1 /* mpn_invertappr and helper functions. Compute I such that 2 floor((B^{2n}-1)/U - 1 <= I + B^n <= floor((B^{2n}-1)/U. 3 4 Contributed to the GNU project by Marco Bodrato. 5 6 The algorithm used here was inspired by ApproximateReciprocal from "Modern 7 Computer Arithmetic", by Richard P. Brent and Paul Zimmermann. Special 8 thanks to Paul Zimmermann for his very valuable suggestions on all the 9 theoretical aspects during the work on this code. 10 11 THE FUNCTIONS IN THIS FILE ARE INTERNAL WITH MUTABLE INTERFACES. IT IS ONLY 12 SAFE TO REACH THEM THROUGH DOCUMENTED INTERFACES. IN FACT, IT IS ALMOST 13 GUARANTEED THAT THEY WILL CHANGE OR DISAPPEAR IN A FUTURE GMP RELEASE. 14 15 Copyright (C) 2007, 2009, 2010, 2012, 2015 Free Software Foundation, Inc. 16 17 This file is part of the GNU MP Library. 18 19 The GNU MP Library is free software; you can redistribute it and/or modify 20 it under the terms of either: 21 22 * the GNU Lesser General Public License as published by the Free 23 Software Foundation; either version 3 of the License, or (at your 24 option) any later version. 25 26 or 27 28 * the GNU General Public License as published by the Free Software 29 Foundation; either version 2 of the License, or (at your option) any 30 later version. 31 32 or both in parallel, as here. 33 34 The GNU MP Library is distributed in the hope that it will be useful, but 35 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 36 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 37 for more details. 38 39 You should have received copies of the GNU General Public License and the 40 GNU Lesser General Public License along with the GNU MP Library. If not, 41 see https://www.gnu.org/licenses/. */ 42 43 #include "gmp.h" 44 #include "gmp-impl.h" 45 #include "longlong.h" 46 47 /* FIXME: The iterative version splits the operand in two slightly unbalanced 48 parts, the use of log_2 (or counting the bits) underestimate the maximum 49 number of iterations. */ 50 51 #if TUNE_PROGRAM_BUILD 52 #define NPOWS \ 53 ((sizeof(mp_size_t) > 6 ? 48 : 8*sizeof(mp_size_t))) 54 #define MAYBE_dcpi1_divappr 1 55 #else 56 #define NPOWS \ 57 ((sizeof(mp_size_t) > 6 ? 48 : 8*sizeof(mp_size_t)) - LOG2C (INV_NEWTON_THRESHOLD)) 58 #define MAYBE_dcpi1_divappr \ 59 (INV_NEWTON_THRESHOLD < DC_DIVAPPR_Q_THRESHOLD) 60 #if (INV_NEWTON_THRESHOLD > INV_MULMOD_BNM1_THRESHOLD) && \ 61 (INV_APPR_THRESHOLD > INV_MULMOD_BNM1_THRESHOLD) 62 #undef INV_MULMOD_BNM1_THRESHOLD 63 #define INV_MULMOD_BNM1_THRESHOLD 0 /* always when Newton */ 64 #endif 65 #endif 66 67 /* All the three functions mpn{,_bc,_ni}_invertappr (ip, dp, n, scratch), take 68 the strictly normalised value {dp,n} (i.e., most significant bit must be set) 69 as an input, and compute {ip,n}: the approximate reciprocal of {dp,n}. 70 71 Let e = mpn*_invertappr (ip, dp, n, scratch) be the returned value; the 72 following conditions are satisfied by the output: 73 0 <= e <= 1; 74 {dp,n}*(B^n+{ip,n}) < B^{2n} <= {dp,n}*(B^n+{ip,n}+1+e) . 75 I.e. e=0 means that the result {ip,n} equals the one given by mpn_invert. 76 e=1 means that the result _may_ be one less than expected. 77 78 The _bc version returns e=1 most of the time. 79 The _ni version should return e=0 most of the time; only about 1% of 80 possible random input should give e=1. 81 82 When the strict result is needed, i.e., e=0 in the relation above: 83 {dp,n}*(B^n+{ip,n}) < B^{2n} <= {dp,n}*(B^n+{ip,n}+1) ; 84 the function mpn_invert (ip, dp, n, scratch) should be used instead. */ 85 86 /* Maximum scratch needed by this branch (at xp): 2*n */ 87 static mp_limb_t 88 mpn_bc_invertappr (mp_ptr ip, mp_srcptr dp, mp_size_t n, mp_ptr xp) 89 { 90 ASSERT (n > 0); 91 ASSERT (dp[n-1] & GMP_NUMB_HIGHBIT); 92 ASSERT (! MPN_OVERLAP_P (ip, n, dp, n)); 93 ASSERT (! MPN_OVERLAP_P (ip, n, xp, mpn_invertappr_itch(n))); 94 ASSERT (! MPN_OVERLAP_P (dp, n, xp, mpn_invertappr_itch(n))); 95 96 /* Compute a base value of r limbs. */ 97 if (n == 1) 98 invert_limb (*ip, *dp); 99 else { 100 mp_size_t i; 101 102 /* n > 1 here */ 103 i = n; 104 do 105 xp[--i] = GMP_NUMB_MAX; 106 while (i); 107 mpn_com (xp + n, dp, n); 108 109 /* Now xp contains B^2n - {dp,n}*B^n - 1 */ 110 111 /* FIXME: if mpn_*pi1_divappr_q handles n==2, use it! */ 112 if (n == 2) { 113 mpn_divrem_2 (ip, 0, xp, 4, dp); 114 } else { 115 gmp_pi1_t inv; 116 invert_pi1 (inv, dp[n-1], dp[n-2]); 117 if (! MAYBE_dcpi1_divappr 118 || BELOW_THRESHOLD (n, DC_DIVAPPR_Q_THRESHOLD)) 119 mpn_sbpi1_divappr_q (ip, xp, 2 * n, dp, n, inv.inv32); 120 else 121 mpn_dcpi1_divappr_q (ip, xp, 2 * n, dp, n, &inv); 122 MPN_DECR_U(ip, n, CNST_LIMB (1)); 123 return 1; 124 } 125 } 126 return 0; 127 } 128 129 /* mpn_ni_invertappr: computes the approximate reciprocal using Newton's 130 iterations (at least one). 131 132 Inspired by Algorithm "ApproximateReciprocal", published in "Modern Computer 133 Arithmetic" by Richard P. Brent and Paul Zimmermann, algorithm 3.5, page 121 134 in version 0.4 of the book. 135 136 Some adaptations were introduced, to allow product mod B^m-1 and return the 137 value e. 138 139 We introduced a correction in such a way that "the value of 140 B^{n+h}-T computed at step 8 cannot exceed B^n-1" (the book reads 141 "2B^n-1"). 142 143 Maximum scratch needed by this branch <= 2*n, but have to fit 3*rn 144 in the scratch, i.e. 3*rn <= 2*n: we require n>4. 145 146 We use a wrapped product modulo B^m-1. NOTE: is there any normalisation 147 problem for the [0] class? It shouldn't: we compute 2*|A*X_h - B^{n+h}| < 148 B^m-1. We may get [0] if and only if we get AX_h = B^{n+h}. This can 149 happen only if A=B^{n}/2, but this implies X_h = B^{h}*2-1 i.e., AX_h = 150 B^{n+h} - A, then we get into the "negative" branch, where X_h is not 151 incremented (because A < B^n). 152 153 FIXME: the scratch for mulmod_bnm1 does not currently fit in the scratch, it 154 is allocated apart. 155 */ 156 157 mp_limb_t 158 mpn_ni_invertappr (mp_ptr ip, mp_srcptr dp, mp_size_t n, mp_ptr scratch) 159 { 160 mp_limb_t cy; 161 mp_size_t rn, mn; 162 mp_size_t sizes[NPOWS], *sizp; 163 mp_ptr tp; 164 TMP_DECL; 165 #define xp scratch 166 167 ASSERT (n > 4); 168 ASSERT (dp[n-1] & GMP_NUMB_HIGHBIT); 169 ASSERT (! MPN_OVERLAP_P (ip, n, dp, n)); 170 ASSERT (! MPN_OVERLAP_P (ip, n, scratch, mpn_invertappr_itch(n))); 171 ASSERT (! MPN_OVERLAP_P (dp, n, scratch, mpn_invertappr_itch(n))); 172 173 /* Compute the computation precisions from highest to lowest, leaving the 174 base case size in 'rn'. */ 175 sizp = sizes; 176 rn = n; 177 do { 178 *sizp = rn; 179 rn = (rn >> 1) + 1; 180 ++sizp; 181 } while (ABOVE_THRESHOLD (rn, INV_NEWTON_THRESHOLD)); 182 183 /* We search the inverse of 0.{dp,n}, we compute it as 1.{ip,n} */ 184 dp += n; 185 ip += n; 186 187 /* Compute a base value of rn limbs. */ 188 mpn_bc_invertappr (ip - rn, dp - rn, rn, scratch); 189 190 TMP_MARK; 191 192 if (ABOVE_THRESHOLD (n, INV_MULMOD_BNM1_THRESHOLD)) 193 { 194 mn = mpn_mulmod_bnm1_next_size (n + 1); 195 tp = TMP_ALLOC_LIMBS (mpn_mulmod_bnm1_itch (mn, n, (n >> 1) + 1)); 196 } 197 /* Use Newton's iterations to get the desired precision.*/ 198 199 while (1) { 200 n = *--sizp; 201 /* 202 v n v 203 +----+--+ 204 ^ rn ^ 205 */ 206 207 /* Compute i_jd . */ 208 if (BELOW_THRESHOLD (n, INV_MULMOD_BNM1_THRESHOLD) 209 || ((mn = mpn_mulmod_bnm1_next_size (n + 1)) > (n + rn))) { 210 /* FIXME: We do only need {xp,n+1}*/ 211 mpn_mul (xp, dp - n, n, ip - rn, rn); 212 mpn_add_n (xp + rn, xp + rn, dp - n, n - rn + 1); 213 cy = CNST_LIMB(1); /* Remember we truncated, Mod B^(n+1) */ 214 /* We computed (truncated) {xp,n+1} <- 1.{ip,rn} * 0.{dp,n} */ 215 } else { /* Use B^mn-1 wraparound */ 216 mpn_mulmod_bnm1 (xp, mn, dp - n, n, ip - rn, rn, tp); 217 /* We computed {xp,mn} <- {ip,rn} * {dp,n} mod (B^mn-1) */ 218 /* We know that 2*|ip*dp + dp*B^rn - B^{rn+n}| < B^mn-1 */ 219 /* Add dp*B^rn mod (B^mn-1) */ 220 ASSERT (n >= mn - rn); 221 cy = mpn_add_n (xp + rn, xp + rn, dp - n, mn - rn); 222 cy = mpn_add_nc (xp, xp, dp - (n - (mn - rn)), n - (mn - rn), cy); 223 /* Subtract B^{rn+n}, maybe only compensate the carry*/ 224 xp[mn] = CNST_LIMB (1); /* set a limit for DECR_U */ 225 MPN_DECR_U (xp + rn + n - mn, 2 * mn + 1 - rn - n, CNST_LIMB (1) - cy); 226 MPN_DECR_U (xp, mn, CNST_LIMB (1) - xp[mn]); /* if DECR_U eroded xp[mn] */ 227 cy = CNST_LIMB(0); /* Remember we are working Mod B^mn-1 */ 228 } 229 230 if (xp[n] < CNST_LIMB (2)) { /* "positive" residue class */ 231 cy = xp[n]; /* 0 <= cy <= 1 here. */ 232 #if HAVE_NATIVE_mpn_sublsh1_n 233 if (cy++) { 234 if (mpn_cmp (xp, dp - n, n) > 0) { 235 mp_limb_t chk; 236 chk = mpn_sublsh1_n (xp, xp, dp - n, n); 237 ASSERT (chk == xp[n]); 238 ++ cy; 239 } else 240 ASSERT_CARRY (mpn_sub_n (xp, xp, dp - n, n)); 241 } 242 #else /* no mpn_sublsh1_n*/ 243 if (cy++ && !mpn_sub_n (xp, xp, dp - n, n)) { 244 ASSERT_CARRY (mpn_sub_n (xp, xp, dp - n, n)); 245 ++cy; 246 } 247 #endif 248 /* 1 <= cy <= 3 here. */ 249 #if HAVE_NATIVE_mpn_rsblsh1_n 250 if (mpn_cmp (xp, dp - n, n) > 0) { 251 ASSERT_NOCARRY (mpn_rsblsh1_n (xp + n, xp, dp - n, n)); 252 ++cy; 253 } else 254 ASSERT_NOCARRY (mpn_sub_nc (xp + 2 * n - rn, dp - rn, xp + n - rn, rn, mpn_cmp (xp, dp - n, n - rn) > 0)); 255 #else /* no mpn_rsblsh1_n*/ 256 if (mpn_cmp (xp, dp - n, n) > 0) { 257 ASSERT_NOCARRY (mpn_sub_n (xp, xp, dp - n, n)); 258 ++cy; 259 } 260 ASSERT_NOCARRY (mpn_sub_nc (xp + 2 * n - rn, dp - rn, xp + n - rn, rn, mpn_cmp (xp, dp - n, n - rn) > 0)); 261 #endif 262 MPN_DECR_U(ip - rn, rn, cy); /* 1 <= cy <= 4 here. */ 263 } else { /* "negative" residue class */ 264 ASSERT (xp[n] >= GMP_NUMB_MAX - CNST_LIMB(1)); 265 MPN_DECR_U(xp, n + 1, cy); 266 if (xp[n] != GMP_NUMB_MAX) { 267 MPN_INCR_U(ip - rn, rn, CNST_LIMB (1)); 268 ASSERT_CARRY (mpn_add_n (xp, xp, dp - n, n)); 269 } 270 mpn_com (xp + 2 * n - rn, xp + n - rn, rn); 271 } 272 273 /* Compute x_ju_j. FIXME:We need {xp+rn,rn}, mulhi? */ 274 mpn_mul_n (xp, xp + 2 * n - rn, ip - rn, rn); 275 cy = mpn_add_n (xp + rn, xp + rn, xp + 2 * n - rn, 2 * rn - n); 276 cy = mpn_add_nc (ip - n, xp + 3 * rn - n, xp + n + rn, n - rn, cy); 277 MPN_INCR_U (ip - rn, rn, cy); 278 if (sizp == sizes) { /* Get out of the cycle */ 279 /* Check for possible carry propagation from below. */ 280 cy = xp[3 * rn - n - 1] > GMP_NUMB_MAX - CNST_LIMB (7); /* Be conservative. */ 281 /* cy = mpn_add_1 (xp + rn, xp + rn, 2*rn - n, 4); */ 282 break; 283 } 284 rn = n; 285 } 286 TMP_FREE; 287 288 return cy; 289 #undef xp 290 } 291 292 mp_limb_t 293 mpn_invertappr (mp_ptr ip, mp_srcptr dp, mp_size_t n, mp_ptr scratch) 294 { 295 ASSERT (n > 0); 296 ASSERT (dp[n-1] & GMP_NUMB_HIGHBIT); 297 ASSERT (! MPN_OVERLAP_P (ip, n, dp, n)); 298 ASSERT (! MPN_OVERLAP_P (ip, n, scratch, mpn_invertappr_itch(n))); 299 ASSERT (! MPN_OVERLAP_P (dp, n, scratch, mpn_invertappr_itch(n))); 300 301 if (BELOW_THRESHOLD (n, INV_NEWTON_THRESHOLD)) 302 return mpn_bc_invertappr (ip, dp, n, scratch); 303 else 304 return mpn_ni_invertappr (ip, dp, n, scratch); 305 } 306