--- /dev/null
+/* 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);
+}
projects / msp430-gcc.git / blobdiff