1 Copyright 1999-2015 Free Software Foundation, Inc.
2 Contributed by the AriC and Caramel projects, INRIA.
4 This file is part of the GNU MPFR Library.
6 The GNU MPFR Library is free software; you can redistribute it and/or modify
7 it under the terms of the GNU Lesser General Public License as published by
8 the Free Software Foundation; either version 3 of the License, or (at your
9 option) any later version.
11 The GNU MPFR Library is distributed in the hope that it will be useful, but
12 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
13 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
14 License for more details.
16 You should have received a copy of the GNU Lesser General Public License
17 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
18 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
19 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
24 3. Changes in existing functions
25 4. New functions to implement
30 ##############################################################################
32 ##############################################################################
34 - add a description of the algorithms used + proof of correctness
36 ##############################################################################
38 ##############################################################################
40 - if we want to distinguish GMP and MPIR, we can check at configure time
41 the following symbols which are only defined in MPIR:
43 #define __MPIR_VERSION 0
44 #define __MPIR_VERSION_MINOR 9
45 #define __MPIR_VERSION_PATCHLEVEL 0
47 There is also a library symbol mpir_version, which should match VERSION, set
48 by configure, for example 0.9.0.
50 ##############################################################################
51 3. Changes in existing functions
52 ##############################################################################
54 - export mpfr_overflow and mpfr_underflow as public functions
56 - many functions currently taking into account the precision of the *input*
57 variable to set the initial working precison (acosh, asinh, cosh, ...).
58 This is nonsense since the "average" working precision should only depend
59 on the precision of the *output* variable (and maybe on the *value* of
60 the input in case of cancellation).
61 -> remove those dependencies from the input precision.
64 change the meaning of the 2nd argument (err). Currently the error is
65 at most 2^(MPFR_EXP(b)-err), i.e. err is the relative shift wrt the
66 most significant bit of the approximation. I propose that the error
67 is now at most 2^err ulps of the approximation, i.e.
68 2^(MPFR_EXP(b)-MPFR_PREC(b)+err).
70 - mpfr_set_q first tries to convert the numerator and the denominator
71 to mpfr_t. But this conversion may fail even if the correctly rounded
72 result is representable. New way to implement:
73 Function q = a/b. nq = PREC(q) na = PREC(a) nb = PREC(b)
76 n <- na-nb+ (HIGH(a,nb) >= b)
79 a = q*bb+r --> q has exactly n bits.
82 aa = q*b+r --> q has exactly n bits.
83 If RNDN, takes nq+1 bits. (See also the new division function).
86 ##############################################################################
87 4. New functions to implement
88 ##############################################################################
90 - implement mpfr_q_sub, mpfr_z_div, mpfr_q_div?
91 - implement functions for random distributions, see for example
92 https://sympa.inria.fr/sympa/arc/mpfr/2010-01/msg00034.html
93 (suggested by Charles Karney <ckarney@Sarnoff.com>, 18 Jan 2010):
94 * a Bernoulli distribution with prob p/q (exact)
95 * a general discrete distribution (i with prob w[i]/sum(w[i]) (Walker
96 algorithm, but make it exact)
97 * a uniform distribution in (a,b)
98 * exponential distribution (mean lambda) (von Neumann's method?)
99 * normal distribution (mean m, s.d. sigma) (ratio method?)
100 - wanted for Magma [John Cannon <john@maths.usyd.edu.au>, Tue, 19 Apr 2005]:
101 HypergeometricU(a,b,s) = 1/gamma(a)*int(exp(-su)*u^(a-1)*(1+u)^(b-a-1),
104 PolylogP, PolylogD, PolylogDold: see http://arxiv.org/abs/math.CA/0702243
105 and the references herein.
106 JBessel(n, x) = BesselJ(n+1/2, x)
107 IncompleteGamma [also wanted by <keith.briggs@bt.com> 4 Feb 2008: Gamma(a,x),
108 gamma(a,x), P(a,x), Q(a,x); see A&S 6.5, ref. [Smith01] in algorithms.bib]
109 KBessel, KBessel2 [2nd kind]
112 ExponentialIntegralE1
113 E1(z) = int(exp(-t)/t, t=z..infinity), |arg z| < Pi
114 mpfr_eint1: implement E1(x) for x > 0, and Ei(-x) for x < 0
121 GammaD(x) = Gamma(x+1/2)
122 - functions defined in the LIA-2 standard
123 + minimum and maximum (5.2.2): max, min, max_seq, min_seq, mmax_seq
124 and mmin_seq (mpfr_min and mpfr_max correspond to mmin and mmax);
125 + rounding_rest, floor_rest, ceiling_rest (5.2.4);
126 + remr (5.2.5): x - round(x/y) y;
127 + error functions from 5.2.7 (if useful in MPFR);
128 + power1pm1 (5.3.6.7): (1 + x)^y - 1;
129 + logbase (5.3.6.12): \log_x(y);
130 + logbase1p1p (5.3.6.13): \log_{1+x}(1+y);
131 + rad (5.3.9.1): x - round(x / (2 pi)) 2 pi = remr(x, 2 pi);
132 + axis_rad (5.3.9.1) if useful in MPFR;
133 + cycle (5.3.10.1): rad(2 pi x / u) u / (2 pi) = remr(x, u);
134 + axis_cycle (5.3.10.1) if useful in MPFR;
135 + sinu, cosu, tanu, cotu, secu, cscu, cossinu, arcsinu, arccosu,
136 arctanu, arccotu, arcsecu, arccscu (5.3.10.{2..14}):
137 sin(x 2 pi / u), etc.;
138 [from which sinpi(x) = sin(Pi*x), ... are trivial to implement, with u=2.]
139 + arcu (5.3.10.15): arctan2(y,x) u / (2 pi);
140 + rad_to_cycle, cycle_to_rad, cycle_to_cycle (5.3.11.{1..3}).
141 - From GSL, missing special functions (if useful in MPFR):
142 (cf http://www.gnu.org/software/gsl/manual/gsl-ref.html#Special-Functions)
143 + The Airy functions Ai(x) and Bi(x) defined by the integral representations:
144 * Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
145 * Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt
146 * Derivatives of Airy Functions
147 + The Bessel functions for n integer and n fractional:
148 * Regular Modified Cylindrical Bessel Functions I_n
149 * Irregular Modified Cylindrical Bessel Functions K_n
150 * Regular Spherical Bessel Functions j_n: j_0(x) = \sin(x)/x,
151 j_1(x)= (\sin(x)/x-\cos(x))/x & j_2(x)= ((3/x^2-1)\sin(x)-3\cos(x)/x)/x
152 Note: the "spherical" Bessel functions are solutions of
153 x^2 y'' + 2 x y' + [x^2 - n (n+1)] y = 0 and satisfy
154 j_n(x) = sqrt(Pi/(2x)) J_{n+1/2}(x). They should not be mixed with the
155 classical Bessel Functions, also noted j0, j1, jn, y0, y1, yn in C99
157 Cf https://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions
158 *Irregular Spherical Bessel Functions y_n: y_0(x) = -\cos(x)/x,
159 y_1(x)= -(\cos(x)/x+\sin(x))/x &
160 y_2(x)= (-3/x^3+1/x)\cos(x)-(3/x^2)\sin(x)
161 * Regular Modified Spherical Bessel Functions i_n:
162 i_l(x) = \sqrt{\pi/(2x)} I_{l+1/2}(x)
163 * Irregular Modified Spherical Bessel Functions:
164 k_l(x) = \sqrt{\pi/(2x)} K_{l+1/2}(x).
166 Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
167 Cl_2(\theta) = \Im Li_2(\exp(i \theta)) (dilogarithm).
168 + Dawson Function: \exp(-x^2) \int_0^x dt \exp(t^2).
169 + Debye Functions: D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
170 + Elliptic Integrals:
171 * Definition of Legendre Forms:
172 F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
173 E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))
174 P(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
175 * Complete Legendre forms are denoted by
178 * Definition of Carlson Forms
179 RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
180 RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
181 RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
182 RJ(x,y,z,p) = 3/2 \int_0^\infty dt
183 (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
184 + Elliptic Functions (Jacobi)
185 + N-relative exponential:
186 exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
187 + exponential integral:
188 E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
189 Ei_3(x) = \int_0^x dt \exp(-t^3) for x >= 0.
190 Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
191 + Hyperbolic/Trigonometric Integrals
192 Shi(x) = \int_0^x dt \sinh(t)/t
193 Chi(x) := Re[ \gamma_E + \log(x) + \int_0^x dt (\cosh[t]-1)/t]
194 Si(x) = \int_0^x dt \sin(t)/t
195 Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0
196 AtanInt(x) = \int_0^x dt \arctan(t)/t
197 [ \gamma_E is the Euler constant ]
198 + Fermi-Dirac Function:
199 F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
200 + Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(a) : see [Smith01] in
202 logarithm of the Pochhammer symbol
203 + Gegenbauer Functions
205 + Eta Function: \eta(s) = (1-2^{1-s}) \zeta(s)
206 Hurwitz zeta function: \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.
207 + Lambert W Functions, W(x) are defined to be solutions of the equation:
209 This function has multiple branches for x < 0 (2 funcs W0(x) and Wm1(x))
210 + Trigamma Function psi'(x).
211 and Polygamma Function: psi^{(m)}(x) for m >= 0, x > 0.
213 - from gnumeric (www.gnome.org/projects/gnumeric/doc/function-reference.html):
220 - mpfr_inp_raw, mpfr_out_raw (cf mail "Serialization of mpfr_t" from Alexey
221 and answer from Granlund on mpfr list, May 2007)
222 - [maybe useful for SAGE] implement companion frac_* functions to the rint_*
223 functions. For example mpfr_frac_floor(x) = x - floor(x). (The current
224 mpfr_frac function corresponds to mpfr_rint_trunc.)
225 - scaled erfc (https://sympa.inria.fr/sympa/arc/mpfr/2009-05/msg00054.html)
226 - asec, acsc, acot, asech, acsch and acoth (mail from Björn Terelius on mpfr
229 ##############################################################################
231 ##############################################################################
233 - implement a mpfr_sqrthigh algorithm based on Mulders' algorithm, with a
235 - use mpn_div_q to speed up mpfr_div. However mpn_div_q, which is new in
236 GMP 5, is not documented in the GMP manual, thus we are not sure it
237 guarantees to return the same quotient as mpn_tdiv_qr.
238 Also mpfr_div uses the remainder computed by mpn_divrem. A workaround would
239 be to first try with mpn_div_q, and if we cannot (easily) compute the
240 rounding, then use the current code with mpn_divrem.
241 - compute exp by using the series for cosh or sinh, which has half the terms
242 (see Exercise 4.11 from Modern Computer Arithmetic, version 0.3)
243 The same method can be used for log, using the series for atanh, i.e.,
244 atanh(x) = 1/2*log((1+x)/(1-x)).
245 - improve mpfr_gamma (see https://code.google.com/p/fastfunlib/). A possible
246 idea is to implement a fast algorithm for the argument reconstruction
247 gamma(x+k). One could also use the series for 1/gamma(x), see for example
248 http://dlmf.nist.gov/5/7/ or formula (36) from
249 http://mathworld.wolfram.com/GammaFunction.html
250 - fix regression with mpfr_mpz_root (from Keith Briggs, 5 July 2006), for
251 example on 3Ghz P4 with gmp-4.2, x=12.345:
252 prec=50000 k=2 k=3 k=10 k=100
253 mpz_root 0.036 0.072 0.476 7.628
254 mpfr_mpz_root 0.004 0.004 0.036 12.20
255 See also mail from Carl Witty on mpfr list, 09 Oct 2007.
256 - implement Mulders algorithm for squaring and division
257 - for sparse input (say x=1 with 2 bits), mpfr_exp is not faster than for
258 full precision when precision <= MPFR_EXP_THRESHOLD. The reason is
259 that argument reduction kills sparsity. Maybe avoid argument reduction
261 - speed up const_euler for large precision [for x=1.1, prec=16610, it takes
262 75% of the total time of eint(x)!]
263 - speed up mpfr_atan for large arguments (to speed up mpc_log)
264 [from Mark Watkins on Fri, 18 Mar 2005]
265 Also mpfr_atan(x) seems slower (by a factor of 2) for x near from 1.
266 Example on a Athlon for 10^5 bits: x=1.1 takes 3s, whereas 2.1 takes 1.8s.
267 The current implementation does not give monotonous timing for the following:
268 mpfr_random (x); for (i = 0; i < k; i++) mpfr_atan (y, x, MPFR_RNDN);
269 for precision 300 and k=1000, we get 1070ms, and 500ms only for p=400!
270 - improve mpfr_sin on values like ~pi (do not compute sin from cos, because
271 of the cancellation). For instance, reduce the input modulo pi/2 in
272 [-pi/4,pi/4], and define auxiliary functions for which the argument is
273 assumed to be already reduced (so that the sin function can avoid
274 unnecessary computations by calling the auxiliary cos function instead of
275 the full cos function). This will require a native code for sin, for
276 example using the reduction sin(3x)=3sin(x)-4sin(x)^3.
277 See https://sympa.inria.fr/sympa/arc/mpfr/2007-08/msg00001.html and
278 the following messages.
279 - improve generic.c to work for number of terms <> 2^k
280 - rewrite mpfr_greater_p... as native code.
282 - mpf_t uses a scheme where the number of limbs actually present can
283 be less than the selected precision, thereby allowing low precision
284 values (for instance small integers) to be stored and manipulated in
285 an mpf_t efficiently.
287 Perhaps mpfr should get something similar, especially if looking to
288 replace mpf with mpfr, though it'd be a major change. Alternately
289 perhaps those mpfr routines like mpfr_mul where optimizations are
290 possible through stripping low zero bits or limbs could check for
291 that (this would be less efficient but easier).
293 - try the idea of the paper "Reduced Cancellation in the Evaluation of Entire
294 Functions and Applications to the Error Function" by W. Gawronski, J. Mueller
295 and M. Reinhard, to be published in SIAM Journal on Numerical Analysis: to
296 avoid cancellation in say erfc(x) for x large, they compute the Taylor
297 expansion of erfc(x)*exp(x^2/2) instead (which has less cancellation),
298 and then divide by exp(x^2/2) (which is simpler to compute).
300 - replace the *_THRESHOLD macros by global (TLS) variables that can be
301 changed at run time (via a function, like other variables)? One benefit
302 is that users could use a single MPFR binary on several machines (e.g.,
303 a library provided by binary packages or shared via NFS) with different
304 thresholds. On the default values, this would be a bit less efficient
305 than the current code, but this isn't probably noticeable (this should
306 be tested). Something like:
307 long *mpfr_tune_get(void) to get the current values (the first value
308 is the size of the array).
309 int mpfr_tune_set(long *array) to set the tune values.
310 int mpfr_tune_run(long level) to find the best values (the support
311 for this feature is optional, this can also be done with an
314 - better distinguish different processors (for example Opteron and Core 2)
315 and use corresponding default tuning parameters (as in GMP). This could be
316 done in configure.ac to avoid hacking config.guess, for example define
318 Note (VL): the effect on cross-compilation (that can be a processor
319 with the same architecture, e.g. compilation on a Core 2 for an
320 Opteron) is not clear. The choice should be consistent with the
321 build target (e.g. -march or -mtune value with gcc).
322 Also choose better default values. For instance, the default value of
323 MPFR_MUL_THRESHOLD is 40, while the best values that have been found
324 are between 11 and 19 for 32 bits and between 4 and 10 for 64 bits!
326 - during the Many Digits competition, we noticed that (our implantation of)
327 Mulders short product was slower than a full product for large sizes.
328 This should be precisely analyzed and fixed if needed.
330 ##############################################################################
332 ##############################################################################
334 - [suggested by Tobias Burnus <burnus(at)net-b.de> and
335 Asher Langton <langton(at)gcc.gnu.org>, Wed, 01 Aug 2007]
336 support quiet and signaling NaNs in mpfr:
337 * functions to set/test a quiet/signaling NaN: mpfr_set_snan, mpfr_snan_p,
338 mpfr_set_qnan, mpfr_qnan_p
339 * correctly convert to/from double (if encoding of s/qNaN is fixed in 754R)
341 - check again coverage: on 2007-07-27, Patrick Pelissier reports that the
342 following files are not tested at 100%: add1.c, atan.c, atan2.c,
343 cache.c, cmp2.c, const_catalan.c, const_euler.c, const_log2.c, cos.c,
344 gen_inverse.h, div_ui.c, eint.c, exp3.c, exp_2.c, expm1.c, fma.c, fms.c,
345 lngamma.c, gamma.c, get_d.c, get_f.c, get_ld.c, get_str.c, get_z.c,
346 inp_str.c, jn.c, jyn_asympt.c, lngamma.c, mpfr-gmp.c, mul.c, mul_ui.c,
347 mulders.c, out_str.c, pow.c, print_raw.c, rint.c, root.c, round_near_x.c,
348 round_raw_generic.c, set_d.c, set_ld.c, set_q.c, set_uj.c, set_z.c, sin.c,
349 sin_cos.c, sinh.c, sqr.c, stack_interface.c, sub1.c, sub1sp.c, subnormal.c,
350 uceil_exp2.c, uceil_log2.c, ui_pow_ui.c, urandomb.c, yn.c, zeta.c, zeta_ui.c.
352 - check the constants mpfr_set_emin (-16382-63) and mpfr_set_emax (16383) in
353 get_ld.c and the other constants, and provide a testcase for large and
356 - from Kevin Ryde <user42@zip.com.au>:
357 Also for pi.c, a pre-calculated compiled-in pi to a few thousand
358 digits would be good value I think. After all, say 10000 bits using
359 1250 bytes would still be small compared to the code size!
360 Store pi in round to zero mode (to recover other modes).
362 - add a new rounding mode: round to nearest, with ties away from zero
363 (this is roundTiesToAway in 754-2008, could be used by mpfr_round)
364 - add a new roundind mode: round to odd. If the result is not exactly
365 representable, then round to the odd mantissa. This rounding
366 has the nice property that for k > 1, if:
367 y = round(x, p+k, TO_ODD)
368 z = round(y, p, TO_NEAREST_EVEN), then
369 z = round(x, p, TO_NEAREST_EVEN)
370 so it avoids the double-rounding problem.
372 - add tests of the ternary value for constants
374 - When doing Extensive Check (--enable-assert=full), since all the
375 functions use a similar use of MACROS (ZivLoop, ROUND_P), it should
376 be possible to do such a scheme:
377 For the first call to ROUND_P when we can round.
378 Mark it as such and save the approximated rounding value in
379 a temporary variable.
380 Then after, if the mark is set, check if:
381 - we still can round.
382 - The rounded value is the same.
383 It should be a complement to tgeneric tests.
385 - in div.c, try to find a case for which cy != 0 after the line
386 cy = mpn_sub_1 (sp + k, sp + k, qsize, cy);
387 (which should be added to the tests), e.g. by having {vp, k} = 0, or
388 prove that this cannot happen.
390 - add a configure test for --enable-logging to ignore the option if
391 it cannot be supported. Modify the "configure --help" description
392 to say "on systems that support it".
394 - add generic bad cases for functions that don't have an inverse
395 function that is implemented (use a single Newton iteration).
397 - add bad cases for the internal error bound (by using a dichotomy
398 between a bad case for the correct rounding and some input value
399 with fewer Ziv iterations?).
401 - add an option to use a 32-bit exponent type (int) on LP64 machines,
402 mainly for developers, in order to be able to test the case where the
403 extended exponent range is the same as the default exponent range, on
405 Tests can be done with the exp-int branch (added on 2010-12-17, and
406 many tests fail at this time).
408 - test underflow/overflow detection of various functions (in particular
409 mpfr_exp) in reduced exponent ranges, including ranges that do not
412 - add an internal macro that does the equivalent of the following?
413 MPFR_IS_ZERO(x) || MPFR_GET_EXP(x) <= value
415 - check whether __gmpfr_emin and __gmpfr_emax could be replaced by
416 a constant (see README.dev). Also check the use of MPFR_EMIN_MIN
420 ##############################################################################
422 ##############################################################################
424 - add a web page with results of builds on different architectures
426 - support the decimal64 function without requiring --with-gmp-build
428 - [Kevin about texp.c long strings]
429 For strings longer than c99 guarantees, it might be cleaner to
430 introduce a "tests_strdupcat" or something to concatenate literal
431 strings into newly allocated memory. I thought I'd done that in a
432 couple of places already. Arrays of chars are not much fun.
434 - use https://gcc.gnu.org/viewcvs/gcc/trunk/config/stdint.m4 for mpfr-gmp.h