[gcc] GCC 4.5.0=>4.5.1

gcc/mpfr/rec_sqrt.c

Index: gcc/mpfr/rec_sqrt.c
diff --git a/gcc/mpfr/rec_sqrt.c b/gcc/mpfr/rec_sqrt.c
deleted file mode 100644
index da12a1de7091373ffc7a92a5682fd9f8b9769c71..0000000000000000000000000000000000000000
--- a/gcc/mpfr/rec_sqrt.c
+++ /dev/null
@@ -1,536 +0,0 @@
-/* mpfr_rec_sqrt -- inverse square root
-
-Copyright 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 as published by
-the Free Software Foundation; either version 2.1 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.LIB.  If not, write to
-the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
-MA 02110-1301, USA. */
-
-#include <stdio.h>
-#include <stdlib.h>
-
-#define MPFR_NEED_LONGLONG_H /* for umul_ppmm */
-#include "mpfr-impl.h"
-
-#define LIMB_SIZE(x) ((((x)-1)>>MPFR_LOG2_BITS_PER_MP_LIMB) + 1)
-
-#define MPFR_COM_N(x,y,n)                               \
-  {                                                     \
-    mp_size_t i;                                        \
-    for (i = 0; i < n; i++)                             \
-      *((x)+i) = ~*((y)+i);                             \
-  }
-
-/* Put in X a p-bit approximation of 1/sqrt(A),
-   where X = {x, n}/B^n, n = ceil(p/GMP_NUMB_BITS),
-   A = 2^(1+as)*{a, an}/B^an, as is 0 or 1, an = ceil(ap/GMP_NUMB_BITS),
-   where B = 2^GMP_NUMB_BITS.
-
-   We have 1 <= A < 4 and 1/2 <= X < 1.
-
-   The error in the approximate result with respect to the true
-   value 1/sqrt(A) is bounded by 1 ulp(X), i.e., 2^{-p} since 1/2 <= X < 1.
-
-   Note: x and a are left-aligned, i.e., the most significant bit of
-   a[an-1] is set, and so is the most significant bit of the output x[n-1].
-
-   If p is not a multiple of GMP_NUMB_BITS, the extra low bits of the input
-   A are taken into account to compute the approximation of 1/sqrt(A), but
-   whether or not they are zero, the error between X and 1/sqrt(A) is bounded
-   by 1 ulp(X) [in precision p].
-   The extra low bits of the output X (if p is not a multiple of GMP_NUMB_BITS)
-   are set to 0.
-
-   Assumptions:
-   (1) A should be normalized, i.e., the most significant bit of a[an-1]
-       should be 1. If as=0, we have 1 <= A < 2; if as=1, we have 2 <= A < 4.
-   (2) p >= 12
-   (3) {a, an} and {x, n} should not overlap
-   (4) GMP_NUMB_BITS >= 12 and is even
-
-   Note: this routine is much more efficient when ap is small compared to p,
-   including the case where ap <= GMP_NUMB_BITS, thus it can be used to
-   implement an efficient mpfr_rec_sqrt_ui function.
-
-   Reference: Modern Computer Algebra, Richard Brent and Paul Zimmermann,
-   http://www.loria.fr/~zimmerma/mca/pub226.html
-*/
-static void
-mpfr_mpn_rec_sqrt (mp_ptr x, mp_prec_t p,
-                   mp_srcptr a, mp_prec_t ap, int as)
-
-{
-  /* the following T1 and T2 are bipartite tables giving initial
-     approximation for the inverse square root, with 13-bit input split in
-     5+4+4, and 11-bit output. More precisely, if 2048 <= i < 8192,
-     with i = a*2^8 + b*2^4 + c, we use for approximation of
-     2048/sqrt(i/2048) the value x = T1[16*(a-8)+b] + T2[16*(a-8)+c].
-     The largest error is obtained for i = 2054, where x = 2044,
-     and 2048/sqrt(i/2048) = 2045.006576...
-  */
-  static short int T1[384] = {
-2040, 2033, 2025, 2017, 2009, 2002, 1994, 1987, 1980, 1972, 1965, 1958, 1951,
-1944, 1938, 1931, /* a=8 */
-1925, 1918, 1912, 1905, 1899, 1892, 1886, 1880, 1874, 1867, 1861, 1855, 1849,
-1844, 1838, 1832, /* a=9 */
-1827, 1821, 1815, 1810, 1804, 1799, 1793, 1788, 1783, 1777, 1772, 1767, 1762,
-1757, 1752, 1747, /* a=10 */
-1742, 1737, 1733, 1728, 1723, 1718, 1713, 1709, 1704, 1699, 1695, 1690, 1686,
-1681, 1677, 1673, /* a=11 */
-1669, 1664, 1660, 1656, 1652, 1647, 1643, 1639, 1635, 1631, 1627, 1623, 1619,
-1615, 1611, 1607, /* a=12 */
-1603, 1600, 1596, 1592, 1588, 1585, 1581, 1577, 1574, 1570, 1566, 1563, 1559,
-1556, 1552, 1549, /* a=13 */
-1545, 1542, 1538, 1535, 1532, 1528, 1525, 1522, 1518, 1515, 1512, 1509, 1505,
-1502, 1499, 1496, /* a=14 */
-1493, 1490, 1487, 1484, 1481, 1478, 1475, 1472, 1469, 1466, 1463, 1460, 1457,
-1454, 1451, 1449, /* a=15 */
-1446, 1443, 1440, 1438, 1435, 1432, 1429, 1427, 1424, 1421, 1419, 1416, 1413,
-1411, 1408, 1405, /* a=16 */
-1403, 1400, 1398, 1395, 1393, 1390, 1388, 1385, 1383, 1380, 1378, 1375, 1373,
-1371, 1368, 1366, /* a=17 */
-1363, 1360, 1358, 1356, 1353, 1351, 1349, 1346, 1344, 1342, 1340, 1337, 1335,
-1333, 1331, 1329, /* a=18 */
-1327, 1325, 1323, 1321, 1319, 1316, 1314, 1312, 1310, 1308, 1306, 1304, 1302,
-1300, 1298, 1296, /* a=19 */
-1294, 1292, 1290, 1288, 1286, 1284, 1282, 1280, 1278, 1276, 1274, 1272, 1270,
-1268, 1266, 1265, /* a=20 */
-1263, 1261, 1259, 1257, 1255, 1253, 1251, 1250, 1248, 1246, 1244, 1242, 1241,
-1239, 1237, 1235, /* a=21 */
-1234, 1232, 1230, 1229, 1227, 1225, 1223, 1222, 1220, 1218, 1217, 1215, 1213,
-1212, 1210, 1208, /* a=22 */
-1206, 1204, 1203, 1201, 1199, 1198, 1196, 1195, 1193, 1191, 1190, 1188, 1187,
-1185, 1184, 1182, /* a=23 */
-1181, 1180, 1178, 1177, 1175, 1174, 1172, 1171, 1169, 1168, 1166, 1165, 1163,
-1162, 1160, 1159, /* a=24 */
-1157, 1156, 1154, 1153, 1151, 1150, 1149, 1147, 1146, 1144, 1143, 1142, 1140,
-1139, 1137, 1136, /* a=25 */
-1135, 1133, 1132, 1131, 1129, 1128, 1127, 1125, 1124, 1123, 1121, 1120, 1119,
-1117, 1116, 1115, /* a=26 */
-1114, 1113, 1111, 1110, 1109, 1108, 1106, 1105, 1104, 1103, 1101, 1100, 1099,
-1098, 1096, 1095, /* a=27 */
-1093, 1092, 1091, 1090, 1089, 1087, 1086, 1085, 1084, 1083, 1081, 1080, 1079,
-1078, 1077, 1076, /* a=28 */
-1075, 1073, 1072, 1071, 1070, 1069, 1068, 1067, 1065, 1064, 1063, 1062, 1061,
-1060, 1059, 1058, /* a=29 */
-1057, 1056, 1055, 1054, 1052, 1051, 1050, 1049, 1048, 1047, 1046, 1045, 1044,
-1043, 1042, 1041, /* a=30 */
-1040, 1039, 1038, 1037, 1036, 1035, 1034, 1033, 1032, 1031, 1030, 1029, 1028,
-1027, 1026, 1025 /* a=31 */
-};
-  static unsigned char T2[384] = {
-    7, 7, 6, 6, 5, 5, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0, /* a=8 */
-    6, 5, 5, 5, 4, 4, 3, 3, 3, 2, 2, 2, 1, 1, 0, 0, /* a=9 */
-    5, 5, 4, 4, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, /* a=10 */
-    4, 4, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, /* a=11 */
-    3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, /* a=12 */
-    3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, /* a=13 */
-    3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, /* a=14 */
-    2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, /* a=15 */
-    2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, /* a=16 */
-    2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, /* a=17 */
-    3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, /* a=18 */
-    2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, /* a=19 */
-    1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, /* a=20 */
-    1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, /* a=21 */
-    1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, /* a=22 */
-    2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, /* a=23 */
-    1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, /* a=24 */
-    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, /* a=25 */
-    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, /* a=26 */
-    1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, /* a=27 */
-    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, /* a=28 */
-    1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, /* a=29 */
-    1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, /* a=30 */
-    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0  /* a=31 */
-};
-  mp_size_t n = LIMB_SIZE(p);   /* number of limbs of X */
-  mp_size_t an = LIMB_SIZE(ap); /* number of limbs of A */
-
-  /* A should be normalized */
-  MPFR_ASSERTD((a[an - 1] & MPFR_LIMB_HIGHBIT) != 0);
-  /* We should have enough bits in one limb and GMP_NUMB_BITS should be even.
-     Since that does not depend on MPFR, we always check this. */
-  MPFR_ASSERTN((GMP_NUMB_BITS >= 12) && ((GMP_NUMB_BITS & 1) == 0));
-  /* {a, an} and {x, n} should not overlap */
-  MPFR_ASSERTD((a + an <= x) || (x + n <= a));
-  MPFR_ASSERTD(p >= 11);
-
-  if (MPFR_UNLIKELY(an > n)) /* we can cut the input to n limbs */
-    {
-      a += an - n;
-      an = n;
-    }
-
-  if (p == 11) /* should happen only from recursive calls */
-    {
-      unsigned long i, ab, ac;
-      mp_limb_t t;
-
-      /* take the 12+as most significant bits of A */
-      i = a[an - 1] >> (GMP_NUMB_BITS - (12 + as));
-      /* if one wants faithful rounding for p=11, replace #if 0 by #if 1 */
-      ab = i >> 4;
-      ac = (ab & 0x3F0) | (i & 0x0F);
-      t = (mp_limb_t) T1[ab - 0x80] + (mp_limb_t) T2[ac - 0x80];
-      x[0] = t << (GMP_NUMB_BITS - p);
-    }
-  else /* p >= 12 */
-    {
-      mp_prec_t h, pl;
-      mp_ptr r, s, t, u;
-      mp_size_t xn, rn, th, ln, tn, sn, ahn, un;
-      mp_limb_t neg, cy, cu;
-      MPFR_TMP_DECL(marker);
-
-      /* h = max(11, ceil((p+3)/2)) is the bitsize of the recursive call */
-      h = (p < 18) ? 11 : (p >> 1) + 2;
-
-      xn = LIMB_SIZE(h);       /* limb size of the recursive Xh */
-      rn = LIMB_SIZE(2 * h);   /* a priori limb size of Xh^2 */
-      ln = n - xn;             /* remaining limbs to be computed */
-
-      /* Since |Xh - A^{-1/2}| <= 2^{-h}, then by multiplying by Xh + A^{-1/2}
-         we get |Xh^2 - 1/A| <= 2^{-h+1}, thus |A*Xh^2 - 1| <= 2^{-h+3},
-         thus the h-3 most significant bits of t should be zero,
-         which is in fact h+1+as-3 because of the normalization of A.
-         This corresponds to th=floor((h+1+as-3)/GMP_NUMB_BITS) limbs. */
-      th = (h + 1 + as - 3) >> MPFR_LOG2_BITS_PER_MP_LIMB;
-      tn = LIMB_SIZE(2 * h + 1 + as);
-
-      /* we need h+1+as bits of a */
-      ahn = LIMB_SIZE(h + 1 + as); /* number of high limbs of A
-                                      needed for the recursive call*/
-      if (MPFR_UNLIKELY(ahn > an))
-        ahn = an;
-      mpfr_mpn_rec_sqrt (x + ln, h, a + an - ahn, ahn * GMP_NUMB_BITS, as);
-      /* the most h significant bits of X are set, X has ceil(h/GMP_NUMB_BITS)
-         limbs, the low (-h) % GMP_NUMB_BITS bits are zero */
-
-      MPFR_TMP_MARK (marker);
-      /* first step: square X in r, result is exact */
-      un = xn + (tn - th);
-      /* We use the same temporary buffer to store r and u: r needs 2*xn
-         limbs where u needs xn+(tn-th) limbs. Since tn can store at least
-         2h bits, and th at most h bits, then tn-th can store at least h bits,
-         thus tn - th >= xn, and reserving the space for u is enough. */
-      MPFR_ASSERTD(2 * xn <= un);
-      u = r = (mp_ptr) MPFR_TMP_ALLOC (un * sizeof (mp_limb_t));
-      if (2 * h <= GMP_NUMB_BITS) /* xn=rn=1, and since p <= 2h-3, n=1,
-                                     thus ln = 0 */
-        {
-          MPFR_ASSERTD(ln == 0);
-          cy = x[0] >> (GMP_NUMB_BITS >> 1);
-          r ++;
-          r[0] = cy * cy;
-        }
-      else if (xn == 1) /* xn=1, rn=2 */
-        umul_ppmm(r[1], r[0], x[ln], x[ln]);
-      else
-        {
-          mpn_mul_n (r, x + ln, x + ln, xn);
-          if (rn < 2 * xn)
-            r ++;
-        }
-      /* now the 2h most significant bits of {r, rn} contains X^2, r has rn
-         limbs, and the low (-2h) % GMP_NUMB_BITS bits are zero */
-
-      /* Second step: s <- A * (r^2), and truncate the low ap bits,
-         i.e., at weight 2^{-2h} (s is aligned to the low significant bits)
-       */
-      sn = an + rn;
-      s = (mp_ptr) MPFR_TMP_ALLOC (sn * sizeof (mp_limb_t));
-      if (rn == 1) /* rn=1 implies n=1, since rn*GMP_NUMB_BITS >= 2h,
-                           and 2h >= p+3 */
-        {
-          /* necessarily p <= GMP_NUMB_BITS-3: we can ignore the two low
-             bits from A */
-          /* since n=1, and we ensured an <= n, we also have an=1 */
-          MPFR_ASSERTD(an == 1);
-          umul_ppmm (s[1], s[0], r[0], a[0]);
-        }
-      else
-        {
-          /* we have p <= n * GMP_NUMB_BITS
-             2h <= rn * GMP_NUMB_BITS with p+3 <= 2h <= p+4
-             thus n <= rn <= n + 1 */
-          MPFR_ASSERTD(rn <= n + 1);
-          /* since we ensured an <= n, we have an <= rn */
-          MPFR_ASSERTD(an <= rn);
-          mpn_mul (s, r, rn, a, an);
-          /* s should be near B^sn/2^(1+as), thus s[sn-1] is either
-             100000... or 011111... if as=0, or
-             010000... or 001111... if as=1.
-             We ignore the bits of s after the first 2h+1+as ones.
-          */
-        }
-
-      /* We ignore the bits of s after the first 2h+1+as ones: s has rn + an
-         limbs, where rn = LIMBS(2h), an=LIMBS(a), and tn = LIMBS(2h+1+as). */
-      t = s + sn - tn; /* pointer to low limb of the high part of t */
-      /* the upper h-3 bits of 1-t should be zero,
-         where 1 corresponds to the most significant bit of t[tn-1] if as=0,
-         and to the 2nd most significant bit of t[tn-1] if as=1 */
-
-      /* compute t <- 1 - t, which is B^tn - {t, tn+1},
-         with rounding towards -Inf, i.e., rounding the input t towards +Inf.
-         We could only modify the low tn - th limbs from t, but it gives only
-         a small speedup, and would make the code more complex.
-      */
-      neg = t[tn - 1] & (MPFR_LIMB_HIGHBIT >> as);
-      if (neg == 0) /* Ax^2 < 1: we have t = th + eps, where 0 <= eps < ulp(th)
-                       is the part truncated above, thus 1 - t rounded to -Inf
-                       is 1 - th - ulp(th) */
-        {
-          /* since the 1+as most significant bits of t are zero, set them
-             to 1 before the one-complement */
-          t[tn - 1] |= MPFR_LIMB_HIGHBIT | (MPFR_LIMB_HIGHBIT >> as);
-          MPFR_COM_N (t, t, tn);
-          /* we should add 1 here to get 1-th complement, and subtract 1 for
-             -ulp(th), thus we do nothing */
-        }
-      else /* negative case: we want 1 - t rounded towards -Inf, i.e.,
-              th + eps rounded towards +Inf, which is th + ulp(th):
-              we discard the bit corresponding to 1,
-              and we add 1 to the least significant bit of t */
-        {
-          t[tn - 1] ^= neg;
-          mpn_add_1 (t, t, tn, 1);
-        }
-      tn -= th; /* we know at least th = floor((h+1+as-3)/GMP_NUMB_LIMBS) of
-                   the high limbs of {t, tn} are zero */
-
-      /* tn = rn - th, where rn * GMP_NUMB_BITS >= 2*h and
-         th * GMP_NUMB_BITS <= h+1+as-3, thus tn > 0 */
-      MPFR_ASSERTD(tn > 0);
-
-      /* u <- x * t, where {t, tn} contains at least h+3 bits,
-         and {x, xn} contains h bits, thus tn >= xn */
-      MPFR_ASSERTD(tn >= xn);
-      if (tn == 1) /* necessarily xn=1 */
-        umul_ppmm (u[1], u[0], t[0], x[ln]);
-      else
-        mpn_mul (u, t, tn, x + ln, xn);
-
-      /* we have already discarded the upper th high limbs of t, thus we only
-         have to consider the upper n - th limbs of u */
-      un = n - th; /* un cannot be zero, since p <= n*GMP_NUMB_BITS,
-                      h = ceil((p+3)/2) <= (p+4)/2,
-                      th*GMP_NUMB_BITS <= h-1 <= p/2+1,
-                      thus (n-th)*GMP_NUMB_BITS >= p/2-1.
-                   */
-      MPFR_ASSERTD(un > 0);
-      u += (tn + xn) - un; /* xn + tn - un = xn + (original_tn - th) - (n - th)
-                                           = xn + original_tn - n
-                              = LIMBS(h) + LIMBS(2h+1+as) - LIMBS(p) > 0
-                              since 2h >= p+3 */
-      MPFR_ASSERTD(tn + xn > un); /* will allow to access u[-1] below */
-
-      /* In case as=0, u contains |x*(1-Ax^2)/2|, which is exactly what we
-         need to add or subtract.
-         In case as=1, u contains |x*(1-Ax^2)/4|, thus we need to multiply
-         u by 2. */
-
-      if (as == 1)
-        /* shift on un+1 limbs to get most significant bit of u[-1] into
-           least significant bit of u[0] */
-        mpn_lshift (u - 1, u - 1, un + 1, 1);
-
-      pl = n * GMP_NUMB_BITS - p;       /* low bits from x */
-      /* We want that the low pl bits are zero after rounding to nearest,
-         thus we round u to nearest at bit pl-1 of u[0] */
-      if (pl > 0)
-        {
-          cu = mpn_add_1 (u, u, un, u[0] & (MPFR_LIMB_ONE << (pl - 1)));
-          /* mask bits 0..pl-1 of u[0] */
-          u[0] &= ~MPFR_LIMB_MASK(pl);
-        }
-      else /* round bit is in u[-1] */
-        cu = mpn_add_1 (u, u, un, u[-1] >> (GMP_NUMB_BITS - 1));
-
-      /* We already have filled {x + ln, xn = n - ln}, and we want to add or
-         subtract cu*B^un + {u, un} at position x.
-         un = n - th, where th contains <= h+1+as-3<=h-1 bits
-         ln = n - xn, where xn contains >= h bits
-         thus un > ln.
-         Warning: ln might be zero.
-      */
-      MPFR_ASSERTD(un > ln);
-      /* we can have un = ln + 2, for example with GMP_NUMB_BITS=32 and
-         p=62, as=0, then h=33, n=2, th=0, xn=2, thus un=2 and ln=0. */
-      MPFR_ASSERTD(un == ln + 1 || un == ln + 2);
-      /* the high un-ln limbs of u will overlap the low part of {x+ln,xn},
-         we need to add or subtract the overlapping part {u + ln, un - ln} */
-      if (neg == 0)
-        {
-          if (ln > 0)
-            MPN_COPY (x, u, ln);
-          cy = mpn_add (x + ln, x + ln, xn, u + ln, un - ln);
-          /* add cu at x+un */
-          cy += mpn_add_1 (x + un, x + un, th, cu);
-        }
-      else /* negative case */
-        {
-          /* subtract {u+ln, un-ln} from {x+ln,un} */
-          cy = mpn_sub (x + ln, x + ln, xn, u + ln, un - ln);
-          /* carry cy is at x+un, like cu */
-          cy = mpn_sub_1 (x + un, x + un, th, cy + cu); /* n - un = th */
-          /* cy cannot be zero, since the most significant bit of Xh is 1,
-             and the correction is bounded by 2^{-h+3} */
-          MPFR_ASSERTD(cy == 0);
-          if (ln > 0)
-            {
-              MPFR_COM_N (x, u, ln);
-              /* we must add one for the 2-complement ... */
-              cy = mpn_add_1 (x, x, n, MPFR_LIMB_ONE);
-              /* ... and subtract 1 at x[ln], where n = ln + xn */
-              cy -= mpn_sub_1 (x + ln, x + ln, xn, MPFR_LIMB_ONE);
-            }
-        }
-
-      /* cy can be 1 when A=1, i.e., {a, n} = B^n. In that case we should
-         have X = B^n, and setting X to 1-2^{-p} satisties the error bound
-         of 1 ulp. */
-      if (MPFR_UNLIKELY(cy != 0))
-        {
-          cy -= mpn_sub_1 (x, x, n, MPFR_LIMB_ONE << pl);
-          MPFR_ASSERTD(cy == 0);
-        }
-
-      MPFR_TMP_FREE (marker);
-    }
-}
-
-int
-mpfr_rec_sqrt (mpfr_ptr r, mpfr_srcptr u, mp_rnd_t rnd_mode)
-{
-  mp_prec_t rp, up, wp;
-  mp_size_t rn, wn;
-  int s, cy, inex;
-  mp_ptr x;
-  int out_of_place;
-  MPFR_TMP_DECL(marker);
-
-  MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", u, u, rnd_mode),
-                 ("y[%#R]=%R inexact=%d", r, r, inex));
-
-  /* special values */
-  if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(u)))
-    {
-      if (MPFR_IS_NAN(u))
-        {
-          MPFR_SET_NAN(r);
-          MPFR_RET_NAN;
-        }
-      else if (MPFR_IS_ZERO(u)) /* 1/sqrt(+0) = 1/sqrt(-0) = +Inf */
-        {
-          /* 0+ or 0- */
-          MPFR_SET_INF(r);
-          MPFR_SET_POS(r);
-          MPFR_RET(0); /* Inf is exact */
-        }
-      else
-        {
-          MPFR_ASSERTD(MPFR_IS_INF(u));
-          /* 1/sqrt(-Inf) = NAN */
-          if (MPFR_IS_NEG(u))
-            {
-              MPFR_SET_NAN(r);
-              MPFR_RET_NAN;
-            }
-          /* 1/sqrt(+Inf) = +0 */
-          MPFR_SET_POS(r);
-          MPFR_SET_ZERO(r);
-          MPFR_RET(0);
-        }
-    }
-
-  /* if u < 0, 1/sqrt(u) is NaN */
-  if (MPFR_UNLIKELY(MPFR_IS_NEG(u)))
-    {
-      MPFR_SET_NAN(r);
-      MPFR_RET_NAN;
-    }
-
-  MPFR_CLEAR_FLAGS(r);
-  MPFR_SET_POS(r);
-
-  rp = MPFR_PREC(r); /* output precision */
-  up = MPFR_PREC(u); /* input precision */
-  wp = rp + 11;      /* initial working precision */
-
-  /* Let u = U*2^e, where e = EXP(u), and 1/2 <= U < 1.
-     If e is even, we compute an approximation of X of (4U)^{-1/2},
-     and the result is X*2^(-(e-2)/2) [case s=1].
-     If e is odd, we compute an approximation of X of (2U)^{-1/2},
-     and the result is X*2^(-(e-1)/2) [case s=0]. */
-
-  /* parity of the exponent of u */
-  s = 1 - ((mpfr_uexp_t) MPFR_GET_EXP (u) & 1);
-
-  rn = LIMB_SIZE(rp);
-
-  /* for the first iteration, if rp + 11 fits into rn limbs, we round up
-     up to a full limb to maximize the chance of rounding, while avoiding
-     to allocate extra space */
-  wp = rp + 11;
-  if (wp < rn * BITS_PER_MP_LIMB)
-    wp = rn * BITS_PER_MP_LIMB;
-  for (;;)
-    {
-      MPFR_TMP_MARK (marker);
-      wn = LIMB_SIZE(wp);
-      out_of_place = (r == u) || (wn > rn);
-      if (out_of_place)
-        x = (mp_ptr) MPFR_TMP_ALLOC (wn * sizeof (mp_limb_t));
-      else
-        x = MPFR_MANT(r);
-      mpfr_mpn_rec_sqrt (x, wp, MPFR_MANT(u), up, s);
-      /* If the input was not truncated, the error is at most one ulp;
-         if the input was truncated, the error is at most two ulps
-         (see algorithms.tex). */
-      if (MPFR_LIKELY (mpfr_round_p (x, wn, wp - (wp < up),
-                                     rp + (rnd_mode == GMP_RNDN))))
-        break;
-
-      /* We detect only now the exact case where u=2^(2e), to avoid
-         slowing down the average case. This can happen only when the
-         mantissa is exactly 1/2 and the exponent is odd. */
-      if (s == 0 && mpfr_cmp_ui_2exp (u, 1, MPFR_EXP(u) - 1) == 0)
-        {
-          mp_prec_t pl = wn * BITS_PER_MP_LIMB - wp;
-
-          /* we should have x=111...111 */
-          mpn_add_1 (x, x, wn, MPFR_LIMB_ONE << pl);
-          x[wn - 1] = MPFR_LIMB_HIGHBIT;
-          s += 2;
-          break; /* go through */
-        }
-      MPFR_TMP_FREE(marker);
-
-      wp += BITS_PER_MP_LIMB;
-    }
-  cy = mpfr_round_raw (MPFR_MANT(r), x, wp, 0, rp, rnd_mode, &inex);
-  MPFR_EXP(r) = - (MPFR_EXP(u) - 1 - s) / 2;
-  if (MPFR_UNLIKELY(cy != 0))
-    {
-      MPFR_EXP(r) ++;
-      MPFR_MANT(r)[rn - 1] = MPFR_LIMB_HIGHBIT;
-    }
-  MPFR_TMP_FREE(marker);
-  return mpfr_check_range (r, inex, rnd_mode);
-}