/* dc_divappr_q - middle-product-based divide and conquer approximate quotient Copyright (C) 2009, David Harvey All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ #include "mpir.h" #include "gmp-impl.h" #include "longlong.h" #include <stdlib.h> #include <stdio.h> #include <string.h> #include <math.h> /* Computes an approximation to N/D, where N = {np,2n}, D = {dp,n}. D must be normalised (i.e. B/2 <= dp[n-1] < B). More precisely, returns Q such that N = Q*D + R, where -D < R < D. Q is n+1 limbs; low limbs written to {qp,n}, high limb returned. The high limb is either 0 or 1. {dip,2} is precomputed inverse of high limbs of dp (see mpn_sb_divappr_q). N is destroyed. None of the buffers may overlap. tp is scratch space. */ #define DC_DIVAPPR_Q_N_THRESHOLD 36 mp_limb_t mpn_dc_divappr_q_n (mp_ptr qp, mp_ptr np, mp_srcptr dp, mp_size_t n, mp_srcptr dip, mp_ptr tp) { mp_limb_t qh, cy; mp_ptr q_hi; mp_size_t m; ASSERT (n >= 6); m = (n + 1) / 2; q_hi = qp + n - m; /* FIXME: we could probably avoid this copy if we could guarantee that sb_div_appr_q/dc_divappr_q_n did not destroy the "bottom half" of N */ MPN_COPY (tp, np, 2*n); /* estimate high m+1 limbs of quotient */ if (m < DC_DIVAPPR_Q_N_THRESHOLD) qh = mpn_sb_divappr_q (q_hi, tp + 2*n - 2*m, 2*m, dp + n - m, m, dip); else qh = mpn_dc_divappr_q_n (q_hi, tp + 2*n - 2*m, dp + n - m, m, dip, tp + 2*n); /* decrease the estimate slightly (FIXME: actually I think 6 would be enough? but let's do 10 to be safe...) */ qh -= mpn_sub_1 (q_hi, q_hi, m, (mp_limb_t) 10); /* don't let the estimate become negative */ if (qh & GMP_NUMB_HIGHBIT) { MPN_ZERO (q_hi, m); qh = 0; } /* we know that {np+n-m, n+m} = q_hi * D + e0, where 0 <= e0 < C*B^n, where C is a small positive constant. Estimate q_hi * D using middle product. */ mpn_mulmid (tp, dp, n, q_hi + 1, m - 2); /* do some parts of the middle product "manually": */ tp[n - m + 2] += mpn_addmul_1 (tp, dp + m - 2, n - m + 2, q_hi[0]); mpn_addmul_1 (tp + 1, dp, n - m + 2, q_hi[m-1]); if (qh) mpn_add_n (tp + 2, tp + 2, dp, n - m + 1); /* subtract that estimate from N */ mpn_sub_n (np + n - 2, np + n - 2, tp, n - m + 3); /* recursively divide to obtain low half of quotient */ if (n - m + 2 < DC_DIVAPPR_Q_N_THRESHOLD) cy = mpn_sb_divappr_q (tp, np + m - 3, 2*n - 2*m + 4, dp + m - 2, n - m + 2, dip); else cy = mpn_dc_divappr_q_n (tp, np + m - 3, dp + m - 2, n - m + 2, dip, tp + n - m + 2); /* FIXME: this copy is annoying. The only reason it happens is that we elected to develop one extra quotient limb in the second recursive quotient. But I don't see how to avoid this and stay within the required error bounds. We inherit the error from the quotient, but there's also an error from the missed terms at the low end of the middle product. */ MPN_COPY (qp, tp + 1, n - m); qh += mpn_add_1 (qp + n - m, qp + n - m, m, tp[n-m+1]); qh += mpn_add_1 (qp + n - m + 1, qp + n - m + 1, m - 1, cy); if (tp[0] >= GMP_NUMB_HIGHBIT) qh += mpn_add_1 (qp, qp, n, 1); /* round quotient up */ /* if qh == 2 (unlikely!), then Q must be 2000.... and we should return instead 1ffff.... */ if (qh >= 2) { /* FIXME: hmmmm my test suite doesn't seem to generate this case, is it actually possible at all? */ qh -= mpn_sub_1 (qp, qp, n, 1); ASSERT (qh == 1); } return qh; }