chromium / native_client / nacl-gcc / f80d6b9ee7f94755c697ffb7194fb01dd0c537dd / . / mpfr-2.4.1 / TODO

Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. | |

Contributed by the Arenaire and Cacao 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 (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 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.LIB. If not, 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 | |

- mpfr_set_prec: add an explanation of how to speed up calculations | |

which increase their precision at each step. | |

############################################################################## | |

2. Installation | |

############################################################################## | |

- nothing to do currently :-) | |

############################################################################## | |

3. 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. | |

############################################################################## | |

4. New functions to implement | |

############################################################################## | |

- 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] | |

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; | |

+ 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 <sgk@troutmask.apl.washington.edu> 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) | |

- [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.) | |

############################################################################## | |

5. 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 | |

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, 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 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 http://websympa.loria.fr/wwsympa/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. | |

- 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). | |

- 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). | |

############################################################################## | |

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 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 <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: 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) / "infinitary" | |

(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. | |

- 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. This would need to rename all mp_exp_t as mpfr_exp_t | |

and add a typedef either to mp_exp_t (default) or to int (when this | |

option is enabled). | |

- test underflow/overflow detection of various functions (in particular | |

mpfr_exp) in reduced exponent ranges, including ranges that do not | |

contain 0. | |

############################################################################## | |

7. 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 |