MPIR, we can check at configure time
the following symbols which are only defined in MPIR:
#define __MPIR_VERSION 0
#define __MPIR_VERSION_MINOR 9
#define __MPIR_VERSION_PATCHLEVEL 0
There is also a library symbol mpir_version, which should match VERSION, set
by configure, for example 0.9.0.
- update ICC detection.
* Use only __INTEL_COMPILER instead of the obsolete macro __ICC?
* Possibly enable some features with ICC. For instance, removing
"&& !defined(__ICC)" from src/mpfr.h works with ICC 15.0.0 20140723.
##############################################################################
3. Changes in existing functions
##############################################################################
- export mpfr_overflow and mpfr_underflow as public functions
- many functions currently taking into account the precision of the *input*
variable to set the initial working precision (acosh, asinh, cosh, ...).
This is nonsense since the "average" working precision should only depend
on the precision of the *output* variable (and maybe on the *value* of
the input in case of cancellation).
-> remove those dependencies from the input precision.
- mpfr_can_round:
change the meaning of the 2nd argument (err). Currently the error is
at most 2^(MPFR_EXP(b)-err), i.e. err is the relative shift wrt the
most significant bit of the approximation. I propose that the error
is now at most 2^err ulps of the approximation, i.e.
2^(MPFR_EXP(b)-MPFR_PREC(b)+err).
- mpfr_set_q first tries to convert the numerator and the denominator
to mpfr_t. But this conversion may fail even if the correctly rounded
result is representable. New way to implement:
Function q = a/b. nq = PREC(q) na = PREC(a) nb = PREC(b)
If na < nb
a <- a*2^(nb-na)
n <- na-nb+ (HIGH(a,nb) >= b)
if (n >= nq)
bb <- b*2^(n-nq)
a = q*bb+r --> q has exactly n bits.
else
aa <- a*2^(nq-n)
aa = q*b+r --> q has exactly n bits.
If RNDN, takes nq+1 bits. (See also the new division function).
- revisit the conversion functions between a MPFR number and a native
floating-point value.
* Consequences if some exception is trapped?
* Specify under which conditions (current rounding direction and
precision of the FPU, whether a format has been recognized...),
correct rounding is guaranteed. Fix the code if need be. Do not
forget subnormals.
* Provide mpfr_buildopt_* functions to tell whether the format of a
native type (float / double / long double) has been recognized and
which format it is?
* For functions that return a native floating-point value (mpfr_get_flt,
mpfr_get_d, mpfr_get_ld, mpfr_get_decimal64), in case of underflow or
overflow, follow the convention used for the functions in <math.h>?
See §7.12.1 "Treatment of error conditions" of ISO C11, which provides
two ways of handling error conditions, depending on math_errhandling:
errno (to be set to ERANGE here) and floating-point exceptions.
If floating-point exceptions need to be generated, do not use
feraiseexcept(), as this function may require the math library (-lm);
use a floating-point expression instead, such as DBL_MIN * DBL_MIN
(underflow) or DBL_MAX * DBL_MAX (overflow), which are probably safe
as used in the GNU libc implementation.
* For testing the lack of subnormal support:
see the -mfpu GCC option for ARM and
https://en.wikipedia.org/wiki/Denormal_number#Disabling_denormal_floats_at_the_code_level
##############################################################################
4. New functions to implement
##############################################################################
- implement new functions from the C++17 standard:
http://en.cppreference.com/w/cpp/numeric/special_math
assoc_laguerre, assoc_legendre, comp_ellint_1, comp_ellint_2, comp_ellint_3,
cyl_bessel_i, cyl_bessel_j, cyl_bessel_k, cyl_neumann, ellint_1, ellint_2,
ellint_3, hermite, legendre, laguerre, sph_bessel, sph_legendre,
sph_neumann.
Already in mpfr4: beta and riemann_zeta.
See also https://isocpp.org/files/papers/P0226R1.pdf and §29.9.5 in the
C++17 draft:
https://github.com/cplusplus/draft/blob/master/source/numerics.tex
- implement mpfr_get_decimal128 and mpfr_set_decimal128
- implement mpfr_log_ui to compute log(n) for an unsigned long n.
We can write for argument reduction n = 2^k * n/2^k, where
2/3 <= n/2^k < 4/3, i.e., k = floor(log2(3n))-1, thus
log(n) = k*log(2) + log(n/2^k), and we can use binary splitting on the
Taylor expansion of log(1+x) to compute log(n/2^k), where at most
p*log(2)/log(3) terms are needed for precision p.
Other idea (from Fredrik Johansson): compute log(m) + log(n/m) where
m=2^a*3^b*5^c*7^d and m is close to n.
- implement mpfr_q_sub, mpfr_z_div, mpfr_q_div?
- implement mpfr_pow_q and variants with two integers (native or mpz)
instead of a rational? See IEEE P1788.
- implement functions for random distributions, see for example
https://sympa.inria.fr/sympa/arc/mpfr/2010-01/msg00034.html
(suggested by Charles Karney <ckarney@Sarnoff.com>, 18 Jan 2010):
* a Bernoulli distribution with prob p/q (exact)
* a general discrete distribution (i with prob w[i]/sum(w[i]) (Walker
algorithm, but make it exact)
* a uniform distribution in (a,b)
* exponential distribution (mean lambda) (von Neumann's method?)
* normal distribution (mean m, s.d. sigma) (ratio method?)
- wanted for Magma [John Cannon <john@maths.usyd.edu.au>, Tue, 19 Apr 2005]:
HypergeometricU(a,b,s) = 1/gamma(a)*int(exp(-su)*u^(a-1)*(1+u)^(b-a-1),
u=0..infinity)
JacobiThetaNullK
PolylogP, PolylogD, PolylogDold: see http://arxiv.org/abs/math.CA/0702243
and the references herein.
JBessel(n, x) = BesselJ(n+1/2, x)
KBessel, KBessel2 [2nd kind]
JacobiTheta
(see http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2017-03245-2/home.html)
LogIntegral
ExponentialIntegralEn (formula 5.1.4 of Abramowitz and Stegun)
DawsonIntegral
GammaD(x) = Gamma(x+1/2)
- new functions of IEEE 754-2008, and more generally functions of the
C binding draft TS 18661-4:
http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1946.pdf
Some propositions about rootn: mpfr_rootn_si, mpfr_rootn_sj, mpfr_rootn_z,
and versions with an unsigned integer: mpfr_rootn_ui (now implemented, as
similar to mpfr_root) and mpfr_rootn_uj.
- functions defined in the LIA-2 standard
+ minimum and maximum (5.2.2): max, min, max_seq, min_seq, mmax_seq
and mmin_seq (mpfr_min and mpfr_max correspond to mmin and mmax);
+ rounding_rest, floor_rest, ceiling_rest (5.2.4);
+ remr (5.2.5): x - round(x/y) y;
+ error functions from 5.2.7 (if useful in MPFR);
+ power1pm1 (5.3.6.7): (1 + x)^y - 1;
+ logbase (5.3.6.12): \log_x(y);
+ logbase1p1p (5.3.6.13): \log_{1+x}(1+y);
+ rad (5.3.9.1): x - round(x / (2 pi)) 2 pi = remr(x, 2 pi);
+ axis_rad (5.3.9.1) if useful in MPFR;
+ cycle (5.3.10.1): rad(2 pi x / u) u / (2 pi) = remr(x, u);
+ axis_cycle (5.3.10.1) if useful in MPFR;
+ sinu, cosu, tanu, cotu, secu, cscu, cossinu, arcsinu, arccosu,
arctanu, arccotu, arcsecu, arccscu (5.3.10.{2..14}):
sin(x 2 pi / u), etc.;
[from which sinpi(x) = sin(Pi*x), ... are trivial to implement, with u=2.]
+ arcu (5.3.10.15): arctan2(y,x) u / (2 pi);
+ rad_to_cycle, cycle_to_rad, cycle_to_cycle (5.3.11.{1..3}).
- From GSL, missing special functions (if useful in MPFR):
(cf https://www.gnu.org/software/gsl/manual/gsl-ref.html#Special-Functions)
+ The Airy functions Ai(x) and Bi(x) defined by the integral representations:
* Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
* Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt
* Derivatives of Airy Functions
+ The Bessel functions for n integer and n fractional:
* Regular Modified Cylindrical Bessel Functions I_n
* Irregular Modified Cylindrical Bessel Functions K_n
* Regular Spherical Bessel Functions j_n: j_0(x) = \sin(x)/x,
j_1(x)= (\sin(x)/x-\cos(x))/x & j_2(x)= ((3/x^2-1)\sin(x)-3\cos(x)/x)/x
Note: the "spherical" Bessel functions are solutions of
x^2 y'' + 2 x y' + [x^2 - n (n+1)] y = 0 and satisfy
j_n(x) = sqrt(Pi/(2x)) J_{n+1/2}(x). They should not be mixed with the
classical Bessel Functions, also noted j0, j1, jn, y0, y1, yn in C99
and mpfr.
Cf https://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions
*Irregular Spherical Bessel Functions y_n: y_0(x) = -\cos(x)/x,
y_1(x)= -(\cos(x)/x+\sin(x))/x &
y_2(x)= (-3/x^3+1/x)\cos(x)-(3/x^2)\sin(x)
* Regular Modified Spherical Bessel Functions i_n:
i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)
* Irregular Modified Spherical Bessel Functions:
k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).
+ Clausen Function:
Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
Cl_2(\theta) = \Im Li_2(\exp(i \theta)) (dilogarithm).
+ Dawson Function: \exp(-x^2) \int_0^x dt \exp(t^2).
+ Debye Functions: D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
+ Elliptic Integrals:
* Definition of Legendre Forms:
F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))
P(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
* Complete Legendre forms are denoted by
K(k) = F(\pi/2, k)
E(k) = E(\pi/2, k)
* Definition of Carlson Forms
RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
RJ(x,y,z,p) = 3/2 \int_0^\infty dt
(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
+ Elliptic Functions (Jacobi)
+ N-relative exponential:
exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
+ exponential integral:
E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
Ei_3(x) = \int_0^x dt \exp(-t^3) for x >= 0.
Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
+ Hyperbolic/Trigonometric Integrals
Shi(x) = \int_0^x dt \sinh(t)/t
Chi(x) := Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t]
Si(x) = \int_0^x dt \sin(t)/t
Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0
AtanInt(x) = \int_0^x dt \arctan(t)/t
[ \gamma_E is the Euler constant ]
+ Fermi-Dirac Function:
F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
+ Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(a) : see [Smith01] in
algorithms.bib
logarithm of the Pochhammer symbol
+ Gegenbauer Functions
+ Laguerre Functions
+ Eta Function: \eta(s) = (1-2^{1-s}) \zeta(s)
Hurwitz zeta function: \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.
+ Lambert W Functions, W(x) are defined to be solutions of the equation:
W(x) \exp(W(x)) = x.
This function has multiple branches for x < 0 (2 funcs W0(x) and Wm1(x))
From Fredrik Johansson:
See https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf, in particular
formulas 5.2 and 5.3 for the error bound: one first computes an
approximation w, and then evaluates the residual w e^w - x. There is an
expression for the error in terms of the residual and the derivative W'(t),
where the derivative can be bounded by piecewise simple functions,
something like min(1, 1/t) when t >= 0.
See https://arxiv.org/abs/1705.03266 for rigorous error bounds.
+ Trigamma Function psi'(x).
and Polygamma Function: psi^{(m)}(x) for m >= 0, x > 0.
- functions from ISO/IEC 24747:2009 (Extensions to the C Library,
to Support Mathematical Special Functions).
Standard: http://www.iso.org/iso/catalogue_detail.htm?csnumber=38857
Draft: http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1292.pdf
Rationale: http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1244.pdf
See also: http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2010/n3060.pdf
(similar, for C++).
Also check whether the functions that are already implemented in MPFR
match this standard.
- from gnumeric (www.gnome.org/projects/gnumeric/doc/function-reference.html):
- incomplete beta function, see message from Martin Maechler
<maechler@stat.math.ethz.ch> on 18 Jan 2016, and Section 6.6 in
Abramowitz & Stegun
- betaln
- degrees
- radians
- sqrtpi
- mpfr_inp_raw, mpfr_out_raw (cf mail "Serialization of mpfr_t" from Alexey
and answer from Granlund on mpfr list, May 2007)
- [maybe useful for SAGE] implement companion frac_* functions to the rint_*
functions. For example mpfr_frac_floor(x) = x - floor(x). (The current
mpfr_frac function corresponds to mpfr_rint_trunc.)
- scaled erfc (https://sympa.inria.fr/sympa/arc/mpfr/2009-05/msg00054.html)
- asec, acsc, acot, asech, acsch and acoth (mail from Björn Terelius on mpfr
list, 18 June 2009)
##############################################################################
5. Efficiency
##############################################################################
- Fredrik Johansson reports that mpfr_ai is slow for large arguments: an
asymptotic expansion should be used (once done, remove REDUCE_EMAX from
tests/tai.c and update the description in mpfr.texi).
- for exp(x), Fredrik Johansson reports a 20% speed improvement starting from
4000 bits, and up to a 75% memory improvement in his Arb implementation, by
using recursive instead of iterative binary splitting:
https://github.com/fredrik-johansson/arb/blob/master/elefun/exp_sum_bs_powtab.c
- improve mpfr_grandom using the algorithm in http://arxiv.org/abs/1303.6257
- use the src/x86_64/corei5/mparam.h file once GMP recognizes correctly the
Core i5 processors (note that gcc -mtune=native gives __tune_corei7__
and not __tune_corei5__ on those processors)
- implement a mpfr_sqrthigh algorithm based on Mulders' algorithm, with a
basecase variant
- use mpn_div_q to speed up mpfr_div. However mpn_div_q, which is new in
GMP 5, is not documented in the GMP manual, thus we are not sure it
guarantees to return the same quotient as mpn_tdiv_qr.
Also mpfr_div uses the remainder computed by mpn_divrem. A workaround would
be to first try with mpn_div_q, and if we cannot (easily) compute the
rounding, then use the current code with mpn_divrem.
- improve atanh(x) for small x by using atanh(x) = log1p(2x/(1-x)),
and log1p should also be improved for small arguments.
- compute exp by using the series for cosh or sinh, which has half the terms
(see Exercise 4.11 from Modern Computer Arithmetic, version 0.3)
The same method can be used for log, using the series for atanh, i.e.,
atanh(x) = 1/2*log((1+x)/(1-x)).
- improve mpfr_gamma (see https://code.google.com/p/fastfunlib/). A possible
idea is to implement a fast algorithm for the argument reconstruction
gamma(x+k): instead of performing k products by x+i, we could precompute
x^2, ..., x^m for m ~ sqrt(k), and perform only sqrt(k) products.
One could also use the series for 1/gamma(x), see for example
http://dlmf.nist.gov/5/7/ or formula (36) from
http://mathworld.wolfram.com/GammaFunction.html
- improve the computation of Bernoulli numbers: instead of computing just one
B[2n] at a time in mpfr_bernoulli_internal, we could compute several at a
time, sharing the expensive computation of the 1/p^(2n) series.
- fix regression with mpfr_mpz_root (from Keith Briggs, 5 July 2006), for
example on 3Ghz P4 with gmp-4.2, x=12.345:
prec=50000 k=2 k=3 k=10 k=100
mpz_root 0.036 0.072 0.476 7.628
mpfr_mpz_root 0.004 0.004 0.036 12.20
See also mail from Carl Witty on mpfr list, 09 Oct 2007.
- for sparse input (say x=1 with 2 bits), mpfr_exp is not faster than for
full precision when precision <= MPFR_EXP_THRESHOLD. The reason is
that argument reduction kills sparsity. Maybe avoid argument reduction
for sparse input?
- speed up mpfr_atan for large arguments (to speed up mpc_log) see FR #6198
- improve mpfr_sin on values like ~pi (do not compute sin from cos, because
of the cancellation). For instance, reduce the input modulo pi/2 in
[-pi/4,pi/4], and define auxiliary functions for which the argument is
assumed to be already reduced (so that the sin function can avoid
unnecessary computations by calling the auxiliary cos function instead of
the full cos function). This will require a native code for sin, for
example using the reduction sin(3x)=3sin(x)-4sin(x)^3.
See https://sympa.inria.fr/sympa/arc/mpfr/2007-08/msg00001.html and
the following messages.
- improve generic.c to work for number of terms <> 2^k
- rewrite mpfr_greater_p... as native code.
- mpf_t uses a scheme where the number of limbs actually present can
be less than the selected precision, thereby allowing low precision
values (for instance small integers) to be stored and manipulated in
an mpf_t efficiently.
Perhaps mpfr should get something similar, especially if looking to
replace mpf with mpfr, though it'd be a major change. Alternately
perhaps those mpfr routines like mpfr_mul where optimizations are
possible through stripping low zero bits or limbs could check for
that (this would be less efficient but easier).
- try the idea of the paper "Reduced Cancellation in the Evaluation of Entire
Functions and Applications to the Error Function" by W. Gawronski, J. Mueller
and M. Reinhard, to be published in SIAM Journal on Numerical Analysis: to
avoid cancellation in say erfc(x) for x large, they compute the Taylor
expansion of erfc(x)*exp(x^2/2) instead (which has less cancellation),
and then divide by exp(x^2/2) (which is simpler to compute).
- replace the *_THRESHOLD macros by global (TLS) variables that can be
changed at run time (via a function, like other variables)? One benefit
is that users could use a single MPFR binary on several machines (e.g.,
a library provided by binary packages or shared via NFS) with different
thresholds. On the default values, this would be a bit less efficient
than the current code, but this isn't probably noticeable (this should
be tested). Something like:
long *mpfr_tune_get(void) to get the current values (the first value
is the size of the array).
int mpfr_tune_set(long *array) to set the tune values.
int mpfr_tune_run(long level) to find the best values (the support
for this feature is optional, this can also be done with an
external function).
- better distinguish different processors (for example Opteron and Core 2)
and use corresponding default tuning parameters (as in GMP). This could be
done in configure.ac to avoid hacking config.guess, for example define
MPFR_HAVE_CORE2.
Note (VL): the effect on cross-compilation (that can be a processor
with the same architecture, e.g. compilation on a Core 2 for an
Opteron) is not clear. The choice should be consistent with the
build target (e.g. -march or -mtune value with gcc).
Also choose better default values. For instance, the default value of
MPFR_MUL_THRESHOLD is 40, while the best values that have been found
are between 11 and 19 for 32 bits and between 4 and 10 for 64 bits!
- during the Many Digits competition, we noticed that (our implantation of)
Mulders short product was slower than a full product for large sizes.
This should be precisely analyzed and fixed if needed.
- for various functions, check the timings as a function of the magnitude
of the input (and the input and/or output precisions?), and use better
thresholds for asymptotic expansions.
- improve the special case of mpfr_{add,sub} (x, x, y, ...) when |x| > |y|
to do the addition in-place and have a complexity of O(prec(y)) in most
cases. The mpfr_{add,sub}_{d,ui} functions should automatically benefit
from this change.
- in gmp_op.c, for functions with mpz_srcptr, check whether mpz_fits_slong_p
is really useful in all cases (see TODO in this file).
- optimize code that uses a test based on the fact that x >> s is
undefined in C for s == width of x but the result is expected to
be 0. ARM and PowerPC could benefit from such an optimization,
but not x86. This needs support from the compiler.
For PowerPC: https://gcc.gnu.org/bugzilla/show_bug.cgi?id=79233
- deal with MPFR_RNDF in mpfr_round_near_x (replaced by MPFR_RNDZ).
- instead of a fixed mparam.h, optionally use function multiversioning
(FMV), currently only available with the GNU C++ front end:
https://gcc.gnu.org/wiki/FunctionMultiVersioning
According to https://lwn.net/Articles/691932/ the dispatch resolution
is now done by the dynamic loader, so that this should be fast enough
(the cost would be the reading of a static variable, initialized at
load time, instead of a constant).
In particular, binary package distributions would benefit from FMV as
only one binary is generated for different processor families.
##############################################################################
6. Miscellaneous
##############################################################################
- [suggested by Tobias Burnus <burnus(at)net-b.de> and
Asher Langton <langton(at)gcc.gnu.org>, Wed, 01 Aug 2007]
support quiet and signaling NaNs in mpfr:
* functions to set/test a quiet/signaling NaN: mpfr_set_snan, mpfr_snan_p,
mpfr_set_qnan, mpfr_qnan_p
* correctly convert to/from double (if encoding of s/qNaN is fixed in 754R)
Note: Signaling NaNs are not specified by the ISO C standard and may
not be supported by the implementation. GCC needs the -fsignaling-nans
option (but this does not affect the C library, which may or may not
accept signaling NaNs).
- check again coverage: on 2007-07-27, Patrick Pelissier reports that the
following files are not tested at 100%: add1.c, atan.c, atan2.c,
cache.c, cmp2.c, const_catalan.c, const_euler.c, const_log2.c, cos.c,
gen_inverse.h, div_ui.c, eint.c, exp3.c, exp_2.c, expm1.c, fma.c, fms.c,
lngamma.c, gamma.c, get_d.c, get_f.c, get_ld.c, get_str.c, get_z.c,
inp_str.c, jn.c, jyn_asympt.c, lngamma.c, mpfr-gmp.c, mul.c, mul_ui.c,
mulders.c, out_str.c, pow.c, print_raw.c, rint.c, root.c, round_near_x.c,
round_raw_generic.c, set_d.c, set_ld.c, set_q.c, set_uj.c, set_z.c, sin.c,
sin_cos.c, sinh.c, sqr.c, stack_interface.c, sub1.c, sub1sp.c, subnormal.c,
uceil_exp2.c, uceil_log2.c, ui_pow_ui.c, urandomb.c, yn.c, zeta.c, zeta_ui.c.
- check the constants mpfr_set_emin (-16382-63) and mpfr_set_emax (16383) in
get_ld.c and the other constants, and provide a testcase for large and
small numbers.
- from Kevin Ryde <user42@zip.com.au>:
Also for pi.c, a pre-calculated compiled-in pi to a few thousand
digits would be good value I think. After all, say 10000 bits using
1250 bytes would still be small compared to the code size!
Store pi in round to zero mode (to recover other modes).
- add other prototypes for round to nearest-away (mpfr_round_nearest_away
only deals with the prototypes of say mpfr_sin) or implement it as a native
rounding mode
- add a new roundind mode: round to odd. If the result is not exactly
representable, then round to the odd mantissa. This rounding
has the nice property that for k > 1, if:
y = round(x, p+k, TO_ODD)
z = round(y, p, TO_NEAREST_EVEN), then
z = round(x, p, TO_NEAREST_EVEN)
so it avoids the double-rounding problem.
VL: I prefer the (original?) term "sticky rounding", as used in
J Strother Moore, Tom Lynch, Matt Kaufmann. A Mechanically Checked
Proof of the Correctness of the Kernel of the AMD5K86 Floating-Point
Division Algorithm. IEEE Transactions on Computers, 1996.
and
http://www.russinoff.com/libman/text/node26.html
- new rounding mode MPFR_RNDE when the result is known to be exact?
* In normal mode, this would allow MPFR to optimize using
this information.
* In debug mode, MPFR would check that the result is exact
(i.e. that the ternary value is 0).
- new "rounding mode" MPFR_RNDF (faithful rounding)?
This is not really a rounding mode since it is non-deterministic. The
goal is to avoid the Table Maker's Dilemma in internal computations.
The definition of faithful rounding of a real number x is: return either
RNDD(x) or RNDU(x). This means that if x is exactly representable, one
returns x exactly. In MPFR, the ternary value should be unspecified for
efficiency reasons.
Note: One typically implements faithful rounding by computing an
approximation to the result with some adequately chosen error bound,
then by rounding this approximation to nearest.
Concerning the choice of the error bound, if the result x is equal to
1 + t, where t is a very small positive number, then the error bound
needs to be at most ulp(x)/4 + t. Since t can be arbitrarily small,
the error bound needs to be at most ulp(x)/4. And this error bound
is sufficient in all cases. Note that with the even rounding rule or
rounding away from zero, it is not needed to relax the condition when
x is exactly representable.
- add tests of the ternary value for constants
- When doing Extensive Check (--enable-assert=full), since all the
functions use a similar use of MACROS (ZivLoop, ROUND_P), it should
be possible to do such a scheme:
For the first call to ROUND_P when we can round.
Mark it as such and save the approximated rounding value in
a temporary variable.
Then after, if the mark is set, check if:
- we still can round.
- The rounded value is the same.
It should be a complement to tgeneric tests.
- in div.c, try to find a case for which cy != 0 after the line
cy = mpn_sub_1 (sp + k, sp + k, qsize, cy);
(which should be added to the tests), e.g. by having {vp, k} = 0, or
prove that this cannot happen.
- add a configure test for --enable-logging to ignore the option if
it cannot be supported. Modify the "configure --help" description
to say "on systems that support it".
- add generic bad cases for functions that don't have an inverse
function that is implemented (use a single Newton iteration).
- add bad cases for the internal error bound (by using a dichotomy
between a bad case for the correct rounding and some input value
with fewer Ziv iterations?).
- add an option to use a 32-bit exponent type (int) on LP64 machines,
mainly for developers, in order to be able to test the case where the
extended exponent range is the same as the default exponent range, on
such platforms.
Tests can be done with the exp-int branch (added on 2010-12-17, and
many tests fail at this time).
- test underflow/overflow detection of various functions (in particular
mpfr_exp) in reduced exponent ranges, including ranges that do not
contain 0.
- add an internal macro that does the equivalent of the following?
MPFR_IS_ZERO(x) || MPFR_GET_EXP(x) <= value
- check whether __gmpfr_emin and __gmpfr_emax could be replaced by
a constant (see README.dev). Also check the use of MPFR_EMIN_MIN
and MPFR_EMAX_MAX.
- add a test checking that no mpfr.h macros depend on mpfr-impl.h
(the current tests cannot check that since mpfr-impl.h is always
included).
- move some macro definitions from acinclude.m4 to the m4 directory
as suggested by the Automake manual? The reason is that the
acinclude.m4 file is big and a bit difficult to read.
- use symbol versioning.
- check whether mpz_t caching (pool) is necessary. Timings with -static
with details about the C / C library implementation should be put
somewhere as a comment in the source or in the doc. Using -static
is important because otherwise the cache saves the dynamic call to
mpz_init and mpz_clear; so, what we're measuring is not clear.
See thread:
https://gmplib.org/list-archives/gmp-devel/2015-September/004147.html
Summary: It will not be integrated in GMP because 1) This yields
problems with threading (in MPFR, we have TLS variables, but this is
not the case of GMP). 2) The gain (if confirmed with -static) would
be due to a poor malloc implementation (timings would depend on the
platform). 3) Applications would use more RAM.
Additional notes [VL]: the major differences in the timings given
by Patrick in 2014-01 under Linux were:
Before:
arccos(x) took 0.054689 ms (32767 eval in 1792 ms)
arctan(x) took 0.042116 ms (32767 eval in 1380 ms)
After:
arccos(x) took 0.043580 ms (32767 eval in 1428 ms)
arctan(x) took 0.035401 ms (32767 eval in 1160 ms)
mpfr_acos doesn't use mpz, but calls mpfr_atan, so that the issue comes
from mpfr_atan, which uses mpz a lot. The problem mainly comes from the
reallocations in GMP because mpz_init is used instead of mpz_init2 with
the estimated maximum size. Other places in the code that uses mpz_init
may be concerned.
Issues with mpz_t caching:
* The pool can take much memory, which may no longer be useful.
For instance:
mpfr_init2 (x, 10000000);
mpfr_log_ui (x, 17, MPFR_RNDN);
/* ... */
mpfr_clear (x);
/* followed by code using only small precision */
while contrary to real caches, they contain no data. This is not
valuable memory: freeing/allocating a large block of memory is
much faster than the actual computations, so that mpz_t caching
has no impact on the performance in such cases. A pool with large
blocks also potentially destroys the data locality.
* It assumes that the real GMP functions are __gmpz_init and
__gmpz_clear, which are not part of the official GMP API, thus
is based on GMP internals, which may change in the future or
may be different in forks / compatible libraries / etc. This
can be solved if MPFR code calls mpfr_mpz_init / mpfr_mpz_clear
directly, avoiding the #define's.
Questions that need to be answered:
* What about the comparisons with other memory allocators?
* Shouldn't the pool be part of the memory allocator?
For the default memory allocator (malloc): RFE?
If it is decided to keep some form of mpz_t caching, a possible solution
for both issues: define mpfr_mpz_init2 and mpfr_mpz_clear2, which both
take 2 arguments like mpz_init2, where mpfr_mpz_init2 behaves in a way
similar to mpz_init2, and mpfr_mpz_clear2 behaves in a way similar to
mpz_clear but where the size argument is a hint for the pool; if it is
too large, then the mpz_t should not be pushed back to the pool. The
size argument of mpfr_mpz_init2 could also be a hint to decide which
element to pull from the pool.
- in tsum, add testcases for mpfr_sum triggering the bug fixed in r9722,
that is, with a large error during the computation of the secondary term
(when the TMD occurs).
- add internal or public variants of some basic functions (+, -, *) with
mpz_t as the exponent for correctly rounded polynomials (like fmma),
in order to avoid internal overflow and underflow in extreme cases?
Alternatively, for fmma, modify add*.c and sub*.c code to define
mpfr_add_raw, which takes arrays of limbs and their precision, and the
positive exponent delta (as mpfr_uexp_t to be always representable).
The exponent delta should be sufficient for now since it can be bounded
by MPFR_PREC_MAX+1 if need be.
- use the keyword "static" in array indices of parameter declarations with
C99 compilers (6.7.5.3p7) when the pointer is expected not to be null?
For instance, if mpfr.h is changed to have:
__MPFR_DECLSPEC void mpfr_dump (const __mpfr_struct [static 1]);
and one calls
mpfr_dump (NULL);
one gets a warning with Clang. This is just an example; this needs to be
done in a clean way.
See:
http://stackoverflow.com/a/3430353/3782797
https://hamberg.no/erlend/posts/2013-02-18-static-array-indices.html
- change most mpfr_urandomb occurrences to mpfr_urandom in the tests?
(The one done in r10573 allowed us to find a bug even without
assertion checking.)
- tzeta has been much slower since r9848 (which increases the precision
of the input for the low output precisions), at least with the x86
32-bit ABI. This seems to come from the fact that the working precision
in the mpfr_zeta implementation depends on the precision of the input.
Once mpfr_zeta has improved, change the last argument of test_generic
in tzeta.c back to 5 (as it was before r10667).
- check the small-precision tables in the tests?
This may require to export some pointer to the tables, but this could
be done only if some debug macro is defined.
- optionally use malloc() for the caches? See mpfr_mp_memory_cleanup.
Note: This can be implemented by adding a TLS flag saying whether we
are under cache generation or not, and by making the MPFR allocation
functions consider this flag. Moreover, this can only work for mpfr_t
caching (floating-point constants), not for mpz_t caching (Bernoulli
constants) because we do not have the control of memory allocation for
mpz_init.
##############################################################################
7. Portability
##############################################################################
- add a web page with results of builds on different architectures
- [Kevin about texp.c long strings]
For strings longer than c99 guarantees, it might be cleaner to
introduce a "tests_strdupcat" or something to concatenate literal
strings into newly allocated memory. I thought I'd done that in a
couple of places already. Arrays of chars are not much fun.
- use https://gcc.gnu.org/viewcvs/gcc/trunk/config/stdint.m4 for mpfr-gmp.h
- with MinGW, build with -D__USE_MINGW_ANSI_STDIO by default?

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This directory contains simple examples.