435 lines
21 KiB
Plaintext
435 lines
21 KiB
Plaintext
Copyright 1999-2016 Free Software Foundation, Inc.
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Contributed by the AriC and Caramba projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
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Table of contents:
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1. Documentation
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2. Installation
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3. Changes in existing functions
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4. New functions to implement
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5. Efficiency
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6. Miscellaneous
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7. Portability
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##############################################################################
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1. Documentation
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##############################################################################
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- add a description of the algorithms used + proof of correctness
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##############################################################################
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2. Installation
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##############################################################################
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- if we want to distinguish GMP and MPIR, we can check at configure time
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the following symbols which are only defined in MPIR:
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#define __MPIR_VERSION 0
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#define __MPIR_VERSION_MINOR 9
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#define __MPIR_VERSION_PATCHLEVEL 0
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There is also a library symbol mpir_version, which should match VERSION, set
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by configure, for example 0.9.0.
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##############################################################################
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3. Changes in existing functions
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##############################################################################
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- export mpfr_overflow and mpfr_underflow as public functions
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- many functions currently taking into account the precision of the *input*
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variable to set the initial working precison (acosh, asinh, cosh, ...).
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This is nonsense since the "average" working precision should only depend
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on the precision of the *output* variable (and maybe on the *value* of
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the input in case of cancellation).
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-> remove those dependencies from the input precision.
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- mpfr_can_round:
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change the meaning of the 2nd argument (err). Currently the error is
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at most 2^(MPFR_EXP(b)-err), i.e. err is the relative shift wrt the
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most significant bit of the approximation. I propose that the error
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is now at most 2^err ulps of the approximation, i.e.
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2^(MPFR_EXP(b)-MPFR_PREC(b)+err).
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- mpfr_set_q first tries to convert the numerator and the denominator
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to mpfr_t. But this conversion may fail even if the correctly rounded
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result is representable. New way to implement:
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Function q = a/b. nq = PREC(q) na = PREC(a) nb = PREC(b)
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If na < nb
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a <- a*2^(nb-na)
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n <- na-nb+ (HIGH(a,nb) >= b)
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if (n >= nq)
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bb <- b*2^(n-nq)
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a = q*bb+r --> q has exactly n bits.
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else
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aa <- a*2^(nq-n)
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aa = q*b+r --> q has exactly n bits.
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If RNDN, takes nq+1 bits. (See also the new division function).
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##############################################################################
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4. New functions to implement
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##############################################################################
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- implement mpfr_q_sub, mpfr_z_div, mpfr_q_div?
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- implement functions for random distributions, see for example
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https://sympa.inria.fr/sympa/arc/mpfr/2010-01/msg00034.html
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(suggested by Charles Karney <ckarney@Sarnoff.com>, 18 Jan 2010):
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* a Bernoulli distribution with prob p/q (exact)
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* a general discrete distribution (i with prob w[i]/sum(w[i]) (Walker
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algorithm, but make it exact)
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* a uniform distribution in (a,b)
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* exponential distribution (mean lambda) (von Neumann's method?)
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* normal distribution (mean m, s.d. sigma) (ratio method?)
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- wanted for Magma [John Cannon <john@maths.usyd.edu.au>, Tue, 19 Apr 2005]:
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HypergeometricU(a,b,s) = 1/gamma(a)*int(exp(-su)*u^(a-1)*(1+u)^(b-a-1),
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u=0..infinity)
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JacobiThetaNullK
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PolylogP, PolylogD, PolylogDold: see http://arxiv.org/abs/math.CA/0702243
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and the references herein.
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JBessel(n, x) = BesselJ(n+1/2, x)
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IncompleteGamma [also wanted by <keith.briggs@bt.com> 4 Feb 2008: Gamma(a,x),
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gamma(a,x), P(a,x), Q(a,x); see A&S 6.5, ref. [Smith01] in algorithms.bib]
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KBessel, KBessel2 [2nd kind]
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JacobiTheta
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LogIntegral
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ExponentialIntegralE1
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E1(z) = int(exp(-t)/t, t=z..infinity), |arg z| < Pi
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mpfr_eint1: implement E1(x) for x > 0, and Ei(-x) for x < 0
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E1(NaN) = NaN
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E1(+Inf) = +0
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E1(-Inf) = -Inf
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E1(+0) = +Inf
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E1(-0) = -Inf
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DawsonIntegral
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GammaD(x) = Gamma(x+1/2)
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- functions defined in the LIA-2 standard
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+ minimum and maximum (5.2.2): max, min, max_seq, min_seq, mmax_seq
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and mmin_seq (mpfr_min and mpfr_max correspond to mmin and mmax);
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+ rounding_rest, floor_rest, ceiling_rest (5.2.4);
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+ remr (5.2.5): x - round(x/y) y;
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+ error functions from 5.2.7 (if useful in MPFR);
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+ power1pm1 (5.3.6.7): (1 + x)^y - 1;
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+ logbase (5.3.6.12): \log_x(y);
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+ logbase1p1p (5.3.6.13): \log_{1+x}(1+y);
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+ rad (5.3.9.1): x - round(x / (2 pi)) 2 pi = remr(x, 2 pi);
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+ axis_rad (5.3.9.1) if useful in MPFR;
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+ cycle (5.3.10.1): rad(2 pi x / u) u / (2 pi) = remr(x, u);
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+ axis_cycle (5.3.10.1) if useful in MPFR;
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+ sinu, cosu, tanu, cotu, secu, cscu, cossinu, arcsinu, arccosu,
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arctanu, arccotu, arcsecu, arccscu (5.3.10.{2..14}):
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sin(x 2 pi / u), etc.;
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[from which sinpi(x) = sin(Pi*x), ... are trivial to implement, with u=2.]
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+ arcu (5.3.10.15): arctan2(y,x) u / (2 pi);
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+ rad_to_cycle, cycle_to_rad, cycle_to_cycle (5.3.11.{1..3}).
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- From GSL, missing special functions (if useful in MPFR):
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(cf http://www.gnu.org/software/gsl/manual/gsl-ref.html#Special-Functions)
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+ The Airy functions Ai(x) and Bi(x) defined by the integral representations:
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* Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
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* Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt
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* Derivatives of Airy Functions
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+ The Bessel functions for n integer and n fractional:
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* Regular Modified Cylindrical Bessel Functions I_n
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* Irregular Modified Cylindrical Bessel Functions K_n
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* Regular Spherical Bessel Functions j_n: j_0(x) = \sin(x)/x,
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j_1(x)= (\sin(x)/x-\cos(x))/x & j_2(x)= ((3/x^2-1)\sin(x)-3\cos(x)/x)/x
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Note: the "spherical" Bessel functions are solutions of
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x^2 y'' + 2 x y' + [x^2 - n (n+1)] y = 0 and satisfy
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j_n(x) = sqrt(Pi/(2x)) J_{n+1/2}(x). They should not be mixed with the
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classical Bessel Functions, also noted j0, j1, jn, y0, y1, yn in C99
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and mpfr.
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Cf https://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions
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*Irregular Spherical Bessel Functions y_n: y_0(x) = -\cos(x)/x,
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y_1(x)= -(\cos(x)/x+\sin(x))/x &
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y_2(x)= (-3/x^3+1/x)\cos(x)-(3/x^2)\sin(x)
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* Regular Modified Spherical Bessel Functions i_n:
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i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)
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* Irregular Modified Spherical Bessel Functions:
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k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).
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+ Clausen Function:
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Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
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Cl_2(\theta) = \Im Li_2(\exp(i \theta)) (dilogarithm).
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+ Dawson Function: \exp(-x^2) \int_0^x dt \exp(t^2).
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+ Debye Functions: D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
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+ Elliptic Integrals:
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* Definition of Legendre Forms:
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F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
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E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))
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P(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
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* Complete Legendre forms are denoted by
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K(k) = F(\pi/2, k)
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E(k) = E(\pi/2, k)
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* Definition of Carlson Forms
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RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
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RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
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RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
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RJ(x,y,z,p) = 3/2 \int_0^\infty dt
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(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
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+ Elliptic Functions (Jacobi)
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+ N-relative exponential:
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exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
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+ exponential integral:
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E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
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Ei_3(x) = \int_0^x dt \exp(-t^3) for x >= 0.
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Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
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+ Hyperbolic/Trigonometric Integrals
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Shi(x) = \int_0^x dt \sinh(t)/t
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Chi(x) := Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t]
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Si(x) = \int_0^x dt \sin(t)/t
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Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0
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AtanInt(x) = \int_0^x dt \arctan(t)/t
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[ \gamma_E is the Euler constant ]
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+ Fermi-Dirac Function:
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F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
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+ Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(a) : see [Smith01] in
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algorithms.bib
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logarithm of the Pochhammer symbol
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+ Gegenbauer Functions
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+ Laguerre Functions
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+ Eta Function: \eta(s) = (1-2^{1-s}) \zeta(s)
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Hurwitz zeta function: \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.
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+ Lambert W Functions, W(x) are defined to be solutions of the equation:
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W(x) \exp(W(x)) = x.
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This function has multiple branches for x < 0 (2 funcs W0(x) and Wm1(x))
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+ Trigamma Function psi'(x).
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and Polygamma Function: psi^{(m)}(x) for m >= 0, x > 0.
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- from gnumeric (www.gnome.org/projects/gnumeric/doc/function-reference.html):
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- beta
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- betaln
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- degrees
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- radians
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- sqrtpi
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- mpfr_inp_raw, mpfr_out_raw (cf mail "Serialization of mpfr_t" from Alexey
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and answer from Granlund on mpfr list, May 2007)
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- [maybe useful for SAGE] implement companion frac_* functions to the rint_*
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functions. For example mpfr_frac_floor(x) = x - floor(x). (The current
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mpfr_frac function corresponds to mpfr_rint_trunc.)
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- scaled erfc (https://sympa.inria.fr/sympa/arc/mpfr/2009-05/msg00054.html)
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- asec, acsc, acot, asech, acsch and acoth (mail from Björn Terelius on mpfr
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list, 18 June 2009)
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##############################################################################
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5. Efficiency
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##############################################################################
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- implement a mpfr_sqrthigh algorithm based on Mulders' algorithm, with a
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basecase variant
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- use mpn_div_q to speed up mpfr_div. However mpn_div_q, which is new in
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GMP 5, is not documented in the GMP manual, thus we are not sure it
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guarantees to return the same quotient as mpn_tdiv_qr.
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Also mpfr_div uses the remainder computed by mpn_divrem. A workaround would
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be to first try with mpn_div_q, and if we cannot (easily) compute the
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rounding, then use the current code with mpn_divrem.
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- compute exp by using the series for cosh or sinh, which has half the terms
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(see Exercise 4.11 from Modern Computer Arithmetic, version 0.3)
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The same method can be used for log, using the series for atanh, i.e.,
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atanh(x) = 1/2*log((1+x)/(1-x)).
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- improve mpfr_gamma (see https://code.google.com/p/fastfunlib/). A possible
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idea is to implement a fast algorithm for the argument reconstruction
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gamma(x+k). One could also use the series for 1/gamma(x), see for example
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http://dlmf.nist.gov/5/7/ or formula (36) from
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http://mathworld.wolfram.com/GammaFunction.html
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- fix regression with mpfr_mpz_root (from Keith Briggs, 5 July 2006), for
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example on 3Ghz P4 with gmp-4.2, x=12.345:
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prec=50000 k=2 k=3 k=10 k=100
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mpz_root 0.036 0.072 0.476 7.628
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mpfr_mpz_root 0.004 0.004 0.036 12.20
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See also mail from Carl Witty on mpfr list, 09 Oct 2007.
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- implement Mulders algorithm for squaring and division
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- for sparse input (say x=1 with 2 bits), mpfr_exp is not faster than for
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full precision when precision <= MPFR_EXP_THRESHOLD. The reason is
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that argument reduction kills sparsity. Maybe avoid argument reduction
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for sparse input?
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- speed up const_euler for large precision [for x=1.1, prec=16610, it takes
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75% of the total time of eint(x)!]
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- speed up mpfr_atan for large arguments (to speed up mpc_log)
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[from Mark Watkins on Fri, 18 Mar 2005]
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Also mpfr_atan(x) seems slower (by a factor of 2) for x near from 1.
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Example on a Athlon for 10^5 bits: x=1.1 takes 3s, whereas 2.1 takes 1.8s.
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The current implementation does not give monotonous timing for the following:
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mpfr_random (x); for (i = 0; i < k; i++) mpfr_atan (y, x, MPFR_RNDN);
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for precision 300 and k=1000, we get 1070ms, and 500ms only for p=400!
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- improve mpfr_sin on values like ~pi (do not compute sin from cos, because
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of the cancellation). For instance, reduce the input modulo pi/2 in
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[-pi/4,pi/4], and define auxiliary functions for which the argument is
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assumed to be already reduced (so that the sin function can avoid
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unnecessary computations by calling the auxiliary cos function instead of
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the full cos function). This will require a native code for sin, for
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example using the reduction sin(3x)=3sin(x)-4sin(x)^3.
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See https://sympa.inria.fr/sympa/arc/mpfr/2007-08/msg00001.html and
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the following messages.
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- improve generic.c to work for number of terms <> 2^k
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- rewrite mpfr_greater_p... as native code.
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- mpf_t uses a scheme where the number of limbs actually present can
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be less than the selected precision, thereby allowing low precision
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values (for instance small integers) to be stored and manipulated in
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an mpf_t efficiently.
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Perhaps mpfr should get something similar, especially if looking to
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replace mpf with mpfr, though it'd be a major change. Alternately
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perhaps those mpfr routines like mpfr_mul where optimizations are
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possible through stripping low zero bits or limbs could check for
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that (this would be less efficient but easier).
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- try the idea of the paper "Reduced Cancellation in the Evaluation of Entire
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Functions and Applications to the Error Function" by W. Gawronski, J. Mueller
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and M. Reinhard, to be published in SIAM Journal on Numerical Analysis: to
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avoid cancellation in say erfc(x) for x large, they compute the Taylor
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expansion of erfc(x)*exp(x^2/2) instead (which has less cancellation),
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and then divide by exp(x^2/2) (which is simpler to compute).
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- replace the *_THRESHOLD macros by global (TLS) variables that can be
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changed at run time (via a function, like other variables)? One benefit
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is that users could use a single MPFR binary on several machines (e.g.,
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a library provided by binary packages or shared via NFS) with different
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thresholds. On the default values, this would be a bit less efficient
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than the current code, but this isn't probably noticeable (this should
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be tested). Something like:
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long *mpfr_tune_get(void) to get the current values (the first value
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is the size of the array).
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int mpfr_tune_set(long *array) to set the tune values.
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int mpfr_tune_run(long level) to find the best values (the support
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for this feature is optional, this can also be done with an
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external function).
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- better distinguish different processors (for example Opteron and Core 2)
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and use corresponding default tuning parameters (as in GMP). This could be
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done in configure.ac to avoid hacking config.guess, for example define
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MPFR_HAVE_CORE2.
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Note (VL): the effect on cross-compilation (that can be a processor
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with the same architecture, e.g. compilation on a Core 2 for an
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Opteron) is not clear. The choice should be consistent with the
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build target (e.g. -march or -mtune value with gcc).
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Also choose better default values. For instance, the default value of
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MPFR_MUL_THRESHOLD is 40, while the best values that have been found
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are between 11 and 19 for 32 bits and between 4 and 10 for 64 bits!
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- during the Many Digits competition, we noticed that (our implantation of)
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Mulders short product was slower than a full product for large sizes.
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This should be precisely analyzed and fixed if needed.
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##############################################################################
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6. Miscellaneous
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##############################################################################
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- [suggested by Tobias Burnus <burnus(at)net-b.de> and
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Asher Langton <langton(at)gcc.gnu.org>, Wed, 01 Aug 2007]
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support quiet and signaling NaNs in mpfr:
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* functions to set/test a quiet/signaling NaN: mpfr_set_snan, mpfr_snan_p,
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mpfr_set_qnan, mpfr_qnan_p
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* correctly convert to/from double (if encoding of s/qNaN is fixed in 754R)
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- check again coverage: on 2007-07-27, Patrick Pelissier reports that the
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following files are not tested at 100%: add1.c, atan.c, atan2.c,
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cache.c, cmp2.c, const_catalan.c, const_euler.c, const_log2.c, cos.c,
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gen_inverse.h, div_ui.c, eint.c, exp3.c, exp_2.c, expm1.c, fma.c, fms.c,
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lngamma.c, gamma.c, get_d.c, get_f.c, get_ld.c, get_str.c, get_z.c,
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inp_str.c, jn.c, jyn_asympt.c, lngamma.c, mpfr-gmp.c, mul.c, mul_ui.c,
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mulders.c, out_str.c, pow.c, print_raw.c, rint.c, root.c, round_near_x.c,
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round_raw_generic.c, set_d.c, set_ld.c, set_q.c, set_uj.c, set_z.c, sin.c,
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sin_cos.c, sinh.c, sqr.c, stack_interface.c, sub1.c, sub1sp.c, subnormal.c,
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uceil_exp2.c, uceil_log2.c, ui_pow_ui.c, urandomb.c, yn.c, zeta.c, zeta_ui.c.
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|
- 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 a new rounding mode: round to nearest, with ties away from zero
|
|
(this is roundTiesToAway in 754-2008, could be used by mpfr_round)
|
|
- 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.
|
|
|
|
- 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.
|
|
|
|
|
|
##############################################################################
|
|
7. Portability
|
|
##############################################################################
|
|
|
|
- add a web page with results of builds on different architectures
|
|
|
|
- support the decimal64 function without requiring --with-gmp-build
|
|
|
|
- [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
|