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Red Bear OS is a full fork. All sources must be available from git clone with zero network access. Removed gitignore rules that excluded fetched source trees under recipes/*/source/, local/recipes/kde/*/source/, local/recipes/qt/*/source/, and vendor source trees. Build artifacts (target/, build/, source.tar, *.o, *.so) remain excluded. 127291 files added — kernel, relibc, base, bootloader, pkgar, all KDE/Qt frameworks, mesa, wayland, DRM drivers, and every other recipe source.
382 lines
14 KiB
C
382 lines
14 KiB
C
/* mpfr_eint, mpfr_eint1 -- the exponential integral
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Copyright 2005-2025 Free Software Foundation, Inc.
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Contributed by the Pascaline and Caramba projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER.
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If not, see <https://www.gnu.org/licenses/>. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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/* eint1(x) = -gamma - log(x) - sum((-1)^k*z^k/k/k!, k=1..infinity) for x > 0
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= - eint(-x) for x < 0
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where
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eint (x) = gamma + log(x) + sum(z^k/k/k!, k=1..infinity) for x > 0
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eint (x) is undefined for x < 0.
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*/
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/* Compute in y an approximation of sum(x^k/k/k!, k=1..infinity),
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assuming x != 0, and return e such that the absolute error is
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bounded by 2^e ulp(y).
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Return PREC(y) when the truncated series does not converge.
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*/
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static mpfr_exp_t
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mpfr_eint_aux (mpfr_ptr y, mpfr_srcptr x)
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{
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mpfr_t eps; /* dynamic (absolute) error bound on t */
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mpfr_t erru, errs;
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mpz_t m, s, t, u;
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mpfr_exp_t e, sizeinbase;
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mpfr_prec_t w = MPFR_PREC(y);
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unsigned long k;
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MPFR_GROUP_DECL (group);
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MPFR_LOG_FUNC (
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("x[%Pd]=%.*Rg", mpfr_get_prec (x), mpfr_log_prec, x),
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("y[%Pd]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y));
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/* for |x| <= 1, we have S := sum(x^k/k/k!, k=1..infinity) = x + R(x)
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where |R(x)| <= (x/2)^2/(1-|x|/2) <= 2*(x/2)^2
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thus |R(x)/x| <= |x|/2
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thus if |x| <= 2^(-PREC(y)) we have |S - o(x)| <= ulp(y) */
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if (MPFR_GET_EXP(x) <= - (mpfr_exp_t) w)
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{
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mpfr_set (y, x, MPFR_RNDN);
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return 0;
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}
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mpz_init (s); /* initializes to 0 */
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mpz_init (t);
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mpz_init (u);
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mpz_init (m);
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MPFR_GROUP_INIT_3 (group, 31, eps, erru, errs);
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e = mpfr_get_z_2exp (m, x); /* x = m * 2^e with m != 0 */
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MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e));
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MPFR_ASSERTD (mpz_sizeinbase (m, 2) == MPFR_PREC (x)); /* since m != 0 */
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if (MPFR_PREC (x) > w)
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{
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e += MPFR_PREC (x) - w;
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mpz_tdiv_q_2exp (m, m, MPFR_PREC (x) - w); /* one still has m != 0 */
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MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e));
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}
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/* Remove trailing zeroes from m: this will speed up much cases where
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x is a small integer divided by a power of 2.
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Note: As shown above, m != 0. This is needed for the "e += ..." below,
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otherwise n would take the largest value of mp_bitcnt_t and could be
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too large. */
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{
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mp_bitcnt_t n = mpz_scan1 (m, 0);
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mpz_tdiv_q_2exp (m, m, n);
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/* Since one initially has mpz_sizeinbase (m, 2) == MPFR_PREC (x)
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and m has not increased, one can deduce that n <= MPFR_PREC (x),
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so that the cast to mpfr_prec_t is valid. This cast is needed to
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ensure that the operand e of the addition below is not converted
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to an unsigned integer type, which could yield incorrect results
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with some C implementations. */
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MPFR_ASSERTD (n <= MPFR_PREC (x));
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e += (mpfr_prec_t) n;
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}
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/* initialize t to 2^w */
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mpz_set_ui (t, 1);
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mpz_mul_2exp (t, t, w);
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mpfr_set_ui (eps, 0, MPFR_RNDN); /* eps[0] = 0 */
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mpfr_set_ui (errs, 0, MPFR_RNDN); /* maximal error on s */
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for (k = 1;; k++)
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{
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/* let t[k] = x^k/k/k!, and eps[k] be the absolute error on t[k]:
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since t[k] = trunc(t[k-1]*m*2^e/k), we have
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eps[k+1] <= 1 + eps[k-1]*|m|*2^e/k + |t[k-1]|*|m|*2^(1-w)*2^e/k
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= 1 + (eps[k-1] + |t[k-1]|*2^(1-w))*|m|*2^e/k
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= 1 + (eps[k-1]*2^(w-1) + |t[k-1]|)*2^(1-w)*|m|*2^e/k */
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mpfr_mul_2ui (eps, eps, w - 1, MPFR_RNDU);
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if (mpz_sgn (t) >= 0)
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mpfr_add_z (eps, eps, t, MPFR_RNDU);
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else
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mpfr_sub_z (eps, eps, t, MPFR_RNDU);
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MPFR_MPZ_SIZEINBASE2 (sizeinbase, m);
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mpfr_mul_2si (eps, eps, sizeinbase - (w - 1) + e, MPFR_RNDU);
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mpfr_div_ui (eps, eps, k, MPFR_RNDU);
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mpfr_add_ui (eps, eps, 1, MPFR_RNDU);
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mpz_mul (t, t, m);
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if (e < 0)
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mpz_tdiv_q_2exp (t, t, -e);
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else
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mpz_mul_2exp (t, t, e);
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mpz_tdiv_q_ui (t, t, k);
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mpz_tdiv_q_ui (u, t, k);
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mpz_add (s, s, u);
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/* the absolute error on u is <= 1 + eps[k]/k */
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mpfr_div_ui (erru, eps, k, MPFR_RNDU);
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mpfr_add_ui (erru, erru, 1, MPFR_RNDU);
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/* and that on s is the sum of all errors on u */
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mpfr_add (errs, errs, erru, MPFR_RNDU);
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/* we are done when t is smaller than errs */
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if (mpz_sgn (t) == 0)
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sizeinbase = 0;
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else
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MPFR_MPZ_SIZEINBASE2 (sizeinbase, t);
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if (sizeinbase < MPFR_GET_EXP (errs))
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break;
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}
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/* the truncation error is bounded by (|t|+eps)/k*(|x|/k + |x|^2/k^2 + ...)
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<= (|t|+eps)/k*|x|/(k-|x|) */
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mpz_abs (t, t);
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mpfr_add_z (eps, eps, t, MPFR_RNDU);
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mpfr_div_ui (eps, eps, k, MPFR_RNDU);
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mpfr_abs (erru, x, MPFR_RNDU); /* |x| */
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mpfr_mul (eps, eps, erru, MPFR_RNDU);
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mpfr_ui_sub (erru, k, erru, MPFR_RNDD);
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if (MPFR_IS_NEG (erru))
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{
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/* the truncated series does not converge, return fail */
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e = w;
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}
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else
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{
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mpfr_div (eps, eps, erru, MPFR_RNDU);
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mpfr_add (errs, errs, eps, MPFR_RNDU);
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mpfr_set_z (y, s, MPFR_RNDN);
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mpfr_div_2ui (y, y, w, MPFR_RNDN);
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/* errs was an absolute error bound on s. We must convert it to an error
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in terms of ulp(y). Since ulp(y) = 2^(EXP(y)-PREC(y)), we must
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divide the error by 2^(EXP(y)-PREC(y)), but since we divided also
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y by 2^w = 2^PREC(y), we must simply divide by 2^EXP(y). */
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e = MPFR_GET_EXP (errs) - MPFR_GET_EXP (y);
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}
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MPFR_GROUP_CLEAR (group);
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mpz_clear (s);
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mpz_clear (t);
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mpz_clear (u);
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mpz_clear (m);
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MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e));
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return e;
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}
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/* Return in y an approximation of Ei(x) using the asymptotic expansion:
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Ei(x) = exp(x)/x * (1 + 1/x + 2/x^2 + ... + k!/x^k + ...)
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Assumes |x| >= PREC(y) * log(2).
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Returns the error bound in terms of ulp(y).
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*/
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static mpfr_exp_t
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mpfr_eint_asympt (mpfr_ptr y, mpfr_srcptr x)
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{
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mpfr_prec_t p = MPFR_PREC(y);
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mpfr_t invx, t, err;
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unsigned long k;
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mpfr_exp_t err_exp;
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MPFR_LOG_FUNC (
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("x[%Pd]=%.*Rg", mpfr_get_prec (x), mpfr_log_prec, x),
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("err_exp=%" MPFR_EXP_FSPEC "d", (mpfr_eexp_t) err_exp));
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mpfr_init2 (t, p);
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mpfr_init2 (invx, p);
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mpfr_init2 (err, 31); /* error in ulps on y */
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mpfr_ui_div (invx, 1, x, MPFR_RNDN); /* invx = 1/x*(1+u) with |u|<=2^(1-p) */
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mpfr_set_ui (t, 1, MPFR_RNDN); /* exact */
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mpfr_set (y, t, MPFR_RNDN);
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mpfr_set_ui (err, 0, MPFR_RNDN);
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for (k = 1; MPFR_GET_EXP(t) + (mpfr_exp_t) p > MPFR_GET_EXP(y); k++)
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{
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mpfr_mul (t, t, invx, MPFR_RNDN); /* 2 more roundings */
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mpfr_mul_ui (t, t, k, MPFR_RNDN); /* 1 more rounding: t = k!/x^k*(1+u)^e
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with u=2^{-p} and |e| <= 3*k */
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/* we use the fact that |(1+u)^n-1| <= 2*|n*u| for |n*u| <= 1, thus
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the error on t is less than 6*k*2^{-p}*t <= 6*k*ulp(t) */
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/* err is in terms of ulp(y): transform it in terms of ulp(t) */
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mpfr_mul_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU);
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mpfr_add_ui (err, err, 6 * k, MPFR_RNDU);
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/* transform back in terms of ulp(y) */
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mpfr_div_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU);
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mpfr_add (y, y, t, MPFR_RNDN);
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}
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/* add the truncation error bounded by ulp(y): 1 ulp */
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mpfr_mul (y, y, invx, MPFR_RNDN); /* err <= 2*err + 3/2 */
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mpfr_exp (t, x, MPFR_RNDN); /* err(t) <= 1/2*ulp(t) */
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mpfr_mul (y, y, t, MPFR_RNDN); /* again: err <= 2*err + 3/2 */
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mpfr_mul_2ui (err, err, 2, MPFR_RNDU);
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mpfr_add_ui (err, err, 8, MPFR_RNDU);
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err_exp = MPFR_GET_EXP(err);
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mpfr_clear (t);
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mpfr_clear (invx);
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mpfr_clear (err);
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return err_exp;
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}
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/* mpfr_eint returns Ei(x) for x >= 0,
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and -E1(-x) for x < 0, following https://dlmf.nist.gov/6.2 */
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int
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mpfr_eint (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
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{
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int inex;
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mpfr_t tmp, ump, x_abs;
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mpfr_exp_t err, te;
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mpfr_prec_t prec;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_ZIV_DECL (loop);
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MPFR_LOG_FUNC (
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("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd),
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("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex));
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if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
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{
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if (MPFR_IS_NAN (x))
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{
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MPFR_SET_NAN (y);
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MPFR_RET_NAN;
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}
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else if (MPFR_IS_INF (x))
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{
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/* eint(+inf) = +inf and eint(-inf) = -0 */
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if (MPFR_IS_POS (x))
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{
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MPFR_SET_INF(y);
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MPFR_SET_POS(y);
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}
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else
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{
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MPFR_SET_ZERO(y);
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MPFR_SET_NEG(y);
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}
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MPFR_RET(0);
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}
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else /* eint(+/-0) = -Inf */
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{
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MPFR_SET_INF(y);
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MPFR_SET_NEG(y);
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MPFR_SET_DIVBY0 ();
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MPFR_RET(0);
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}
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}
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MPFR_TMP_INIT_ABS (x_abs, x);
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MPFR_SAVE_EXPO_MARK (expo);
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/* Init stuff */
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prec = MPFR_PREC (y) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 6;
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mpfr_init2 (tmp, 64);
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mpfr_init2 (ump, 64);
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/* Since eint(x) >= exp(x)/x, we have log2(eint(x)) >= (x-log(x))/log(2).
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Let's compute k <= (x-log(x))/log(2) in a low precision. If k >= emax,
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then log2(eint(x)) >= emax, and eint(x) >= 2^emax, i.e. it overflows. */
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if (MPFR_IS_POS(x))
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{
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mpfr_log (tmp, x, MPFR_RNDU);
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mpfr_sub (ump, x, tmp, MPFR_RNDD);
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mpfr_div (ump, ump, __gmpfr_const_log2_RNDU, MPFR_RNDD);
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/* FIXME: We really need a mpfr_cmp_exp_t function. */
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MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
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if (mpfr_cmp_ui (ump, __gmpfr_emax) >= 0)
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{
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mpfr_clear (tmp);
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mpfr_clear (ump);
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_overflow (y, rnd, 1);
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}
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}
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/* Since E1(x) <= exp(-x) for x >= 1, we have log2(E1(x)) <= -x/log(2).
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Let's compute k >= -x/log(2) in a low precision. If k < emin
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then log2(E1(x)) <= emin-1, and E1(x) <= 2^(emin-1): it underflows. */
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if (MPFR_IS_NEG(x) && MPFR_GET_EXP(x) >= 1)
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{
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mpfr_div (ump, x, __gmpfr_const_log2_RNDD, MPFR_RNDU);
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MPFR_ASSERTN (MPFR_EMIN_MIN >= LONG_MIN);
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if (mpfr_cmp_si (ump, __gmpfr_emin) < 0)
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{
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mpfr_clear (tmp);
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mpfr_clear (ump);
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_underflow (y, rnd, -1);
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}
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}
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/* eint() has a root 0.37250741078136663446...,
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so if x is near, already take more bits */
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if (MPFR_IS_POS(x) && MPFR_GET_EXP(x) == -1) /* 1/4 <= x < 1/2 */
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{
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mpfr_t y;
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mpfr_init2 (y, 32);
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/* 1599907147/2^32 is a 32-bit approximation of 0.37250741078136663446 */
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mpfr_set_ui_2exp (y, 1599907147UL, -32, MPFR_RNDN);
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mpfr_sub (y, x, y, MPFR_RNDN);
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prec += (mpfr_zero_p (y)) ? 32
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: mpfr_get_exp (y) < 0 ? -mpfr_get_exp (y) : 0;
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mpfr_clear (y);
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}
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mpfr_set_prec (tmp, prec);
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mpfr_set_prec (ump, prec);
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MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controller */
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for (;;) /* Infinite loop */
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{
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/* For the asymptotic expansion to work, we need that the smallest
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value of k!/|x|^k is smaller than 2^(-p). The minimum is obtained for
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x=k, and it is smaller than e*sqrt(x)/e^x for x>=1. */
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if (MPFR_GET_EXP (x) > 0 &&
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mpfr_cmp_d (x_abs, ((double) prec +
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0.5 * (double) MPFR_GET_EXP (x)) * LOG2 + 1.0) > 0)
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err = mpfr_eint_asympt (tmp, x);
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else
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{
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err = mpfr_eint_aux (tmp, x); /* error <= 2^err ulp(tmp) */
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te = MPFR_GET_EXP(tmp);
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mpfr_const_euler (ump, MPFR_RNDN); /* 0.577 -> EXP(ump)=0 */
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mpfr_add (tmp, tmp, ump, MPFR_RNDN);
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/* If tmp <> 0:
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error <= 1/2 + 1/2*2^(EXP(ump)-EXP(tmp)) + 2^(te-EXP(tmp)+err)
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<= 1/2 + 2^(MAX(EXP(ump), te+err+1) - EXP(tmp))
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<= 2^(MAX(0, 1 + MAX(EXP(ump), te+err+1) - EXP(tmp))).
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If tmp = 0 we can use the same bound, replacing
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EXP(tmp) by EXP(ump). */
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err = MAX(1, te + err + 2);
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te = MPFR_IS_ZERO(tmp) ? MPFR_GET_EXP(ump) : MPFR_GET_EXP(tmp);
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err = err - te;
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err = MAX(0, err);
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mpfr_log (ump, x_abs, MPFR_RNDN);
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mpfr_add (tmp, tmp, ump, MPFR_RNDN);
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/* same formula as above, except now EXP(ump) is not 0 */
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err += te + 1;
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if (MPFR_LIKELY (!MPFR_IS_ZERO (ump)))
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err = MAX (MPFR_GET_EXP (ump), err);
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/* if tmp is zero, we surely cannot round correctly */
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err = (MPFR_IS_ZERO(tmp)) ? prec : MAX(0, err - MPFR_GET_EXP (tmp));
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}
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/* Note: we assume here that MPFR_CAN_ROUND returns the same result
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for rnd and MPFR_INVERT_RND(rnd) */
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if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - err, MPFR_PREC (y), rnd)))
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break;
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MPFR_ZIV_NEXT (loop, prec); /* Increase used precision */
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mpfr_set_prec (tmp, prec);
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|
mpfr_set_prec (ump, prec);
|
|
}
|
|
MPFR_ZIV_FREE (loop); /* Free the ZivLoop Controller */
|
|
|
|
/* Set y to the computed value */
|
|
inex = mpfr_set (y, tmp, rnd);
|
|
mpfr_clear (tmp);
|
|
mpfr_clear (ump);
|
|
|
|
MPFR_SAVE_EXPO_FREE (expo);
|
|
return mpfr_check_range (y, inex, rnd);
|
|
}
|