/* mpfr_rec_sqrt -- inverse square root Copyright 2008-2023 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 /* for umul_ppmm */ #include "mpfr-impl.h" #define LIMB_SIZE(x) ((((x)-1)>>MPFR_LOG2_GMP_NUMB_BITS) + 1) #define MPFR_COM_N(x,y,n) \ do \ { \ mp_size_t i; \ for (i = 0; i < n; i++) \ *((x)+i) = ~*((y)+i); \ } \ while (0) /* 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. References: [1] Modern Computer Algebra, Richard Brent and Paul Zimmermann, https://members.loria.fr/PZimmermann/mca/pub226.html */ static void mpfr_mpn_rec_sqrt (mpfr_limb_ptr x, mpfr_prec_t p, mpfr_limb_srcptr a, mpfr_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_STAT_STATIC_ASSERT ((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; unsigned int t; /* take the 12+as most significant bits of A */ #if GMP_NUMB_BITS >= 16 i = a[an - 1] >> (GMP_NUMB_BITS - (12 + as)); #else MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 8); { unsigned int a12 = a[an - 1] << 8; if (an >= 2) a12 |= a[an - 2]; MPFR_ASSERTN(GMP_NUMB_BITS >= 4 + as); i = a12 >> (GMP_NUMB_BITS - (4 + as)); } #endif /* if one wants faithful rounding for p=11, replace #if 0 by #if 1 */ ab = i >> 4; ac = (ab & 0x3F0) | (i & 0x0F); t = T1[ab - 0x80] + T2[ac - 0x80]; /* fits on 16 bits */ #if GMP_NUMB_BITS >= 16 /* x has only one limb */ x[0] = (mp_limb_t) t << (GMP_NUMB_BITS - p); #else MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 8); MPFR_ASSERTD (1024 <= t && t <= 2047); x[1] = t >> 3; /* 128 <= x[1] <= 255 */ x[0] = MPFR_LIMB_LSHIFT(t, 5); #endif } else /* p >= 12 */ { mpfr_prec_t h, pl; mpfr_limb_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); /* compared to Algorithm 3.9 of [1], we have {a, an} = A/2 if as=0, and A/4 if as=1. */ /* 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. More precisely we have |Xh^2 - 1/A| <= 2^{-h} * (Xh + A^{-1/2}) <= 2^{-h} * (2 A^{-1/2} + 2^{-h}) <= 2.001 * 2^{-h} * A^{-1/2} since A < 4 and h >= 11, thus |A*Xh^2 - 1| <= 2.001 * 2^{-h} * A^{1/2} <= 1.001 * 2^{2-h}. This is sufficient to prove that the upper limb of {t,tn} below is less that 0.501 * 2^GMP_NUMB_BITS, thus cu = 0 below. */ th = (h + 1 + as - 3) >> MPFR_LOG2_GMP_NUMB_BITS; 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 */ /* compared to Algorithm 3.9 of [1], we have {x+ln,xn} = X_h */ 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 = MPFR_TMP_LIMBS_ALLOC (un); 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_sqr (r, x + ln, xn); /* we have {r, 2*xn} = X_h^2 */ 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 = MPFR_TMP_LIMBS_ALLOC (sn); 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 have {s, rn+an} = A*X_h^2/2 if as=0, A*X_h^2/4 if as=1. */ } /* 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 toward -Inf, i.e., rounding the input t toward +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 toward -Inf, i.e., th + eps rounded toward +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 {u, tn+xn} = T_l X_h/2 if as=0, T_l X_h/4 if as=1 */ /* 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); /* now {u,un} represents U / 2 from Algorithm 3.9 */ 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_LSHIFT(MPFR_LIMB_ONE, pl - 1)); /* mask bits 0..pl-1 of u[0] */ u[0] &= MPFR_LIMB(~MPFR_LIMB_MASK(pl)); } else /* round bit is in u[-1] */ cu = mpn_add_1 (u, u, un, u[-1] >> (GMP_NUMB_BITS - 1)); MPFR_ASSERTN(cu == 0); /* We already have filled {x + ln, xn = n - ln}, and we want to add or subtract {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} */ /* Warning! th may be 0, in which case the mpn_add_1 and mpn_sub_1 below (with size = th) mustn't be used. */ if (neg == 0) { if (ln > 0) MPN_COPY (x, u, ln); cy = mpn_add (x + ln, x + ln, xn, u + ln, un - ln); /* cy is the carry at x + (ln + xn) = x + n */ } else /* negative case */ { /* subtract {u+ln, un-ln} from {x+ln,un} */ cy = mpn_sub (x + ln, x + ln, xn, u + ln, un - ln); /* cy is the borrow at x + (ln + xn) = x + n */ /* cy cannot be non-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} satisfies 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, mpfr_rnd_t rnd_mode) { mpfr_prec_t rp, up, wp; mp_size_t rn, wn; int s, cy, inex; mpfr_limb_ptr x; MPFR_TMP_DECL(marker); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (u), mpfr_log_prec, u, rnd_mode), ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (r), mpfr_log_prec, 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_SET_DIVBY0 (); 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_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 * GMP_NUMB_BITS) wp = rn * GMP_NUMB_BITS; MPFR_ZIV_INIT (loop, wp); for (;;) { MPFR_TMP_MARK (marker); wn = LIMB_SIZE(wp); if (r == u || wn > rn) /* out of place, i.e., we cannot write to r */ x = MPFR_TMP_LIMBS_ALLOC (wn); 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 == MPFR_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) { mpfr_prec_t pl = wn * GMP_NUMB_BITS - 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); MPFR_ZIV_NEXT (loop, wp); } MPFR_ZIV_FREE (loop); 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); }