/* Mulders' short product, square and division. Copyright 2005-2021 Free Software Foundation, Inc. Contributed by the AriC and Caramba projects, INRIA. This file is part of the GNU MPFR Library. The GNU MPFR Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. The GNU MPFR Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ /* References: [1] Short Division of Long Integers, David Harvey and Paul Zimmermann, Proceedings of the 20th Symposium on Computer Arithmetic (ARITH-20), July 25-27, 2011, pages 7-14. */ #define MPFR_NEED_LONGLONG_H #include "mpfr-impl.h" /* Don't use MPFR_MULHIGH_SIZE since it is handled by tuneup */ #ifdef MPFR_MULHIGH_TAB_SIZE static short mulhigh_ktab[MPFR_MULHIGH_TAB_SIZE]; #else static short mulhigh_ktab[] = {MPFR_MULHIGH_TAB}; #define MPFR_MULHIGH_TAB_SIZE (numberof_const (mulhigh_ktab)) #endif /* Put in rp[n..2n-1] an approximation of the n high limbs of {up, n} * {vp, n}. The error is less than n ulps of rp[n] (and the approximation is always less or equal to the truncated full product). Assume 2n limbs are allocated at rp. Implements Algorithm ShortMulNaive from [1]. */ static void mpfr_mulhigh_n_basecase (mpfr_limb_ptr rp, mpfr_limb_srcptr up, mpfr_limb_srcptr vp, mp_size_t n) { mp_size_t i; rp += n - 1; umul_ppmm (rp[1], rp[0], up[n-1], vp[0]); /* we neglect up[0..n-2]*vp[0], which is less than B^n */ for (i = 1 ; i < n ; i++) /* here, we neglect up[0..n-i-2] * vp[i], which is less than B^n too */ rp[i + 1] = mpn_addmul_1 (rp, up + (n - i - 1), i + 1, vp[i]); /* in total, we neglect less than n*B^n, i.e., n ulps of rp[n]. */ } /* Put in rp[n..2n-1] an approximation of the n high limbs of {np, n} * {mp, n}. The error is less than n ulps of rp[n] (and the approximation is always less or equal to the truncated full product). Implements Algorithm ShortMul from [1]. */ void mpfr_mulhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mpfr_limb_srcptr mp, mp_size_t n) { mp_size_t k; MPFR_STAT_STATIC_ASSERT (MPFR_MULHIGH_TAB_SIZE >= 8); /* so that 3*(n/4) > n/2 */ k = MPFR_LIKELY (n < MPFR_MULHIGH_TAB_SIZE) ? mulhigh_ktab[n] : 3*(n/4); /* Algorithm ShortMul from [1] requires k >= (n+3)/2, which translates into k >= (n+4)/2 in the C language. */ MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n)); if (k < 0) mpn_mul_basecase (rp, np, n, mp, n); /* result is exact, no error */ else if (k == 0) mpfr_mulhigh_n_basecase (rp, np, mp, n); /* basecase error < n ulps */ else if (n > MUL_FFT_THRESHOLD) mpn_mul_n (rp, np, mp, n); /* result is exact, no error */ else { mp_size_t l = n - k; mp_limb_t cy; mpn_mul_n (rp + 2 * l, np + l, mp + l, k); /* fills rp[2l..2n-1] */ mpfr_mulhigh_n (rp, np + k, mp, l); /* fills rp[l-1..2l-1] */ cy = mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1); mpfr_mulhigh_n (rp, np, mp + k, l); /* fills rp[l-1..2l-1] */ cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1); mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */ } } #ifdef MPFR_SQRHIGH_TAB_SIZE static short sqrhigh_ktab[MPFR_SQRHIGH_TAB_SIZE]; #else static short sqrhigh_ktab[] = {MPFR_SQRHIGH_TAB}; #define MPFR_SQRHIGH_TAB_SIZE (numberof_const (sqrhigh_ktab)) #endif /* Put in rp[n..2n-1] an approximation of the n high limbs of {np, n}^2. The error is less than n ulps of rp[n]. */ void mpfr_sqrhigh_n (mpfr_limb_ptr rp, mpfr_limb_srcptr np, mp_size_t n) { mp_size_t k; MPFR_STAT_STATIC_ASSERT (MPFR_SQRHIGH_TAB_SIZE > 2); /* ensures k < n */ k = MPFR_LIKELY (n < MPFR_SQRHIGH_TAB_SIZE) ? sqrhigh_ktab[n] : (n+4)/2; /* ensures that k >= (n+3)/2 */ MPFR_ASSERTD (k == -1 || k == 0 || (k >= (n+4)/2 && k < n)); if (k < 0) /* we can't use mpn_sqr_basecase here, since it requires n <= SQR_KARATSUBA_THRESHOLD, where SQR_KARATSUBA_THRESHOLD is not exported by GMP */ mpn_sqr (rp, np, n); else if (k == 0) mpfr_mulhigh_n_basecase (rp, np, np, n); else { mp_size_t l = n - k; mp_limb_t cy; mpn_sqr (rp + 2 * l, np + l, k); /* fills rp[2l..2n-1] */ mpfr_mulhigh_n (rp, np, np + k, l); /* fills rp[l-1..2l-1] */ /* {rp+n-1,l+1} += 2 * {rp+l-1,l+1} */ cy = mpn_lshift (rp + l - 1, rp + l - 1, l + 1, 1); cy += mpn_add_n (rp + n - 1, rp + n - 1, rp + l - 1, l + 1); mpn_add_1 (rp + n + l, rp + n + l, k, cy); /* propagate carry */ } } #ifdef MPFR_DIVHIGH_TAB_SIZE static short divhigh_ktab[MPFR_DIVHIGH_TAB_SIZE]; #else static short divhigh_ktab[] = {MPFR_DIVHIGH_TAB}; #define MPFR_DIVHIGH_TAB_SIZE (numberof_const (divhigh_ktab)) #endif /* Put in Q={qp, n} an approximation of N={np, 2*n} divided by D={dp, n}, with the most significant limb of the quotient as return value (0 or 1). Assumes the most significant bit of D is set. Clobbers N. The approximate quotient Q satisfies - 2(n-1) < N/D - Q <= 4. Assumes n >= 2. */ static mp_limb_t mpfr_divhigh_n_basecase (mpfr_limb_ptr qp, mpfr_limb_ptr np, mpfr_limb_srcptr dp, mp_size_t n) { mp_limb_t qh, d1, d0, q2, q1, q0; mpfr_pi1_t dinv2; MPFR_ASSERTD(n >= 2); np += n; if ((qh = (mpn_cmp (np, dp, n) >= 0))) mpn_sub_n (np, np, dp, n); /* now {np, n} is less than D={dp, n}, which implies np[n-1] <= dp[n-1] */ d1 = dp[n - 1]; /* we assumed n >= 2 */ d0 = dp[n - 2]; invert_pi1 (dinv2, d1, d0); /* dinv2.inv32 = floor ((B^3 - 1) / (d0 + d1 B)) - B */ while (n > 1) { /* Invariant: it remains to reduce n limbs from N (in addition to the initial low n limbs). Since n >= 2 here, necessarily we had n >= 2 initially, which means that in addition to the limb np[n-1] to reduce, we have at least 2 extra limbs, thus accessing np[n-3] is valid. */ /* Warning: we can have np[n-1]>d1 or (np[n-1]=d1 and np[n-2]>=d0) here, since we truncate the divisor at each step, but since {np,n} < D originally, the largest possible partial quotient is B-1. */ if (MPFR_UNLIKELY(np[n-1] > d1 || (np[n-1] == d1 && np[n-2] >= d0))) q2 = MPFR_LIMB_MAX; else udiv_qr_3by2 (q2, q1, q0, np[n - 1], np[n - 2], np[n - 3], d1, d0, dinv2.inv32); /* since q2 = floor((np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0)), we have q2 <= (np[n-1]*B^2+np[n-2]*B+np[n-3])/(d1*B+d0), thus np[n-1]*B^2+np[n-2]*B+np[n-3] >= q2*(d1*B+d0) and {np-1, n} >= q2*D - q2*B^(n-2) >= q2*D - B^(n-1) thus {np-1, n} - (q2-1)*D >= D - B^(n-1) >= 0 which proves that at most one correction is needed */ q0 = mpn_submul_1 (np - 1, dp, n, q2); if (MPFR_UNLIKELY(q0 > np[n - 1])) { mpn_add_n (np - 1, np - 1, dp, n); q2 --; } qp[--n] = q2; dp ++; } /* we have B+dinv2 = floor((B^3-1)/(d1*B+d0)) < B^2/d1 q1 = floor(np[0]*(B+dinv2)/B) <= floor(np[0]*B/d1) <= floor((np[0]*B+np[1])/d1) thus q1 is not larger than the true quotient. q1 > np[0]*(B+dinv2)/B - 1 > np[0]*(B^3-1)/(d1*B+d0)/B - 2 For d1*B+d0 <> B^2/2, we have B+dinv2 = floor(B^3/(d1*B+d0)) thus q1 > np[0]*B^2/(d1*B+d0) - 2, i.e., (d1*B+d0)*q1 > np[0]*B^2 - 2*(d1*B+d0) d1*B*q1 > np[0]*B^2 - 2*d1*B - 2*d0 - d0*q1 >= np[0]*B^2 - 2*d1*B - B^2 thus q1 > np[0]*B/d1 - 2 - B/d1 > np[0]*B/d1 - 4. For d1*B+d0 = B^2/2, dinv2 = B-1 thus q1 > np[0]*(2B-1)/B - 1 > np[0]*B/d1 - 2. In all cases, if q = floor((np[0]*B+np[1])/d1), we have: q - 4 <= q1 <= q */ umul_ppmm (q1, q0, np[0], dinv2.inv32); qp[0] = np[0] + q1; return qh; } /* Put in {qp, n} an approximation of N={np, 2*n} divided by D={dp, n}, with the most significant limb of the quotient as return value (0 or 1). Assumes the most significant bit of D is set. Clobbers N. This implements the ShortDiv algorithm from reference [1]. Assumes n >= 2 (which should be fulfilled also in the recursive calls). */ mp_limb_t mpfr_divhigh_n (mpfr_limb_ptr qp, mpfr_limb_ptr np, mpfr_limb_ptr dp, mp_size_t n) { mp_size_t k, l; mp_limb_t qh, cy; mpfr_limb_ptr tp; MPFR_TMP_DECL(marker); MPFR_STAT_STATIC_ASSERT (MPFR_DIVHIGH_TAB_SIZE >= 15); /* so that 2*(n/3) >= (n+4)/2 */ MPFR_ASSERTD(n >= 2); k = MPFR_LIKELY (n < MPFR_DIVHIGH_TAB_SIZE) ? divhigh_ktab[n] : 2*(n/3); if (k == 0) { #if defined(WANT_GMP_INTERNALS) && defined(HAVE___GMPN_SBPI1_DIVAPPR_Q) mpfr_pi1_t dinv2; invert_pi1 (dinv2, dp[n - 1], dp[n - 2]); if (n > 2) /* sbpi1_divappr_q wants n > 2 */ return __gmpn_sbpi1_divappr_q (qp, np, n + n, dp, n, dinv2.inv32); else return mpfr_divhigh_n_basecase (qp, np, dp, n); #else /* use our own code for base-case short division */ return mpfr_divhigh_n_basecase (qp, np, dp, n); #endif } /* Check the bounds from [1]. In addition, we forbid k=n-1, which would give l=1 in the recursive call. It follows n >= 5. */ MPFR_ASSERTD ((n+4)/2 <= k && k < n-1); MPFR_TMP_MARK (marker); l = n - k; /* first divide the most significant 2k limbs from N by the most significant k limbs of D */ qh = mpn_divrem (qp + l, 0, np + 2 * l, 2 * k, dp + l, k); /* exact */ /* it remains {np,2l+k} = {np,n+l} as remainder */ /* now we have to subtract high(Q1)*D0 where Q1=qh*B^k+{qp+l,k} and D0={dp,l} */ tp = MPFR_TMP_LIMBS_ALLOC (2 * l); mpfr_mulhigh_n (tp, qp + k, dp, l); /* we are only interested in the upper l limbs from {tp,2l} */ cy = mpn_sub_n (np + n, np + n, tp + l, l); if (qh) cy += mpn_sub_n (np + n, np + n, dp, l); while (cy > 0) /* Q1 was too large: subtract 1 to Q1 and add D to np+l */ { qh -= mpn_sub_1 (qp + l, qp + l, k, MPFR_LIMB_ONE); cy -= mpn_add_n (np + l, np + l, dp, n); } /* now it remains {np,n+l} to divide by D */ cy = mpfr_divhigh_n (qp, np + k, dp + k, l); qh += mpn_add_1 (qp + l, qp + l, k, cy); MPFR_TMP_FREE(marker); return qh; }