/* 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,
http://www.loria.fr/~zimmerma/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_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);
}
int
mpfr_set_f (mpfr_ptr y, mpf_srcptr x, mpfr_rnd_t rnd_mode)
{
mp_limb_t *my, *mx, *tmp;
unsigned long cnt, sx, sy;
int inexact, carry = 0;
MPFR_TMP_DECL(marker);
sx = ABS(SIZ(x)); /* number of limbs of the mantissa of x */
if (sx == 0) /* x is zero */
{
MPFR_SET_ZERO(y);
MPFR_SET_POS(y);
return 0; /* 0 is exact */
}
if (SIZ(x) * MPFR_FROM_SIGN_TO_INT(MPFR_SIGN(y)) < 0)
MPFR_CHANGE_SIGN (y);
sy = MPFR_LIMB_SIZE (y);
my = MPFR_MANT(y);
mx = PTR(x);
count_leading_zeros(cnt, mx[sx - 1]);
if (sy <= sx) /* we may have to round even when sy = sx */
{
unsigned long xprec = sx * GMP_NUMB_BITS;
MPFR_TMP_MARK(marker);
tmp = MPFR_TMP_LIMBS_ALLOC (sx);
if (cnt)
mpn_lshift (tmp, mx, sx, cnt);
else
/* FIXME: we may avoid the copy here, and directly call mpfr_round_raw
on mx instead of tmp */
MPN_COPY (tmp, mx, sx);
carry = mpfr_round_raw (my, tmp, xprec, (SIZ(x) < 0), MPFR_PREC(y),
rnd_mode, &inexact);
if (MPFR_UNLIKELY(carry)) /* result is a power of two */
my[sy - 1] = MPFR_LIMB_HIGHBIT;
MPFR_TMP_FREE(marker);
}
else
{
if (cnt)
mpn_lshift (my + sy - sx, mx, sx, cnt);
else
MPN_COPY (my + sy - sx, mx, sx);
MPN_ZERO(my, sy - sx);
/* no rounding necessary, since y has a larger mantissa */
inexact = 0;
}
/* warning: EXP(x) * GMP_NUMB_BITS may exceed the maximal exponent */
if (EXP(x) > 1 + (__gmpfr_emax - 1) / GMP_NUMB_BITS)
{
/* EXP(x) >= 2 + floor((__gmpfr_emax-1)/GMP_NUMB_BITS)
EXP(x) >= 2 + (__gmpfr_emax - GMP_NUMB_BITS) / GMP_NUMB_BITS
>= 1 + __gmpfr_emax / GMP_NUMB_BITS
EXP(x) * GMP_NUMB_BITS >= __gmpfr_emax + GMP_NUMB_BITS
Since 0 <= cnt <= GMP_NUMB_BITS-1, and 0 <= carry <= 1,
we have then EXP(x) * GMP_NUMB_BITS - cnt + carry > __gmpfr_emax */
return mpfr_overflow (y, rnd_mode, MPFR_SIGN (y));
}
else
{
/* Do not use MPFR_SET_EXP as the exponent may be out of range. */
MPFR_EXP (y) = EXP (x) * GMP_NUMB_BITS - (mpfr_exp_t) cnt + carry;
}
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_acosh (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
{
MPFR_SAVE_EXPO_DECL (expo);
int inexact;
int comp;
MPFR_LOG_FUNC (
("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
/* Deal with special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* Nan, or zero or -Inf */
if (MPFR_IS_INF (x) && MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else /* Nan, or zero or -Inf */
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
comp = mpfr_cmp_ui (x, 1);
if (MPFR_UNLIKELY (comp < 0))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_UNLIKELY (comp == 0))
{
MPFR_SET_ZERO (y); /* acosh(1) = 0 */
MPFR_SET_POS (y);
MPFR_RET (0);
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variables */
mpfr_t t;
/* Declaration of the size variables */
mpfr_prec_t Ny = MPFR_PREC(y); /* Precision of output variable */
mpfr_prec_t Nt; /* Precision of the intermediary variable */
mpfr_exp_t err, exp_te, d; /* Precision of error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialization of intermediary variables */
mpfr_init2 (t, Nt);
/* First computation of acosh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* compute acosh */
MPFR_BLOCK (flags, mpfr_mul (t, x, x, MPFR_RNDD)); /* x^2 */
if (MPFR_OVERFLOW (flags))
{
mpfr_t ln2;
mpfr_prec_t pln2;
/* As x is very large and the precision is not too large, we
assume that we obtain the same result by evaluating ln(2x).
We need to compute ln(x) + ln(2) as 2x can overflow. TODO:
write a proof and add an MPFR_ASSERTN. */
mpfr_log (t, x, MPFR_RNDN); /* err(log) < 1/2 ulp(t) */
pln2 = Nt - MPFR_PREC_MIN < MPFR_GET_EXP (t) ?
MPFR_PREC_MIN : Nt - MPFR_GET_EXP (t);
mpfr_init2 (ln2, pln2);
mpfr_const_log2 (ln2, MPFR_RNDN); /* err(ln2) < 1/2 ulp(t) */
mpfr_add (t, t, ln2, MPFR_RNDN); /* err <= 3/2 ulp(t) */
mpfr_clear (ln2);
err = 1;
}
else
{
exp_te = MPFR_GET_EXP (t);
mpfr_sub_ui (t, t, 1, MPFR_RNDD); /* x^2-1 */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
{
/* This means that x is very close to 1: x = 1 + t with
t < 2^(-Nt). We have: acosh(x) = sqrt(2t) (1 - eps(t))
with 0 < eps(t) < t / 12. */
mpfr_sub_ui (t, x, 1, MPFR_RNDD); /* t = x - 1 */
mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* 2t */
mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(2t) */
err = 1;
}
else
{
d = exp_te - MPFR_GET_EXP (t);
mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(x^2-1) */
mpfr_add (t, t, x, MPFR_RNDN); /* sqrt(x^2-1)+x */
mpfr_log (t, t, MPFR_RNDN); /* ln(sqrt(x^2-1)+x) */
/* error estimate -- see algorithms.tex */
err = 3 + MAX (1, d) - MPFR_GET_EXP (t);
/* error is bounded by 1/2 + 2^err <= 2^(max(0,1+err)) */
err = MAX (0, 1 + err);
}
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_erf (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t xf;
int inex, large;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
("y[%#R]=%R inexact=%d", y, y, inex));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x)) /* erf(+inf) = +1, erf(-inf) = -1 */
return mpfr_set_si (y, MPFR_INT_SIGN (x), GMP_RNDN);
else /* erf(+0) = +0, erf(-0) = -0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
return mpfr_set (y, x, GMP_RNDN); /* should keep the sign of x */
}
}
/* now x is neither NaN, Inf nor 0 */
/* first try expansion at x=0 when x is small, or asymptotic expansion
where x is large */
MPFR_SAVE_EXPO_MARK (expo);
/* around x=0, we have erf(x) = 2x/sqrt(Pi) (1 - x^2/3 + ...),
with 1 - x^2/3 <= sqrt(Pi)*erf(x)/2/x <= 1 for x >= 0. This means that
if x^2/3 < 2^(-PREC(y)-1) we can decide of the correct rounding,
unless we have a worst-case for 2x/sqrt(Pi). */
if (MPFR_EXP(x) < - (mp_exp_t) (MPFR_PREC(y) / 2))
{
/* we use 2x/sqrt(Pi) (1 - x^2/3) <= erf(x) <= 2x/sqrt(Pi) for x > 0
and 2x/sqrt(Pi) <= erf(x) <= 2x/sqrt(Pi) (1 - x^2/3) for x < 0.
In both cases |2x/sqrt(Pi) (1 - x^2/3)| <= |erf(x)| <= |2x/sqrt(Pi)|.
We will compute l and h such that l <= |2x/sqrt(Pi) (1 - x^2/3)|
and |2x/sqrt(Pi)| <= h. If l and h round to the same value to
precision PREC(y) and rounding rnd_mode, then we are done. */
mpfr_t l, h; /* lower and upper bounds for erf(x) */
int ok, inex2;
mpfr_init2 (l, MPFR_PREC(y) + 17);
mpfr_init2 (h, MPFR_PREC(y) + 17);
/* first compute l */
mpfr_mul (l, x, x, GMP_RNDU);
mpfr_div_ui (l, l, 3, GMP_RNDU); /* upper bound on x^2/3 */
mpfr_ui_sub (l, 1, l, GMP_RNDZ); /* lower bound on 1 - x^2/3 */
mpfr_const_pi (h, GMP_RNDU); /* upper bound of Pi */
mpfr_sqrt (h, h, GMP_RNDU); /* upper bound on sqrt(Pi) */
mpfr_div (l, l, h, GMP_RNDZ); /* lower bound on 1/sqrt(Pi) (1 - x^2/3) */
mpfr_mul_2ui (l, l, 1, GMP_RNDZ); /* 2/sqrt(Pi) (1 - x^2/3) */
mpfr_mul (l, l, x, GMP_RNDZ); /* |l| is a lower bound on
|2x/sqrt(Pi) (1 - x^2/3)| */
/* now compute h */
mpfr_const_pi (h, GMP_RNDD); /* lower bound on Pi */
mpfr_sqrt (h, h, GMP_RNDD); /* lower bound on sqrt(Pi) */
mpfr_div_2ui (h, h, 1, GMP_RNDD); /* lower bound on sqrt(Pi)/2 */
/* since sqrt(Pi)/2 < 1, the following should not underflow */
mpfr_div (h, x, h, MPFR_IS_POS(x) ? GMP_RNDU : GMP_RNDD);
/* round l and h to precision PREC(y) */
inex = mpfr_prec_round (l, MPFR_PREC(y), rnd_mode);
inex2 = mpfr_prec_round (h, MPFR_PREC(y), rnd_mode);
/* Caution: we also need inex=inex2 (inex might be 0). */
ok = SAME_SIGN (inex, inex2) && mpfr_cmp (l, h) == 0;
if (ok)
mpfr_set (y, h, rnd_mode);
mpfr_clear (l);
mpfr_clear (h);
if (ok)
goto end;
/* this test can still fail for small precision, for example
for x=-0.100E-2 with a target precision of 3 bits, since
the error term x^2/3 is not that small. */
}
mpfr_init2 (xf, 53);
mpfr_const_log2 (xf, GMP_RNDU);
mpfr_div (xf, x, xf, GMP_RNDZ); /* round to zero ensures we get a lower
bound of |x/log(2)| */
mpfr_mul (xf, xf, x, GMP_RNDZ);
large = mpfr_cmp_ui (xf, MPFR_PREC (y) + 1) > 0;
mpfr_clear (xf);
/* when x goes to infinity, we have erf(x) = 1 - 1/sqrt(Pi)/exp(x^2)/x + ...
and |erf(x) - 1| <= exp(-x^2) is true for any x >= 0, thus if
exp(-x^2) < 2^(-PREC(y)-1) the result is 1 or 1-epsilon.
This rewrites as x^2/log(2) > p+1. */
if (MPFR_UNLIKELY (large))
/* |erf x| = 1 or 1- */
{
mp_rnd_t rnd2 = MPFR_IS_POS (x) ? rnd_mode : MPFR_INVERT_RND(rnd_mode);
if (rnd2 == GMP_RNDN || rnd2 == GMP_RNDU)
{
inex = MPFR_INT_SIGN (x);
mpfr_set_si (y, inex, rnd2);
}
else /* round to zero */
{
inex = -MPFR_INT_SIGN (x);
mpfr_setmax (y, 0); /* warning: setmax keeps the old sign of y */
MPFR_SET_SAME_SIGN (y, x);
}
}
else /* use Taylor */
{
double xf2;
/* FIXME: get rid of doubles/mpfr_get_d here */
xf2 = mpfr_get_d (x, GMP_RNDN);
xf2 = xf2 * xf2; /* xf2 ~ x^2 */
inex = mpfr_erf_0 (y, x, xf2, rnd_mode);
}
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
int
mpfr_sqr (mpfr_ptr a, mpfr_srcptr b, mpfr_rnd_t rnd_mode)
{
int cc, inexact;
mpfr_exp_t ax;
mp_limb_t *tmp;
mp_limb_t b1;
mpfr_prec_t bq;
mp_size_t bn, tn;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", b, b, rnd_mode),
("y[%#R]=%R inexact=%d", a, a, inexact));
/* deal with special cases */
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(b)))
{
if (MPFR_IS_NAN(b))
{
MPFR_SET_NAN(a);
MPFR_RET_NAN;
}
MPFR_SET_POS (a);
if (MPFR_IS_INF(b))
MPFR_SET_INF(a);
else
( MPFR_ASSERTD(MPFR_IS_ZERO(b)), MPFR_SET_ZERO(a) );
MPFR_RET(0);
}
ax = 2 * MPFR_GET_EXP (b);
bq = MPFR_PREC(b);
MPFR_ASSERTD (2 * bq > bq); /* PREC_MAX is /2 so no integer overflow */
bn = MPFR_LIMB_SIZE(b); /* number of limbs of b */
tn = 1 + (2 * bq - 1) / GMP_NUMB_BITS; /* number of limbs of square,
2*bn or 2*bn-1 */
MPFR_TMP_MARK(marker);
tmp = (mp_limb_t *) MPFR_TMP_ALLOC((size_t) 2 * bn * BYTES_PER_MP_LIMB);
/* Multiplies the mantissa in temporary allocated space */
mpn_sqr_n (tmp, MPFR_MANT(b), bn);
b1 = tmp[2 * bn - 1];
/* now tmp[0]..tmp[2*bn-1] contains the product of both mantissa,
with tmp[2*bn-1]>=2^(GMP_NUMB_BITS-2) */
b1 >>= GMP_NUMB_BITS - 1; /* msb from the product */
/* if the mantissas of b and c are uniformly distributed in ]1/2, 1],
then their product is in ]1/4, 1/2] with probability 2*ln(2)-1 ~ 0.386
and in [1/2, 1] with probability 2-2*ln(2) ~ 0.614 */
tmp += 2 * bn - tn; /* +0 or +1 */
if (MPFR_UNLIKELY(b1 == 0))
mpn_lshift (tmp, tmp, tn, 1); /* tn <= k, so no stack corruption */
cc = mpfr_round_raw (MPFR_MANT (a), tmp, 2 * bq, 0,
MPFR_PREC (a), rnd_mode, &inexact);
/* cc = 1 ==> result is a power of two */
if (MPFR_UNLIKELY(cc))
MPFR_MANT(a)[MPFR_LIMB_SIZE(a)-1] = MPFR_LIMB_HIGHBIT;
MPFR_TMP_FREE(marker);
{
mpfr_exp_t ax2 = ax + (mpfr_exp_t) (b1 - 1 + cc);
if (MPFR_UNLIKELY( ax2 > __gmpfr_emax))
return mpfr_overflow (a, rnd_mode, MPFR_SIGN_POS);
if (MPFR_UNLIKELY( ax2 < __gmpfr_emin))
{
/* In the rounding to the nearest mode, if the exponent of the exact
result (i.e. before rounding, i.e. without taking cc into account)
is < __gmpfr_emin - 1 or the exact result is a power of 2 (i.e. if
both arguments are powers of 2), then round to zero. */
if (rnd_mode == MPFR_RNDN &&
(ax + (mpfr_exp_t) b1 < __gmpfr_emin || mpfr_powerof2_raw (b)))
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (a, rnd_mode, MPFR_SIGN_POS);
}
MPFR_SET_EXP (a, ax2);
MPFR_SET_POS (a);
}
MPFR_RET (inexact);
}
/* computes tan(x) = sign(x)*sqrt(1/cos(x)^2-1) */
int
mpfr_tan (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t precy, m;
int inexact;
mpfr_t s, c;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_GROUP_DECL (group);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN(y, x);
MPFR_RET(0);
}
}
/* tan(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2 * MPFR_GET_EXP (x), 1, 1,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
precy = MPFR_PREC (y);
m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
MPFR_ASSERTD (m >= 2); /* needed for the error analysis in algorithms.tex */
MPFR_GROUP_INIT_2 (group, m, s, c);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* The only way to get an overflow is to get ~ Pi/2
But the result will be ~ 2^Prec(y). */
mpfr_sin_cos (s, c, x, MPFR_RNDN); /* err <= 1/2 ulp on s and c */
mpfr_div (c, s, c, MPFR_RNDN); /* err <= 4 ulps */
MPFR_ASSERTD (!MPFR_IS_SINGULAR (c));
if (MPFR_LIKELY (MPFR_CAN_ROUND (c, m - 2, precy, rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, m);
MPFR_GROUP_REPREC_2 (group, m, s, c);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, c, rnd_mode);
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
MPFR_HOT_FUNCTION_ATTR int
mpfr_add (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode)
{
MPFR_LOG_FUNC
(("b[%Pu]=%.*Rg c[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (b), mpfr_log_prec, b,
mpfr_get_prec (c), mpfr_log_prec, c, rnd_mode),
("a[%Pu]=%.*Rg", mpfr_get_prec (a), mpfr_log_prec, a));
if (MPFR_ARE_SINGULAR(b,c))
{
if (MPFR_IS_NAN(b) || MPFR_IS_NAN(c))
{
MPFR_SET_NAN(a);
MPFR_RET_NAN;
}
/* neither b nor c is NaN here */
else if (MPFR_IS_INF(b))
{
if (!MPFR_IS_INF(c) || MPFR_SIGN(b) == MPFR_SIGN(c))
{
MPFR_SET_INF(a);
MPFR_SET_SAME_SIGN(a, b);
MPFR_RET(0); /* exact */
}
else
{
MPFR_SET_NAN(a);
MPFR_RET_NAN;
}
}
else if (MPFR_IS_INF(c))
{
MPFR_SET_INF(a);
MPFR_SET_SAME_SIGN(a, c);
MPFR_RET(0); /* exact */
}
/* now either b or c is zero */
else if (MPFR_IS_ZERO(b))
{
if (MPFR_IS_ZERO(c))
{
/* for round away, we take the same convention for 0 + 0
as for round to zero or to nearest: it always gives +0,
except (-0) + (-0) = -0. */
MPFR_SET_SIGN(a,
(rnd_mode != MPFR_RNDD ?
(MPFR_IS_NEG(b) && MPFR_IS_NEG(c) ?
MPFR_SIGN_NEG : MPFR_SIGN_POS) :
(MPFR_IS_POS(b) && MPFR_IS_POS(c) ?
MPFR_SIGN_POS : MPFR_SIGN_NEG)));
MPFR_SET_ZERO(a);
MPFR_RET(0); /* 0 + 0 is exact */
}
return mpfr_set (a, c, rnd_mode);
}
else
{
MPFR_ASSERTD(MPFR_IS_ZERO(c));
return mpfr_set (a, b, rnd_mode);
}
}
MPFR_ASSERTD (MPFR_IS_PURE_FP (b));
MPFR_ASSERTD (MPFR_IS_PURE_FP (c));
if (MPFR_UNLIKELY(MPFR_SIGN(b) != MPFR_SIGN(c)))
{ /* signs differ, it is a subtraction */
if (MPFR_LIKELY(MPFR_PREC(a) == MPFR_PREC(b)
&& MPFR_PREC(b) == MPFR_PREC(c)))
return mpfr_sub1sp(a, b, c, rnd_mode);
else
return mpfr_sub1(a, b, c, rnd_mode);
}
else
{ /* signs are equal, it's an addition */
if (MPFR_LIKELY(MPFR_PREC(a) == MPFR_PREC(b)
&& MPFR_PREC(b) == MPFR_PREC(c)))
if (MPFR_GET_EXP(b) < MPFR_GET_EXP(c))
return mpfr_add1sp(a, c, b, rnd_mode);
else
return mpfr_add1sp(a, b, c, rnd_mode);
else
if (MPFR_GET_EXP(b) < MPFR_GET_EXP(c))
return mpfr_add1(a, c, b, rnd_mode);
else
return mpfr_add1(a, b, c, rnd_mode);
}
}
int
mpfr_rint (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
{
int sign;
int rnd_away;
mpfr_exp_t exp;
if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(u) ))
{
if (MPFR_IS_NAN(u))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
MPFR_SET_SAME_SIGN(r, u);
if (MPFR_IS_INF(u))
{
MPFR_SET_INF(r);
MPFR_RET(0); /* infinity is exact */
}
else /* now u is zero */
{
MPFR_ASSERTD(MPFR_IS_ZERO(u));
MPFR_SET_ZERO(r);
MPFR_RET(0); /* zero is exact */
}
}
MPFR_SET_SAME_SIGN (r, u); /* Does nothing if r==u */
sign = MPFR_INT_SIGN (u);
exp = MPFR_GET_EXP (u);
rnd_away =
rnd_mode == MPFR_RNDD ? sign < 0 :
rnd_mode == MPFR_RNDU ? sign > 0 :
rnd_mode == MPFR_RNDZ ? 0 :
rnd_mode == MPFR_RNDA ? 1 :
-1; /* round to nearest-even (RNDN) or nearest-away (RNDNA) */
/* rnd_away:
1 if round away from zero,
0 if round to zero,
-1 if not decided yet.
*/
if (MPFR_UNLIKELY (exp <= 0)) /* 0 < |u| < 1 ==> round |u| to 0 or 1 */
{
/* Note: in the MPFR_RNDN mode, 0.5 must be rounded to 0. */
if (rnd_away != 0 &&
(rnd_away > 0 ||
(exp == 0 && (rnd_mode == MPFR_RNDNA ||
!mpfr_powerof2_raw (u)))))
{
mp_limb_t *rp;
mp_size_t rm;
rp = MPFR_MANT(r);
rm = (MPFR_PREC(r) - 1) / GMP_NUMB_BITS;
rp[rm] = MPFR_LIMB_HIGHBIT;
MPN_ZERO(rp, rm);
MPFR_SET_EXP (r, 1); /* |r| = 1 */
MPFR_RET(sign > 0 ? 2 : -2);
}
else
{
MPFR_SET_ZERO(r); /* r = 0 */
MPFR_RET(sign > 0 ? -2 : 2);
}
}
else /* exp > 0, |u| >= 1 */
{
mp_limb_t *up, *rp;
mp_size_t un, rn, ui;
int sh, idiff;
int uflags;
/*
* uflags will contain:
* _ 0 if u is an integer representable in r,
* _ 1 if u is an integer not representable in r,
* _ 2 if u is not an integer.
*/
up = MPFR_MANT(u);
rp = MPFR_MANT(r);
un = MPFR_LIMB_SIZE(u);
rn = MPFR_LIMB_SIZE(r);
MPFR_UNSIGNED_MINUS_MODULO (sh, MPFR_PREC (r));
MPFR_SET_EXP (r, exp); /* Does nothing if r==u */
if ((exp - 1) / GMP_NUMB_BITS >= un)
{
ui = un;
idiff = 0;
uflags = 0; /* u is an integer, representable or not in r */
}
else
{
mp_size_t uj;
ui = (exp - 1) / GMP_NUMB_BITS + 1; /* #limbs of the int part */
MPFR_ASSERTD (un >= ui);
uj = un - ui; /* lowest limb of the integer part */
idiff = exp % GMP_NUMB_BITS; /* #int-part bits in up[uj] or 0 */
uflags = idiff == 0 || (up[uj] << idiff) == 0 ? 0 : 2;
if (uflags == 0)
while (uj > 0)
if (up[--uj] != 0)
{
uflags = 2;
break;
}
}
if (ui > rn)
{
/* More limbs in the integer part of u than in r.
Just round u with the precision of r. */
MPFR_ASSERTD (rp != up && un > rn);
MPN_COPY (rp, up + (un - rn), rn); /* r != u */
if (rnd_away < 0)
{
/* This is a rounding to nearest mode (MPFR_RNDN or MPFR_RNDNA).
Decide the rounding direction here. */
if (rnd_mode == MPFR_RNDN &&
(rp[0] & (MPFR_LIMB_ONE << sh)) == 0)
{ /* halfway cases rounded toward zero */
mp_limb_t a, b;
/* a: rounding bit and some of the following bits */
/* b: boundary for a (weight of the rounding bit in a) */
if (sh != 0)
{
a = rp[0] & ((MPFR_LIMB_ONE << sh) - 1);
b = MPFR_LIMB_ONE << (sh - 1);
}
else
{
a = up[un - rn - 1];
b = MPFR_LIMB_HIGHBIT;
}
rnd_away = a > b;
if (a == b)
{
mp_size_t i;
for (i = un - rn - 1 - (sh == 0); i >= 0; i--)
if (up[i] != 0)
{
rnd_away = 1;
break;
}
}
}
else /* halfway cases rounded away from zero */
rnd_away = /* rounding bit */
((sh != 0 && (rp[0] & (MPFR_LIMB_ONE << (sh - 1))) != 0) ||
(sh == 0 && (up[un - rn - 1] & MPFR_LIMB_HIGHBIT) != 0));
}
if (uflags == 0)
{ /* u is an integer; determine if it is representable in r */
if (sh != 0 && rp[0] << (GMP_NUMB_BITS - sh) != 0)
uflags = 1; /* u is not representable in r */
else
{
mp_size_t i;
for (i = un - rn - 1; i >= 0; i--)
if (up[i] != 0)
{
uflags = 1; /* u is not representable in r */
break;
}
}
}
}
else /* ui <= rn */
{
mp_size_t uj, rj;
int ush;
uj = un - ui; /* lowest limb of the integer part in u */
rj = rn - ui; /* lowest limb of the integer part in r */
if (MPFR_LIKELY (rp != up))
MPN_COPY(rp + rj, up + uj, ui);
/* Ignore the lowest rj limbs, all equal to zero. */
rp += rj;
rn = ui;
/* number of fractional bits in whole rp[0] */
ush = idiff == 0 ? 0 : GMP_NUMB_BITS - idiff;
if (rj == 0 && ush < sh)
{
/* If u is an integer (uflags == 0), we need to determine
if it is representable in r, i.e. if its sh - ush bits
in the non-significant part of r are all 0. */
if (uflags == 0 && (rp[0] & ((MPFR_LIMB_ONE << sh) -
(MPFR_LIMB_ONE << ush))) != 0)
uflags = 1; /* u is an integer not representable in r */
}
else /* The integer part of u fits in r, we'll round to it. */
sh = ush;
if (rnd_away < 0)
{
/* This is a rounding to nearest mode.
Decide the rounding direction here. */
if (uj == 0 && sh == 0)
rnd_away = 0; /* rounding bit = 0 (not represented in u) */
else if (rnd_mode == MPFR_RNDN &&
(rp[0] & (MPFR_LIMB_ONE << sh)) == 0)
{ /* halfway cases rounded toward zero */
mp_limb_t a, b;
/* a: rounding bit and some of the following bits */
/* b: boundary for a (weight of the rounding bit in a) */
if (sh != 0)
{
a = rp[0] & ((MPFR_LIMB_ONE << sh) - 1);
b = MPFR_LIMB_ONE << (sh - 1);
}
else
{
MPFR_ASSERTD (uj >= 1); /* see above */
a = up[uj - 1];
b = MPFR_LIMB_HIGHBIT;
}
rnd_away = a > b;
if (a == b)
{
mp_size_t i;
for (i = uj - 1 - (sh == 0); i >= 0; i--)
if (up[i] != 0)
{
rnd_away = 1;
break;
}
}
}
else /* halfway cases rounded away from zero */
rnd_away = /* rounding bit */
((sh != 0 && (rp[0] & (MPFR_LIMB_ONE << (sh - 1))) != 0) ||
(sh == 0 && (MPFR_ASSERTD (uj >= 1),
up[uj - 1] & MPFR_LIMB_HIGHBIT) != 0));
}
/* Now we can make the low rj limbs to 0 */
MPN_ZERO (rp-rj, rj);
}
if (sh != 0)
rp[0] &= MP_LIMB_T_MAX << sh;
/* If u is a representable integer, there is no rounding. */
if (uflags == 0)
MPFR_RET(0);
MPFR_ASSERTD (rnd_away >= 0); /* rounding direction is defined */
if (rnd_away && mpn_add_1(rp, rp, rn, MPFR_LIMB_ONE << sh))
{
if (exp == __gmpfr_emax)
return mpfr_overflow(r, rnd_mode, MPFR_SIGN(r)) >= 0 ?
uflags : -uflags;
else
{
MPFR_SET_EXP(r, exp + 1);
rp[rn-1] = MPFR_LIMB_HIGHBIT;
}
}
MPFR_RET (rnd_away ^ (sign < 0) ? uflags : -uflags);
} /* exp > 0, |u| >= 1 */
}
int
mpfr_cmp2 (mpfr_srcptr b, mpfr_srcptr c, mpfr_prec_t *cancel)
{
mp_limb_t *bp, *cp, bb, cc = 0, lastc = 0, dif, high_dif = 0;
mp_size_t bn, cn;
mpfr_uexp_t diff_exp;
mpfr_prec_t res = 0;
int sign;
/* b=c should not happen, since cmp2 is called only from agm
(with different variables), and from sub1 (if same b=c, then
sub1sp would be called instead */
MPFR_ASSERTD (b != c);
/* the cases b=0 or c=0 are also treated apart in agm and sub
(which calls sub1) */
MPFR_ASSERTD (MPFR_IS_PURE_FP(b));
MPFR_ASSERTD (MPFR_IS_PURE_FP(c));
if (MPFR_GET_EXP (b) >= MPFR_GET_EXP (c))
{
sign = 1;
diff_exp = (mpfr_uexp_t) MPFR_GET_EXP (b) - MPFR_GET_EXP (c);
bp = MPFR_MANT(b);
cp = MPFR_MANT(c);
bn = (MPFR_PREC(b) - 1) / GMP_NUMB_BITS;
cn = (MPFR_PREC(c) - 1) / GMP_NUMB_BITS; /* # of limbs of c minus 1 */
if (MPFR_UNLIKELY( diff_exp == 0 ))
{
while (bn >= 0 && cn >= 0 && bp[bn] == cp[cn])
{
bn--;
cn--;
res += GMP_NUMB_BITS;
}
if (MPFR_UNLIKELY (bn < 0))
{
if (MPFR_LIKELY (cn < 0)) /* b = c */
return 0;
bp = cp;
bn = cn;
cn = -1;
sign = -1;
}
if (MPFR_UNLIKELY (cn < 0))
/* c discards exactly the upper part of b */
{
unsigned int z;
MPFR_ASSERTD (bn >= 0);
while (bp[bn] == 0)
{
if (--bn < 0) /* b = c */
return 0;
res += GMP_NUMB_BITS;
}
count_leading_zeros(z, bp[bn]); /* bp[bn] <> 0 */
*cancel = res + z;
return sign;
}
MPFR_ASSERTD (bn >= 0);
MPFR_ASSERTD (cn >= 0);
MPFR_ASSERTD (bp[bn] != cp[cn]);
if (bp[bn] < cp[cn])
{
mp_limb_t *tp;
mp_size_t tn;
tp = bp; bp = cp; cp = tp;
tn = bn; bn = cn; cn = tn;
sign = -1;
}
}
} /* MPFR_EXP(b) >= MPFR_EXP(c) */
else /* MPFR_EXP(b) < MPFR_EXP(c) */
{
sign = -1;
diff_exp = (mpfr_uexp_t) MPFR_GET_EXP (c) - MPFR_GET_EXP (b);
bp = MPFR_MANT(c);
cp = MPFR_MANT(b);
bn = (MPFR_PREC(c) - 1) / GMP_NUMB_BITS;
cn = (MPFR_PREC(b) - 1) / GMP_NUMB_BITS;
}
/* now we have removed the identical upper limbs of b and c
(can happen only when diff_exp = 0), and after the possible
swap, we have |b| > |c|: bp[bn] > cc, bn >= 0, cn >= 0,
diff_exp = EXP(b) - EXP(c).
*/
if (MPFR_LIKELY (diff_exp < GMP_NUMB_BITS))
{
cc = cp[cn] >> diff_exp;
/* warning: a shift by GMP_NUMB_BITS may give wrong results */
if (diff_exp)
lastc = cp[cn] << (GMP_NUMB_BITS - diff_exp);
cn--;
}
/* set f to the rational q */
int
mpfr_set_q (mpfr_ptr f, mpq_srcptr q, mpfr_rnd_t rnd)
{
mpz_srcptr num, den;
mpfr_t n, d;
int inexact;
int cn, cd;
long shift;
mp_size_t sn, sd;
MPFR_SAVE_EXPO_DECL (expo);
num = mpq_numref (q);
den = mpq_denref (q);
/* NAN and INF for mpq are not really documented, but could be found */
if (MPFR_UNLIKELY (mpz_sgn (num) == 0))
{
if (MPFR_UNLIKELY (mpz_sgn (den) == 0))
{
MPFR_SET_NAN (f);
MPFR_RET_NAN;
}
else
{
MPFR_SET_ZERO (f);
MPFR_SET_POS (f);
MPFR_RET (0);
}
}
if (MPFR_UNLIKELY (mpz_sgn (den) == 0))
{
MPFR_SET_INF (f);
MPFR_SET_SIGN (f, mpz_sgn (num));
MPFR_RET (0);
}
MPFR_SAVE_EXPO_MARK (expo);
cn = set_z (n, num, &sn);
cd = set_z (d, den, &sd);
sn -= sd;
if (MPFR_UNLIKELY (sn > MPFR_EMAX_MAX / GMP_NUMB_BITS))
{
MPFR_SAVE_EXPO_FREE (expo);
inexact = mpfr_overflow (f, rnd, MPFR_SIGN (f));
goto end;
}
if (MPFR_UNLIKELY (sn < MPFR_EMIN_MIN / GMP_NUMB_BITS -1))
{
MPFR_SAVE_EXPO_FREE (expo);
if (rnd == MPFR_RNDN)
rnd = MPFR_RNDZ;
inexact = mpfr_underflow (f, rnd, MPFR_SIGN (f));
goto end;
}
inexact = mpfr_div (f, n, d, rnd);
shift = GMP_NUMB_BITS*sn+cn-cd;
MPFR_ASSERTD (shift == GMP_NUMB_BITS*sn+cn-cd);
cd = mpfr_mul_2si (f, f, shift, rnd);
MPFR_SAVE_EXPO_FREE (expo);
if (MPFR_UNLIKELY (cd != 0))
inexact = cd;
else
inexact = mpfr_check_range (f, inexact, rnd);
end:
mpfr_clear (d);
mpfr_clear (n);
MPFR_RET (inexact);
}
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_t x, t, te;
mpfr_prec_t Nx, Ny, Nt;
mpfr_exp_t err;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
{
/* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
between -1 and 1 */
if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* necessarily xt is 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (xt));
MPFR_SET_ZERO (y); /* atanh(0) = 0 */
MPFR_SET_SAME_SIGN (y,xt);
MPFR_RET (0);
}
}
/* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
{
if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, xt);
MPFR_SET_DIVBY0 ();
MPFR_RET (0);
}
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
Nx = MPFR_PREC (xt);
MPFR_TMP_INIT_ABS (x, xt);
Ny = MPFR_PREC (y);
Nt = MAX (Nx, Ny);
/* the optimal number of bits : see algorithms.ps */
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
mpfr_init2 (te, Nt);
/* First computation of cosh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute atanh */
mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/
mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/
mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/
mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/
mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/
/* error estimate: see algorithms.tex */
/* FIXME: this does not correspond to the value in algorithms.tex!!! */
/* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
if (MPFR_LIKELY (MPFR_IS_ZERO (t)
|| MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
mpfr_set_prec (te, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
mpfr_clear(t);
mpfr_clear(te);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
size_t
mpfr_out_str (FILE *stream, int base, size_t n_digits, mpfr_srcptr op,
mpfr_rnd_t rnd_mode)
{
char *s, *s0;
size_t l;
mpfr_exp_t e;
int err;
MPFR_ASSERTN (base >= 2 && base <= 62);
/* when stream=NULL, output to stdout */
if (stream == NULL)
stream = stdout;
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (op)))
{
if (MPFR_IS_NAN (op))
OUT_STR_RET ("@NaN@");
else if (MPFR_IS_INF (op))
OUT_STR_RET (MPFR_IS_POS (op) ? "@Inf@" : "-@Inf@");
else
{
MPFR_ASSERTD (MPFR_IS_ZERO (op));
OUT_STR_RET (MPFR_IS_POS (op) ? "0" : "-0");
}
}
s = mpfr_get_str (NULL, &e, base, n_digits, op, rnd_mode);
s0 = s;
/* for op=3.1416 we have s = "31416" and e = 1 */
l = strlen (s) + 1; /* size of allocated block returned by mpfr_get_str
- may be incorrect, as only an upper bound? */
/* outputs possible sign and significand */
err = (*s == '-' && fputc (*s++, stream) == EOF)
|| fputc (*s++, stream) == EOF /* leading digit */
|| fputc ((unsigned char) MPFR_DECIMAL_POINT, stream) == EOF
|| fputs (s, stream) == EOF; /* trailing significand */
mpfr_free_func (s0, l);
if (MPFR_UNLIKELY (err))
return 0;
e--; /* due to the leading digit */
/* outputs exponent */
if (e)
{
int r;
MPFR_ASSERTN(e >= LONG_MIN);
MPFR_ASSERTN(e <= LONG_MAX);
r = fprintf (stream, (base <= 10 ? "e%ld" : "@%ld"), (long) e);
if (MPFR_UNLIKELY (r < 0))
return 0;
l += r;
}
return l;
}
int
mpfr_frac (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
{
mpfr_exp_t re, ue;
mpfr_prec_t uq;
mp_size_t un, tn, t0;
mp_limb_t *up, *tp, k;
int sh;
mpfr_t tmp;
mpfr_ptr t;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
/* Special cases */
if (MPFR_UNLIKELY(MPFR_IS_NAN(u)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
else if (MPFR_UNLIKELY(MPFR_IS_INF(u) || mpfr_integer_p (u)))
{
MPFR_SET_SAME_SIGN(r, u);
MPFR_SET_ZERO(r);
MPFR_RET(0); /* zero is exact */
}
ue = MPFR_GET_EXP (u);
if (ue <= 0) /* |u| < 1 */
return mpfr_set (r, u, rnd_mode);
/* Now |u| >= 1, meaning that an overflow is not possible. */
uq = MPFR_PREC(u);
un = (uq - 1) / GMP_NUMB_BITS; /* index of most significant limb */
un -= (mp_size_t) (ue / GMP_NUMB_BITS);
/* now the index of the MSL containing bits of the fractional part */
up = MPFR_MANT(u);
sh = ue % GMP_NUMB_BITS;
k = up[un] << sh;
/* the first bit of the fractional part is the MSB of k */
if (k != 0)
{
int cnt;
count_leading_zeros(cnt, k);
/* first bit 1 of the fractional part -> MSB of the number */
re = -cnt;
sh += cnt;
MPFR_ASSERTN (sh < GMP_NUMB_BITS);
k <<= cnt;
}
else
{
re = sh - GMP_NUMB_BITS;
/* searching for the first bit 1 (exists since u isn't an integer) */
while (up[--un] == 0)
re -= GMP_NUMB_BITS;
MPFR_ASSERTN(un >= 0);
k = up[un];
count_leading_zeros(sh, k);
re -= sh;
k <<= sh;
}
/* The exponent of r will be re */
/* un: index of the limb of u that contains the first bit 1 of the FP */
t = (mp_size_t) (MPFR_PREC(r) - 1) / GMP_NUMB_BITS < un ?
(mpfr_init2 (tmp, (un + 1) * GMP_NUMB_BITS), tmp) : r;
/* t has enough precision to contain the fractional part of u */
/* If we use a temporary variable, we take the non-significant bits
of u into account, because of the mpn_lshift below. */
MPFR_SET_SAME_SIGN(t, u);
/* Put the fractional part of u into t */
tn = (MPFR_PREC(t) - 1) / GMP_NUMB_BITS;
MPFR_ASSERTN(tn >= un);
t0 = tn - un;
tp = MPFR_MANT(t);
if (sh == 0)
MPN_COPY_DECR(tp + t0, up, un + 1);
else /* warning: un may be 0 here */
tp[tn] = k | ((un) ? mpn_lshift (tp + t0, up, un, sh) : (mp_limb_t) 0);
if (t0 > 0)
MPN_ZERO(tp, t0);
MPFR_SAVE_EXPO_MARK (expo);
if (t != r)
{ /* t is tmp */
MPFR_EXP (t) = 0; /* should be re, but not necessarily in the range */
inex = mpfr_set (r, t, rnd_mode); /* no underflow */
mpfr_clear (t);
MPFR_EXP (r) += re;
}
else
{ /* There may be remaining non-significant bits in t (= r). */
int carry;
MPFR_EXP (r) = re;
carry = mpfr_round_raw (tp, tp,
(mpfr_prec_t) (tn + 1) * GMP_NUMB_BITS,
MPFR_IS_NEG (r), MPFR_PREC (r), rnd_mode,
&inex);
if (carry)
{
tp[tn] = MPFR_LIMB_HIGHBIT;
MPFR_EXP (r) ++;
}
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inex, rnd_mode);
}
int
mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t c, xr;
mpfr_srcptr xx;
mpfr_exp_t expx, err;
mpfr_prec_t precy, m;
int inexact, sign, reduce;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
/* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2 * MPFR_GET_EXP (x), 2, 0,
rnd_mode, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
precy = MPFR_PREC (y);
if (precy >= MPFR_SINCOS_THRESHOLD)
return mpfr_sin_fast (y, x, rnd_mode);
m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
expx = MPFR_GET_EXP (x);
mpfr_init (c);
mpfr_init (xr);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* first perform argument reduction modulo 2*Pi (if needed),
also helps to determine the sign of sin(x) */
if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
the sign of sin(x). For 2 <= |x| < Pi, we could avoid
the reduction. */
{
reduce = 1;
/* As expx + m - 1 will silently be converted into mpfr_prec_t
in the mpfr_set_prec call, the assert below may be useful to
avoid undefined behavior. */
MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
mpfr_set_prec (c, expx + m - 1);
mpfr_set_prec (xr, m);
mpfr_const_pi (c, MPFR_RNDN);
mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
mpfr_remainder (xr, x, c, MPFR_RNDN);
/* The analysis is similar to that of cos.c:
|xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
of sin(x) if xr is at distance at least 2^(2-m) of both
0 and +/-Pi. */
mpfr_div_2ui (c, c, 1, MPFR_RNDN);
/* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
it suffices to check that c - |xr| >= 2^(2-m). */
if (MPFR_SIGN (xr) > 0)
mpfr_sub (c, c, xr, MPFR_RNDZ);
else
mpfr_add (c, c, xr, MPFR_RNDZ);
if (MPFR_IS_ZERO(xr)
|| MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
|| MPFR_IS_ZERO(c)
|| MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
goto ziv_next;
/* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
xx = xr;
}
else /* the input argument is already reduced */
{
reduce = 0;
xx = x;
}
sign = MPFR_SIGN(xx);
/* now that the argument is reduced, precision m is enough */
mpfr_set_prec (c, m);
mpfr_cos (c, xx, MPFR_RNDZ); /* can't be exact */
mpfr_nexttoinf (c); /* now c = cos(x) rounded away */
mpfr_mul (c, c, c, MPFR_RNDU); /* away */
mpfr_ui_sub (c, 1, c, MPFR_RNDZ);
mpfr_sqrt (c, c, MPFR_RNDZ);
if (MPFR_IS_NEG_SIGN(sign))
MPFR_CHANGE_SIGN(c);
/* Warning: c may be 0! */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
{
/* Huge cancellation: increase prec a lot! */
m = MAX (m, MPFR_PREC (x));
m = 2 * m;
}
else
{
/* the absolute error on c is at most 2^(3-m-EXP(c)),
plus 2^(2-m) if there was an argument reduction.
Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
is at most 2^(3-m-EXP(c)) in case of argument reduction. */
err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0);
if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
break;
/* check for huge cancellation (Near 0) */
if (err < (mpfr_exp_t) MPFR_PREC (y))
m += MPFR_PREC (y) - err;
/* Check if near 1 */
if (MPFR_GET_EXP (c) == 1)
m += m;
}
ziv_next:
/* Else generic increase */
MPFR_ZIV_NEXT (loop, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, c, rnd_mode);
/* inexact cannot be 0, since this would mean that c was representable
within the target precision, but in that case mpfr_can_round will fail */
mpfr_clear (c);
mpfr_clear (xr);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_atan2 (mpfr_ptr dest, mpfr_srcptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t tmp, pi;
int inexact;
mpfr_prec_t prec;
mpfr_exp_t e;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("y[%Pu]=%.*Rg x[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (y), mpfr_log_prec, y,
mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("atan[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (dest), mpfr_log_prec, dest, inexact));
/* Special cases */
if (MPFR_ARE_SINGULAR (x, y))
{
/* atan2(0, 0) does not raise the "invalid" floating-point
exception, nor does atan2(y, 0) raise the "divide-by-zero"
floating-point exception.
-- atan2(±0, -0) returns ±pi.313)
-- atan2(±0, +0) returns ±0.
-- atan2(±0, x) returns ±pi, for x < 0.
-- atan2(±0, x) returns ±0, for x > 0.
-- atan2(y, ±0) returns -pi/2 for y < 0.
-- atan2(y, ±0) returns pi/2 for y > 0.
-- atan2(±oo, -oo) returns ±3pi/4.
-- atan2(±oo, +oo) returns ±pi/4.
-- atan2(±oo, x) returns ±pi/2, for finite x.
-- atan2(±y, -oo) returns ±pi, for finite y > 0.
-- atan2(±y, +oo) returns ±0, for finite y > 0.
*/
if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
{
MPFR_SET_NAN (dest);
MPFR_RET_NAN;
}
if (MPFR_IS_ZERO (y))
{
if (MPFR_IS_NEG (x)) /* +/- PI */
{
set_pi:
if (MPFR_IS_NEG (y))
{
inexact = mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (dest);
return -inexact;
}
else
return mpfr_const_pi (dest, rnd_mode);
}
else /* +/- 0 */
{
set_zero:
MPFR_SET_ZERO (dest);
MPFR_SET_SAME_SIGN (dest, y);
return 0;
}
}
if (MPFR_IS_ZERO (x))
{
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
}
if (MPFR_IS_INF (y))
{
if (!MPFR_IS_INF (x)) /* +/- PI/2 */
return pi_div_2ui (dest, 1, MPFR_IS_NEG (y), rnd_mode);
else if (MPFR_IS_POS (x)) /* +/- PI/4 */
return pi_div_2ui (dest, 2, MPFR_IS_NEG (y), rnd_mode);
else /* +/- 3*PI/4: Ugly since we have to round properly */
{
mpfr_t tmp2;
MPFR_ZIV_DECL (loop2);
mpfr_prec_t prec2 = MPFR_PREC (dest) + 10;
MPFR_SAVE_EXPO_MARK (expo);
mpfr_init2 (tmp2, prec2);
MPFR_ZIV_INIT (loop2, prec2);
for (;;)
{
mpfr_const_pi (tmp2, MPFR_RNDN);
mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDN); /* Error <= 2 */
mpfr_div_2ui (tmp2, tmp2, 2, MPFR_RNDN);
if (mpfr_round_p (MPFR_MANT (tmp2), MPFR_LIMB_SIZE (tmp2),
MPFR_PREC (tmp2) - 2,
MPFR_PREC (dest) + (rnd_mode == MPFR_RNDN)))
break;
MPFR_ZIV_NEXT (loop2, prec2);
mpfr_set_prec (tmp2, prec2);
}
MPFR_ZIV_FREE (loop2);
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp2);
inexact = mpfr_set (dest, tmp2, rnd_mode);
mpfr_clear (tmp2);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
}
MPFR_ASSERTD (MPFR_IS_INF (x));
if (MPFR_IS_NEG (x))
goto set_pi;
else
goto set_zero;
}
/* When x is a power of two, we call directly atan(y/x) since y/x is
exact. */
if (MPFR_UNLIKELY (MPFR_IS_POWER_OF_2 (x)))
{
int r;
mpfr_t yoverx;
unsigned int saved_flags = __gmpfr_flags;
mpfr_init2 (yoverx, MPFR_PREC (y));
if (MPFR_LIKELY (mpfr_div_2si (yoverx, y, MPFR_GET_EXP (x) - 1,
MPFR_RNDN) == 0))
{
/* Here the flags have not changed due to mpfr_div_2si. */
r = mpfr_atan (dest, yoverx, rnd_mode);
mpfr_clear (yoverx);
return r;
}
else
{
/* Division is inexact because of a small exponent range */
mpfr_clear (yoverx);
__gmpfr_flags = saved_flags;
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* Set up initial prec */
prec = MPFR_PREC (dest) + 3 + MPFR_INT_CEIL_LOG2 (MPFR_PREC (dest));
mpfr_init2 (tmp, prec);
MPFR_ZIV_INIT (loop, prec);
if (MPFR_IS_POS (x))
/* use atan2(y,x) = atan(y/x) */
for (;;)
{
int div_inex;
MPFR_BLOCK_DECL (flags);
MPFR_BLOCK (flags, div_inex = mpfr_div (tmp, y, x, MPFR_RNDN));
if (div_inex == 0)
{
/* Result is exact. */
inexact = mpfr_atan (dest, tmp, rnd_mode);
goto end;
}
/* Error <= ulp (tmp) except in case of underflow or overflow. */
/* If the division underflowed, since |atan(z)/z| < 1, we have
an underflow. */
if (MPFR_UNDERFLOW (flags))
{
int sign;
/* In the case MPFR_RNDN with 2^(emin-2) < |y/x| < 2^(emin-1):
The smallest significand value S > 1 of |y/x| is:
* 1 / (1 - 2^(-px)) if py <= px,
* (1 - 2^(-px) + 2^(-py)) / (1 - 2^(-px)) if py >= px.
Therefore S - 1 > 2^(-pz), where pz = max(px,py). We have:
atan(|y/x|) > atan(z), where z = 2^(emin-2) * (1 + 2^(-pz)).
> z - z^3 / 3.
> 2^(emin-2) * (1 + 2^(-pz) - 2^(2 emin - 5))
Assuming pz <= -2 emin + 5, we can round away from zero
(this is what mpfr_underflow always does on MPFR_RNDN).
In the case MPFR_RNDN with |y/x| <= 2^(emin-2), we round
toward zero, as |atan(z)/z| < 1. */
MPFR_ASSERTN (MPFR_PREC_MAX <=
2 * (mpfr_uexp_t) - MPFR_EMIN_MIN + 5);
if (rnd_mode == MPFR_RNDN && MPFR_IS_ZERO (tmp))
rnd_mode = MPFR_RNDZ;
sign = MPFR_SIGN (tmp);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (dest, rnd_mode, sign);
}
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
/* TODO: check that the error bound is correct in case of overflow. */
/* FIXME: Error <= ulp(tmp) ? */
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 2, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
}
else /* x < 0 */
/* Use sign(y)*(PI - atan (|y/x|)) */
{
mpfr_init2 (pi, prec);
for (;;)
{
mpfr_div (tmp, y, x, MPFR_RNDN); /* Error <= ulp (tmp) */
/* If tmp is 0, we have |y/x| <= 2^(-emin-2), thus
atan|y/x| < 2^(-emin-2). */
MPFR_SET_POS (tmp); /* no error */
mpfr_atan (tmp, tmp, MPFR_RNDN); /* Error <= 2*ulp (tmp) since
abs(D(arctan)) <= 1 */
mpfr_const_pi (pi, MPFR_RNDN); /* Error <= ulp(pi) /2 */
e = MPFR_NOTZERO(tmp) ? MPFR_GET_EXP (tmp) : __gmpfr_emin - 1;
mpfr_sub (tmp, pi, tmp, MPFR_RNDN); /* see above */
if (MPFR_IS_NEG (y))
MPFR_CHANGE_SIGN (tmp);
/* Error(tmp) <= (1/2+2^(EXP(pi)-EXP(tmp)-1)+2^(e-EXP(tmp)+1))*ulp
<= 2^(MAX (MAX (EXP(PI)-EXP(tmp)-1, e-EXP(tmp)+1),
-1)+2)*ulp(tmp) */
e = MAX (MAX (MPFR_GET_EXP (pi)-MPFR_GET_EXP (tmp) - 1,
e - MPFR_GET_EXP (tmp) + 1), -1) + 2;
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - e, MPFR_PREC (dest),
rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
mpfr_set_prec (pi, prec);
}
mpfr_clear (pi);
}
inexact = mpfr_set (dest, tmp, rnd_mode);
end:
MPFR_ZIV_FREE (loop);
mpfr_clear (tmp);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (dest, inexact, rnd_mode);
}
/* Compute the real part of the dilogarithm defined by
Li2(x) = -\Int_{t=0}^x log(1-t)/t dt */
int
mpfr_li2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inexact;
mp_exp_t err;
mpfr_prec_t yp, m;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode), ("y[%#R]=%R", y));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
MPFR_SET_NEG (y);
MPFR_SET_INF (y);
MPFR_RET (0);
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_SAME_SIGN (y, x);
MPFR_SET_ZERO (y);
MPFR_RET (0);
}
}
/* Li2(x) = x + x^2/4 + x^3/9 + ..., more precisely for 0 < x <= 1/2
we have |Li2(x) - x| < x^2/2 <= 2^(2EXP(x)-1) and for -1/2 <= x < 0
we have |Li2(x) - x| < x^2/4 <= 2^(2EXP(x)-2) */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -MPFR_GET_EXP (x), 1, 1, rnd_mode,
{});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -MPFR_GET_EXP (x), 2, 0, rnd_mode,
{});
MPFR_SAVE_EXPO_MARK (expo);
yp = MPFR_PREC (y);
m = yp + MPFR_INT_CEIL_LOG2 (yp) + 13;
if (MPFR_LIKELY ((mpfr_cmp_ui (x, 0) > 0) && (mpfr_cmp_d (x, 0.5) <= 0)))
/* 0 < x <= 1/2: Li2(x) = S(-log(1-x))-log^2(1-x)/4 */
{
mpfr_t s, u;
mp_exp_t expo_l;
int k;
mpfr_init2 (u, m);
mpfr_init2 (s, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_ui_sub (u, 1, x, GMP_RNDN);
mpfr_log (u, u, GMP_RNDU);
if (MPFR_IS_ZERO(u))
goto next_m;
mpfr_neg (u, u, GMP_RNDN); /* u = -log(1-x) */
expo_l = MPFR_GET_EXP (u);
k = li2_series (s, u, GMP_RNDU);
err = 1 + MPFR_INT_CEIL_LOG2 (k + 1);
mpfr_sqr (u, u, GMP_RNDU);
mpfr_div_2ui (u, u, 2, GMP_RNDU); /* u = log^2(1-x) / 4 */
mpfr_sub (s, s, u, GMP_RNDN);
/* error(s) <= (0.5 + 2^(d-EXP(s))
+ 2^(3 + MAX(1, - expo_l) - EXP(s))) ulp(s) */
err = MAX (err, MAX (1, - expo_l) - 1) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err);
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
next_m:
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (u, m);
mpfr_set_prec (s, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clear (u);
mpfr_clear (s);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else if (!mpfr_cmp_ui (x, 1))
/* Li2(1)= pi^2 / 6 */
{
mpfr_t u;
mpfr_init2 (u, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_const_pi (u, GMP_RNDU);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_ui (u, u, 6, GMP_RNDN);
err = m - 4; /* error(u) <= 19/2 ulp(u) */
if (MPFR_CAN_ROUND (u, err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (u, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, u, rnd_mode);
mpfr_clear (u);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else if (mpfr_cmp_ui (x, 2) >= 0)
/* x >= 2: Li2(x) = -S(-log(1-1/x))-log^2(x)/2+log^2(1-1/x)/4+pi^2/3 */
{
int k;
mp_exp_t expo_l;
mpfr_t s, u, xx;
if (mpfr_cmp_ui (x, 38) >= 0)
{
inexact = mpfr_li2_asympt_pos (y, x, rnd_mode);
if (inexact != 0)
goto end_of_case_gt2;
}
mpfr_init2 (u, m);
mpfr_init2 (s, m);
mpfr_init2 (xx, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_ui_div (xx, 1, x, GMP_RNDN);
mpfr_neg (xx, xx, GMP_RNDN);
mpfr_log1p (u, xx, GMP_RNDD);
mpfr_neg (u, u, GMP_RNDU); /* u = -log(1-1/x) */
expo_l = MPFR_GET_EXP (u);
k = li2_series (s, u, GMP_RNDN);
mpfr_neg (s, s, GMP_RNDN);
err = MPFR_INT_CEIL_LOG2 (k + 1) + 1; /* error(s) <= 2^err ulp(s) */
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u= log^2(1-1/x)/4 */
mpfr_add (s, s, u, GMP_RNDN);
err =
MAX (err,
3 + MAX (1, -expo_l) + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
err += MPFR_GET_EXP (s);
mpfr_log (u, x, GMP_RNDU);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_2ui (u, u, 1, GMP_RNDN); /* u = log^2(x)/2 */
mpfr_sub (s, s, u, GMP_RNDN);
err = MAX (err, 3 + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
err += MPFR_GET_EXP (s);
mpfr_const_pi (u, GMP_RNDU);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_ui (u, u, 3, GMP_RNDN); /* u = pi^2/3 */
mpfr_add (s, s, u, GMP_RNDN);
err = MAX (err, 2) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err); /* error(s) <= 2^err ulp(s) */
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (u, m);
mpfr_set_prec (s, m);
mpfr_set_prec (xx, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clears (s, u, xx, (mpfr_ptr) 0);
end_of_case_gt2:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else if (mpfr_cmp_ui (x, 1) > 0)
/* 2 > x > 1: Li2(x) = S(log(x))+log^2(x)/4-log(x)log(x-1)+pi^2/6 */
{
int k;
mp_exp_t e1, e2;
mpfr_t s, u, v, xx;
mpfr_init2 (s, m);
mpfr_init2 (u, m);
mpfr_init2 (v, m);
mpfr_init2 (xx, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_log (v, x, GMP_RNDU);
k = li2_series (s, v, GMP_RNDN);
e1 = MPFR_GET_EXP (s);
mpfr_sqr (u, v, GMP_RNDN);
mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(x)/4 */
mpfr_add (s, s, u, GMP_RNDN);
mpfr_sub_ui (xx, x, 1, GMP_RNDN);
mpfr_log (u, xx, GMP_RNDU);
e2 = MPFR_GET_EXP (u);
mpfr_mul (u, v, u, GMP_RNDN); /* u = log(x) * log(x-1) */
mpfr_sub (s, s, u, GMP_RNDN);
mpfr_const_pi (u, GMP_RNDU);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_ui (u, u, 6, GMP_RNDN); /* u = pi^2/6 */
mpfr_add (s, s, u, GMP_RNDN);
/* error(s) <= (31 + (k+1) * 2^(1-e1) + 2^(1-e2)) ulp(s)
see algorithms.tex */
err = MAX (MPFR_INT_CEIL_LOG2 (k + 1) + 1 - e1, 1 - e2);
err = 2 + MAX (5, err);
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (s, m);
mpfr_set_prec (u, m);
mpfr_set_prec (v, m);
mpfr_set_prec (xx, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clears (s, u, v, xx, (mpfr_ptr) 0);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else if (mpfr_cmp_ui_2exp (x, 1, -1) > 0) /* 1/2 < x < 1 */
/* 1 > x > 1/2: Li2(x) = -S(-log(x))+log^2(x)/4-log(x)log(1-x)+pi^2/6 */
{
int k;
mpfr_t s, u, v, xx;
mpfr_init2 (s, m);
mpfr_init2 (u, m);
mpfr_init2 (v, m);
mpfr_init2 (xx, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_log (u, x, GMP_RNDD);
mpfr_neg (u, u, GMP_RNDN);
k = li2_series (s, u, GMP_RNDN);
mpfr_neg (s, s, GMP_RNDN);
err = 1 + MPFR_INT_CEIL_LOG2 (k + 1) - MPFR_GET_EXP (s);
mpfr_ui_sub (xx, 1, x, GMP_RNDN);
mpfr_log (v, xx, GMP_RNDU);
mpfr_mul (v, v, u, GMP_RNDN); /* v = - log(x) * log(1-x) */
mpfr_add (s, s, v, GMP_RNDN);
err = MAX (err, 1 - MPFR_GET_EXP (v));
err = 2 + MAX (3, err) - MPFR_GET_EXP (s);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(x)/4 */
mpfr_add (s, s, u, GMP_RNDN);
err = MAX (err, 2 + MPFR_GET_EXP (u)) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
mpfr_const_pi (u, GMP_RNDU);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_ui (u, u, 6, GMP_RNDN); /* u = pi^2/6 */
mpfr_add (s, s, u, GMP_RNDN);
err = MAX (err, 3) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err);
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (s, m);
mpfr_set_prec (u, m);
mpfr_set_prec (v, m);
mpfr_set_prec (xx, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clears (s, u, v, xx, (mpfr_ptr) 0);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else if (mpfr_cmp_si (x, -1) >= 0)
/* 0 > x >= -1: Li2(x) = -S(log(1-x))-log^2(1-x)/4 */
{
int k;
mp_exp_t expo_l;
mpfr_t s, u, xx;
mpfr_init2 (s, m);
mpfr_init2 (u, m);
mpfr_init2 (xx, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_neg (xx, x, GMP_RNDN);
mpfr_log1p (u, xx, GMP_RNDN);
k = li2_series (s, u, GMP_RNDN);
mpfr_neg (s, s, GMP_RNDN);
expo_l = MPFR_GET_EXP (u);
err = 1 + MPFR_INT_CEIL_LOG2 (k + 1) - MPFR_GET_EXP (s);
mpfr_sqr (u, u, GMP_RNDN);
mpfr_div_2ui (u, u, 2, GMP_RNDN); /* u = log^2(1-x)/4 */
mpfr_sub (s, s, u, GMP_RNDN);
err = MAX (err, - expo_l);
err = 2 + MAX (err, 3);
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (s, m);
mpfr_set_prec (u, m);
mpfr_set_prec (xx, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clears (s, u, xx, (mpfr_ptr) 0);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
else
/* x < -1: Li2(x)
= S(log(1-1/x))-log^2(-x)/4-log(1-x)log(-x)/2+log^2(1-x)/4-pi^2/6 */
{
int k;
mpfr_t s, u, v, w, xx;
if (mpfr_cmp_si (x, -7) <= 0)
{
inexact = mpfr_li2_asympt_neg (y, x, rnd_mode);
if (inexact != 0)
goto end_of_case_ltm1;
}
mpfr_init2 (s, m);
mpfr_init2 (u, m);
mpfr_init2 (v, m);
mpfr_init2 (w, m);
mpfr_init2 (xx, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_ui_div (xx, 1, x, GMP_RNDN);
mpfr_neg (xx, xx, GMP_RNDN);
mpfr_log1p (u, xx, GMP_RNDN);
k = li2_series (s, u, GMP_RNDN);
mpfr_ui_sub (xx, 1, x, GMP_RNDN);
mpfr_log (u, xx, GMP_RNDU);
mpfr_neg (xx, x, GMP_RNDN);
mpfr_log (v, xx, GMP_RNDU);
mpfr_mul (w, v, u, GMP_RNDN);
mpfr_div_2ui (w, w, 1, GMP_RNDN); /* w = log(-x) * log(1-x) / 2 */
mpfr_sub (s, s, w, GMP_RNDN);
err = 1 + MAX (3, MPFR_INT_CEIL_LOG2 (k+1) + 1 - MPFR_GET_EXP (s))
+ MPFR_GET_EXP (s);
mpfr_sqr (w, v, GMP_RNDN);
mpfr_div_2ui (w, w, 2, GMP_RNDN); /* w = log^2(-x) / 4 */
mpfr_sub (s, s, w, GMP_RNDN);
err = MAX (err, 3 + MPFR_GET_EXP(w)) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
mpfr_sqr (w, u, GMP_RNDN);
mpfr_div_2ui (w, w, 2, GMP_RNDN); /* w = log^2(1-x) / 4 */
mpfr_add (s, s, w, GMP_RNDN);
err = MAX (err, 3 + MPFR_GET_EXP (w)) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
mpfr_const_pi (w, GMP_RNDU);
mpfr_sqr (w, w, GMP_RNDN);
mpfr_div_ui (w, w, 6, GMP_RNDN); /* w = pi^2 / 6 */
mpfr_sub (s, s, w, GMP_RNDN);
err = MAX (err, 3) - MPFR_GET_EXP (s);
err = 2 + MAX (-1, err) + MPFR_GET_EXP (s);
if (MPFR_CAN_ROUND (s, (mp_exp_t) m - err, yp, rnd_mode))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (s, m);
mpfr_set_prec (u, m);
mpfr_set_prec (v, m);
mpfr_set_prec (w, m);
mpfr_set_prec (xx, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clears (s, u, v, w, xx, (mpfr_ptr) 0);
end_of_case_ltm1:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
MPFR_ASSERTN (0); /* should never reach this point */
}
/* returns 0 if result exact, non-zero otherwise */
int
mpfr_div_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int u, mpfr_rnd_t rnd_mode)
{
long i;
int sh;
mp_size_t xn, yn, dif;
mp_limb_t *xp, *yp, *tmp, c, d;
mpfr_exp_t exp;
int inexact, middle = 1, nexttoinf;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg u=%lu rnd=%d",
mpfr_get_prec(x), mpfr_log_prec, x, u, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec(y), mpfr_log_prec, y, inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
else
{
MPFR_ASSERTD (MPFR_IS_ZERO(x));
if (u == 0) /* 0/0 is NaN */
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else
{
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET(0);
}
}
}
else if (MPFR_UNLIKELY (u <= 1))
{
if (u < 1)
{
/* x/0 is Inf since x != 0*/
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, x);
mpfr_set_divby0 ();
MPFR_RET (0);
}
else /* y = x/1 = x */
return mpfr_set (y, x, rnd_mode);
}
else if (MPFR_UNLIKELY (IS_POW2 (u)))
return mpfr_div_2si (y, x, MPFR_INT_CEIL_LOG2 (u), rnd_mode);
MPFR_SET_SAME_SIGN (y, x);
MPFR_TMP_MARK (marker);
xn = MPFR_LIMB_SIZE (x);
yn = MPFR_LIMB_SIZE (y);
xp = MPFR_MANT (x);
yp = MPFR_MANT (y);
exp = MPFR_GET_EXP (x);
dif = yn + 1 - xn;
/* we need to store yn+1 = xn + dif limbs of the quotient */
/* don't use tmp=yp since the mpn_lshift call below requires yp >= tmp+1 */
tmp = MPFR_TMP_LIMBS_ALLOC (yn + 1);
c = (mp_limb_t) u;
MPFR_ASSERTN (u == c);
if (dif >= 0)
c = mpn_divrem_1 (tmp, dif, xp, xn, c); /* used all the dividend */
else /* dif < 0 i.e. xn > yn, don't use the (-dif) low limbs from x */
c = mpn_divrem_1 (tmp, 0, xp - dif, yn + 1, c);
inexact = (c != 0);
/* First pass in estimating next bit of the quotient, in case of RNDN *
* In case we just have the right number of bits (postpone this ?), *
* we need to check whether the remainder is more or less than half *
* the divisor. The test must be performed with a subtraction, so as *
* to prevent carries. */
if (MPFR_LIKELY (rnd_mode == MPFR_RNDN))
{
if (c < (mp_limb_t) u - c) /* We have u > c */
middle = -1;
else if (c > (mp_limb_t) u - c)
middle = 1;
else
middle = 0; /* exactly in the middle */
}
/* If we believe that we are right in the middle or exact, we should check
that we did not neglect any word of x (division large / 1 -> small). */
for (i=0; ((inexact == 0) || (middle == 0)) && (i < -dif); i++)
if (xp[i])
inexact = middle = 1; /* larger than middle */
/*
If the high limb of the result is 0 (xp[xn-1] < u), remove it.
Otherwise, compute the left shift to be performed to normalize.
In the latter case, we discard some low bits computed. They
contain information useful for the rounding, hence the updating
of middle and inexact.
*/
if (tmp[yn] == 0)
{
MPN_COPY(yp, tmp, yn);
exp -= GMP_NUMB_BITS;
}
else
{
int shlz;
count_leading_zeros (shlz, tmp[yn]);
/* shift left to normalize */
if (MPFR_LIKELY (shlz != 0))
{
mp_limb_t w = tmp[0] << shlz;
mpn_lshift (yp, tmp + 1, yn, shlz);
yp[0] += tmp[0] >> (GMP_NUMB_BITS - shlz);
if (w > (MPFR_LIMB_ONE << (GMP_NUMB_BITS - 1)))
{ middle = 1; }
else if (w < (MPFR_LIMB_ONE << (GMP_NUMB_BITS - 1)))
{ middle = -1; }
else
{ middle = (c != 0); }
inexact = inexact || (w != 0);
exp -= shlz;
}
else
{ /* this happens only if u == 1 and xp[xn-1] >=
int
mpfr_round_raw_generic(
#if flag == 0
mp_limb_t *yp,
#endif
const mp_limb_t *xp, mpfr_prec_t xprec,
int neg, mpfr_prec_t yprec, mpfr_rnd_t rnd_mode
#if use_inexp != 0
, int *inexp
#endif
)
{
mp_size_t xsize, nw;
mp_limb_t himask, lomask, sb;
int rw;
#if flag == 0
int carry;
#endif
#if use_inexp == 0
int *inexp;
#endif
if (use_inexp)
MPFR_ASSERTD(inexp != ((int*) 0));
MPFR_ASSERTD(neg == 0 || neg == 1);
if (flag && !use_inexp &&
(xprec <= yprec || MPFR_IS_LIKE_RNDZ (rnd_mode, neg)))
return 0;
xsize = (xprec-1)/GMP_NUMB_BITS + 1;
nw = yprec / GMP_NUMB_BITS;
rw = yprec & (GMP_NUMB_BITS - 1);
if (MPFR_UNLIKELY(xprec <= yprec))
{ /* No rounding is necessary. */
/* if yp=xp, maybe an overlap: MPN_COPY_DECR is ok when src <= dst */
if (MPFR_LIKELY(rw))
nw++;
MPFR_ASSERTD(nw >= 1);
MPFR_ASSERTD(nw >= xsize);
if (use_inexp)
*inexp = 0;
#if flag == 0
MPN_COPY_DECR(yp + (nw - xsize), xp, xsize);
MPN_ZERO(yp, nw - xsize);
#endif
return 0;
}
if (use_inexp || !MPFR_IS_LIKE_RNDZ(rnd_mode, neg))
{
mp_size_t k = xsize - nw - 1;
if (MPFR_LIKELY(rw))
{
nw++;
lomask = MPFR_LIMB_MASK (GMP_NUMB_BITS - rw);
himask = ~lomask;
}
else
{
lomask = ~(mp_limb_t) 0;
himask = ~(mp_limb_t) 0;
}
MPFR_ASSERTD(k >= 0);
sb = xp[k] & lomask; /* First non-significant bits */
/* Rounding to nearest ? */
if (MPFR_LIKELY( rnd_mode == MPFR_RNDN) )
{
/* Rounding to nearest */
mp_limb_t rbmask = MPFR_LIMB_ONE << (GMP_NUMB_BITS - 1 - rw);
if (sb & rbmask) /* rounding bit */
sb &= ~rbmask; /* it is 1, clear it */
else
{
/* Rounding bit is 0, behave like rounding to 0 */
goto rnd_RNDZ;
}
while (MPFR_UNLIKELY(sb == 0) && k > 0)
sb = xp[--k];
/* rounding to nearest, with rounding bit = 1 */
if (MPFR_UNLIKELY(sb == 0)) /* Even rounding. */
{
/* sb == 0 && rnd_mode == MPFR_RNDN */
sb = xp[xsize - nw] & (himask ^ (himask << 1));
if (sb == 0)
{
if (use_inexp)
*inexp = 2*MPFR_EVEN_INEX*neg-MPFR_EVEN_INEX;
/* ((neg!=0)^(sb!=0)) ? MPFR_EVEN_INEX : -MPFR_EVEN_INEX;*/
/* Since neg = 0 or 1 and sb=0*/
#if flag == 1
return 0 /*sb != 0 && rnd_mode != MPFR_RNDZ */;
#else
MPN_COPY_INCR(yp, xp + xsize - nw, nw);
yp[0] &= himask;
return 0;
#endif
}
else
{
/* sb != 0 && rnd_mode == MPFR_RNDN */
if (use_inexp)
*inexp = MPFR_EVEN_INEX-2*MPFR_EVEN_INEX*neg;
/*((neg!=0)^(sb!=0))? MPFR_EVEN_INEX : -MPFR_EVEN_INEX; */
/*Since neg= 0 or 1 and sb != 0 */
goto rnd_RNDN_add_one_ulp;
}
}
else /* sb != 0 && rnd_mode == MPFR_RNDN*/
{
if (use_inexp)
/* *inexp = (neg == 0) ? 1 : -1; but since neg = 0 or 1 */
*inexp = 1-2*neg;
rnd_RNDN_add_one_ulp:
#if flag == 1
return 1; /*sb != 0 && rnd_mode != MPFR_RNDZ;*/
#else
carry = mpn_add_1 (yp, xp + xsize - nw, nw,
rw ?
MPFR_LIMB_ONE << (GMP_NUMB_BITS - rw)
: MPFR_LIMB_ONE);
yp[0] &= himask;
return carry;
#endif
}
}
/* Rounding to Zero ? */
else if (MPFR_IS_LIKE_RNDZ(rnd_mode, neg))
{
/* rnd_mode == MPFR_RNDZ */
rnd_RNDZ:
while (MPFR_UNLIKELY(sb == 0) && k > 0)
sb = xp[--k];
if (use_inexp)
/* rnd_mode == MPFR_RNDZ and neg = 0 or 1 */
/* (neg != 0) ^ (rnd_mode != MPFR_RNDZ)) ? 1 : -1);*/
*inexp = MPFR_UNLIKELY(sb == 0) ? 0 : (2*neg-1);
#if flag == 1
return 0; /*sb != 0 && rnd_mode != MPFR_RNDZ;*/
#else
MPN_COPY_INCR(yp, xp + xsize - nw, nw);
yp[0] &= himask;
return 0;
#endif
}
else
{
/* rnd_mode = Away */
while (MPFR_UNLIKELY(sb == 0) && k > 0)
sb = xp[--k];
if (MPFR_UNLIKELY(sb == 0))
{
/* sb = 0 && rnd_mode != MPFR_RNDZ */
if (use_inexp)
/* (neg != 0) ^ (rnd_mode != MPFR_RNDZ)) ? 1 : -1);*/
*inexp = 0;
#if flag == 1
return 0;
#else
MPN_COPY_INCR(yp, xp + xsize - nw, nw);
yp[0] &= himask;
return 0;
#endif
}
else
{
/* sb != 0 && rnd_mode != MPFR_RNDZ */
if (use_inexp)
/* (neg != 0) ^ (rnd_mode != MPFR_RNDZ)) ? 1 : -1);*/
*inexp = 1-2*neg;
#if flag == 1
return 1;
#else
carry = mpn_add_1(yp, xp + xsize - nw, nw,
rw ? MPFR_LIMB_ONE << (GMP_NUMB_BITS - rw)
: 1);
yp[0] &= himask;
return carry;
#endif
}
}
}
else
{
/* Roundind mode = Zero / No inexact flag */
#if flag == 1
return 0 /*sb != 0 && rnd_mode != MPFR_RNDZ*/;
#else
if (MPFR_LIKELY(rw))
{
nw++;
himask = ~MPFR_LIMB_MASK (GMP_NUMB_BITS - rw);
}
else
himask = ~(mp_limb_t) 0;
MPN_COPY_INCR(yp, xp + xsize - nw, nw);
yp[0] &= himask;
return 0;
#endif
}
}
static int
mpfr_rem1 (mpfr_ptr rem, long *quo, mpfr_rnd_t rnd_q,
mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd)
{
mpfr_exp_t ex, ey;
int compare, inex, q_is_odd, sign, signx = MPFR_SIGN (x);
mpz_t mx, my, r;
MPFR_ASSERTD (rnd_q == MPFR_RNDN || rnd_q == MPFR_RNDZ);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x) || MPFR_IS_SINGULAR (y)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y) || MPFR_IS_INF (x)
|| MPFR_IS_ZERO (y))
{
/* for remquo, quo is undefined */
MPFR_SET_NAN (rem);
MPFR_RET_NAN;
}
else /* either y is Inf and x is 0 or non-special,
or x is 0 and y is non-special,
in both cases the quotient is zero. */
{
if (quo)
*quo = 0;
return mpfr_set (rem, x, rnd);
}
}
/* now neither x nor y is NaN, Inf or zero */
mpz_init (mx);
mpz_init (my);
mpz_init (r);
ex = mpfr_get_z_2exp (mx, x); /* x = mx*2^ex */
ey = mpfr_get_z_2exp (my, y); /* y = my*2^ey */
/* to get rid of sign problems, we compute it separately:
quo(-x,-y) = quo(x,y), rem(-x,-y) = -rem(x,y)
quo(-x,y) = -quo(x,y), rem(-x,y) = -rem(x,y)
thus quo = sign(x/y)*quo(|x|,|y|), rem = sign(x)*rem(|x|,|y|) */
sign = (signx == MPFR_SIGN (y)) ? 1 : -1;
mpz_abs (mx, mx);
mpz_abs (my, my);
q_is_odd = 0;
/* divide my by 2^k if possible to make operations mod my easier */
{
unsigned long k = mpz_scan1 (my, 0);
ey += k;
mpz_fdiv_q_2exp (my, my, k);
}
if (ex <= ey)
{
/* q = x/y = mx/(my*2^(ey-ex)) */
mpz_mul_2exp (my, my, ey - ex); /* divide mx by my*2^(ey-ex) */
if (rnd_q == MPFR_RNDZ)
/* 0 <= |r| <= |my|, r has the same sign as mx */
mpz_tdiv_qr (mx, r, mx, my);
else
/* 0 <= |r| <= |my|, r has the same sign as my */
mpz_fdiv_qr (mx, r, mx, my);
if (rnd_q == MPFR_RNDN)
q_is_odd = mpz_tstbit (mx, 0);
if (quo) /* mx is the quotient */
{
mpz_tdiv_r_2exp (mx, mx, WANTED_BITS);
*quo = mpz_get_si (mx);
}
}
else /* ex > ey */
{
if (quo) /* remquo case */
/* for remquo, to get the low WANTED_BITS more bits of the quotient,
we first compute R = X mod Y*2^WANTED_BITS, where X and Y are
defined below. Then the low WANTED_BITS of the quotient are
floor(R/Y). */
mpz_mul_2exp (my, my, WANTED_BITS); /* 2^WANTED_BITS*Y */
else if (rnd_q == MPFR_RNDN) /* remainder case */
/* Let X = mx*2^(ex-ey) and Y = my. Then both X and Y are integers.
Assume X = R mod Y, then x = X*2^ey = R*2^ey mod (Y*2^ey=y).
To be able to perform the rounding, we need the least significant
bit of the quotient, i.e., one more bit in the remainder,
which is obtained by dividing by 2Y. */
mpz_mul_2exp (my, my, 1); /* 2Y */
mpz_set_ui (r, 2);
mpz_powm_ui (r, r, ex - ey, my); /* 2^(ex-ey) mod my */
mpz_mul (r, r, mx);
mpz_mod (r, r, my);
if (quo) /* now 0 <= r < 2^WANTED_BITS*Y */
{
mpz_fdiv_q_2exp (my, my, WANTED_BITS); /* back to Y */
mpz_tdiv_qr (mx, r, r, my);
/* oldr = mx*my + newr */
*quo = mpz_get_si (mx);
q_is_odd = *quo & 1;
}
else if (rnd_q == MPFR_RNDN) /* now 0 <= r < 2Y in the remainder case */
{
mpz_fdiv_q_2exp (my, my, 1); /* back to Y */
/* least significant bit of q */
q_is_odd = mpz_cmpabs (r, my) >= 0;
if (q_is_odd)
mpz_sub (r, r, my);
}
/* now 0 <= |r| < |my|, and if needed,
q_is_odd is the least significant bit of q */
}
if (mpz_cmp_ui (r, 0) == 0)
{
inex = mpfr_set_ui (rem, 0, MPFR_RNDN);
/* take into account sign of x */
if (signx < 0)
mpfr_neg (rem, rem, MPFR_RNDN);
}
else
{
if (rnd_q == MPFR_RNDN)
{
/* FIXME: the comparison 2*r < my could be done more efficiently
at the mpn level */
mpz_mul_2exp (r, r, 1);
compare = mpz_cmpabs (r, my);
mpz_fdiv_q_2exp (r, r, 1);
compare = ((compare > 0) ||
((rnd_q == MPFR_RNDN) && (compare == 0) && q_is_odd));
/* if compare != 0, we need to subtract my to r, and add 1 to quo */
if (compare)
{
mpz_sub (r, r, my);
if (quo && (rnd_q == MPFR_RNDN))
*quo += 1;
}
}
/* take into account sign of x */
if (signx < 0)
mpz_neg (r, r);
inex = mpfr_set_z_2exp (rem, r, ex > ey ? ey : ex, rnd);
}
if (quo)
*quo *= sign;
mpz_clear (mx);
mpz_clear (my);
mpz_clear (r);
return inex;
}
long double
mpfr_get_ld (mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
return (long double) mpfr_get_d (x, rnd_mode);
else /* now x is a normal non-zero number */
{
long double r; /* result */
long double m;
double s; /* part of result */
mpfr_exp_t sh; /* exponent shift, so that x/2^sh is in the double range */
mpfr_t y, z;
int sign;
/* first round x to the target long double precision, so that
all subsequent operations are exact (this avoids double rounding
problems) */
mpfr_init2 (y, MPFR_LDBL_MANT_DIG);
mpfr_init2 (z, IEEE_DBL_MANT_DIG);
mpfr_set (y, x, rnd_mode);
sh = MPFR_GET_EXP (y);
sign = MPFR_SIGN (y);
MPFR_SET_EXP (y, 0);
MPFR_SET_POS (y);
r = 0.0;
do {
s = mpfr_get_d (y, MPFR_RNDN); /* high part of y */
r += (long double) s;
mpfr_set_d (z, s, MPFR_RNDN); /* exact */
mpfr_sub (y, y, z, MPFR_RNDN); /* exact */
} while (!MPFR_IS_ZERO (y));
mpfr_clear (z);
mpfr_clear (y);
/* we now have to multiply back by 2^sh */
MPFR_ASSERTD (r > 0);
if (sh != 0)
{
/* An overflow may occurs (example: 0.5*2^1024) */
while (r < 1.0)
{
r += r;
sh--;
}
if (sh > 0)
m = 2.0;
else
{
m = 0.5;
sh = -sh;
}
for (;;)
{
if (sh % 2)
r = r * m;
sh >>= 1;
if (sh == 0)
break;
m = m * m;
}
}
if (sign < 0)
r = -r;
return r;
}
}
int
mpfr_digamma (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inex));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x))
{
if (MPFR_IS_POS(x)) /* Digamma(+Inf) = +Inf */
{
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
MPFR_RET(0);
}
else /* Digamma(-Inf) = NaN */
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
else /* Zero case */
{
/* the following works also in case of overlap */
MPFR_SET_INF(y);
MPFR_SET_OPPOSITE_SIGN(y, x);
MPFR_SET_DIVBY0 ();
MPFR_RET(0);
}
}
/* Digamma is undefined for negative integers */
if (MPFR_IS_NEG(x) && mpfr_integer_p (x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
/* now x is a normal number */
MPFR_SAVE_EXPO_MARK (expo);
/* for x very small, we have Digamma(x) = -1/x - gamma + O(x), more precisely
-1 < Digamma(x) + 1/x < 0 for -0.2 < x < 0.2, thus:
(i) either x is a power of two, then 1/x is exactly representable, and
as long as 1/2*ulp(1/x) > 1, we can conclude;
(ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then
|y + 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place.
Since |Digamma(x) + 1/x| <= 1, if 2^(-2n) ufp(y) >= 2, then
|y - Digamma(x)| >= 2^(-2n-1)ufp(y), and rounding -1/x gives the correct result.
If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */
if (MPFR_EXP(x) < -2)
{
if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y)))
{
int signx = MPFR_SIGN(x);
inex = mpfr_si_div (y, -1, x, rnd_mode);
if (inex == 0) /* x is a power of two */
{ /* result always -1/x, except when rounding down */
if (rnd_mode == MPFR_RNDA)
rnd_mode = (signx > 0) ? MPFR_RNDD : MPFR_RNDU;
if (rnd_mode == MPFR_RNDZ)
rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD;
if (rnd_mode == MPFR_RNDU)
inex = 1;
else if (rnd_mode == MPFR_RNDD)
{
mpfr_nextbelow (y);
inex = -1;
}
else /* nearest */
inex = 1;
}
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
goto end;
}
}
if (MPFR_IS_NEG(x))
inex = mpfr_digamma_reflection (y, x, rnd_mode);
/* if x < 1/2 we use the reflection formula */
else if (MPFR_EXP(x) < 0)
inex = mpfr_digamma_reflection (y, x, rnd_mode);
else
inex = mpfr_digamma_positive (y, x, rnd_mode);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
int
mpfr_ui_div (mpfr_ptr y, unsigned long int u, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
MPFR_LOG_FUNC
(("u=%lu x[%Pu]=%.*Rg rnd=%d",
u, mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y));
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(x)))
{
if (MPFR_IS_NAN(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF(x)) /* u/Inf = 0 */
{
MPFR_SET_ZERO(y);
MPFR_SET_SAME_SIGN(y,x);
MPFR_RET(0);
}
else /* u / 0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
if (u)
{
/* u > 0, so y = sign(x) * Inf */
MPFR_SET_SAME_SIGN(y, x);
MPFR_SET_INF(y);
mpfr_set_divby0 ();
MPFR_RET(0);
}
else
{
/* 0 / 0 */
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
}
}
else if (MPFR_LIKELY(u != 0))
{
mpfr_t uu;
mp_limb_t up[1];
int cnt;
int inex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT1(up, uu, GMP_NUMB_BITS);
MPFR_ASSERTN(u == (mp_limb_t) u);
count_leading_zeros(cnt, (mp_limb_t) u);
up[0] = (mp_limb_t) u << cnt;
/* Optimization note: Exponent save/restore operations may be
removed if mpfr_div works even when uu is out-of-range. */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt);
inex = mpfr_div (y, uu, x, rnd_mode);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
else /* u = 0, and x != 0 */
{
MPFR_SET_ZERO(y); /* if u=0, then set y to 0 */
MPFR_SET_SAME_SIGN(y, x); /* u considered as +0: sign(+0/x) = sign(x) */
MPFR_RET(0);
}
}
int
mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
{
mpfr_t z_pre, s1, y, p;
long add;
mpfr_prec_t precz, prec1, precs, precs1;
int inex;
MPFR_GROUP_DECL (group);
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (
("s[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (s), mpfr_log_prec, s, rnd_mode),
("z[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (z), mpfr_log_prec, z, inex));
/* Zero, Nan or Inf ? */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (s)))
{
if (MPFR_IS_NAN (s))
{
MPFR_SET_NAN (z);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (s))
{
if (MPFR_IS_POS (s))
return mpfr_set_ui (z, 1, MPFR_RNDN); /* Zeta(+Inf) = 1 */
MPFR_SET_NAN (z); /* Zeta(-Inf) = NaN */
MPFR_RET_NAN;
}
else /* s iz zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (s));
return mpfr_set_si_2exp (z, -1, -1, rnd_mode);
}
}
/* s is neither Nan, nor Inf, nor Zero */
/* check tiny s: we have zeta(s) = -1/2 - 1/2 log(2 Pi) s + ... around s=0,
and for |s| <= 2^(-4), we have |zeta(s) + 1/2| <= |s|.
EXP(s) + 1 < -PREC(z) is a sufficient condition to be able to round
correctly, for any PREC(z) >= 1 (see algorithms.tex for details). */
if (MPFR_GET_EXP (s) + 1 < - (mpfr_exp_t) MPFR_PREC(z))
{
int signs = MPFR_SIGN(s);
MPFR_SAVE_EXPO_MARK (expo);
mpfr_set_si_2exp (z, -1, -1, rnd_mode); /* -1/2 */
if (rnd_mode == MPFR_RNDA)
rnd_mode = MPFR_RNDD; /* the result is around -1/2, thus negative */
if ((rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDZ) && signs < 0)
{
mpfr_nextabove (z); /* z = -1/2 + epsilon */
inex = 1;
}
else if (rnd_mode == MPFR_RNDD && signs > 0)
{
mpfr_nextbelow (z); /* z = -1/2 - epsilon */
inex = -1;
}
else
{
if (rnd_mode == MPFR_RNDU) /* s > 0: z = -1/2 */
inex = 1;
else if (rnd_mode == MPFR_RNDD)
inex = -1; /* s < 0: z = -1/2 */
else /* (MPFR_RNDZ and s > 0) or MPFR_RNDN: z = -1/2 */
inex = (signs > 0) ? 1 : -1;
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inex, rnd_mode);
}
/* Check for case s= -2n */
if (MPFR_IS_NEG (s))
{
mpfr_t tmp;
tmp[0] = *s;
MPFR_EXP (tmp) = MPFR_GET_EXP (s) - 1;
if (mpfr_integer_p (tmp))
{
MPFR_SET_ZERO (z);
MPFR_SET_POS (z);
MPFR_RET (0);
}
}
/* Check for case s=1 before changing the exponent range */
if (mpfr_cmp (s, __gmpfr_one) == 0)
{
MPFR_SET_INF (z);
MPFR_SET_POS (z);
MPFR_SET_DIVBY0 ();
MPFR_RET (0);
}
MPFR_SAVE_EXPO_MARK (expo);
/* Compute Zeta */
if (MPFR_IS_POS (s) && MPFR_GET_EXP (s) >= 0) /* Case s >= 1/2 */
inex = mpfr_zeta_pos (z, s, rnd_mode);
else /* use reflection formula
zeta(s) = 2^s*Pi^(s-1)*sin(Pi*s/2)*gamma(1-s)*zeta(1-s) */
{
int overflow = 0;
precz = MPFR_PREC (z);
precs = MPFR_PREC (s);
/* Precision precs1 needed to represent 1 - s, and s + 2,
without any truncation */
precs1 = precs + 2 + MAX (0, - MPFR_GET_EXP (s));
/* Precision prec1 is the precision on elementary computations;
it ensures a final precision prec1 - add for zeta(s) */
add = compute_add (s, precz);
prec1 = precz + add;
/* FIXME: To avoid that the working precision (prec1) depends on the
input precision, one would need to take into account the error made
when s1 is not exactly 1-s when computing zeta(s1) and gamma(s1)
below, and also in the case y=Inf (i.e. when gamma(s1) overflows).
Make sure that underflows do not occur in intermediate computations.
Due to the limited precision, they are probably not possible
in practice; add some MPFR_ASSERTN's to be sure that problems
do not remain undetected? */
prec1 = MAX (prec1, precs1) + 10;
MPFR_GROUP_INIT_4 (group, prec1, z_pre, s1, y, p);
MPFR_ZIV_INIT (loop, prec1);
for (;;)
{
mpfr_exp_t ey;
mpfr_t z_up;
mpfr_const_pi (p, MPFR_RNDD); /* p is Pi */
mpfr_sub (s1, __gmpfr_one, s, MPFR_RNDN); /* s1 = 1-s */
mpfr_gamma (y, s1, MPFR_RNDN); /* gamma(1-s) */
if (MPFR_IS_INF (y)) /* zeta(s) < 0 for -4k-2 < s < -4k,
zeta(s) > 0 for -4k < s < -4k+2 */
{
/* FIXME: An overflow in gamma(s1) does not imply that
zeta(s) will overflow. A solution:
1. Compute
log(|zeta(s)|/2) = (s-1)*log(2*pi) + lngamma(1-s)
+ log(abs(sin(Pi*s/2)) * zeta(1-s))
(possibly sharing computations with the normal case)
with a rather good accuracy (see (2)).
Memorize the sign of sin(...) for the final sign.
2. Take the exponential, ~= |zeta(s)|/2. If there is an
overflow, then this means an overflow on the final result
(due to the multiplication by 2, which has not been done
yet).
3. Ziv test.
4. Correct the sign from the sign of sin(...).
5. Round then multiply by 2. Here, an overflow in either
operation means a real overflow. */
mpfr_reflection_overflow (z_pre, s1, s, y, p, MPFR_RNDD);
/* z_pre is a lower bound of |zeta(s)|/2, thus if it overflows,
or has exponent emax, then |zeta(s)| overflows too. */
if (MPFR_IS_INF (z_pre) || MPFR_GET_EXP(z_pre) == __gmpfr_emax)
{ /* determine the sign of overflow */
mpfr_div_2ui (s1, s, 2, MPFR_RNDN); /* s/4, exact */
mpfr_frac (s1, s1, MPFR_RNDN); /* exact, -1 < s1 < 0 */
overflow = (mpfr_cmp_si_2exp (s1, -1, -1) > 0) ? -1 : 1;
break;
}
else /* EXP(z_pre) < __gmpfr_emax */
{
int ok = 0;
mpfr_t z_down;
mpfr_init2 (z_up, mpfr_get_prec (z_pre));
mpfr_reflection_overflow (z_up, s1, s, y, p, MPFR_RNDU);
/* if the lower approximation z_pre does not overflow, but
z_up does, we need more precision */
if (MPFR_IS_INF (z_up) || MPFR_GET_EXP(z_up) == __gmpfr_emax)
goto next_loop;
/* check if z_pre and z_up round to the same number */
mpfr_init2 (z_down, precz);
mpfr_set (z_down, z_pre, rnd_mode);
/* Note: it might be that EXP(z_down) = emax here, in that
case we will have overflow below when we multiply by 2 */
mpfr_prec_round (z_up, precz, rnd_mode);
ok = mpfr_cmp (z_down, z_up) == 0;
mpfr_clear (z_up);
mpfr_clear (z_down);
if (ok)
{
/* get correct sign and multiply by 2 */
mpfr_div_2ui (s1, s, 2, MPFR_RNDN); /* s/4, exact */
mpfr_frac (s1, s1, MPFR_RNDN); /* exact, -1 < s1 < 0 */
if (mpfr_cmp_si_2exp (s1, -1, -1) > 0)
mpfr_neg (z_pre, z_pre, rnd_mode);
mpfr_mul_2ui (z_pre, z_pre, 1, rnd_mode);
break;
}
else
goto next_loop;
}
}
mpfr_zeta_pos (z_pre, s1, MPFR_RNDN); /* zeta(1-s) */
mpfr_mul (z_pre, z_pre, y, MPFR_RNDN); /* gamma(1-s)*zeta(1-s) */
/* multiply z_pre by 2^s*Pi^(s-1) where p=Pi, s1=1-s */
mpfr_mul_2ui (y, p, 1, MPFR_RNDN); /* 2*Pi */
mpfr_neg (s1, s1, MPFR_RNDN); /* s-1 */
mpfr_pow (y, y, s1, MPFR_RNDN); /* (2*Pi)^(s-1) */
mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
mpfr_mul_2ui (z_pre, z_pre, 1, MPFR_RNDN);
/* multiply z_pre by sin(Pi*s/2) */
mpfr_mul (y, s, p, MPFR_RNDN);
mpfr_div_2ui (p, y, 1, MPFR_RNDN); /* p = s*Pi/2 */
/* FIXME: sinpi will be available, we should replace the mpfr_sin
call below by mpfr_sinpi(s/2), where s/2 will be exact.
Can mpfr_sin underflow? Moreover, the code below should be
improved so that the "if" condition becomes unlikely, e.g.
by taking a slightly larger working precision. */
mpfr_sin (y, p, MPFR_RNDN); /* y = sin(Pi*s/2) */
ey = MPFR_GET_EXP (y);
if (ey < 0) /* take account of cancellation in sin(p) */
{
mpfr_t t;
MPFR_ASSERTN (- ey < MPFR_PREC_MAX - prec1);
mpfr_init2 (t, prec1 - ey);
mpfr_const_pi (t, MPFR_RNDD);
mpfr_mul (t, s, t, MPFR_RNDN);
mpfr_div_2ui (t, t, 1, MPFR_RNDN);
mpfr_sin (y, t, MPFR_RNDN);
mpfr_clear (t);
}
mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz,
rnd_mode)))
break;
next_loop:
MPFR_ZIV_NEXT (loop, prec1);
MPFR_GROUP_REPREC_4 (group, prec1, z_pre, s1, y, p);
}
MPFR_ZIV_FREE (loop);
if (overflow != 0)
{
inex = mpfr_overflow (z, rnd_mode, overflow);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
}
else
inex = mpfr_set (z, z_pre, rnd_mode);
MPFR_GROUP_CLEAR (group);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (z, inex, rnd_mode);
}
int
mpfr_exp_3 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t t, x_copy, tmp;
mpz_t uk;
mpfr_exp_t ttt, shift_x;
unsigned long twopoweri;
mpz_t *P;
mpfr_prec_t *mult;
int i, k, loop;
int prec_x;
mpfr_prec_t realprec, Prec;
int iter;
int inexact = 0;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (ziv_loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
inexact));
MPFR_SAVE_EXPO_MARK (expo);
/* decompose x */
/* we first write x = 1.xxxxxxxxxxxxx
----- k bits -- */
prec_x = MPFR_INT_CEIL_LOG2 (MPFR_PREC (x)) - MPFR_LOG2_GMP_NUMB_BITS;
if (prec_x < 0)
prec_x = 0;
ttt = MPFR_GET_EXP (x);
mpfr_init2 (x_copy, MPFR_PREC(x));
mpfr_set (x_copy, x, MPFR_RNDD);
/* we shift to get a number less than 1 */
if (ttt > 0)
{
shift_x = ttt;
mpfr_div_2ui (x_copy, x, ttt, MPFR_RNDN);
ttt = MPFR_GET_EXP (x_copy);
}
else
shift_x = 0;
MPFR_ASSERTD (ttt <= 0);
/* Init prec and vars */
realprec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (prec_x + MPFR_PREC (y));
Prec = realprec + shift + 2 + shift_x;
mpfr_init2 (t, Prec);
mpfr_init2 (tmp, Prec);
mpz_init (uk);
/* Main loop */
MPFR_ZIV_INIT (ziv_loop, realprec);
for (;;)
{
int scaled = 0;
MPFR_BLOCK_DECL (flags);
k = MPFR_INT_CEIL_LOG2 (Prec) - MPFR_LOG2_GMP_NUMB_BITS;
/* now we have to extract */
twopoweri = GMP_NUMB_BITS;
/* Allocate tables */
P = (mpz_t*) (*__gmp_allocate_func) (3*(k+2)*sizeof(mpz_t));
for (i = 0; i < 3*(k+2); i++)
mpz_init (P[i]);
mult = (mpfr_prec_t*) (*__gmp_allocate_func) (2*(k+2)*sizeof(mpfr_prec_t));
/* Particular case for i==0 */
mpfr_extract (uk, x_copy, 0);
MPFR_ASSERTD (mpz_cmp_ui (uk, 0) != 0);
mpfr_exp_rational (tmp, uk, shift + twopoweri - ttt, k + 1, P, mult);
for (loop = 0; loop < shift; loop++)
mpfr_sqr (tmp, tmp, MPFR_RNDD);
twopoweri *= 2;
/* General case */
iter = (k <= prec_x) ? k : prec_x;
for (i = 1; i <= iter; i++)
{
mpfr_extract (uk, x_copy, i);
if (MPFR_LIKELY (mpz_cmp_ui (uk, 0) != 0))
{
mpfr_exp_rational (t, uk, twopoweri - ttt, k - i + 1, P, mult);
mpfr_mul (tmp, tmp, t, MPFR_RNDD);
}
MPFR_ASSERTN (twopoweri <= LONG_MAX/2);
twopoweri *=2;
}
/* Clear tables */
for (i = 0; i < 3*(k+2); i++)
mpz_clear (P[i]);
(*__gmp_free_func) (P, 3*(k+2)*sizeof(mpz_t));
(*__gmp_free_func) (mult, 2*(k+2)*sizeof(mpfr_prec_t));
if (shift_x > 0)
{
MPFR_BLOCK (flags, {
for (loop = 0; loop < shift_x - 1; loop++)
mpfr_sqr (tmp, tmp, MPFR_RNDD);
mpfr_sqr (t, tmp, MPFR_RNDD);
} );
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
{
/* tmp <= exact result, so that it is a real overflow. */
inexact = mpfr_overflow (y, rnd_mode, 1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
{
/* This may be a spurious underflow. So, let's scale
the result. */
mpfr_mul_2ui (tmp, tmp, 1, MPFR_RNDD); /* no overflow, exact */
mpfr_sqr (t, tmp, MPFR_RNDD);
if (MPFR_IS_ZERO (t))
{
/* approximate result < 2^(emin - 3), thus
exact result < 2^(emin - 2). */
inexact = mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ?
MPFR_RNDZ : rnd_mode, 1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
break;
}
scaled = 1;
}
}
/* Since MPFR-3.0, return the usual inexact value.
The erange flag is set if an error occurred in the conversion
(y is NaN, +Inf, or -Inf that have no equivalent in mpf)
*/
int
mpfr_get_f (mpf_ptr x, mpfr_srcptr y, mpfr_rnd_t rnd_mode)
{
int inex;
mp_size_t sx, sy;
mpfr_prec_t precx, precy;
mp_limb_t *xp;
int sh;
if (MPFR_UNLIKELY(MPFR_IS_SINGULAR(y)))
{
if (MPFR_IS_ZERO(y))
{
mpf_set_ui (x, 0);
return 0;
}
else if (MPFR_IS_NAN (y))
{
MPFR_SET_ERANGEFLAG ();
return 0;
}
else /* y is plus infinity (resp. minus infinity), set x to the maximum
value (resp. the minimum value) in precision PREC(x) */
{
int i;
mp_limb_t *xp;
MPFR_SET_ERANGEFLAG ();
/* To this day, [mp_exp_t] and mp_size_t are #defined as the same
type */
EXP (x) = MP_SIZE_T_MAX;
sx = PREC (x);
SIZ (x) = sx;
xp = PTR (x);
for (i = 0; i < sx; i++)
xp[i] = MP_LIMB_T_MAX;
if (MPFR_IS_POS (y))
return -1;
else
{
mpf_neg (x, x);
return +1;
}
}
}
sx = PREC(x); /* number of limbs of the mantissa of x */
precy = MPFR_PREC(y);
precx = (mpfr_prec_t) sx * GMP_NUMB_BITS;
sy = MPFR_LIMB_SIZE (y);
xp = PTR (x);
/* since mpf numbers are represented in base 2^GMP_NUMB_BITS,
we loose -EXP(y) % GMP_NUMB_BITS bits in the most significant limb */
sh = MPFR_GET_EXP(y) % GMP_NUMB_BITS;
sh = sh <= 0 ? - sh : GMP_NUMB_BITS - sh;
MPFR_ASSERTD (sh >= 0);
if (precy + sh <= precx) /* we can copy directly */
{
mp_size_t ds;
MPFR_ASSERTN (sx >= sy);
ds = sx - sy;
if (sh != 0)
{
mp_limb_t out;
out = mpn_rshift (xp + ds, MPFR_MANT(y), sy, sh);
MPFR_ASSERTN (ds > 0 || out == 0);
if (ds > 0)
xp[--ds] = out;
}
else
MPN_COPY (xp + ds, MPFR_MANT (y), sy);
if (ds > 0)
MPN_ZERO (xp, ds);
EXP(x) = (MPFR_GET_EXP(y) + sh) / GMP_NUMB_BITS;
inex = 0;
}
else /* we have to round to precx - sh bits */
{
mpfr_t z;
mp_size_t sz;
/* Recall that precx = (mpfr_prec_t) sx * GMP_NUMB_BITS, thus removing
sh bits (sh < GMP_NUMB_BITSS) won't reduce the number of limbs. */
mpfr_init2 (z, precx - sh);
sz = MPFR_LIMB_SIZE (z);
MPFR_ASSERTN (sx == sz);
inex = mpfr_set (z, y, rnd_mode);
/* warning, sh may change due to rounding, but then z is a power of two,
thus we can safely ignore its last bit which is 0 */
sh = MPFR_GET_EXP(z) % GMP_NUMB_BITS;
sh = sh <= 0 ? - sh : GMP_NUMB_BITS - sh;
MPFR_ASSERTD (sh >= 0);
if (sh != 0)
{
mp_limb_t out;
out = mpn_rshift (xp, MPFR_MANT(z), sz, sh);
/* If sh hasn't changed, it is the number of the non-significant
bits in the lowest limb of z. Therefore out == 0. */
MPFR_ASSERTD (out == 0); (void) out; /* avoid a warning */
}
else
MPN_COPY (xp, MPFR_MANT(z), sz);
EXP(x) = (MPFR_GET_EXP(z) + sh) / GMP_NUMB_BITS;
mpfr_clear (z);
}
/* set size and sign */
SIZ(x) = (MPFR_FROM_SIGN_TO_INT(MPFR_SIGN(y)) < 0) ? -sx : sx;
return inex;
}
int
mpfr_eint (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
int inex;
mpfr_t tmp, ump;
mpfr_exp_t err, te;
mpfr_prec_t prec;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC (
("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* exp(NaN) = exp(-Inf) = NaN */
if (MPFR_IS_NAN (x) || (MPFR_IS_INF (x) && MPFR_IS_NEG(x)))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* eint(+inf) = +inf */
else if (MPFR_IS_INF (x))
{
MPFR_SET_INF(y);
MPFR_SET_POS(y);
MPFR_RET(0);
}
else /* eint(+/-0) = -Inf */
{
MPFR_SET_INF(y);
MPFR_SET_NEG(y);
mpfr_set_divby0 ();
MPFR_RET(0);
}
}
/* eint(x) = NaN for x < 0 */
if (MPFR_IS_NEG(x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SAVE_EXPO_MARK (expo);
/* Since eint(x) >= exp(x)/x, we have log2(eint(x)) >= (x-log(x))/log(2).
Let's compute k <= (x-log(x))/log(2) in a low precision. If k >= emax,
then log2(eint(x)) >= emax, and eint(x) >= 2^emax, i.e. it overflows. */
mpfr_init2 (tmp, 64);
mpfr_init2 (ump, 64);
mpfr_log (tmp, x, MPFR_RNDU);
mpfr_sub (ump, x, tmp, MPFR_RNDD);
mpfr_const_log2 (tmp, MPFR_RNDU);
mpfr_div (ump, ump, tmp, MPFR_RNDD);
/* FIXME: We really need mpfr_set_exp_t and mpfr_cmpfr_exp_t functions. */
MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX);
if (mpfr_cmp_ui (ump, __gmpfr_emax) >= 0)
{
mpfr_clear (tmp);
mpfr_clear (ump);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_overflow (y, rnd, 1);
}
/* Init stuff */
prec = MPFR_PREC (y) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 6;
/* eint() has a root 0.37250741078136663446..., so if x is near,
already take more bits */
/* FIXME: do not use native floating-point here. */
if (MPFR_GET_EXP(x) == -1) /* 1/4 <= x < 1/2 */
{
double d;
d = mpfr_get_d (x, MPFR_RNDN) - 0.37250741078136663;
d = (d == 0.0) ? -53 : __gmpfr_ceil_log2 (d);
prec += -d;
}
mpfr_set_prec (tmp, prec);
mpfr_set_prec (ump, prec);
MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controler */
for (;;) /* Infinite loop */
{
/* We need that the smallest value of k!/x^k is smaller than 2^(-p).
The minimum is obtained for x=k, and it is smaller than e*sqrt(x)/e^x
for x>=1. */
if (MPFR_GET_EXP (x) > 0 && mpfr_cmp_d (x, ((double) prec +
0.5 * (double) MPFR_GET_EXP (x)) * LOG2 + 1.0) > 0)
err = mpfr_eint_asympt (tmp, x);
else
{
err = mpfr_eint_aux (tmp, x); /* error <= 2^err ulp(tmp) */
te = MPFR_GET_EXP(tmp);
mpfr_const_euler (ump, MPFR_RNDN); /* 0.577 -> EXP(ump)=0 */
mpfr_add (tmp, tmp, ump, MPFR_RNDN);
/* error <= 1/2 + 1/2*2^(EXP(ump)-EXP(tmp)) + 2^(te-EXP(tmp)+err)
<= 1/2 + 2^(MAX(EXP(ump), te+err+1) - EXP(tmp))
<= 2^(MAX(0, 1 + MAX(EXP(ump), te+err+1) - EXP(tmp))) */
err = MAX(1, te + err + 2) - MPFR_GET_EXP(tmp);
err = MAX(0, err);
te = MPFR_GET_EXP(tmp);
mpfr_log (ump, x, MPFR_RNDN);
mpfr_add (tmp, tmp, ump, MPFR_RNDN);
/* same formula as above, except now EXP(ump) is not 0 */
err += te + 1;
if (MPFR_LIKELY (!MPFR_IS_ZERO (ump)))
err = MAX (MPFR_GET_EXP (ump), err);
err = MAX(0, err - MPFR_GET_EXP (tmp));
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - err, MPFR_PREC (y), rnd)))
break;
MPFR_ZIV_NEXT (loop, prec); /* Increase used precision */
mpfr_set_prec (tmp, prec);
mpfr_set_prec (ump, prec);
}
MPFR_ZIV_FREE (loop); /* Free the ZivLoop Controler */
inex = mpfr_set (y, tmp, rnd); /* Set y to the computed value */
mpfr_clear (tmp);
mpfr_clear (ump);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
