Merged the latest changes from the trunk.

git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/branches/faithful@11102 280ebfd0-de03-0410-8827-d642c229c3f4

6 files changed, 336 insertions, 782 deletions

diff --git a/src/log.c b/src/log.c
index 2cc222d64..4b428070f 100644
--- a/src/log.c
+++ b/src/log.c
@@ -125,17 +125,24 @@ mpfr_log (mpfr_ptr r, mpfr_srcptr a, mpfr_rnd_t rnd_mode)
       mpfr_exp_t cancel;
 
       /* Calculus of m (depends on p)
-         If mpfr_exp_t has N bits, then both (p + 1) / 2 and |exp_a| fit
-         on N-2 bits, so that there cannot be an overflow.
-         Note that exp_a - 1 is a constant in the loop (thus grouped). */
-      m = (p + 1) / 2 - (exp_a - 1);
+         If mpfr_exp_t has N bits, then both (p + 3) / 2 and |exp_a| fit
+         on N-2 bits, so that there cannot be an overflow. */
+      m = (p + 3) / 2 - exp_a;
 
       /* In standard configuration (_MPFR_EXP_FORMAT <= 3), one has
          mpfr_exp_t <= long, so that the following assertion is always
          true. */
       MPFR_ASSERTN (m >= LONG_MIN && m <= LONG_MAX);
 
+      /* FIXME: Why 1 ulp and not 1/2 ulp? Ditto with some other ones
+         below. The error concerning the AGM should be explained since
+         4/s is inexact (one needs a bound on its derivative). */
       mpfr_mul_2si (tmp2, a, m, MPFR_RNDN);    /* s=a*2^m,        err<=1 ulp  */
+      MPFR_ASSERTD (MPFR_EXP (tmp2) >= (p + 3) / 2);
+      /* [FIXME] and one can have the equality, even if p is even.
+         This means that if a is a power of 2 and p is even, then
+         s = (1/2) * 2^((p+2)/2) = 2^(p/2), so that the condition
+         s > 2^(p/2) from algorithms.tex is not satisfied. */
       mpfr_div (tmp1, __gmpfr_four, tmp2, MPFR_RNDN);/* 4/s,      err<=2 ulps */
       mpfr_agm (tmp2, __gmpfr_one, tmp1, MPFR_RNDN); /* AG(1,4/s),err<=3 ulps */
       mpfr_mul_2ui (tmp2, tmp2, 1, MPFR_RNDN); /* 2*AG(1,4/s),    err<=3 ulps */
diff --git a/src/mpfr-impl.h b/src/mpfr-impl.h
index 5f2a550fe..0521a2193 100644
--- a/src/mpfr-impl.h
+++ b/src/mpfr-impl.h
@@ -2099,6 +2099,10 @@ __MPFR_DECLSPEC int mpfr_add1sp (mpfr_ptr, mpfr_srcptr,
                                  mpfr_srcptr, mpfr_rnd_t);
 __MPFR_DECLSPEC int mpfr_sub1sp (mpfr_ptr, mpfr_srcptr,
                                  mpfr_srcptr, mpfr_rnd_t);
+__MPFR_DECLSPEC int mpfr_mul_1 (mpfr_ptr, mpfr_srcptr, mpfr_srcptr, mpfr_rnd_t,
+                                mpfr_prec_t);
+__MPFR_DECLSPEC int mpfr_mul_2 (mpfr_ptr, mpfr_srcptr, mpfr_srcptr, mpfr_rnd_t,
+                                mpfr_prec_t);
 __MPFR_DECLSPEC int mpfr_can_round_raw (const mp_limb_t *,
              mp_size_t, int, mpfr_exp_t, mpfr_rnd_t, mpfr_rnd_t, mpfr_prec_t);
 
diff --git a/src/mul.c b/src/mul.c
index 1c3ba3b4c..49ee9a636 100644
--- a/src/mul.c
+++ b/src/mul.c
@@ -208,9 +208,10 @@ mpfr_mul (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode)
 
 /* Multiply 2 mpfr_t */
 
-/* special code for prec(a) < GMP_NUMB_BITS and
-   prec(b), prec(c) <= GMP_NUMB_BITS */
-static int
+/* Special code for prec(a) < GMP_NUMB_BITS and
+   prec(b), prec(c) <= GMP_NUMB_BITS.
+   We export it since it is called from mpfr_sqr too. */
+int
 mpfr_mul_1 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode,
               mpfr_prec_t p)
 {
@@ -304,9 +305,10 @@ mpfr_mul_1 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode,
     }
 }
 
-/* special code for GMP_NUMB_BITS < prec(a) < 2*GMP_NUMB_BITS and
-   GMP_NUMB_BITS < prec(b), prec(c) <= 2*GMP_NUMB_BITS */
-static int
+/* Special code for GMP_NUMB_BITS < prec(a) < 2*GMP_NUMB_BITS and
+   GMP_NUMB_BITS < prec(b), prec(c) <= 2*GMP_NUMB_BITS.
+   It is exported since called from mpfr_sqr too. */
+int
 mpfr_mul_2 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode,
               mpfr_prec_t p)
 {
diff --git a/src/sqr.c b/src/sqr.c
index 4096be731..098bec75a 100644
--- a/src/sqr.c
+++ b/src/sqr.c
@@ -53,9 +53,16 @@ mpfr_sqr (mpfr_ptr a, mpfr_srcptr b, mpfr_rnd_t rnd_mode)
         ( MPFR_ASSERTD(MPFR_IS_ZERO(b)), MPFR_SET_ZERO(a) );
       MPFR_RET(0);
     }
-  ax = 2 * MPFR_GET_EXP (b);
   bq = MPFR_PREC(b);
 
+  if (MPFR_GET_PREC(a) < GMP_NUMB_BITS && bq <= GMP_NUMB_BITS)
+    return mpfr_mul_1 (a, b, b, rnd_mode, MPFR_GET_PREC(a));
+
+  if (GMP_NUMB_BITS < MPFR_GET_PREC(a) && MPFR_GET_PREC(a) < 2 * GMP_NUMB_BITS
+      && GMP_NUMB_BITS < bq && bq <= 2 * GMP_NUMB_BITS)
+    return mpfr_mul_2 (a, b, b, rnd_mode, MPFR_GET_PREC(a));
+
+  ax = 2 * MPFR_GET_EXP (b);
   MPFR_ASSERTN (2 * (mpfr_uprec_t) bq <= MPFR_PREC_MAX);
 
   bn = MPFR_LIMB_SIZE (b); /* number of limbs of b */
diff --git a/src/sqrt.c b/src/sqrt.c
index 2ace08ff4..8154dd299 100644
--- a/src/sqrt.c
+++ b/src/sqrt.c
@@ -20,194 +20,11 @@ along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
 
-/*
-*** Note ***
-
-  The code of mpfr_sqrt1 and/or functions it calls depends on
-  implementation-defined features of the C standard. See lines with a
-  cast to mp_limb_signed_t, associated with a shift to the right '>>'.
-  Such features are known to behave as the code below expects with GCC,
-  according to the GCC manual (excerpt from the GCC 4.9.3 manual):
-
-    4 C Implementation-defined behavior
-    ***********************************
-
-    4.5 Integers
-    ============
-
-    * 'The result of, or the signal raised by, converting an integer to
-      a signed integer type when the value cannot be represented in an
-      object of that type (C90 6.2.1.2, C99 and C11 6.3.1.3).'
-
-      For conversion to a type of width N, the value is reduced modulo
-      2^N to be within range of the type; no signal is raised.
-
-    * 'The results of some bitwise operations on signed integers (C90
-      6.3, C99 and C11 6.5).'
-
-      Bitwise operators act on the representation of the value including
-      both the sign and value bits, where the sign bit is considered
-      immediately above the highest-value value bit.  Signed '>>' acts
-      on negative numbers by sign extension.
-
-  It is not known whether it works with other compilers. Thus this code
-  is currently enabled only when __GNUC__ is defined (which includes
-  compilers that declare a compatibility with GCC). A configure test
-  might be an alternative solution (but without any guarantee, in case
-  the result may also depend on the context).
-
-  Warning! The right shift of a negative value corresponds to an integer
-  division by a power of two, with rounding toward negative.
-
-  TODO: Complete the comments when a right shift of a negative value
-  may be involved, so that the rounding toward negative appears in the
-  proof. There has been at least an error with a proof of a bound!
-*/
-
 #define MPFR_NEED_LONGLONG_H
 #include "mpfr-impl.h"
 
-#if !defined(MPFR_GENERIC_ABI) && (GMP_NUMB_BITS == 32 || GMP_NUMB_BITS == 64)
-
-/* The tables T1[] and T2[] below were generated using the Sage code below,
-   with T1,T2 = bipartite(4,4,4,16,16). Note: we would get a slightly smaller
-   error using an approximation of the form T1[a,b] * (1 + T2[a,b]), but this
-   would make the code more complex, and the approximation would not always fit
-   on p2 bits (assuming p2 >= p1).
-# bi-partite method, using a table T1 of p1 bits, and T2 of p2 bits
-# approximate sqrt(a:b:c) with T1[a,b] + T2[a,c]
-def bipartite(pa,pb,pc,p1,p2):
-   T1 = dict()
-   T2 = dict()
-   maxerr = maxnum = 0
-   for a in range(2^(pa-2),2^pa):
-      for b in range(2^pb):
-         A1 = (a*2^pb+b)/2^(pa+pb)
-         A2 = (a*2^pb+b+1)/2^(pa+pb)
-         X = (1/sqrt(A1) + 1/sqrt(A2))/2
-         X = round(X*2^p1)/2^p1
-         T1[a,b] = X*2^p1
-         maxnum = max(maxnum, abs(T1[a,b]))
-         # print "a=", a, "b=", b, "T1=", x
-         maxerr = max(maxerr, n(abs(1/sqrt(A1) - X)))
-         maxerr = max(maxerr, n(abs(1/sqrt(A2) - X)))
-   print "maxerr1 =", maxerr, log(maxerr)/log(2.0), "maxnum=", maxnum
-   maxerr = maxnum = 0
-   for a in range(2^(pa-2),2^pa):
-      for c in range(2^pc):
-         Xmin = infinity
-         Xmax = -infinity
-         for b in range(2^pb):
-            A = (a*2^(pb+pc)+b*2^pc+c)/2^(pa+pb+pc)
-            X = 1/sqrt(A) - T1[a,b]/2^p1
-            X = round(X*2^p2)/2^p2
-            Xmin = min (Xmin, X)
-            Xmax = max (Xmax, X)
-            A = (a*2^(pb+pc)+b*2^pc+c+1)/2^(pa+pb+pc)
-            X = 1/sqrt(A) - T1[a,b]/2^p1
-            X = round(X*2^p2)/2^p2
-            Xmin = min (Xmin, X)
-            Xmax = max (Xmax, X)
-         T2[a,c] = round((Xmin + Xmax)/2*2^p2)
-         maxnum = max(maxnum, abs(T2[a,c]))
-         # print "a=", a, "c=", c, "T2=", T2[a,c]
-         for b in range(2^pb):
-            A = (a*2^(pb+pc)+b*2^pc+c)/2^(pa+pb+pc)
-            X = 1/sqrt(A)
-            maxerr = max(maxerr, n(abs(X - (T1[a,b]/2^p1 + T2[a,c]/2^p2))))
-            A = (a*2^(pb+pc)+b*2^pc+c+1)/2^(pa+pb+pc)
-            X = 1/sqrt(A)
-            maxerr = max(maxerr, n(abs(X - (T1[a,b]/2^p1 + T2[a,c]/2^p2))))
-   print "maxerr2 =", maxerr, log(maxerr)/log(2.0), "maxnum=", maxnum
-   return [T1[a,b] for a in range(2^(pa-2),2^pa) for b in range(2^pb)], \
-          [T2[a,c] for a in range(2^(pa-2),2^pa) for c in range(2^pc)]
-*/
-
-static const mp_limb_t T1[] = {130566, 129565, 128587, 127631, 126696, 125781, 124886, 124009, 123151, 122311, 121487, 120680, 119888, 119112, 118351, 117604, 116871, 116152, 115446, 114753, 114072, 113402, 112745, 112099, 111464, 110839, 110225, 109621, 109027, 108442, 107867, 107301, 106743, 106194, 105654, 105122, 104597, 104081, 103572, 103070, 102576, 102089, 101608, 101134, 100667, 100207, 99752, 99304, 98861, 98425, 97994, 97569, 97149, 96735, 96326, 95922, 95523, 95129, 94740, 94356, 93976, 93601, 93230, 92864, 92502, 92144, 91790, 91441, 91095, 90753, 90415, 90081, 89750, 89423, 89100, 88780, 88463, 88150, 87840, 87534, 87230, 86930, 86633, 86339, 86048, 85759, 85474, 85191, 84912, 84635, 84360, 84088, 83819, 83553, 83289, 83027, 82768, 82512, 82257, 82005, 81756, 81508, 81263, 81020, 80780, 80541, 80304, 80070, 79837, 79607, 79379, 79152, 78928, 78705, 78484, 78265, 78048, 77833, 77619, 77408, 77197, 76989, 76782, 76577, 76374, 76172, 75972, 75773, 75576, 75381, 75187, 74994, 74803, 74613, 74425, 74239, 74053, 73869, 73687, 73505, 73325, 73147, 72969, 72793, 72619, 72445, 72273, 72102, 71932, 71763, 71596, 71429, 71264, 71100, 70937, 70776, 70615, 70455, 70297, 70139, 69983, 69828, 69673, 69520, 69368, 69216, 69066, 68917, 68768, 68621, 68475, 68329, 68184, 68041, 67898, 67756, 67615, 67475, 67336, 67197, 67060, 66923, 66787, 66652, 66518, 66384, 66252, 66120, 65989, 65858, 65729, 65600};
-
-static const mp_limb_signed_t T2[] = {415, 360, 304, 249, 194, 139, 85, 29, -34, -89, -143, -198, -252, -307, -361, -415, 304, 264, 224, 183, 143, 102, 62, 21, -24, -64, -105, -145, -185, -225, -265, -305, 236, 204, 173, 142, 111, 79, 48, 17, -18, -50, -81, -112, -143, -174, -205, -236, 190, 164, 139, 114, 89, 64, 39, 13, -14, -39, -65, -90, -114, -140, -165, -189, 157, 136, 115, 94, 74, 53, 32, 11, -12, -33, -53, -74, -95, -116, -137, -157, 133, 115, 97, 80, 63, 45, 27, 9, -10, -27, -45, -62, -80, -97, -115, -133, 114, 99, 84, 69, 54, 38, 23, 8, -9, -24, -39, -53, -68, -84, -99, -114, 99, 86, 73, 60, 46, 33, 20, 7, -8, -21, -34, -47, -60, -73, -86, -99, 88, 76, 65, 53, 41, 30, 18, 6, -7, -18, -30, -41, -53, -64, -76, -87, 78, 68, 57, 47, 36, 26, 16, 5, -6, -16, -26, -37, -47, -58, -68, -78, 70, 61, 51, 42, 33, 24, 14, 5, -5, -15, -24, -33, -42, -52, -61, -70, 63, 55, 47, 38, 30, 21, 13, 4, -5, -13, -22, -30, -38, -47, -55, -63};
-
-/* For GMP_NUMB_BITS=32: return a (1+20)-bit approximation x0 of 2^36/sqrt(a0).
-   For GMP_NUMB_BITS=64: return a (1+40)-bit approximation x0 of 2^72/sqrt(a0).
-   Assume a0 >= 2^(GMP_NUMB_BITS-2), and GMP_NUMB_BITS = 32 or 64.
-   Must ensure: x0 <= 2^36/sqrt(a0) for GMP_NUMB_BITS=32,
-                x0 <= 2^72/sqrt(a0) for GMP_NUMB_BITS=64. */
-static mp_limb_t
-mpn_rsqrtrem1 (mp_limb_t a0)
-{
-  mp_limb_t a = a0 >> (GMP_NUMB_BITS - 4);
-  mp_limb_t b = (a0 >> (GMP_NUMB_BITS - 8)) & 0xf;
-  mp_limb_t c = (a0 >> (GMP_NUMB_BITS - 12)) & 0xf;
-  mp_limb_t x0, t;
-
-
-  MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 32 || GMP_NUMB_BITS == 64);
-
-  x0 = T1[((a - 4) << 4) + b] + T2[((a - 4) << 4) + c];
-  /* now x0 is a 16-bit approximation, with maximal error 2^(-9.48):
-     -2^(-9.48) <= x0/2^16 - 1/sqrt(a0/2^64) <= 2^(-9.48)
-     The worst case is obtained for floor(a0/2^52) = 1265. */
-
-#if GMP_NUMB_BITS == 32
-  x0 >>= 1; /* reduce approximation to 1+15 bits */
-  /* In principle, a0>>10 can have up to 22 bits, and (x0^2)>>12 can have up to
-     20 bits, thus the product can have up to 42 bits, but since a0*x0^2 is very
-     near 2^62, thus (a0>>10)*(x0^2)>>12 is very near 2^40, and when we reduce
-     it mod 2^32 and interpret as a signed number in [-2^31, 2^31-1], we get
-     the correct remainder (a0>>10)*(x0^2)>>12 - 2^40 */
-  t = (mp_limb_signed_t) ((a0 >> 10) * ((x0 * x0) >> 12)) >> 8;
-  /* |t| < 6742843 <= 2^23 - 256 (by exhaustive search) */
-  t = (mp_limb_signed_t) t >> 8; /* now |t| < 2^15, thus |x0*t| < 2^31 */
-  x0 = (x0 << 5) - ((mp_limb_signed_t) (x0 * t) >> 20);
-
-  /* by exhaustive search on all possible values of a0, we get:
-     -1.61 <= x0 - 2^36/sqrt(a0) <= 3.11 thus
-     -2^(-19.3) < -1.54e-6 <= x0/2^20 - 2^16/sqrt(a0) <= 2.97e-6 < 2^(-18.3)
-  */
-
-  return x0 - 4; /* ensures x0 <= 2^36/sqrt(a0) */
-#else /* GMP_NUMB_BITS = 64 */
-  {
-    mp_limb_t a1 = a0 >> (GMP_NUMB_BITS - 32);
-    /* a1 has 32 bits, thus a1*x0^2 has 64 bits */
-    t = (mp_limb_signed_t) (-a1 * (x0 * x0)) >> 32;
-    /* t has 32 bits now */
-  }
-  x0 = (x0 << 15) + ((mp_limb_signed_t) (x0 * t) >> (17+1));
-  /* now x0 is a 31-bit approximation (32 bits, 1 <= x0/2^31 <= 2),
-     with maximal error 2^(-19.19):
-     -2^(-19.22) <= x0/2^31 - 1/sqrt(a0/2^64) <= 2^(-19.22).
-     The worst case is attained for a1 = 1326448625, with x0 = 3864240837. */
-
-  {
-    mp_limb_t a1 = a0 >> (GMP_NUMB_BITS - 41); /* 2^39 <= a1 < 2^41 */
-    t = (x0 * x0) >> 23; /* t < 2^41 */
-    /* a1 * t has at most 82 bits, but we know the upper 19 bits cancel with 1 */
-    t = (mp_limb_signed_t) (-a1 * t) >> 31;
-    /* it remains 49 bits in theory, but t has only 31 bits at most */
-    /* the error on t is at most 1 (from the truncation >> 31,
-       plus 2^(41-31) from the truncation of a0 into a1 multiplied by t,
-       and another 2^(41-31) from the truncation of x0^2 multiplied by a1,
-       thus at most 2^11+1 (in fact since t/2^41 is about 2^41/a1, the maximal
-       error is when a0 is near 2^64 or 2^62, and is 1.25*2^10 + 1) */
-  }
-  x0 = (x0 << 9) + ((mp_limb_signed_t) (x0 * t) >> (31+1+49-40));
-  /* The error on x0 is at most (1.25*2^10+1)*2^(32-41) from the previous error on t,
-     plus 1 from the truncation by >> (31+1+49-40), thus at most 2.5+2^(-9)+1.
-     Since the mathematical error is bounded by 2^(-37.85) <= 4.44 ulps [but this
-     error can only make x0 smaller], we have at the end:
-     -(3.5+2^(-9))/2^40 - 2^(-37.85) <= x0/2^40 - 2^32/sqrt(a0) <= (3.5+2^(-9))/2^40:
-
-     -2^(-37.01) < x0/2^40 - 2^32/sqrt(a0) < 2^(-38.19)
-  */
-  x0 = x0 - 4; /* ensures x0 <= 2^72/sqrt(a0) */
-  /* x0-2 fails for example for a0=4967732205162787840 (gives x0=2118754313105) */
-  /* now -2^(-36.30) < x0/2^40 - 2^32/sqrt(a0) < 0 */
-  return x0;
-#endif
-}
+#if !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64
 
-#if GMP_NUMB_BITS == 64
 /* The Taylor coefficient of order 0 of sqrt(i/2^8+x) is
    U[i-64][0]/2^64 + U[i-64][1]/2^128, then the Taylor coefficient of order j is
    (up to sign) U[i-64][j+1]/2^(64-8*j).
@@ -242,9 +59,9 @@ mpn_sqrtrem2_approx (mp_limb_t n)
   /* the truncation error on h is at most 5 */
   umul_ppmm (h, l, u[3] - h, x);
   /* the truncation error on h is at most 6 */
-  umul_ppmm (h, l, u[2] - h, x >> 8); /* here we shift by 8 since u[0] has the
-                                         same weight 1/2^64 as u[2], the truncation
-                                         error on h + l/2^64 is at most 6/2^8 */
+  umul_ppmm (h, l, u[2] - h, x >> 8); /* here we shift by 8 since u[0] has weight
+                                         1/2^64 and u[2] has weight 1/2^72, the
+                                         truncation error on h+l/2^64 is <= 6/2^8 */
   add_ssaaaa (h, l, h, l, u[0], u[1]);
   /* Since the above addition is exact, the truncation error on h + l/2^64
      is still at most 6/2^8. Together with the mathematical error < .927e-21*2^64,
@@ -270,215 +87,82 @@ mpn_sqrtrem2_approx (mp_limb_t n)
 
 /* Put in rp[1]*2^64+rp[0] an approximation of sqrt(2^128*n),
    with 2^126 <= n := np[1]*2^64 + np[0] < 2^128.
-   We use a Taylor polynomial of degree 14.
-   The coefficients of degree 0 to 9 are represented by two 64-bit limbs
-   (most significant first), the remaining coefficients by one 64-bit limb,
-   thus the degree-0 coefficient is u[0]/2^64 + u[1]/2^128,
-   the degree-1 coefficient is u[2]/2^64 + u[3]/2^128,
-   ...,
-   the degree-9 coefficient is u[18]/2^64 + u[19]/2^128,
-   the degree-10 coefficient is u[20]/2^64,
-   ...,
-   the degree-14 coefficient is u[24]/2^64. */
+   The error on {rp, 2} is less than 43 ulps (in unknown direction).
+*/
 static void
 mpn_sqrtrem4_approx (mpfr_limb_ptr rp, mpfr_limb_srcptr np)
 {
   int i = np[1] >> 56;
-  mp_limb_t xh, xl, h, l, yh, yl;
+  mp_limb_t x, r1, r0, h, l, t;
   const mp_limb_t *u;
 
-  xh = (np[1] << 8) | (np[0] >> 56);
-  xl = np[0] << 8;
+  x = np[1] << 8 | (np[0] >> 56);
   u = V[i - 64];
-  umul_ppmm (h, l, u[24],     xh);
-  umul_ppmm (h, l, u[23] - h, xh);
-  umul_ppmm (h, l, u[22] - h, xh);
-  umul_ppmm (h, l, u[21] - h, xh);
-  umul_ppmm (h, l, u[20] - h, xh);
-  /* now we have to deal with two limbs */
-  sub_ddmmss (yh, yl, u[18], u[19], 0, h);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[16], u[17], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[14], u[15], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[12], u[13], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[10], u[11], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[8], u[9], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[6], u[7], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[4], u[5], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  sub_ddmmss (yh, yl, u[2], u[3], h, l);
-  umul_ppmm2 (h, l, yh, yl, xh, xl);
-  add_ssaaaa (rp[1], rp[0], u[0], u[1], h, l);
-}
-#endif /* GMP_NUMB_BITS == 64 */
-
-#if GMP_NUMB_BITS == 32
-/* Given as input np[0] and np[1], with B/4 <= np[1] (where B = 2^GMP_NUMB_BITS),
-   mpn_sqrtrem2 returns a value x, 0 <= x <= 1, and stores values s in sp[0] and
-   r in rp[0] such that:
-
-   n := np[1]*B + np[0] = s^2 + x*B + r, with n < (s+1)^2
-
-   or equivalently x*B + r <= 2*s.
-
-   This comment is for GMP_NUMB_BITS=64 for simplicity, but the code is valid
-   for any even value of GMP_NUMB_BITS.
-   The algorithm used is the following, and uses Karp-Markstein's trick:
-   - start from x, a 41-bit approximation of 2^72/sqrt(n1), with x <= 2^72/sqrt(n1)
-   - y = floor(n1*x/2^72), which is a 32-bit approximation of sqrt(n1)
-   - t = n1 - y^2
-   - u = (x * t) >> 33
-   - y = (y << 32) + u
-
-   Proof:
-   * we know that Newton's iteration for the reciprocal square root,
-
-                x' = x + (x/2) (1 - a*x^2),                         (1)
-
-     if evaluated with infinite precision, always produces x' <= 1/sqrt(a).
-     See for example Lemma 3.14 in "Modern Computer Arithmetic" by Brent and
-     Zimmermann.
-
-   * if we multiply both sides of equation (1) by a, and write y0 = a*x
-     and y0' = a*x', we get:
-
-                y0' = y0 + (x/2)*(a - y0^2)                         (2)
-
-     and since x' <= 1/sqrt(a), y0' <= sqrt(a).
-
-   * now assume we replace y0 in (2) by y = y0 - e for some e >= 0.
-     Equation (2) becomes:
-
-                y' = y0 + (x/2)*(a - y0^2) - e*(1-x*y0+x*e/2)       (3)
-
-     Since y0 = a*x, the term 1-x*y0+x*e/2 equals 1-a*x^2+x*e/2,
-     which is non-negative since x <= 1/sqrt(a). Thus y' <= y0' <= sqrt(a).
-
-     In practice, we should ensure y <= n1*x/2^64, so that t and u are
-     non-negative, so that all right shifts round towards -infinity.
-*/
-static mp_limb_t
-mpn_sqrtrem2 (mpfr_limb_ptr sp, mpfr_limb_ptr rp, mpfr_limb_srcptr np)
-{
-  mp_limb_t x, y, t, high, low;
-
-  x = mpn_rsqrtrem1 (np[1]);
-  /* we must have x^2*n1 <= 2^72 for GMP_NUMB_BITS=32
-                         <= 2^144 for GMP_NUMB_BITS=64 */
-
-#if GMP_NUMB_BITS == 32
-  MPFR_ASSERTD ((double) x * (double) x * (double) np[1]
-                < 4722366482869645213696.0);
-  /* x is an approximation of 2^36/sqrt(n1), x has 1+20 bits */
-
-  /* We know x/2^20 <= 2^16/sqrt(n1) + 2^(-18)
-     thus n1*x/2^36 <= sqrt(n1) + 2^(-18)*n1/2^16 <= sqrt(n1) + 2^(-2). */
-
-  /* compute y = floor(np[1]*x/2^36), cutting the upper 24 bits of n1 in two
-     parts of 12 and 11 bits, which can be multiplied by x without overflow
-     (warning: if we take 12 bits from low, it might overflow with x */
-  high = np[1] >> 20; /* upper 12 bits from n1 */
-  MPFR_ASSERTD((double) high * (double) x < 4294967296.0);
-  low = (np[1] >> 9) & 0x7ff; /* next 11 bits */
-  MPFR_ASSERTD((double) low * (double) x < 4294967296.0);
-  y = high * x + ((low * x) >> 11); /* y approximates n1*x/2^20 */
-  y = (y - 0x4000) >> 16; /* the constant 0x4000 = 2^(36-2-20) takes
-                             into account the 2^(-2) error above, to ensure
-                             y <= sqrt(n1) */
-#else /* GMP_NUMB_BITS = 64 */
-  MPFR_ASSERTD ((double) x * (double) x * (double) np[1]
-                < 2.2300745198530623e43);
-  /* x is an approximation of 2^72/sqrt(n1), x < 2^41 */
-
-  /* We know 2^32/sqrt(n1) - 2^(-36.30) <= x/2^40 <= 2^32/sqrt(n1) */
-
-  /* compute y = floor(np[1]*x/2^72): we cut the upper 48 bits of n1 in two parts of
-     24 and 23 bits, which can be multiplied by x without overflow (warning: if we
-     take 24 bits from low, it might overflow with x). We could simply write
-     umul_ppmm (high, low, np[1], x) followed by y = high >> 8 but the following
-     code is faster. */
-  high = np[1] >> 40; /* upper 24 bits from n1 */
-  MPFR_ASSERTD((double) high * (double) x < 18446744073709551616.0);
-  low = (np[1] >> 17) & 0x7fffff; /* next 23 bits */
-  MPFR_ASSERTD((double) low * (double) x < 18446744073709551616.0);
-  y = high * x + ((low * x) >> 23); /* y approximates n1*x/2^40 */
-  y = y >> 32;
-#endif
-
-  /* Now y is an approximation of sqrt(n1), with y <= sqrt(n1). The errors are:
-     (a) mathematical error: according to Lemma 3.14 from "Modern Computer
-         Arithmetic", it is bounded by 3/2*x^3/theta^4*2^(-2*36.30), where
-         x <= theta + 2^(-36.30), thus x/theta <= 1 + 2^(-36.30), thus the
-         mathematical error is bounded by 2^(-72.01), thus 2^(-40.01) for a
-         value normalized by 2^32;
-     (b) truncation error in Newton's floor(np[1]*x/2^72): it is bounded by
-         2^17*x/2^72 for the neglected part of np[1], and 1 in (low * x) >> 23.
-     Thus we have:
-     sqrt(np[1]) - 1 - 2^(-13.99) < y <= sqrt(np[1]) */
-
-  t = np[1] - y * y;
-  MPFR_ASSERTD((mp_limb_signed_t) t >= 0);
-
-#if GMP_NUMB_BITS == 32
-  /* we now compute t = (x * t) >> (20 + 1), but x * t might have
-     more than 32 bits thus we cut t in two parts of 12 and 11 bits */
-  high = t >> 11;
-  low = t & 0x7ff;
-  MPFR_ASSERTD((double) high * (double) x < 4294967296.0);
-  MPFR_ASSERTD((double) low * (double) x < 4294967296.0);
-  t = high * x + ((low * x) >> 11); /* approximates t*x/2^11 */
-
-  y = (y << (GMP_NUMB_BITS / 2)) + (t >> 10);
-#else
-  /* we now compute t = (x * t) >> (40 + 1), but x * t might have
-     more than 64 bits thus we cut t in two parts of 24 and 23 bits */
-  high = t >> 23;
-  low = t & 0x7fffff;
-  MPFR_ASSERTD((double) high * (double) x < 18446744073709551616.0);
-  MPFR_ASSERTD((double) low * (double) x < 18446744073709551616.0);
-  t = high * x + ((low * x) >> 23); /* approximates t*x/2^23 */
-
-  y = (y << (GMP_NUMB_BITS / 2)) + (t >> 18);
-#endif
-
-  /* the correction code below assumes y >= 2^(GMP_NUMB_BITS - 1) */
-  if (MPFR_UNLIKELY(y < MPFR_LIMB_HIGHBIT))
-    y = MPFR_LIMB_HIGHBIT;
-
-  umul_ppmm (x, t, y, y);
-  MPFR_ASSERTD(x < np[1] || (x == np[1] && t <= np[0])); /* y should not be too large */
-  sub_ddmmss (x, t, np[1], np[0], x, t);
-
-  /* Remainder x*2^GMP_NUMB_BITS+t should be <= 2*y (which implies x <= 1).
-     If x = 1, it suffices to check t > 2*y mod 2^GMP_NUMB_BITS. */
-  while (x > 1 || (x == 1 && t > 2 * y))
+  umul_ppmm (h, l, u[8], x);
+  /* the truncation error on h is at most 1 here */
+  umul_ppmm (h, l, u[7] - h, x);
+  /* the truncation error on h is at most 2 */
+  umul_ppmm (h, l, u[6] - h, x);
+  /* the truncation error on h is at most 3 */
+  umul_ppmm (h, l, u[5] - h, x);
+  /* the truncation error on h is at most 4 */
+  umul_ppmm (h, l, u[4] - h, x);
+  /* the truncation error on h is at most 5 */
+  umul_ppmm (h, l, u[3] - h, x);
+  /* the truncation error on h is at most 6 */
+  umul_ppmm (h, l, u[2] - h, x >> 6); /* here we shift by 6 since u[0] has weight
+                                         1/2^64 and u[2] has weight 1/2^70, the
+                                         truncation error on h+l/2^64 is <= 6/2^6 */
+  sub_ddmmss (h, l, u[0], u[1], h, l);
+  /* Since the mathematical error is < 0.412e-19*2^64, the total error on
+     h + l/2^64 is less than 0.854 */
+  const mp_limb_t magic = 0xda9fbe76c8b43800UL; /* ceil(0.854*2^64) */
+  x = h - (l < magic && h != 0);
+
+  /* now 2^64 + x is an approximation of 2^96/sqrt(np[1]),
+     with 2^64 + x < 2^96/sqrt(np[1]) */
+
+  umul_ppmm (r1, l, np[1], x);
+  r1 += np[1];
+
+  /* now r1 is an approximation of sqrt(2^64*np[1]), with r1 < sqrt(2^64*np[1]) */
+
+  /* make sure r1 >= 2^63 */
+  if (r1 < MPFR_LIMB_HIGHBIT)
+    r1 = MPFR_LIMB_HIGHBIT;
+
+  umul_ppmm (h, l, r1, r1);
+  sub_ddmmss (h, l, np[1], np[0], h, l);
+  l = (h << 63) | (l >> 1);
+  h = h >> 1;
+
+  /* now add (2^64+x) * (h*2^64+l) / 2^64 to r0 */
+
+  umul_ppmm (r0, t, x, l); /* x * l */
+  r0 += l;
+  r1 += h + (r0 < l); /* now we have added 2^64 * (h*2^64+l) */
+  while (h--)
     {
-      /* (y+1)^2 = y^2 + 2*y + 1 */
-      x -= 1 + (t < (2 * y + 1));
-      t -= 2 * y + 1;
-      y ++;
-      /* GMP_NUMB_BITS=32: average number of loops observed is 0.982,
-         max is 3 (for n1=1277869843, n0=3530774227);
-         GMP_NUMB_BITS=64: average number of loops observed is 0.593,
-         max is 3 (for n1=4651405965214438987, n0=18443926066120424952).
-      */
+      r0 += x;
+      r1 += (r0 < x); /* add x */
     }
 
-  sp[0] = y;
-  rp[0] = t;
-  return x;
+  if (r1 & MPFR_LIMB_HIGHBIT)
+    {
+      rp[0] = r0;
+      rp[1] = r1;
+    }
+  else /* overflow occurred in r1 */
+    {
+      rp[0] = ~MPFR_LIMB_ZERO;
+      rp[1] = ~MPFR_LIMB_ZERO;
+    }
 }
-#endif
 
 /* Special code for prec(r), prec(u) < GMP_NUMB_BITS. We cannot have
    prec(u) = GMP_NUMB_BITS here, since when the exponent of u is odd,
-   we need to shift u by one bit to the right without losing any bit. */
+   we need to shift u by one bit to the right without losing any bit.
+   Assumes GMP_NUMB_BITS = 64. */
 static int
 mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
 {
@@ -487,6 +171,8 @@ mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
   mp_limb_t u0, r0, rb, sb, mask = MPFR_LIMB_MASK(sh);
   mpfr_limb_ptr rp = MPFR_MANT(r);
 
+  MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
+
   /* first make the exponent even */
   u0 = MPFR_MANT(u)[0];
   if (((unsigned int) exp_u & 1) != 0)
@@ -498,7 +184,6 @@ mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
   exp_r = exp_u / 2;
 
   /* then compute the integer square root of u0*2^GMP_NUMB_BITS */
-#if GMP_NUMB_BITS == 64
   r0 = mpn_sqrtrem2_approx (u0);
   sb = 1; /* when we can round correctly with the approximation, the sticky bit
              is non-zero */
@@ -526,15 +211,6 @@ mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
       MPFR_ASSERTN(rb == 0 || (rb == 1 && sb <= 2 * r0));
       sb = rb | sb;
     }
-#else /* 32-bit variant. FIXME: write mpn_sqrtrem2_approx for GMP_NUMB_BITS=32,
-         then use the above code and get rid of mpn_sqrtrem2 */
-  {
-    mp_limb_t sp[2];
-    sp[1] = u0;
-    sp[0] = 0;
-    sb |= mpn_sqrtrem2 (&r0, &sb, sp);
-  }
-#endif
 
   rb = r0 & (MPFR_LIMB_ONE << (sh - 1));
   sb |= (r0 & mask) ^ rb;
@@ -605,185 +281,19 @@ mpfr_sqrt1 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
     }
 }
 
-#if GMP_NUMB_BITS == 64
-/* For GMP_NUMB_BITS=64: return a (1+80)-bit approximation x = xp[1]*B+xp[0]
-   of 2^144/sqrt(ap[1]*B+ap[0]).
-   Assume ap[1] >= B/4, thus sqrt(ap[1]*B+ap[0]) >= B/2, thus x <= 2^81. */
-static void
-mpn_rsqrtrem2 (mpfr_limb_ptr xp, mpfr_limb_srcptr ap)
-{
-  mp_limb_t t1, t0, u1, u0, r0, r1, r2;
-
-  MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
-
-  xp[1] = mpn_rsqrtrem1 (ap[1]);
-
-  /* now we should compute x + (x/2) * (1 - a*x^2), where the upper ~40 bits of
-     a*x^2 should cancel with 1, thus we only need the following 40 bits */
-
-  /* xp[1] has 1+40 bits, with xp[1] <= 2^72/sqrt(ap[1]) */
-  umul_ppmm (t1, t0, xp[1], xp[1]);
-
-  /* now t1 has at most 18 bits, with least 16 bits being its fractional value */
-
-  u1 = ap[1] >> 48;
-  u0 = (ap[1] << 16) | (ap[0] >> 48);
-
-  /* u1 has the 16 most significant bits of a, and u0 the next 64 bits */
-
-  /* we want t1*u1 << 48 + (t1*u0+t0*u1) >> 16 + t0*u0 >> 80
-              [32 bits]       [64 bits]            [48 bits]
-     but since we know the upper ~40 bits cancel with 1, we can ignore t1*u1. */
-  umul_ppmm (r2, r1, t0, u0);
-  r0 = t1 * u0; /* we ignore the upper 16 bits of the product */
-  r1 = t0 * u1; /* we ignore the upper 16 bits of the product */
-  r0 = r0 + r1 + r2;
-
-  /* the upper ~8 bits of r0 should cancel with 1, we are interested in the next
-     40 bits */
-
-  umul_ppmm (t1, t0, xp[1], -r0);
-
-  /* we should now add t1 >> 33 at xp[1] */
-  xp[1] += t1 >> 33;
-  xp[0] = t1 << 31;
-  /* shift by 24 bits to the right, since xp[1] has 24 leading zeros,
-     and we now expect 48 */
-  xp[0] = (xp[1] << 40) | (xp[0] >> 24);
-  xp[1] = xp[1] >> 24;
-}
-
-/* Given as input ap[0-3], with B/4 <= ap[3] (where B = 2^GMP_NUMB_BITS),
-   mpn_sqrtrem4 returns a value x, 0 <= x <= 1, and stores values s in sp[0-1] and
-   r in rp[0-1] such that:
-
-   n := ap[3]*B^3 + ap[2]*B^2 + ap[1]*B + ap[0] = s^2 + x*B + r, with n < (s+1)^2
-
-   or equivalently x*B + r <= 2*s.
-
-   This code currently assumes GMP_NUMB_BITS = 64, and takes on average
-   135.68 cycles on an Intel i5-6500 for precision 113 bits */
-static mp_limb_t
-mpn_sqrtrem4 (mpfr_limb_ptr sp, mpfr_limb_ptr rp, mpfr_limb_srcptr ap)
-{
-  mp_limb_t x[2], t1, t0, r2, r1, h, l, u2, u1, b[4];
-
-  MPFR_STAT_STATIC_ASSERT(GMP_NUMB_BITS == 64);
-
-  mpn_rsqrtrem2 (x, ap + 2);
-
-  /* x[1]*B+x[0] is a 80-bit approximation of 2^144/sqrt(ap[3]*B+ap[2]),
-     and should be smaller */
-
-  /* first compute y0 = a*x with at least 80 bits of precision */
-
-  t1 = ap[3] >> 48;
-  t0 = (ap[3] << 16) | (ap[2] >> 48);
-
-  /* now t1:t0 is a (16+64)-bit approximation of a,
-     (x1*B+x0) * (t1*B+t0) = (x1*t1)*B^2 + (x1*t0+x0*t1)*B + x0*t0 */
-  r2 = x[1] * t1; /* r2 has 32 bits */
-  umul_ppmm (h, r1, x[1], t0);
-  r2 += h;
-  umul_ppmm (h, l, x[0], t1);
-  r1 += l;
-  r2 += h + (r1 < l);
-  umul_ppmm (h, l, x[0], t0);
-  r1 += h;
-  r2 += (r1 < h);
-
-  /* r2 has 32 bits, r1 has 64 bits, thus we have 96 bits in total, we put 64
-     bits in r2 and 16 bits in r1 */
-  r2 = (r2 << 32) | (r1 >> 32);
-  r1 = (r1 << 32) >> 48;
-
-  /* we consider y0 = r2*2^16 + r1, which has 80 bits, and should be smaller than
-     2^16*sqrt(ap[3]*B+ap[2]) */
-
-  /* Now compute y0 + (x/2)*(a - y0^2), which should give ~160 correct bits.
-     Since a - y0^2 has its ~80 most significant bits that cancel, it remains
-     only ~48 bits. */
-
-  /* now r2:r1 represents y0, with r2 of 64 bits and r1 of 16 bits,
-     and we compute y0^2, whose upper ~80 bits should cancel with a:
-     y0^2 = r2^2*2^32 + 2*r2*r1*2^16 + r1^2. */
-  t1 = r2 * r2; /* we can simply ignore the upper 64 bits of r2^2 */
-  umul_ppmm (h, l, r2, r1);
-  t0 = l << 49; /* takes into account the factor 2 in 2*r2*r1 */
-  u1 = (r1 * r1) << 32; /* temporary variable */
-  t0 += u1;
-  t1 += ((h << 49) | (l >> 15)) + (t0 < u1); /* takes into account the factor 2 */
-
-  /* now t1:t0 >> 32 equals y0^2 mod 2^96, since y0 has 160 bits, we should shift
-     t1:t0 by 64 bits to the right */
-  t0 = ap[2] - t1 - (t0 != 0); /* we round downwards to get a lower approximation
-                                  of sqrt(a) at the end */
-
-  /* now t0 equals ap[3]*B+ap[2] - ceil(y0^2/2^32) */
-
-  umul_ppmm (u2, u1, x[1], t0);
-  umul_ppmm (h,  l,  x[0], t0);
-  u1 += h;
-  u2 += (u1 < h);
-
-  /* divide by 2 to take into account the factor 1/2 in (x/2)*(a - y0^2) */
-  u1 = (u2 << 63) | (u1 >> 1);
-  u2 = u2 >> 1;
-
-  /* u2:u1 approximates (x/2)*(ap[3]*B+ap[2] - y0^2/2^32) / 2^64,
-     and should be smaller */
-
-  r1 <<= 48; /* put back the most significant bits of r1 in place */
-
-  /* add u2:u1 >> 16 to y0 */
-  sp[0] = r1 + ((u2 << 48) | (u1 >> 16));
-  sp[1] = r2 + (u2 >> 16) + (sp[0] < r1);
-
-  mpn_mul_n (b, sp, sp, 2);
-  b[2] = ap[2] - b[2] - mpn_sub_n (rp, ap, b, 2);
-
-  /* invariant: the remainder {ap, 4} - {sp, 2}^2 is b[2]*B^2 + {rp, 2} */
-
-  t0 = mpn_lshift (b, sp, 2, 1);
-
-  /* Warning: the initial {sp, 2} might be < 2^127, thus t0 might be 0. */
-
-  /* invariant: 2*{sp,2} = t0*B + {b, 2} */
-
-  /* While the remainder is greater than 2*s we should subtract 2*s+1 to the
-     remainder, and add 1 to the square root. This loop seems to be executed
-     at most twice. */
-  while (b[2] > t0 || (b[2] == t0 &&
-                       (rp[1] > b[1] || (rp[1] == b[1] && rp[0] > b[0]))))
-    {
-      /* subtract 2*s to b[2]*B^2 + {rp, 2} */
-      b[2] -= t0 + mpn_sub_n (rp, rp, b, 2);
-      /* subtract 1 to b[2]*B^2 + {rp, 2}: b[2] -= mpn_sub_1 (rp, rp, 2, 1) */
-      if (rp[0]-- == 0)
-        b[2] -= (rp[1]-- == 0);
-      /* add 1 to s */
-      mpn_add_1 (sp, sp, 2, 1);
-      /* add 2 to t0*B + {b, 2}: t0 += mpn_add_1 (b, b, 2, 2) */
-      b[0] += 2;
-      if (b[0] == 0)
-        t0 += (++b[1] == 0);
-    }
-
-  return b[2];
-}
-
 /* Special code for GMP_NUMB_BITS < prec(r) < 2*GMP_NUMB_BITS,
    and GMP_NUMB_BITS < prec(u) <= 2*GMP_NUMB_BITS.
-   This code should work for any value of GMP_NUMB_BITS, but since mpn_sqrtrem4
-   currently assumes GMP_NUMB_BITS=64, it only works for GMP_NUMB_BITS=64. */
+   Assumes GMP_NUMB_BITS=64. */
 static int
 mpfr_sqrt2 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
 {
   mpfr_prec_t p = MPFR_GET_PREC(r);
   mpfr_limb_ptr up = MPFR_MANT(u), rp = MPFR_MANT(r);
-  mp_limb_t np[4], tp[2], rb, sb, mask;
+  mp_limb_t np[4], rb, sb, mask;
   mpfr_prec_t exp_u = MPFR_EXP(u), exp_r, sh = 2 * GMP_NUMB_BITS - p;
 
+  MPFR_STAT_STATIC_ASSERT (GMP_NUMB_BITS == 64);
+
   if (((unsigned int) exp_u & 1) != 0)
     {
       np[3] = up[1] >> 1;
@@ -797,20 +307,48 @@ mpfr_sqrt2 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
       np[2] = up[0];
       np[1] = 0;
     }
-  MPFR_ASSERTD (((unsigned int) exp_u & 1) == 0);
   exp_r = exp_u / 2;
 
-#if 1
-  np[0] = 0;
-  sb = mpn_sqrtrem4 (rp, tp, np);
-  sb |= tp[0] | tp[1];
-  rb = rp[0] & (MPFR_LIMB_ONE << (sh - 1));
   mask = MPFR_LIMB_MASK(sh);
+
+  mpn_sqrtrem4_approx (rp, np + 2);
+  /* the error is less than 43 ulps on rp[0] */
+  if (((rp[0] + 42) & (mask >> 1)) > 84)
+    sb = 1;
+  else
+    {
+      mp_limb_t tp[4], cy;
+
+      np[0] = 0;
+      mpn_mul_n (tp, rp, rp, 2);
+      while (mpn_cmp (tp, np, 4) > 0) /* approximation is too large */
+        {
+          mpn_sub_1 (rp, rp, 2, 1);
+          /* subtract 2*{rp,2}+1 to {tp,4} */
+          cy = mpn_submul_1 (tp, rp, 2, 2);
+          mpn_sub_1 (tp + 2, tp + 2, 2, cy);
+          mpn_sub_1 (tp, tp, 4, 1);
+        }
+      /* now {tp, 4} <= {np, 4} */
+      mpn_sub_n (tp, np, tp, 4);
+      /* now we want {tp, 4} <= 2 * {rp, 2}, which implies tp[2] <= 1 */
+      MPFR_ASSERTD(tp[3] == 0);
+      while (tp[2] > 1 ||
+             (tp[2] == 1 && tp[1] > ((rp[1] << 1) | (rp[0] >> 63))) ||
+             (tp[2] == 1 && tp[1] == ((rp[1] << 1) | (rp[0] >> 63)) &&
+              tp[0] > (rp[0] << 1)))
+        {
+          /* subtract 2*{rp,2}+1 to {tp,3} (we know tp[3] = 0) */
+          tp[2] -= mpn_submul_1 (tp, rp, 2, 2);
+          mpn_sub_1 (tp, tp, 3, 1);
+          mpn_add_1 (rp, rp, 2, 1);
+        }
+      sb = tp[2] | tp[0] | tp[1];
+    }
+
+  rb = rp[0] & (MPFR_LIMB_ONE << (sh - 1));
   sb |= (rp[0] & mask) ^ rb;
   rp[0] = rp[0] & ~mask;
-#else
-  mpn_sqrtrem4_approx (rp, np);
-#endif
 
   /* rounding */
   if (MPFR_UNLIKELY (exp_r > __gmpfr_emax))
@@ -877,9 +415,7 @@ mpfr_sqrt2 (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
       MPFR_RET(1);
     }
 }
-#endif /* GMP_NUMB_BITS == 64 */
-
-#endif /* !defined(MPFR_GENERIC_ABI) && (GMP_NUMB_BITS == 32 || GMP_NUMB_BITS == 64) */
+#endif /* !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64 */
 
 int
 mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
@@ -943,13 +479,10 @@ mpfr_sqrt (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode)
   MPFR_SET_POS(r);
 
   /* See the note at the beginning of this file about __GNUC__. */
-#if !defined(MPFR_GENERIC_ABI) && defined(__GNUC__) && \
-    (GMP_NUMB_BITS == 32 || GMP_NUMB_BITS == 64)
+#if !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64
   if (MPFR_GET_PREC (r) < GMP_NUMB_BITS && MPFR_GET_PREC (u) < GMP_NUMB_BITS)
     return mpfr_sqrt1 (r, u, rnd_mode);
-#endif
 
-#if !defined(MPFR_GENERIC_ABI) && GMP_NUMB_BITS == 64
   if (GMP_NUMB_BITS < MPFR_GET_PREC (r) && MPFR_GET_PREC (r) < 2*GMP_NUMB_BITS
       && MPFR_LIMB_SIZE(u) == 2)
     return mpfr_sqrt2 (r, u, rnd_mode);
diff --git a/src/sqrt_tab.h b/src/sqrt_tab.h
index 6cf027055..beba183b7 100644
--- a/src/sqrt_tab.h
+++ b/src/sqrt_tab.h
@@ -20,6 +20,7 @@ along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
 
+/* the following table is for mpn_sqrtrem2_approx */
 static const mp_limb_t U[192][9] = {
 /*64*/ {9223372036854775808UL,216172782113783808UL,18446744073709551211UL,72057594037919409UL,562949953352959UL,5497557869691UL,60128965900UL,703958331UL,8224883UL},
 /*65*/ {9295150333696574040UL,9655717601082343424UL,18304296041740945442UL,70401138622072963UL,541547220108952UL,5207184569501UL,56076861514UL,646436672UL,7442144UL},
@@ -216,197 +217,197 @@ static const mp_limb_t U[192][9] = {
 };
 
 /* the following table is for mpn_sqrtrem4_approx */
-static const mp_limb_t V[192][25] = {
-/*64*/ {9223372036854775808UL,1UL,72057594037927935UL,18446744073709551103UL,281474976710655UL,18446744073709513258UL,2199023255551UL,18446744073708420215UL,21474836479UL,18446744073692077217UL,234881023UL,18446744073547044945UL,2752511UL,18446744072723368698UL,33791UL,18446744069606842859UL,428UL,18446744061652873832UL,5UL,10808639080257074370UL,1368531298102989537UL,18467355284921015UL,252452092435139UL,3473825080059UL,43840374765UL},
-/*65*/ {9295150333696574040UL,9458326124125302122UL,71501156413050569UL,10005618548336875860UL,275004447742502UL,3514984916000372795UL,2115418828788UL,8824716288352884128UL,20340565661UL,7889244764712694208UL,219052245UL,10784072660087824819UL,2527525UL,16797450288835695670UL,30552UL,9426418135974882399UL,381UL,16719899883434876553UL,4UL,16532631274260000784UL,1181102529859048277UL,15692937483194843UL,211226058287873UL,2862197357143UL,35618047960UL},
-/*66*/ {9366378580956700130UL,497841112589747147UL,70957413492096213UL,1960244379882753229UL,268778081409455UL,6505710012253274813UL,2036197586435UL,4940467822666603783UL,19282174113UL,17899599546918546651UL,204507907UL,4969227929339347615UL,2323953UL,9070218443673058456UL,27666UL,2084415653409670841UL,340UL,10821198137222165005UL,4UL,5540222971740967432UL,1021638147762529586UL,13368516612456454UL,177214036346802UL,2365231680439UL,29029050710UL},
-/*67*/ {9437069233485010492UL,11580737367997358151UL,70425889802126943UL,17982518410076410691UL,262783170903458UL,13764494735366686066UL,1961068439578UL,928693725205564225UL,18293548876UL,12191774012526806229UL,191126630UL,1008415266253956997UL,2139477UL,3728169498666643994UL,25089UL,15088239956171525593UL,304UL,4811866855040326489UL,3UL,14468499920810387993UL,885633149145958028UL,11415873953324167UL,149071731123549UL,1960161144626UL,23731304361UL},
-/*68*/ {9507234283080136621UL,7239691060711272113UL,69906134434412769UL,5071832513146798917UL,257007847185341UL,1171567947669870650UL,1889763582245UL,2857008775702251295UL,17369150572UL,1959098264845562898UL,178800079UL,7724395595207815645UL,1972059UL,12902969913988497715UL,22786UL,7454154604217825213UL,272UL,4870299479029626572UL,3UL,6208599723312366913UL,769360383294037689UL,9771270956313800UL,125720198537938UL,1628984855653UL,19457965684UL},
-/*69*/ {9576885282238646259UL,14892033443755400406UL,69397719436511929UL,7727220620617389989UL,251441012451130UL,3302962609537615341UL,1822036322109UL,11787075660115400712UL,16503952193UL,407529121305613589UL,167431399UL,1100245233091749818UL,1819906UL,9435839311595429927UL,20723UL,10572060024308967019UL,244UL,508797001380621895UL,2UL,17472627194102320215UL,669727494854789722UL,8382609405995363UL,106290616600718UL,1357420212434UL,16000110967UL},
-/*70*/ {9646033366360708950UL,12341196150107550567UL,68900238331147921UL,1405775977765736050UL,246072279754099UL,13247147624262080908UL,1757659141100UL,13139105792296172147UL,15693385188UL,7364248616383732798UL,156933851UL,16306777270983566949UL,1681434UL,2348796022033977839UL,18873UL,4430994763660334295UL,219UL,1187882383315513513UL,2UL,11213950351883727968UL,584161742658916298UL,7207178913428601UL,90081128929856UL,1134096209640UL,13193577802UL},
-/*71*/ {9714689274533129758UL,15665938063020445587UL,68413304750233308UL,2968269772426975808UL,240891918134624UL,5986157480822898872UL,1696421958694UL,9915061317483892798UL,14933291889UL,16910191437969615063UL,147229638UL,6480183309790632934UL,1555242UL,12149770942171227483UL,17210UL,17245015609066222284UL,196UL,17637313675195771051UL,2UL,5749694383556965296UL,510516965364486831UL,6209864150291768UL,76522999118861UL,949929919395UL,10908925958UL},
-/*72*/ {9782863368999586659UL,4836346233993301752UL,67936551173608240UL,12715722288411381237UL,235890802686139UL,13366800311180541981UL,1638130574209UL,5601227746393481791UL,14219883456UL,12506579137852669460UL,138248866UL,17338552877050484273UL,1440092UL,6713831785060858935UL,15715UL,5416965692859666466UL,177UL,6322133232891599432UL,2UL,969889559513375692UL,446998202011183167UL,5361713247938772UL,65153973241679UL,797642751399UL,9043748007UL},
-/*73*/ {9850565653418023827UL,7527659077098538722UL,67469627763137149UL,9274931345602025996UL,231060369051839UL,10202742216482322436UL,1582605267478UL,6513607465586606955UL,13549702632UL,9595008129429252986UL,129928655UL,7015853332992605759UL,1334883UL,8221498443370854688UL,14367UL,11622217167660319864UL,159UL,16854615406789148400UL,1UL,15227764177360188422UL,392100432777594597UL,4638790911783048UL,55597243312936UL,671383790702UL,7516757311UL},
-/*74*/ {9917805789996281964UL,11920651794986616114UL,67012201283758661UL,17031607066347875855UL,226392571904590UL,1366260920994077891UL,1529679539895UL,16212452638535237092UL,12919590708UL,10700183284086040599UL,122212344UL,9972718832652346008UL,1238638UL,11567969817650339136UL,13151UL,10485765172171133286UL,144UL,7391000361405278305UL,1UL,11550126947666480894UL,344558648887395940UL,4021256745099154UL,47544778530065UL,566435474043UL,6263225762UL},
-/*75*/ {9984593115589071938UL,8913835544056271515UL,66563954103927146UL,4732600735633461501UL,221879847013090UL,8993190784990756082UL,1479198980087UL,4979086358222370207UL,12326658167UL,7266467148519648057UL,115048809UL,10373401382548715390UL,1150488UL,1763940980359071879UL,12052UL,13510955046624137439UL,130UL,10538034506638660398UL,1UL,8315642191370249477UL,303307040434007501UL,3492622300302313UL,40744074543638UL,478982814954UL,5231450460UL},
-/*76*/ {10050936656832218159UL,3517289075780633165UL,66124583268633014UL,3785304969478794255UL,217515076541555UL,17852394942895019696UL,1431020240404UL,17957393248540688360UL,11768258555UL,17744898896370640546UL,108391855UL,2226562550696738777UL,1069656UL,8577863822434044924UL,11058UL,8826612325968090096UL,118UL,4129588719256363862UL,1UL,5466034316893420177UL,267445544564346414UL,3039150423891209UL,34987586614392UL,405931851673UL,4380007706UL},
-/*77*/ {10116845144383624606UL,11656753199936621788UL,65693799638854705UL,4387919089957665419UL,213291557269008UL,14448224807967398639UL,1385010112136UL,7759998996917964630UL,11241965195UL,16832754344579851504UL,102199683UL,11053373810313233491UL,995451UL,8552438427931328027UL,10157UL,12322355134603097633UL,107UL,3379167272494952795UL,1UL,2951316348331781566UL,236212352636756748UL,2649368241397547UL,30104277165836UL,344766354462UL,3675611624UL},
-/*78*/ {10182327026334571527UL,6685538979505051362UL,65271327091888279UL,397601097439959625UL,209202971448359UL,16023927067540794085UL,1341044688771UL,9917331315291175381UL,10745550390UL,14712700277343033536UL,96434426UL,10774389403978678365UL,927254UL,1877325289761038265UL,9340UL,8701718893603937581UL,97UL,5470943259278797689UL,1UL,728524283899193650UL,208961255525037192UL,2313671173886719UL,25952834439644UL,293434371788UL,3091438760UL},
-/*79*/ {10247390480849666243UL,17360278906661226917UL,64856901777529533UL,3495669981292646961UL,205243360055473UL,3805487451811433284UL,1299008607946UL,607843087470573275UL,10276966835UL,296687747110627342UL,91061731UL,8291988557456311564UL,864510UL,2005121899018347417UL,8598UL,3722634290266505168UL,88UL,7948203744456534684UL,0UL,17207401710078876714UL,185142927945931745UL,2024000090699914UL,22416216034923UL,250258125004UL,2605813022UL},
-/*80*/ {10312043428088987146UL,18000430520700691482UL,64450271425556169UL,12333470639586957247UL,201407098204863UL,557356772821788838UL,1258794363780UL,7266888958853226650UL,9834330967UL,633233322293762249UL,86050395UL,17737473532416852204UL,806722UL,8524969722726487247UL,7923UL,3081323293018753618UL,80UL,8663794454956077593UL,0UL,15462534585981475639UL,164289428436130044UL,1773577389927029UL,19397246044529UL,213862231714UL,2201170236UL},
-/*81*/ {10376293541461622784UL,0UL,64051194700380387UL,10248191152060861992UL,197688872532038UL,4301709866297150665UL,1220301682296UL,9819269754353780431UL,9415908042UL,7591106148504182357UL,81372044UL,14959639959556984512UL,753444UL,15852408284604292508UL,7308UL,10109156846635829230UL,73UL,5737908712710299833UL,0UL,13913068168837198886UL,146001331348509627UL,1556690691367327UL,16815052539158UL,183116362325UL,1863240356UL},
-/*82*/ {10440148258255871223UL,6974987521162618580UL,63659440599121165UL,18376794338633045912UL,194083660363174UL,5286353637785429853UL,1183436953433UL,18254017643772729064UL,9020098730UL,8153341690892931738UL,77000842UL,15141941367338905037UL,704276UL,26013219346059930UL,6748UL,5335021931839870907UL,66UL,15968803030571468962UL,0UL,12535100729932897670UL,129937018682798041UL,1368515104410653UL,14602177251331UL,157089296969UL,1580399667UL},
-/*83*/ {10503614789687787167UL,10161280856418332040UL,63274787889685464UL,15952082430101712115UL,190586710511100UL,14716543728582599511UL,1148112713922UL,5422652164256847206UL,8645427062UL,12320141779217819030UL,72913240UL,5304553139396836571UL,658854UL,10715929281815357762UL,6237UL,101441671245788533UL,61UL,1015008327357060907UL,0UL,11307895549840790652UL,115803748701064857UL,1204966831829958UL,12702224473078UL,135012014022UL,1343156132UL},
-/*84*/ {10566700130406508953UL,9367671343862619820UL,62897024585753029UL,8949725841156763685UL,187193525552836UL,7328472284656329976UL,1114247175909UL,13659075818003791361UL,8290529582UL,12207305826779312065UL,69087746UL,9632545319969866635UL,616854UL,16226905933404580212UL,5769UL,16622089491978568738UL,55UL,14942879858973237649UL,0UL,10213430593872362857UL,103350189771138838UL,1062582294943162UL,11067943680243UL,116247954437UL,1143739387UL},
-/*85*/ {10629411067491832079UL,17636675944714852323UL,62525947455834306UL,6505850448785755272UL,183899845458336UL,3599973409745929244UL,1081763796813UL,13693468980571448273UL,7954145564UL,14885210095697593596UL,65504728UL,3334487757148920552UL,577982UL,16522981357863345916UL,5342UL,12894938170961441305UL,51UL,1289745725211588596UL,0UL,9236016602364451099UL,92360165381110313UL,938418097074161UL,9659661968558UL,100269003555UL,975773172UL},
-/*86*/ {10691754188976791336UL,8465351934666085528UL,62161361563818554UL,5197145741120491639UL,180701632452960UL,16853124374668370674UL,1050590886354UL,7819876149312519355UL,7635108185UL,13194773222186152033UL,62146229UL,7722043672930483397UL,541972UL,17173481077114362994UL,4951UL,10697897188115161445UL,46UL,14405340544956101770UL,0UL,8361972557157459652UL,82647402638399265UL,829968043537654UL,8443999360258UL,86636041022UL,834012938UL},
-/*87*/ {10753735891925544260UL,5002486305294571376UL,61803079838652553UL,4057349201760100757UL,177595057007622UL,5153423920550535768UL,1020661247170UL,4482279741473272081UL,7332336545UL,14079289881858165084UL,58995811UL,5308054399115118866UL,508584UL,10700344028761083769UL,4593UL,2368403444427797483UL,42UL,16520823000248268491UL,0UL,7579349368433081504UL,74051114059723211UL,735094156305409UL,7392813434241UL,74983150336UL,714135101UL},
-/*88*/ {10815362390094591593UL,717462364347603762UL,61450922670991997UL,12686213041381837207UL,174576484860772UL,13294637680937210962UL,991911845799UL,15587572503351981956UL,7044828450UL,5220245677400564475UL,56038408UL,2347367690736820009UL,477600UL,1277738525107147142UL,4264UL,5281906686421014471UL,39UL,6861485585994722992UL,0UL,6877694170772068803UL,66432272850780288UL,651969198132212UL,6482330332616UL,65004767929UL,612567338UL},
-/*89*/ {10876639721203305200UL,16778322265980900536UL,61104717534850029UL,4032329197005302591UL,171642464985533UL,14571818578397160642UL,964283511154UL,12621504783692422278UL,6771653870UL,8638389140790037446UL,53260198UL,18100152885518642262UL,448821UL,16630238972171812715UL,3962UL,5757823942685209758UL,36UL,3187474562461331607UL,0UL,6247848887571979596UL,59670466572741907UL,579028683975575UL,5692427608248UL,56445199470UL,526351599UL},
-/*90*/ {10937573753837842284UL,2004219560527036286UL,60764298632432457UL,2470700429608645969UL,168789718423423UL,9076512226322775501UL,937720657907UL,16754976645783212170UL,6511949013UL,4599937633511104277UL,50648492UL,6000224543203313978UL,422070UL,14192505661030110817UL,3684UL,13739356786335959619UL,33UL,4889444745496671446UL,0UL,5681777769980017220UL,53661234212494415UL,514930730359387UL,5006041071445UL,49090047251UL,453033243UL},
-/*91*/ {10998170194010790285UL,13839704208191330737UL,60429506560498847UL,13353643834418365889UL,166015127913458UL,6878198059849471831UL,912171032491UL,9767943017230577666UL,6264910937UL,8150207520054148478UL,48191622UL,10988842316888902015UL,397183UL,12962745938655620354UL,3429UL,6873798812963024176UL,30UL,11425885177593679397UL,0UL,5172419481424698878UL,48313808102731135UL,458522394249676UL,4408673134307UL,42759184241UL,390571115UL},
-/*92*/ {11058434591397295067UL,2084711249241425044UL,60100187996724429UL,13144609700571699382UL,163315728251968UL,10311758545682145937UL,887585479630UL,4868236272194236456UL,6029792660UL,9807841535472319884UL,45878857UL,3683770460886114604UL,374012UL,7799719138058963230UL,3194UL,3790334137930832325UL,28UL,3868191596639666640UL,0UL,4713560014905483532UL,43549195584683227UL,408811396493693UL,3887984423108UL,37300982838UL,337264424UL},
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-/*245*/ {18046075999155727507UL,3830637300661882670UL,36828726528889239UL,14953444968496640560UL,37580333192744UL,2255220397693809472UL,76694557536UL,3919829804211198321UL,195649381UL,8668675355527471162UL,558998UL,4293871272360003291UL,1711UL,4041311066929514339UL,5UL,8999697891809004477UL,0UL,335723070761387177UL,0UL,1141915206671289UL,3961746635244UL,13965340486UL,49876097UL,180027UL,638UL},
-/*246*/ {18082867221850889450UL,2250945240292675374UL,36753795166363596UL,8178132425424745788UL,37351417851995UL,9681603734206863191UL,75917515959UL,6281072894397768258UL,192879867UL,14441536836336884542UL,548845UL,2808105415062269405UL,1673UL,5688808709938192017UL,5UL,6354551432929070738UL,0UL,325621832677641922UL,0UL,1103054988745309UL,3811368863413UL,13380637029UL,47593617UL,171091UL,604UL},
-/*247*/ {18119583741324584319UL,14183079625609595301UL,36679319314422235UL,8579934154868617238UL,37124817119860UL,10370974590145414741UL,75151451659UL,11746252254524179397UL,190160555UL,15106388127689041881UL,538916UL,10237064930164923582UL,1636UL,7125985772064903220UL,5UL,3788833765046748794UL,0UL,315863664913139785UL,0UL,1065666885671776UL,3667274707648UL,12822638697UL,45424222UL,162631UL,572UL},
-/*248*/ {18156226010783715083UL,15617014587789905349UL,36605294376580070UL,13531824825291042625UL,36900498363487UL,17976901814545078532UL,74396166055UL,7734784123129883595UL,187490337UL,15518848937242677400UL,529206UL,11044905817571421605UL,1600UL,7806324334942849634UL,5UL,1299852576432794841UL,0UL,306435596845954394UL,0UL,1029689505530680UL,3529177740608UL,12290024991UL,43361883UL,154622UL,542UL},
-/*249*/ {18192794478871134995UL,7620897419213095314UL,36531715821026375UL,9090508424654725384UL,36678429539183UL,1990855546497567016UL,73651464938UL,2189455333123215085UL,184868134UL,16322344545452733947UL,519709UL,11365815805801752613UL,1565UL,7201794582272090091UL,4UL,17331761217874049422UL,0UL,297325197707143369UL,0UL,995064249354487UL,3396805670372UL,11781545647UL,41400920UL,147037UL,513UL},
-/*250*/ {18229289589729737140UL,3340365934211727143UL,36458579179459474UL,5171769072507097907UL,36458579179459UL,8748928460010834564UL,72917158358UL,16951608916585390053UL,182292895UL,16552214968261512171UL,510420UL,2001701073724344705UL,1531UL,4802158562385656451UL,4UL,14988578187884308092UL,0UL,288520552068848138UL,0UL,961735173562755UL,3269899590004UL,11296016648UL,39535969UL,139852UL,486UL},
-/*251*/ {18265711783065391511UL,12063159118716609369UL,36385880045946995UL,795706742363779270UL,36240916380425UL,5420901602078368034UL,72193060518UL,14305187940787039734UL,179763596UL,16773652212152758089UL,501332UL,13657683501990103147UL,1498UL,114302815538653898UL,4UL,12714647546242304829UL,0UL,280010236537362181UL,0UL,929648859685730UL,3148213269749UL,10832316463UL,37761975UL,133045UL,461UL},
-/*252*/ {18302061494208754842UL,12501700421524245584UL,36313614075811021UL,9467781094124183656UL,36025410789495UL,1125713461895244873UL,71478989661UL,12849073181218150520UL,177279240UL,4286697103264400920UL,492442UL,6160822183190029430UL,1465UL,11108342634144403140UL,4UL,10507658772283395772UL,0UL,271783297587445612UL,0UL,898754290963711UL,3031512489265UL,10389382524UL,36074166UL,126594UL,437UL},
-/*253*/ {18338339154175977771UL,16248650814870867764UL,36241776984537505UL,8818011803515558908UL,35812032593416UL,9359681543099323654UL,70774767971UL,3299538830389246974UL,174838853UL,12612818551428980296UL,483743UL,15937011286411278631UL,1434UL,430032272141437171UL,4UL,8365387164701043385UL,0UL,263829230477769301UL,0UL,869002735433962UL,2919574407493UL,9966207903UL,34468036UL,120480UL,414UL},
-/*254*/ {18374545189728333292UL,11556360322123820026UL,36170364546709317UL,9318745754314505972UL,35600752506603UL,12155699341600398166UL,70080221469UL,12769611947270970503UL,172441489UL,15436994464971192886UL,475232UL,8343577731301108601UL,1403UL,4527385101844362327UL,4UL,6285690332180859352UL,0UL,256137959190759780UL,0UL,840347635140231UL,2812186967902UL,9561838201UL,32939333UL,114684UL,393UL},
-/*255*/ {18410680023430789819UL,15898606764561736782UL,36099372594962332UL,18080046861854172536UL,35391541759766UL,18319789264172470988UL,69395179921UL,2061445994886092866UL,170086225UL,5475778134592504506UL,466903UL,6706497535342542265UL,1373UL,4522900751921280370UL,4UL,4266504840597785723UL,0UL,248699817343300930UL,0UL,812744501121844UL,2709148336991UL,9175368611UL,31484041UL,109187UL,372UL},
+static const mp_limb_t V[192][9] = {
+/*64*/ {18446744073709551615UL,4899916394579099648UL,18446744073709545581UL,216172782113656827UL,2814749766088467UL,38482902961355UL,541157289571UL,7740859288UL,106097935UL},
+/*65*/ {18161848009772339989UL,1152921504606846976UL,18022691487252618265UL,207954132545111171UL,2666078621481441UL,35889516397918UL,496924278946UL,6999053007UL,94535144UL},
+/*66*/ {17883451634243709493UL,15564440312192434176UL,17614640343250061338UL,200166367536834629UL,2527353124681123UL,33506572433969UL,456901224663UL,6338067531UL,84379904UL},
+/*67*/ {17611311504979443698UL,11240984669916758016UL,17221757888329068289UL,192780871884194396UL,2397772037670121UL,31314184344002UL,420632500590UL,5748067474UL,75443438UL},
+/*68*/ {17345196756709786252UL,6052837899185946624UL,16843266273138508320UL,185771319188951581UL,2276609303176843UL,29294602608218UL,387717566591UL,5220536111UL,67564560UL},
+/*69*/ {17084888277784556246UL,1441151880758558720UL,16478438191997264216UL,179113458608598653UL,2163206021304308UL,27431958297051UL,357803629953UL,4748082719UL,60605319UL},
+/*70*/ {16830177951838183974UL,12393906174523604992UL,16126592925964676294UL,172784924206704820UL,2056963382936107UL,25712040410991UL,330579372350UL,4324280312UL,54447381UL},
+/*71*/ {16580867958409902162UL,1441151880758558720UL,15787092746870737218UL,166765064227454943UL,1957336434171347UL,24122102278300UL,305769575214UL,3943528643UL,48989016UL},
+/*72*/ {16336770127177867616UL,12105675798371893248UL,15459339644838850759UL,161034787967024404UL,1863828564057104UL,22650692876300UL,283130504507UL,3600938245UL,44142578UL},
+/*73*/ {16097705341016668929UL,3458764513820540928UL,15142772346181354964UL,155576428214150208UL,1775986623114507UL,21287509577542UL,262445938931UL,3292232046UL,39832399UL},
+/*74*/ {15863502983574883288UL,9223372036854775808UL,14836863592339213322UL,150373617489887201UL,1693396593056225UL,20023269352926UL,243523744638UL,3013661709UL,35993019UL},
+/*75*/ {15634000427501147267UL,2882303761517117440UL,14541117653849896608UL,145411176538465701UL,1615679739049908UL,18849595908266UL,226192915139UL,2761936326UL,32567697UL},
+/*76*/ {15409042559830551656UL,16429131440647569408UL,14255068056227410490UL,140675013712740755UL,1542489185208541UL,17758920603794UL,210301008110UL,2534161504UL,29507154UL},
+/*77*/ {15188481341384057465UL,11817445422220181504UL,13978275497181758352UL,136152034063432051UL,1473506861945082UL,16744395319954UL,195711921547UL,2327787232UL,26768514UL},
+/*78*/ {14972175397337247242UL,16429131440647569408UL,13710325936839711216UL,131830057084973340UL,1408440780631120UL,15799815697494UL,182303960675UL,2140563155UL,24314402UL},
+/*79*/ {14759989636385569376UL,16717361816799281152UL,13450828844595491028UL,127697742195505367UL,1347022596827274UL,14919553403629UL,169968154499UL,1970500133UL,22112181UL},
+/*80*/ {14551794896175207254UL,3746994889972252672UL,13199415587953902631UL,123744521137048554UL,1289005428356513UL,14098496265585UL,158606787110UL,1815837137UL,20133302UL},
+/*81*/ {14347467612885206812UL,6341068275337658368UL,12955737950259656825UL,119960536576460942UL,1234161898795762UL,13331995273766UL,148132114111UL,1675012683UL,18352749UL},
+/*82*/ {14146889513040485373UL,17870283321406128128UL,12719466765560989300UL,116336586270359250UL,1182282380671007UL,12615817593715UL,138465238901UL,1546640138UL,16748569UL},
+/*83*/ {13949947325809406394UL,12393906174523604992UL,12490290660055501212UL,112864072229403052UL,1133173415844740UL,11946104842791UL,129535127283UL,1429486335UL,15301462UL},
+/*84*/ {13756532514195999480UL,6052837899185946624UL,12267914890630685525UL,109534954380618353UL,1086656293356284UL,11319335987220UL,121277741950UL,1322453016UL,13994435UL},
+/*85*/ {13566541023677613236UL,9223372036854775808UL,12052060271957520336UL,106341708281966568UL,1042565767377762UL,10732294300540UL,113635281073UL,1224560707UL,12812505UL},
+/*86*/ {13379873046965548176UL,3458764513820540928UL,11842462184437245937UL,103277286492174819UL,1000748900034183UL,10182037897687UL,106555507420UL,1134934679UL,11742440UL},
+/*87*/ {13196432803680555632UL,10376293541461622784UL,11638869656051533249UL,100335083241814072UL,961064015650201UL,9665873421891UL,99991156365UL,1052792710UL,10772537UL},
+/*88*/ {13016128333838351200UL,1152921504606846976UL,11441044511835600614UL,97508902089499343UL,923379754566422UL,9181332515670UL,93899412725UL,977434385UL,9892434UL},
+/*89*/ {12838871304133663343UL,16140901064495857664UL,11248760585291942085UL,94792926280542242UL,887574216047244UL,8726150753968UL,88241447762UL,908231749UL,9092939UL},
+/*90*/ {12664576826095866436UL,9799832789158199296UL,11061802986597481635UL,92181691554971888UL,853534181007505UL,8298248757746UL,82982008866UL,844621112UL,8365887UL},
+/*91*/ {12493163285265858418UL,10952754293765046272UL,10879967422936407616UL,89670061178040826UL,821154406340355UL,7895715241317UL,78089055398UL,786095852UL,7704022UL},
+/*92*/ {12324552180613356396UL,14699749183737298944UL,10703059566721011201UL,87253202989567544UL,790336983553782UL,7516791776919UL,73533435097UL,732200104UL,7100879UL},
+/*93*/ {12158667973476925828UL,3170534137668829184UL,10530894467849110262UL,84926568289100328UL,760990755235443UL,7159859086292UL,69288596134UL,682523195UL,6550696UL},
+/*94*/ {11995437945366476098UL,3170534137668829184UL,10363296006493966647UL,82685872392234163UL,733030783579603UL,6823424691896UL,65330330557UL,636694743UL,6048328UL},
+/*95*/ {11834792064020221342UL,17582052945254416384UL,10200096383235291729UL,80527076709747792UL,706377865838868UL,6506111780239UL,61636545434UL,594380334UL,5589180UL},
+/*96*/ {11676662857155731392UL,4323455642275676160UL,10041135643621760806UL,78446372215790878UL,680958092117911UL,6206649147189UL,58187058433UL,555277704UL,5169136UL},
+/*97*/ {11520985293398139352UL,17582052945254416384UL,9886261234509753542UL,76440164184349983UL,656702441416028UL,5923862110246UL,54963415020UL,519113352UL,4784512UL},
+/*98*/ {11367696669908236297UL,17005592192950992896UL,9735327589752746909UL,74505058084838983UL,633546412258284UL,5656664286034UL,51948724801UL,485639546UL,4432003UL},
+/*99*/ {11216736506269433213UL,288230376151711744UL,9588195743023510096UL,72637846538053712UL,611429684638275UL,5404050142868UL,49127514817UL,454631670UL,4108641UL},
+/*100*/ {11068046444225730969UL,10664523917613334528UL,9444732965739290288UL,70835497243041754UL,590295810335281UL,5165088248451UL,46485597908UL,425885859UL,3811758UL},
+/*101*/ {10921570152893201920UL,288230376151711744UL,9304812428230575250UL,69095141793788714UL,570091928970146UL,4938915141671UL,44009954451UL,399216903UL,3538955UL},
+/*102*/ {10777253239095312990UL,1729382256910270464UL,9168312882448584857UL,67414065312119475UL,550768507432166UL,4724729765376UL,41688626009UL,374456377UL,3288069UL},
+/*103*/ {10635043162497944398UL,6052837899185946624UL,9035118364646988945UL,65789696829951527UL,532279100547729UL,4521788403899UL,39510619581UL,351450977UL,3057151UL},
+/*104*/ {10494889155243391163UL,17005592192950992896UL,8905117916600905371UL,64219600360100586UL,514580131073758UL,4329400075241UL,37465821313UL,330061033UL,2844440UL},
+/*105*/ {10356742145804166038UL,10664523917613334528UL,8778205324042275765UL,62701466600300035UL,497630687288483UL,4146922333201UL,35544918653UL,310159182UL,2648349UL},
+/*106*/ {10220554686797223244UL,6052837899185946624UL,8654278871096384779UL,61233105220019805UL,481392336621084UL,3973757439516UL,33739330057UL,291629178UL,2467437UL},
+/*107*/ {10086280886517446449UL,12393906174523604992UL,8533241109600597474UL,59812437684114689UL,465828953912927UL,3809348870299UL,32041141439UL,274364832UL,2300405UL},
+/*108*/ {9953876343966030786UL,14411518807585587200UL,8414998642274246565UL,58437490571347410UL,450906563038320UL,3653178124807UL,30443048684UL,258269060UL,2146073UL},
+/*109*/ {9823298087164862999UL,15852670688344145920UL,8299461918788818903UL,57106389349462941UL,436593190733845UL,3504761807863UL,28938305574UL,243253024UL,2003370UL},
+/*110*/ {9694504514562279595UL,13546827679130451968UL,8186545043860896290UL,55817352571777534UL,422858731593930UL,3363648960257UL,27520676589UL,229235368UL,1871327UL},
+/*111*/ {9567455339348762955UL,8358680908399640576UL,8076165596557351891UL,54568686463224142UL,409674823287803UL,3229418614013UL,26184394080UL,216141526UL,1749061UL},
+/*112*/ {9442111536513312906UL,3170534137668829184UL,7968244460063675524UL,53358779866496707UL,397014731140561UL,3101677551786UL,24924119379UL,203903102UL,1635769UL},
+/*113*/ {9318435292482490724UL,5764607523034234880UL,7862705661222525215UL,52186099521387403UL,384853241299849UL,2980058251707UL,23734907446UL,192457312UL,1530723UL},
+/*114*/ {9196389957194550617UL,15852670688344145920UL,7759476219201151458UL,51049185652638190UL,373166561780571UL,2864217000866UL,22612174716UL,181746474UL,1433257UL},
+/*115*/ {9075939998470720079UL,4323455642275676160UL,7658486002693640777UL,49946647843653283UL,361932230743941UL,2753832162261UL,21551669817UL,171717562UL,1342765UL},
+/*116*/ {8957050958554628451UL,3458764513820540928UL,7559667595107359979UL,48877161175262271UL,351129031424819UL,2648602581540UL,20549446891UL,162321787UL,1258694UL},
+/*117*/ {8839689412699170251UL,7782220156096217088UL,7462956167222898252UL,47839462610402420UL,340736913173308UL,2548246121173UL,19601841262UL,153514228UL,1180539UL},
+/*118*/ {8723822929687778758UL,11240984669916758016UL,7368289356853513288UL,46832347607119068UL,330736918123607UL,2452498310883UL,18705447227UL,145253499UL,1107839UL},
+/*119*/ {8609420034184223938UL,2594073385365405696UL,7275607155063872386UL,45854666943679197UL,321111113045607UL,2361111104237UL,17857097776UL,137501440UL,1040171UL},
+/*120*/ {8496450170811680640UL,4323455642275676160UL,7184851798538995239UL,44905323740868096UL,311842525973250UL,2273851732233UL,17053846043UL,130222842UL,977149UL},
+/*121*/ {8384883669867978007UL,4899916394579099648UL,7095967667722983013UL,43983270667703858UL,302915087238478UL,2190501645593UL,16292948345UL,123385200UL,918420UL},
+/*122*/ {8274691714589675607UL,3458764513820540928UL,7008901190373567770UL,43087507317869751UL,294313574571266UL,2110855538239UL,15571848649UL,116958480UL,863659UL},
+/*123*/ {8165846309882949281UL,288230376151711744UL,6923600750202927039UL,42217077745139292UL,286023561954881UL,2034720445115UL,14888164346UL,110914920UL,812570UL},
+/*124*/ {8058320252444240538UL,12970366926827028480UL,6840016600297752792UL,41371068146961741UL,278031371951600UL,1961914908156UL,14239673201UL,105228833UL,764878UL},
+/*125*/ {7952087102198255480UL,4611686018427387904UL,6758100781032398596UL,40548604686193949UL,270324031237754UL,1892268204754UL,13624301387UL,99876446UL,720335UL},
+/*126*/ {7847121154985218247UL,2017612633061982208UL,6677807042208195501UL,39748851441715036UL,262889229108564UL,1825619633584UL,13040112490UL,94835736UL,678708UL},
+/*127*/ {7743397416433313389UL,3746994889972252672UL,6599090769169855731UL,38971008479349155UL,255715278732793UL,1761817853124UL,12485297415UL,90086294UL,639787UL},
+/*128*/ {7640891576956012808UL,12970366926827028480UL,6521908912666391089UL,38214310035154274UL,248791080955140UL,1700720268588UL,11958165103UL,85609191UL,603376UL},
+/*129*/ {7539579987817495892UL,3458764513820540928UL,6446219922239267583UL,37478022803716333UL,242106090460573UL,1642192463401UL,11457133985UL,81386864UL,569296UL},
+/*130*/ {7439439638212654254UL,1441151880758558720UL,6371983682934696814UL,36761444324622934UL,235650284129659UL,1586107671663UL,10980724123UL,77403006UL,537381UL},
+/*131*/ {7340448133311241904UL,8646911284551352320UL,6299161455150117487UL,36063901460782818UL,229414131427537UL,1532346288351UL,10527549962UL,73642468UL,507479UL},
+/*132*/ {7242583673218603016UL,10088063165309911040UL,6227715817437128383UL,35384748962710678UL,223388566681549UL,1480795414310UL,10096313657UL,70091167UL,479448UL},
+/*133*/ {7145825032808096971UL,9799832789158199296UL,6157610612094471828UL,34723368113314429UL,217564963113912UL,1431348433314UL,9685798912UL,66736007UL,453159UL},
+/*134*/ {7050151542382857064UL,5764607523034234880UL,6088810893395202061UL,34079165448107229UL,211935108506165UL,1383904618720UL,9294865293UL,63564801UL,428492UL},
+/*135*/ {6955543069126877200UL,15564440312192434176UL,6021282878301968301UL,33451571546121816UL,206491182381624UL,1338368767447UL,8922442981UL,60566202UL,405336UL},
+/*136*/ {6861979999307631219UL,17582052945254416384UL,5954993899533454775UL,32840039887132806UL,201225734600797UL,1294650859189UL,8567527919UL,57729639UL,383588UL},
+/*137*/ {6769443221194502384UL,16140901064495857664UL,5889912360853501655UL,32244045771095609UL,196131665272676UL,1252665738955UL,8229177329UL,55045260UL,363153UL},
+/*138*/ {6677914108659245588UL,3458764513820540928UL,5826007694462329777UL,31663085295990731UL,191202205892133UL,1212332821198UL,7906505558UL,52503878UL,343943UL},
+/*139*/ {6587374505426530656UL,16140901064495857664UL,5763250320376652025UL,31096674390521324UL,186430901620390UL,1173575813893UL,7598680241UL,50096921UL,325876UL},
+/*140*/ {6497806709944329931UL,17582052945254416384UL,5701611607692315774UL,30544347898351522UL,181811594631690UL,1136322461122UL,7304918740UL,47816387UL,308878UL},
+/*141*/ {6409193460845524524UL,16717361816799281152UL,5641063837629520819UL,30005658710795164UL,177338408454960UL,1100504302773UL,7024484848UL,45654803UL,292877UL},
+/*142*/ {6321517922973618262UL,18158513697557839872UL,5581580168266629825UL,29480176945070078UL,173005733244515UL,1066056450128UL,6756685731UL,43605187UL,277810UL},
+/*143*/ {6234763673946872755UL,6917529027641081856UL,5523134600874164888UL,28967489165423800UL,168808211918587UL,1032917376170UL,6500869093UL,41661010UL,263615UL},
+/*144*/ {6148914691236517205UL,6052837899185946624UL,5465701947765793065UL,28467197644613372UL,164740727108964UL,1001028719557UL,6256420544UL,39816167UL,250236UL},
+/*145*/ {6063955339735948084UL,15276209936040722432UL,5409257801587972342UL,27978919663385938UL,160798388869027UL,970335101288UL,6022761151UL,38064945UL,237622UL},
+/*146*/ {5979870359799021710UL,5188146770730811392UL,5353778505974481819UL,27502286845759206UL,156976523091283UL,940783953150UL,5799345170UL,36401997UL,225723UL},
+/*147*/ {5896644855726661872UL,2017612633061982208UL,5299241127496318577UL,27036944528042330UL,153270660588913UL,912325357125UL,5585657934UL,34822313UL,214495UL},
+/*148*/ {5814264284682059401UL,0UL,5245623428841429404UL,26582551159669300UL,149676526799093UL,884911894979UL,5381213885UL,33321199UL,203896UL},
+/*149*/ {5732714446015735013UL,9223372036854775808UL,5192903843162477661UL,26138777734039217UL,146190032068771UL,858498507329UL,5185554749UL,31894257UL,193887UL},
+/*150*/ {5651981470982674790UL,11240984669916758016UL,5141061449534341629UL,25705307247671614UL,142807262486309UL,833042361535UL,4998247834UL,30537359UL,184431UL},
+/*151*/ {5572051812834632764UL,2017612633061982208UL,5090075949466317215UL,25281834186090890UL,139524471224953UL,808502727802UL,4818884442UL,29246634UL,175494UL},
+/*152*/ {5492912237271530626UL,17582052945254416384UL,5039927644417069942UL,24868064034952563UL,136338070366389UL,784840862943UL,4647078394UL,28018450UL,167045UL},
+/*153*/ {5414549813236673629UL,15564440312192434176UL,4990597414263262793UL,24463712815015914UL,133244623174829UL,762019901268UL,4482464651UL,26849393UL,159054UL},
+/*154*/ {5336951904041247149UL,6341068275337658368UL,4942066696675490649UL,24068506639653288UL,130240836794056UL,740004752137UL,4324698028UL,25736258UL,151493UL},
+/*155*/ {5260106158804262890UL,15564440312192434176UL,4894317467357690733UL,23682181293666174UL,127323555341719UL,718762003713UL,4173451998UL,24676033UL,144337UL},
+/*156*/ {5184000504194789721UL,864691128455135232UL,4847332221108582835UL,23304481832252734UL,124489753376877UL,698259832517UL,4028417567UL,23665886UL,137562UL},
+/*157*/ {5108623136463934038UL,3170534137668829184UL,4801093953665933379UL,22935162199041019UL,121736529718388UL,678467918388UL,3889302234UL,22703152UL,131145UL},
+/*158*/ {5033962513754630560UL,4323455642275676160UL,4755586144296543223UL,22573984862167075UL,119061101593222UL,659357364507UL,3755829010UL,21785326UL,125065UL},
+/*159*/ {4960007348677868472UL,9799832789158199296UL,4710792739096839260UL,22220720467437863UL,116460799095125UL,640900622139UL,3627735504UL,20910047UL,119303UL},
+/*160*/ {4886746601144511923UL,8070450532247928832UL,4666698134970812705UL,21875147507675630UL,113933059935373UL,623071419804UL,3504773070UL,20075095UL,113840UL},
+/*161*/ {4814169471442379666UL,4611686018427387904UL,4623287164253800004UL,21537052007393426UL,111475424468495UL,605844696569UL,3386706006UL,19278377UL,108658UL},
+/*162*/ {4742265393548727872UL,11529215046068469760UL,4580545079952252736UL,21206227222001121UL,109085530976979UL,589196539224UL,3273310807UL,18517920UL,103742UL},
+/*163*/ {4671024028668734341UL,7205759403792793600UL,4538457541571197241UL,20882473350787671UL,106761111199973UL,573104123072UL,3164375462UL,17791865UL,99077UL},
+/*164*/ {4600435258991012948UL,9223372036854775808UL,4497010601502549181UL,20565597262968931UL,104499986091967UL,557545656109UL,3059698798UL,17098460UL,94648UL},
+/*165*/ {4530489181651595587UL,12105675798371893248UL,4456190691948828547UL,20255412236130997UL,102300061798305UL,542500326395UL,2959089865UL,16436049UL,90442UL},
+/*166*/ {4461176102898206310UL,14987979559889010688UL,4415984612358122008UL,19951737706437258UL,100159325835208UL,527948252392UL,2862367360UL,15803073UL,86447UL},
+/*167*/ {4392486532447020087UL,7205759403792793600UL,4376379517347367031UL,19654399030003107UL,98075843462785UL,513870436110UL,2769359082UL,15198058UL,82650UL},
+/*168*/ {4324411178024447676UL,16140901064495857664UL,4337362905092190340UL,19363227254875814UL,96047754240166UL,500248718865UL,2679901427UL,14619612UL,79042UL},
+/*169*/ {4256940940086819603UL,12682136550675316736UL,4298922606162626502UL,19078058903088545UL,94073268752627UL,487065739503UL,2593838911UL,14066421UL,75611UL},
+/*170*/ {4190066906711157116UL,8646911284551352320UL,4261046772785074583UL,18798735762287061UL,92150665501146UL,474304894934UL,2511023721UL,13537241UL,72347UL},
+/*171*/ {4123780348650517232UL,12970366926827028480UL,4223723868511825748UL,18525104686455345UL,90278287945440UL,461950302837UL,2431315297UL,13030900UL,69243UL},
+/*172*/ {4058072714547683378UL,15852670688344145920UL,4186942658280415812UL,18257017405292481UL,88454541692072UL,449986766398UL,2354579931UL,12546286UL,66288UL},
+/*173*/ {3992935626301243551UL,13258597302978740224UL,4150692198845927429UL,17994330341817576UL,86677891819705UL,438399740977UL,2280690403UL,12082348UL,63476UL},
+/*174*/ {3928360874578355112UL,8646911284551352320UL,4114961829570189742UL,17736904437802515UL,84946860334089UL,427175302560UL,2209525627UL,11638093UL,60798UL},
+/*175*/ {3864340414468740001UL,13546827679130451968UL,4079741163552601895UL,17484604986653983UL,83260023745767UL,416300117925UL,2140970320UL,11212579UL,58247UL},
+/*176*/ {3800866361274687005UL,14123288431433875456UL,4045020079088043385UL,17237301473386524UL,81616010763946UL,405761416384UL,2074914700UL,10804916UL,55817UL},
+/*177*/ {3737930986432059349UL,2882303761517117440UL,4010788711438031360UL,16994867421347567UL,80013500100321UL,395546963043UL,2011254187UL,10414259UL,53502UL},
+/*178*/ {3675526713557516935UL,1729382256910270464UL,3977037444901944907UL,16757180245373341UL,78451218377039UL,385645033461UL,1949889134UL,10039809UL,51294UL},
+/*179*/ {3613646114617363563UL,14699749183737298944UL,3943756905175761372UL,16524121111071605UL,76927938133312UL,376044389653UL,1890724565UL,9680808UL,49189UL},
+/*180*/ {3552281906213620964UL,4035225266123964416UL,3910937951986341791UL,16295574799943071UL,75442475925501UL,366734257338UL,1833669936UL,9336537UL,47182UL},
+/*181*/ {3491426945983113972UL,10952754293765046272UL,3878571671989863529UL,16071429580068476UL,73993690515815UL,357704304376UL,1778638899UL,9006316UL,45267UL},
+/*182*/ {3431074229105525154UL,13835058055282163712UL,3846649371923529981UL,15851577082102440UL,72580481145011UL,348944620313UL,1725549090UL,8689498UL,43439UL},
+/*183*/ {3371216884916543041UL,3170534137668829184UL,3815162572000191415UL,15635912180328636UL,71201785884781UL,340445696986UL,1674321926UL,8385471UL,41695UL},
+/*184*/ {3311848173622386342UL,6629298651489370112UL,3784102999535989209UL,15424332878543418UL,69856580065733UL,332198410113UL,1624882407UL,8093651UL,40030UL},
+/*185*/ {3252961483112137433UL,576460752303423488UL,3753462582801589456UL,15216740200546968UL,68543874777111UL,324194001822UL,1577158938UL,7813486UL,38439UL},
+/*186*/ {3194550325864462406UL,17870283321406128128UL,3723233445088002412UL,15013038085032253UL,67262715434610UL,316424064065UL,1531083158UL,7544452UL,36920UL},
+/*187*/ {3136608335945432464UL,5476377146882523136UL,3693407898978392997UL,14813133284672685UL,66012180412863UL,308880522864UL,1486589774UL,7286049UL,35469UL},
+/*188*/ {3079129266094292628UL,5188146770730811392UL,3663978440817675615UL,14616935269219437UL,64791379739336UL,301555623349UL,1443616412UL,7037806UL,34081UL},
+/*189*/ {3022106984894149114UL,17293822569102704640UL,3634937745372055150UL,14424356132428777UL,63599453846581UL,294441915538UL,1402103467UL,6799270UL,32756UL},
+/*190*/ {2965535474024666390UL,8070450532247928832UL,3606278660671026190UL,14235310502648775UL,62435572379937UL,287532240823UL,1361993971UL,6570014UL,31488UL},
+/*191*/ {2909408825593979328UL,6341068275337658368UL,3577994203024675341UL,14049715456903163UL,61298933057944UL,280819719123UL,1323233462UL,6349631UL,30275UL},
+/*192*/ {2853721239547135186UL,1729382256910270464UL,3550077552209447800UL,13867490438318144UL,60188760582885UL,274297736665UL,1285769859UL,6137734UL,29115UL},
+/*193*/ {2798467021148484606UL,2882303761517117440UL,3522522046815840202UL,13688557176745482UL,59104305598985UL,267959934360UL,1249553350UL,5933955UL,28006UL},
+/*194*/ {2743640578535540750UL,14123288431433875456UL,3495321179751767813UL,13512839612442391UL,58044843695973UL,261800196747UL,1214536279UL,5737942UL,26944UL},
+/*195*/ {2689236420341921220UL,11529215046068469760UL,3468468593895626311UL,13340263822675476UL,57009674455796UL,255812641468UL,1180673045UL,5549362UL,25927UL},
+/*196*/ {2635249153387078802UL,5188146770730811392UL,3441958077893327415UL,13170757951122417UL,55998120540408UL,249991609248UL,1147919999UL,5367898UL,24954UL},
+/*197*/ {2581673480430614533UL,6052837899185946624UL,3415783562093834095UL,13004252139951136UL,55009526818669UL,244331654357UL,1116235356UL,5193246UL,24021UL},
+/*198*/ {2528504197989050261UL,7493989779944505344UL,3389939114617955859UL,12840678464461945UL,54043259530492UL,238827535522UL,1085579105UL,5025118UL,23129UL},
+/*199*/ {2475736194213017980UL,3170534137668829184UL,3364418937555388075UL,12679970870183615UL,53098705486462UL,233474207271UL,1055912923UL,4863239UL,22273UL},
+/*200*/ {2423364446822899923UL,14123288431433875456UL,3339217363285192246UL,12522065112319463UL,52175271301265UL,228266811684UL,1027200100UL,4707347UL,21453UL},
+/*201*/ {2371384021101026852UL,6341068275337658368UL,3314328850915116970UL,12366898697444458UL,51272382659325UL,223200670532UL,999405458UL,4557193UL,20668UL},
+/*202*/ {2319790067938612317UL,7493989779944505344UL,3289747982835352702UL,12214410827358975UL,50389483611156UL,218271277781UL,972495284UL,4412538UL,19914UL},
+/*203*/ {2268577821935668102UL,6629298651489370112UL,3265469461382497689UL,12064542345009221UL,49526035898995UL,213474292440UL,946437263UL,4273156UL,19192UL},
+/*204*/ {2217742599552210628UL,10664523917613334528UL,3241488105609688195UL,11917235682388553UL,48681518310355UL,208805531749UL,921200408UL,4138829UL,18499UL},
+/*205*/ {2167279797309130029UL,13258597302978740224UL,3217798848159013720UL,11772434810337848UL,47855426058230UL,204260964673UL,896755006UL,4009350UL,17835UL},
+/*206*/ {2117184890037152952UL,15564440312192434176UL,3194396732232497797UL,11630085190166854UL,47047270186708UL,199836705689UL,873072556UL,3884523UL,17197UL},
+/*207*/ {2067453429172387059UL,4035225266123964416UL,3171276908658077476UL,11490133727022014UL,46256577000845UL,195529008868UL,850125713UL,3764159UL,16585UL},
+/*208*/ {2018081041096989784UL,10664523917613334528UL,3148434633047160215UL,11352528724929658UL,45482887519703UL,191334262217UL,827888238UL,3648077UL,15998UL},
+/*209*/ {1969063425523556281UL,7782220156096217088UL,3125865263040475850UL,11217219843446678UL,44725756951496UL,187248982274UL,806334950UL,3536107UL,15434UL},
+/*210*/ {1920396353921871708UL,8070450532247928832UL,3103564255639074030UL,11084158055853830UL,43984754189853UL,183269808953UL,785441673UL,3428083UL,14892UL},
+/*211*/ {1872075667986721246UL,4323455642275676160UL,3081527164617444225UL,10953295608829773UL,43259461330249UL,179393500613UL,765185197UL,3323849UL,14372UL},
+/*212*/ {1824097278145497499UL,4035225266123964416UL,3059749638015856470UL,10824585983546657UL,42549473205725UL,175616929347UL,745543231UL,3223255UL,13873UL},
+/*213*/ {1776457162104389373UL,11240984669916758016UL,3038227415709136674UL,10697983858130758UL,41854396941005UL,171937076484UL,726494365UL,3126158UL,13393UL},
+/*214*/ {1729151363431979185UL,9511602413006487552UL,3016956327049200867UL,10573445071434111UL,41173851524239UL,168351028283UL,708018032UL,3032419UL,12931UL},
+/*215*/ {1682175990179115728UL,6917529027641081856UL,2995932288578778395UL,10450926588065502UL,40507467395567UL,164855971818UL,690094466UL,2941909UL,12488UL},
+/*216*/ {1635527213533970389UL,9223372036854775808UL,2975151301813855112UL,10330386464631437UL,39854886051784UL,161449191047UL,672704675UL,2854502UL,12062UL},
+/*217*/ {1589201266511221223UL,2882303761517117440UL,2954609451092464197UL,10211783817139849UL,39215759666403UL,158128063041UL,655830399UL,2770078UL,11652UL},
+/*218*/ {1543194442674346228UL,1729382256910270464UL,2934302901487544637UL,10095078789521365UL,38589750724438UL,154890054388UL,639454087UL,2688521UL,11258UL},
+/*219*/ {1497503094890042007UL,1441151880758558720UL,2914227896781675780UL,9980232523224913UL,37976531671297UL,151732717745UL,623558858UL,2609723UL,10879UL},
+/*220*/ {1452123634113817562UL,14699749183737298944UL,2894380757501580971UL,9867207127846295UL,37375784575146UL,148653688536UL,608128480UL,2533578UL,10514UL},
+/*221*/ {1407052528205845270UL,10376293541461622784UL,2874757879010374210UL,9755965652750135UL,36787200802197UL,145650681798UL,593147336UL,2459984UL,10164UL},
+/*222*/ {1362286300776182114UL,8935141660703064064UL,2855355729655601258UL,9646472059647298UL,36210480704355UL,142721489156UL,578600404UL,2388846UL,9826UL},
+/*223*/ {1317821530058504137UL,4899916394579099648UL,2836170848971200825UL,9538691196091479UL,35645333318701UL,139863975923UL,564473226UL,2320071UL,9501UL},
+/*224*/ {1273654847811525781UL,6052837899185946624UL,2817199845931582485UL,9432588769860206UL,35091476078324UL,137076078332UL,550751889UL,2253571UL,9188UL},
+/*225*/ {1229782938247303441UL,1152921504606846976UL,2798439397256086052UL,9328131324186950UL,34548634534001UL,134355800870UL,537423000UL,2189260UL,8887UL},
+/*226*/ {1186202536985649146UL,7205759403792793600UL,2779886245762152320UL,9225286213812449UL,34016542086306UL,131701213738UL,524473664UL,2127058UL,8597UL},
+/*227*/ {1142910430033905901UL,18158513697557839872UL,2761537198765597535UL,9124021581824659UL,33494939727674UL,129110450404UL,511891465UL,2066886UL,8317UL},
+/*228*/ {1099903452791360854UL,8070450532247928832UL,2743389126526443855UL,9024306337258036UL,32983575794050UL,126581705265UL,499664444UL,2008670UL,8048UL},
+/*229*/ {1057178489077596191UL,9511602413006487552UL,2725438960738815414UL,8926110133424065UL,32482205725685UL,124113231402UL,487781083UL,1952337UL,7789UL},
+/*230*/ {1014732470184100532UL,12105675798371893248UL,2707683693063464647UL,8829403346946078UL,31990591836741UL,121703338430UL,476230286UL,1897820UL,7539UL},
+/*231*/ {972562373948485581UL,6052837899185946624UL,2690120373701546278UL,8734157057472550UL,31508503093316UL,119350390429UL,465001358UL,1845051UL,7298UL},
+/*232*/ {930665223850674015UL,6917529027641081856UL,2672746110008306984UL,8640343028044094UL,31035714899565UL,117052803966UL,454083996UL,1793968UL,7066UL},
+/*233*/ {889038088130445021UL,16429131440647569408UL,2655558065145407263UL,8547933686090364UL,30572008891578UL,114809046196UL,443468267UL,1744510UL,6842UL},
+/*234*/ {847678078925743570UL,11529215046068469760UL,2638553456770638658UL,8456902105034096UL,30117172738708UL,112617633036UL,433144596UL,1696619UL,6626UL},
+/*235*/ {806582351431178495UL,576460752303423488UL,2621729555763844100UL,8367221986480351UL,29670999952041UL,110477127415UL,423103751UL,1650238UL,6418UL},
+/*236*/ {765748103076152740UL,4323455642275676160UL,2605083684987892116UL,8278867642969994UL,29233289699737UL,108386137594UL,413336830UL,1605314UL,6218UL},
+/*237*/ {725172572722086771UL,9799832789158199296UL,2588613218083596744UL,8191813981277203UL,28803846628948UL,106343315552UL,403835245UL,1561796UL,6024UL},
+/*238*/ {684853039878213140UL,2594073385365405696UL,2572315578297514589UL,8106036486231662UL,28382480694073UL,104347355434UL,394590713UL,1519633UL,5837UL},
+/*239*/ {644786823935436601UL,5764607523034234880UL,2556188237341588380UL,8021511205046824UL,27969006991083UL,102396992065UL,385595246UL,1478778UL,5657UL},
+/*240*/ {604971283417769996UL,11529215046068469760UL,2540228714283642882UL,7938214732136382UL,27563245597682UL,100490999520UL,376841130UL,1439185UL,5483UL},
+/*241*/ {565403815250871370UL,4035225266123964416UL,2524434574467774006UL,7856124194401785UL,27165021419080UL,98628189746UL,368320927UL,1400810UL,5315UL},
+/*242*/ {526081854047222511UL,864691128455135232UL,2508803428463705670UL,7775217236974293UL,26774164039155UL,96807411247UL,360027453UL,1363611UL,5152UL},
+/*243*/ {487002871407503319UL,3746994889972252672UL,2493332931044221226UL,7695472009395743UL,26390507576790UL,95027547809UL,351953775UL,1327547UL,4996UL},
+/*244*/ {448164375237730105UL,6052837899185946624UL,2478020780189807439UL,7616867152222767UL,26013890547197UL,93287517283UL,344093199UL,1292579UL,4845UL},
+/*245*/ {409563909081739167UL,864691128455135232UL,2462864716119678796UL,7539381784039832UL,25644155728015UL,91586270411UL,336439262UL,1258670UL,4699UL},
+/*246*/ {371199051468609762UL,18158513697557839872UL,2447862520348378716UL,7462995488867007UL,25281150030026UL,89922789696UL,328985720UL,1225782UL,4557UL},
+/*247*/ {333067415274632942UL,2594073385365405696UL,2433012014767181805UL,7387688303948931UL,24924724372286UL,88296088320UL,321726545UL,1193883UL,4421UL},
+/*248*/ {295166647099444599UL,10664523917613334528UL,2418311060749547899UL,7313440707911938UL,24574733561521UL,86705209097UL,314655912UL,1162937UL,4289UL},
+/*249*/ {257494426655952636UL,5764607523034234880UL,2403757558279904161UL,7240233609276818UL,24231036175615UL,85149223468UL,307768191UL,1132914UL,4162UL},
+/*250*/ {220048466173699215UL,10088063165309911040UL,2389349445105056106UL,7168048335315167UL,23893494451041UL,83627230540UL,301057947UL,1103781UL,4039UL},
+/*251*/ {182826509815309846UL,1441151880758558720UL,2375084695907552059UL,7096866621237704UL,23561974174086UL,82138356147UL,294519922UL,1075509UL,3920UL},
+/*252*/ {145826333105691398UL,14411518807585587200UL,2360961321500348319UL,7026670599703416UL,23236344575730UL,80681751963UL,288149036UL,1048069UL,3805UL},
+/*253*/ {109045742373651188UL,13835058055282163712UL,2346977368042144228UL,6957442790638766UL,22916478230026UL,79256594634UL,281940380UL,1021434UL,3694UL},
+/*254*/ {72482574205618946UL,11817445422220181504UL,2333130916272777394UL,6889166091356625UL,22602250955886UL,77862084952UL,275889205UL,995577UL,3587UL},
+/*255*/ {36134694911162869UL,15276209936040722432UL,2319420080768089661UL,6821823766964969UL,22293541722099UL,76497447053UL,269990919UL,970472UL,3483UL},
 };