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Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 Free Software Foundation, Inc. Contributed by the AriC and Caramel 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 http://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. Installation 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 + proof of correctness ############################################################################## 2. Installation ############################################################################## - 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. ############################################################################## 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 precison (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 convertion 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). ############################################################################## 4. New functions to implement ############################################################################## - implement mpfr_q_sub, mpfr_z_div, mpfr_q_div? - implement functions for random distributions, see for example https://sympa.inria.fr/sympa/arc/mpfr/2010-01/msg00034.html (suggested by Charles Karney , 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 , 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) IncompleteGamma [also wanted by 4 Feb 2008: Gamma(a,x), gamma(a,x), P(a,x), Q(a,x); see A&S 6.5, ref. [Smith01] in algorithms.bib] KBessel, KBessel2 [2nd kind] JacobiTheta LogIntegral ExponentialIntegralE1 E1(z) = int(exp(-t)/t, t=z..infinity), |arg z| < Pi mpfr_eint1: implement E1(x) for x > 0, and Ei(-x) for x < 0 E1(NaN) = NaN E1(+Inf) = +0 E1(-Inf) = -Inf E1(+0) = +Inf E1(-0) = -Inf DawsonIntegral GammaD(x) = Gamma(x+1/2) - 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 http://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 http://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)) + Trigamma Function psi'(x). and Polygamma Function: psi^{(m)}(x) for m >= 0, x > 0. - from gnumeric (www.gnome.org/projects/gnumeric/doc/function-reference.html): - beta - 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 ############################################################################## - 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. - 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 http://code.google.com/p/fastfunlib/). A possible idea is to implement a fast algorithm for the argument reconstruction gamma(x+k). 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 - 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. - implement Mulders algorithm for squaring and division - 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 const_euler for large precision [for x=1.1, prec=16610, it takes 75% of the total time of eint(x)!] - speed up mpfr_atan for large arguments (to speed up mpc_log) [from Mark Watkins on Fri, 18 Mar 2005] Also mpfr_atan(x) seems slower (by a factor of 2) for x near from 1. Example on a Athlon for 10^5 bits: x=1.1 takes 3s, whereas 2.1 takes 1.8s. The current implementation does not give monotonous timing for the following: mpfr_random (x); for (i = 0; i < k; i++) mpfr_atan (y, x, MPFR_RNDN); for precision 300 and k=1000, we get 1070ms, and 500ms only for p=400! - 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. ############################################################################## 6. Miscellaneous ############################################################################## - [suggested by Tobias Burnus and Asher Langton , 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) - 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 : 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 http://gcc.gnu.org/viewcvs/trunk/config/stdint.m4 for mpfr-gmp.h