Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 Free Software Foundation, Inc. Contributed by the Arenaire and Cacao projects, INRIA. This file is part of the MPFR Library. The MPFR Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License (either version 2.1 of the License, or, at your option, any later version) and the GNU General Public License as published by the Free Software Foundation (most of MPFR is under the former, some under the latter). The 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 MPFR Library; see the file COPYING.LIB. If not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. ############################################################################## Documentation: ############################################################################## - add a description of the algorithms used + proof of correctness - mpfr_set_prec: add an explanation of how to speed up calculations which increase their precision at each step. ############################################################################## Installation: ############################################################################## - nothing to do currently :-) ############################################################################## Changes in existing 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_get_str should support base up to 62 too. - 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 exaclty n bits. If RNDN, takes nq+1 bits. (See also the new division function). - random functions: get rid of _gmp_rand. ############################################################################## New functions to implement: ############################################################################## - functions operating on mpfr_t and double: mpfr_add_d, mpfr_sub_d, mpfr_d_sub, mpfr_mul_d, mpfr_div_d, mpfr_d_div [suggested by Keith Briggs, 3 Jan 2006] - modf (to extract integer and fractional parts), suggested by Dmitry Antipov Thu, 13 Jun 2002 - mpfr_fmod (mpfr_t, mpfr_srcptr, mpfr_srcptr, mp_rnd_t) [suggested by Tomas Zahradnicky , 29 Nov 2003] Kevin: Might want to be called mpfr_mod, to match mpz_mod. -> we probably want to allow both mpfr_fmod and mpfr_mod. Proposed implementation (apart from special cases): int mpfr_fmod (mpfr_t r, mpfr_t x, mpfr_t y, mpfr_rnd_t r) { mpfr_t q; int inexact; /* if 2^(ex-1) <= |x| < 2^ex, and 2^(ey-1) <= |y| < 2^ey, then |x/y| < 2^(ex-ey+1) */ mpfr_init2 (q, MAX(MPFR_PREC_MIN, mpfr_get_exp (x)-mpfr_get_exp (y) + 1)); mpfr_div (q, x, y, GMP_RNDZ); mpfr_trunc (q, q); mpfr_prec_round (q, mpfr_get_prec (q) + mpfr_get_prec (y), GMP_RNDZ); mpfr_mul (q, q, y, GMP_RNDZ); /* exact */ inexact = mpfr_sub (r, x, q, r); mpfr_clear (q); return inexact; } - 1/sqrt(x) [Regis Dupont , 15 Sep 2004] - dilog() [the dilogarithm: dilog(x) = int(ln(t)/(1-t), t=1..x)] - mpfr_printf [Sisyphus Tue, 04 Jan 2005] for example mpfr_printf ("%.2Ff\n", x) - 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 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 Psi = LogDerivative 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; + rec_sqrt (5.2.6): 1 / sqrt(x); + 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) 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_frexp(mpfr_t rop, mp_exp_t *n, mpfr_t op, mp_rnd_t rnd) suggested by Steve Kargl Sun, 7 Aug 2005 - mpfr_inp_raw, mpfr_out_raw (cf mail "Serialization of mpfr_t" from Alexey and answer from Granlund on mpfr list, May 2007) ############################################################################## Efficiency: ############################################################################## - 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 - 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, GMP_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 to [0,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). - combined mpfr_sinh_cosh() [Geoff Bailey, 20 Apr 2005, and Kaveh R. Ghazi, 17 Jan 2007] - improve generic.c to work for number of terms <> 2^k - rewrite mpfr_greater_p... as native code. - inline mpfr_neg? Problems with NAN flags: #define mpfr_neg(_d,_x,_r) \ (__builtin_constant_p ((_d)==(_x)) && (_d)==(_x) ? \ ((_d)->_mpfr_sign = -(_d)->_mpfr_sign, 0) : \ mpfr_neg ((_d), (_x), (_r))) */ - 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). ############################################################################## 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 July 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. - rename mpf2mpfr.h to gmp-mpf2mpfr.h? (will wait until mpfr is fully integrated into gmp :-) - 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: rounding away from 0. This can be easily implemented as follows: round to zero, and if the result is inexact, add one ulp to the mantissa. - add a new rounding mode: round to nearest, with ties away from zero (will be in 754r, 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. - add a new exception "division by zero" (IEEE-754 terminology) / "pole" (LIA-2 terminology). In IEEE 754R (2006 February 14 8:00): "The division by zero exception shall be signaled iff an exact infinite result is defined for an operation on finite operands. [such as a pole or logarithmic singularity.] In particular, the division by zero exception shall be signaled if the divisor is zero and the dividend is a finite nonzero number." - 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". - allow generic tests to run with a restricted exponent range. ############################################################################## Portability: ############################################################################## - [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 ############################################################################## Possible future MPF / MPFR integration: ############################################################################## - mpf routines can become "extern inline"s calling mpfr equivalents, probably just with GMP_RNDZ hard coded, since that's what mpf has always done. - Want to preserve the mpf_t structure size, for binary compatibility. Breaking compatibility would cause lots of pain and potential subtle breakage for users. If the fields in mpf_t are not enough then extra space under _mp_d can be used. - mpf_sgn has been a macro directly accessing the _mp_size field, so a compatible representation would be required. At worst that field could be maintained for mpf_sgn, but not otherwise used internally. mpf_sgn should probably throw an exception if called with NaN, since there's no useful value it can return, so it might want to become a function. Inlined copies in existing binaries would hopefully never see a NaN, if they only do old-style mpf things. - mpfr routines replacing mpf routines must be reentrant and thread safe, since of course that's what has been documented for mpf. - mpfr_random will not be wanted since there's no corresponding mpf_random and new routines should not use the old style global random state.