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@@ -0,0 +1,434 @@ +Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 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 <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) + IncompleteGamma [also wanted by <keith.briggs@bt.com> 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 <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) + +- 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 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 |