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buildtools/gcc/mpfr/src/gamma.c
Fredrik Holmqvist 88896a1144 Update MPFR to 4.1.0 
Change-Id: I42dc2a783b3e0082a7583cc7ecaa6b043ba162f4
Reviewed-on: https://review.haiku-os.org/c/buildtools/+/3021
Reviewed-by: Fredrik Holmqvist <fredrik.holmqvist@gmail.com>
2020-07-17 10:34:03 +00:00

482 lines
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/* mpfr_gamma -- gamma function
Copyright 2001-2020 Free Software Foundation, Inc.
Contributed by the AriC and Caramba 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
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
#define IS_GAMMA
#include "lngamma.c"
#undef IS_GAMMA
/* return a sufficient precision such that 2-x is exact, assuming x < 0
and x is not an integer */
static mpfr_prec_t
mpfr_gamma_2_minus_x_exact (mpfr_srcptr x)
{
/* Since x < 0, 2-x = 2+y with y := -x.
If y < 2, a precision w >= PREC(y) + EXP(2)-EXP(y) = PREC(y) + 2 - EXP(y)
is enough, since no overlap occurs in 2+y, so no carry happens.
If y >= 2, either ULP(y) <= 2, and we need w >= PREC(y)+1 since a
carry can occur, or ULP(y) > 2, and we need w >= EXP(y)-1:
(a) if EXP(y) <= 1, w = PREC(y) + 2 - EXP(y)
(b) if EXP(y) > 1 and EXP(y)-PREC(y) <= 1, w = PREC(y) + 1
(c) if EXP(y) > 1 and EXP(y)-PREC(y) > 1, w = EXP(y) - 1.
Note: case (c) cannot happen in practice since this would imply that
y is integer, thus x is negative integer */
return (MPFR_GET_EXP(x) <= 1) ? MPFR_PREC(x) + 2 - MPFR_GET_EXP(x)
: MPFR_PREC(x) + 1;
}
/* return a sufficient precision such that 1-x is exact, assuming x < 1
and x is not an integer */
static mpfr_prec_t
mpfr_gamma_1_minus_x_exact (mpfr_srcptr x)
{
if (MPFR_IS_POS(x))
return MPFR_PREC(x) - MPFR_GET_EXP(x);
else if (MPFR_GET_EXP(x) <= 0)
return MPFR_PREC(x) + 1 - MPFR_GET_EXP(x);
else /* necessarily MPFR_PREC(x) > MPFR_GET_EXP(x) since otherwise
x would be an integer */
return MPFR_PREC(x) + 1;
}
/* returns a lower bound of the number of significant bits of n!
(not counting the low zero bits).
We know n! >= (n/e)^n*sqrt(2*Pi*n) for n >= 1, and the number of zero bits
is floor(n/2) + floor(n/4) + floor(n/8) + ...
This approximation is exact for n <= 500000, except for n = 219536, 235928,
298981, 355854, 464848, 493725, 498992 where it returns a value 1 too small.
*/
static unsigned long
bits_fac (unsigned long n)
{
mpfr_t x, y;
unsigned long r, k;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ASSERTD (n >= 1);
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (x, 38);
mpfr_init2 (y, 38);
mpfr_set_ui (x, n, MPFR_RNDZ);
mpfr_set_str_binary (y, "10.101101111110000101010001011000101001"); /* upper bound of e */
mpfr_div (x, x, y, MPFR_RNDZ);
mpfr_pow_ui (x, x, n, MPFR_RNDZ);
mpfr_const_pi (y, MPFR_RNDZ);
mpfr_mul_ui (y, y, 2 * n, MPFR_RNDZ);
mpfr_sqrt (y, y, MPFR_RNDZ);
mpfr_mul (x, x, y, MPFR_RNDZ);
mpfr_log2 (x, x, MPFR_RNDZ);
r = mpfr_get_ui (x, MPFR_RNDU); /* lower bound on ceil(x) */
for (k = 2; k <= n; k *= 2)
{
/* Note: the approximation is accurate enough so that the
subtractions do not wrap. */
MPFR_ASSERTD (r >= n / k);
r -= n / k;
}
mpfr_clear (x);
mpfr_clear (y);
MPFR_SAVE_EXPO_FREE (expo);
return r;
}
/* We use the reflection formula
Gamma(1+t) Gamma(1-t) = - Pi t / sin(Pi (1 + t))
in order to treat the case x <= 1,
i.e. with x = 1-t, then Gamma(x) = -Pi*(1-x)/sin(Pi*(2-x))/GAMMA(2-x)
*/
int
mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t xp, GammaTrial, tmp, tmp2;
mpz_t fact;
mpfr_prec_t realprec;
int compared, is_integer;
int inex = 0; /* 0 means: result gamma not set yet */
MPFR_GROUP_DECL (group);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("gamma[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (gamma), mpfr_log_prec, gamma, inex));
/* Trivial cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_NEG (x))
{
/* gamma(x) has a pole at negative integers, thus even if it goes to zero
for other values, we return NaN */
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
else
{
MPFR_SET_INF (gamma);
MPFR_SET_POS (gamma);
MPFR_RET (0); /* exact */
}
}
else /* x is zero */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
MPFR_SET_INF(gamma);
MPFR_SET_SAME_SIGN(gamma, x);
MPFR_SET_DIVBY0 ();
MPFR_RET (0); /* exact */
}
}
/* Check for tiny arguments, where gamma(x) ~ 1/x - euler + ... can be
approximated by 1/x, with some error term ~= - euler.
We need to make sure that there are no breakpoints (discontinuity
points of the rounding function) between gamma(x) and 1/x (included),
where the possible breakpoints (for all rounding modes) are the numbers
that fit on PREC(gamma)+1 bits. There will be a special case when |x|
is a power of two, since such values are breakpoints. We will choose n
minimum such that x fits on n bits and the breakpoints fit on n+1 bits,
thus
n = MAX(MPFR_PREC(x), MPFR_PREC(gamma)).
We know from "Bound on Runs of Zeros and Ones for Algebraic Functions",
Proceedings of Arith15, T. Lang and J.-M. Muller, 2001, that the maximal
number of consecutive zeroes or ones after the round bit for 1/x is n-1
for an input x of n bits [this is an actually much older result!].
But we need a more precise lower bound. Assume that 1/x is near a
breakpoint y. From the definition of n, the input x fits on n bits
and the breakpoint y fits on of n+1 bits. We can write x = X*2^e,
y = Y/2^f with X, Y integers of n and n+1 bits respectively.
Thus X*Y^2^(e-f) is near 1, i.e., X*Y is near the integer 2^(f-e).
Two cases can happen:
(i) either X*Y is exactly 2^(f-e), but this can happen only if X and Y
are themselves powers of two, i.e., x is a power of two;
(ii) or X*Y is at distance at least one from 2^(f-e), thus
|xy-1| >= 2^(e-f), or |y-1/x| >= 2^(e-f)/x = 2^(-f)/X >= 2^(-f-n).
Since ufp(y) = 2^(n-f) [ufp = unit in first place], this means
that the distance |y-1/x| >= 2^(-2n) ufp(y).
Now, assuming |gamma(x)-1/x| < 1, which is true for 0 < x <= 1,
if 2^(-2n) ufp(y) >= 1, then gamma(x) and 1/x round in the same
way, so that rounding 1/x gives the correct result and correct
(nonzero) ternary value.
If x < 2^E, then y >= 2^(-E), thus ufp(y) >= 2^(-E).
A sufficient condition is thus EXP(x) <= -2n, where
n = MAX(MPFR_PREC(x), MPFR_PREC(gamma)).
*/
/* TODO: The above proof uses the same precision for input and output.
Without this assumption, one might obtain a bound like
PREC(x) + PREC(y) instead of 2 MAX(PREC(x),PREC(y)). */
/* TODO: Handle the very small arguments that do not satisfy the condition,
by using the approximation 1/x - euler and a Ziv loop. Otherwise, after
some tests, even Gamma(1+x)/x would be faster than the generic code. */
if (MPFR_GET_EXP (x)
<= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(gamma)))
{
int sign = MPFR_SIGN (x); /* retrieve sign before possible override */
int special;
MPFR_BLOCK_DECL (flags);
MPFR_SAVE_EXPO_MARK (expo);
/* for overflow cases, see below; this needs to be done
before x possibly gets overridden. */
special =
MPFR_GET_EXP (x) == 1 - MPFR_EMAX_MAX &&
MPFR_IS_POS_SIGN (sign) &&
MPFR_IS_LIKE_RNDD (rnd_mode, sign) &&
mpfr_powerof2_raw (x);
MPFR_BLOCK (flags, inex = mpfr_ui_div (gamma, 1, x, rnd_mode));
if (inex == 0) /* |x| is a power of two */
{
/* return RND(1/x - euler) = RND(+/- 2^k - eps) with eps > 0 */
if (rnd_mode == MPFR_RNDN || MPFR_IS_LIKE_RNDU (rnd_mode, sign))
inex = 1;
else
{
mpfr_nextbelow (gamma);
inex = -1;
}
}
else if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
{
/* Overflow in the division 1/x. This is a real overflow, except
in RNDZ or RNDD when 1/x = 2^emax, i.e. x = 2^(-emax): due to
the "- euler", the rounded value in unbounded exponent range
is 0.111...11 * 2^emax (not an overflow). */
if (!special)
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, flags);
}
MPFR_SAVE_EXPO_FREE (expo);
/* Note: an overflow is possible with an infinite result;
in this case, the overflow flag will automatically be
restored by mpfr_check_range. */
return mpfr_check_range (gamma, inex, rnd_mode);
}
is_integer = mpfr_integer_p (x);
/* gamma(x) for x a negative integer gives NaN */
if (is_integer && MPFR_IS_NEG(x))
{
MPFR_SET_NAN (gamma);
MPFR_RET_NAN;
}
compared = mpfr_cmp_ui (x, 1);
if (compared == 0)
return mpfr_set_ui (gamma, 1, rnd_mode);
/* if x is an integer that fits into an unsigned long, use mpfr_fac_ui
if argument is not too large.
If precision is p, fac_ui costs O(u*p), whereas gamma costs O(p*M(p)),
so for u <= M(p), fac_ui should be faster.
We approximate here M(p) by p*log(p)^2, which is not a bad guess.
Warning: since the generic code does not handle exact cases,
we want all cases where gamma(x) is exact to be treated here.
*/
if (is_integer && mpfr_fits_ulong_p (x, MPFR_RNDN))
{
unsigned long int u;
mpfr_prec_t p = MPFR_PREC(gamma);
u = mpfr_get_ui (x, MPFR_RNDN);
MPFR_ASSERTD (u >= 2);
if (u < 44787929UL && bits_fac (u - 1) <= p + (rnd_mode == MPFR_RNDN))
/* bits_fac: lower bound on the number of bits of m,
where gamma(x) = (u-1)! = m*2^e with m odd. */
return mpfr_fac_ui (gamma, u - 1, rnd_mode);
/* if bits_fac(...) > p (resp. p+1 for rounding to nearest),
then gamma(x) cannot be exact in precision p (resp. p+1).
FIXME: remove the test u < 44787929UL after changing bits_fac
to return a mpz_t or mpfr_t. */
}
MPFR_SAVE_EXPO_MARK (expo);
/* check for overflow: according to (6.1.37) in Abramowitz & Stegun,
gamma(x) >= exp(-x) * x^(x-1/2) * sqrt(2*Pi)
>= 2 * (x/e)^x / x for x >= 1 */
if (compared > 0)
{
mpfr_t yp, zp;
mpfr_exp_t expxp;
MPFR_BLOCK_DECL (flags);
MPFR_GROUP_DECL (group);
/* quick test for the default exponent range */
if (mpfr_get_emax () >= 1073741823UL && MPFR_GET_EXP(x) <= 25)
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_gamma_aux (gamma, x, rnd_mode);
}
MPFR_GROUP_INIT_3 (group, 53, xp, yp, zp);
/* 1/e rounded down to 53 bits */
mpfr_set_str_binary (zp,
"0.010111100010110101011000110110001011001110111100111");
mpfr_mul (xp, x, zp, MPFR_RNDZ);
mpfr_sub_ui (yp, x, 2, MPFR_RNDZ);
mpfr_pow (xp, xp, yp, MPFR_RNDZ); /* (x/e)^(x-2) */
mpfr_mul (xp, xp, zp, MPFR_RNDZ); /* x^(x-2) / e^(x-1) */
mpfr_mul (xp, xp, zp, MPFR_RNDZ); /* x^(x-2) / e^x */
mpfr_mul (xp, xp, x, MPFR_RNDZ); /* lower bound on x^(x-1) / e^x */
MPFR_BLOCK (flags, mpfr_mul_2ui (xp, xp, 1, MPFR_RNDZ));
expxp = MPFR_GET_EXP (xp);
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
return MPFR_OVERFLOW (flags) || expxp > __gmpfr_emax ?
mpfr_overflow (gamma, rnd_mode, 1) :
mpfr_gamma_aux (gamma, x, rnd_mode);
}
/* now compared < 0 */
/* check for underflow: for x < 1,
gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x).
Since gamma(2-x) >= 2 * ((2-x)/e)^(2-x) / (2-x), we have
|gamma(x)| <= Pi*(1-x)*(2-x)/2/((2-x)/e)^(2-x) / |sin(Pi*(2-x))|
<= 12 * ((2-x)/e)^x / |sin(Pi*(2-x))|.
To avoid an underflow in ((2-x)/e)^x, we compute the logarithm.
*/
if (MPFR_IS_NEG(x))
{
int underflow = 0, sgn, ck;
mpfr_prec_t w;
mpfr_init2 (xp, 53);
mpfr_init2 (tmp, 53);
mpfr_init2 (tmp2, 53);
/* we want an upper bound for x * [log(2-x)-1].
since x < 0, we need a lower bound on log(2-x) */
mpfr_ui_sub (xp, 2, x, MPFR_RNDD);
mpfr_log (xp, xp, MPFR_RNDD);
mpfr_sub_ui (xp, xp, 1, MPFR_RNDD);
mpfr_mul (xp, xp, x, MPFR_RNDU);
/* we need an upper bound on 1/|sin(Pi*(2-x))|,
thus a lower bound on |sin(Pi*(2-x))|.
If 2-x is exact, then the error of Pi*(2-x) is (1+u)^2 with u = 2^(-p)
thus the error on sin(Pi*(2-x)) is less than 1/2ulp + 3Pi(2-x)u,
assuming u <= 1, thus <= u + 3Pi(2-x)u */
w = mpfr_gamma_2_minus_x_exact (x); /* 2-x is exact for prec >= w */
w += 17; /* to get tmp2 small enough */
mpfr_set_prec (tmp, w);
mpfr_set_prec (tmp2, w);
MPFR_DBGRES (ck = mpfr_ui_sub (tmp, 2, x, MPFR_RNDN));
MPFR_ASSERTD (ck == 0); /* tmp = 2-x exactly */
mpfr_const_pi (tmp2, MPFR_RNDN);
mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); /* Pi*(2-x) */
mpfr_sin (tmp, tmp2, MPFR_RNDN); /* sin(Pi*(2-x)) */
sgn = mpfr_sgn (tmp);
mpfr_abs (tmp, tmp, MPFR_RNDN);
mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDU); /* 3Pi(2-x) */
mpfr_add_ui (tmp2, tmp2, 1, MPFR_RNDU); /* 3Pi(2-x)+1 */
mpfr_div_2ui (tmp2, tmp2, mpfr_get_prec (tmp), MPFR_RNDU);
/* if tmp2<|tmp|, we get a lower bound */
if (mpfr_cmp (tmp2, tmp) < 0)
{
mpfr_sub (tmp, tmp, tmp2, MPFR_RNDZ); /* low bnd on |sin(Pi*(2-x))| */
mpfr_ui_div (tmp, 12, tmp, MPFR_RNDU); /* upper bound */
mpfr_log2 (tmp, tmp, MPFR_RNDU);
mpfr_add (xp, tmp, xp, MPFR_RNDU);
/* The assert below checks that expo.saved_emin - 2 always
fits in a long. FIXME if we want to allow mpfr_exp_t to
be a long long, for instance. */
MPFR_ASSERTN (MPFR_EMIN_MIN - 2 >= LONG_MIN);
underflow = mpfr_cmp_si (xp, expo.saved_emin - 2) <= 0;
}
mpfr_clear (xp);
mpfr_clear (tmp);
mpfr_clear (tmp2);
if (underflow) /* the sign is the opposite of that of sin(Pi*(2-x)) */
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (gamma, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, -sgn);
}
}
realprec = MPFR_PREC (gamma);
/* we want both 1-x and 2-x to be exact */
{
mpfr_prec_t w;
w = mpfr_gamma_1_minus_x_exact (x);
if (realprec < w)
realprec = w;
w = mpfr_gamma_2_minus_x_exact (x);
if (realprec < w)
realprec = w;
}
realprec = realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20;
MPFR_ASSERTD(realprec >= 5);
MPFR_GROUP_INIT_4 (group, realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20,
xp, tmp, tmp2, GammaTrial);
mpz_init (fact);
MPFR_ZIV_INIT (loop, realprec);
for (;;)
{
mpfr_exp_t err_g;
int ck;
MPFR_GROUP_REPREC_4 (group, realprec, xp, tmp, tmp2, GammaTrial);
/* reflection formula: gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x) */
ck = mpfr_ui_sub (xp, 2, x, MPFR_RNDN); /* 2-x, exact */
MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */
mpfr_gamma (tmp, xp, MPFR_RNDN); /* gamma(2-x), error (1+u) */
mpfr_const_pi (tmp2, MPFR_RNDN); /* Pi, error (1+u) */
mpfr_mul (GammaTrial, tmp2, xp, MPFR_RNDN); /* Pi*(2-x), error (1+u)^2 */
err_g = MPFR_GET_EXP(GammaTrial);
mpfr_sin (GammaTrial, GammaTrial, MPFR_RNDN); /* sin(Pi*(2-x)) */
/* If tmp is +Inf, we compute exp(lngamma(x)). */
if (mpfr_inf_p (tmp))
{
inex = mpfr_explgamma (gamma, x, &expo, tmp, tmp2, rnd_mode);
if (inex)
goto end;
else
goto ziv_next;
}
err_g = err_g + 1 - MPFR_GET_EXP(GammaTrial);
/* let g0 the true value of Pi*(2-x), g the computed value.
We have g = g0 + h with |h| <= |(1+u^2)-1|*g.
Thus sin(g) = sin(g0) + h' with |h'| <= |(1+u^2)-1|*g.
The relative error is thus bounded by |(1+u^2)-1|*g/sin(g)
<= |(1+u^2)-1|*2^err_g. <= 2.25*u*2^err_g for |u|<=1/4.
With the rounding error, this gives (0.5 + 2.25*2^err_g)*u. */
ck = mpfr_sub_ui (xp, x, 1, MPFR_RNDN); /* x-1, exact */
MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */
mpfr_mul (xp, tmp2, xp, MPFR_RNDN); /* Pi*(x-1), error (1+u)^2 */
mpfr_mul (GammaTrial, GammaTrial, tmp, MPFR_RNDN);
/* [1 + (0.5 + 2.25*2^err_g)*u]*(1+u)^2 = 1 + (2.5 + 2.25*2^err_g)*u
+ (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2.
For err_g <= realprec-2, we have (0.5 + 2.25*2^err_g)*u <=
0.5*u + 2.25/4 <= 0.6875 and u^2 <= u/4, thus
(0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2 <= 0.6875*(2u+u/4) + u/4
<= 1.8*u, thus the rel. error is bounded by (4.5 + 2.25*2^err_g)*u. */
mpfr_div (GammaTrial, xp, GammaTrial, MPFR_RNDN);
/* the error is of the form (1+u)^3/[1 + (4.5 + 2.25*2^err_g)*u].
For realprec >= 5 and err_g <= realprec-2, [(4.5 + 2.25*2^err_g)*u]^2
<= 0.71, and for |y|<=0.71, 1/(1-y) can be written 1+a*y with a<=4.
(1+u)^3 * (1+4*(4.5 + 2.25*2^err_g)*u)
= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (55+27*2^err_g)*u^3
+ (18+9*2^err_g)*u^4
<= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (56+28*2^err_g)*u^3
<= 1 + (21 + 9*2^err_g)*u + (59+28*2^err_g)*u^2
<= 1 + (23 + 10*2^err_g)*u.
The final error is thus bounded by (23 + 10*2^err_g) ulps,
which is <= 2^6 for err_g<=2, and <= 2^(err_g+4) for err_g >= 2. */
err_g = (err_g <= 2) ? 6 : err_g + 4;
if (MPFR_LIKELY (MPFR_CAN_ROUND (GammaTrial, realprec - err_g,
MPFR_PREC(gamma), rnd_mode)))
break;
ziv_next:
MPFR_ZIV_NEXT (loop, realprec);
}
end:
MPFR_ZIV_FREE (loop);
if (inex == 0)
inex = mpfr_set (gamma, GammaTrial, rnd_mode);
MPFR_GROUP_CLEAR (group);
mpz_clear (fact);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (gamma, inex, rnd_mode);
}
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