1/* mpfr_agm -- arithmetic-geometric mean of two floating-point numbers 2 3Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc. 4Contributed by the Arenaire and Cacao projects, INRIA. 5 6This file is part of the GNU MPFR Library. 7 8The GNU MPFR Library is free software; you can redistribute it and/or modify 9it under the terms of the GNU Lesser General Public License as published by 10the Free Software Foundation; either version 3 of the License, or (at your 11option) any later version. 12 13The GNU MPFR Library is distributed in the hope that it will be useful, but 14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16License for more details. 17 18You should have received a copy of the GNU Lesser General Public License 19along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23#define MPFR_NEED_LONGLONG_H 24#include "mpfr-impl.h" 25 26/* agm(x,y) is between x and y, so we don't need to save exponent range */ 27int 28mpfr_agm (mpfr_ptr r, mpfr_srcptr op2, mpfr_srcptr op1, mpfr_rnd_t rnd_mode) 29{ 30 int compare, inexact; 31 mp_size_t s; 32 mpfr_prec_t p, q; 33 mp_limb_t *up, *vp, *ufp, *vfp; 34 mpfr_t u, v, uf, vf, sc1, sc2; 35 mpfr_exp_t scaleop = 0, scaleit; 36 unsigned long n; /* number of iterations */ 37 MPFR_ZIV_DECL (loop); 38 MPFR_TMP_DECL(marker); 39 MPFR_SAVE_EXPO_DECL (expo); 40 41 MPFR_LOG_FUNC (("op2[%#R]=%R op1[%#R]=%R rnd=%d", op2,op2,op1,op1,rnd_mode), 42 ("r[%#R]=%R inexact=%d", r, r, inexact)); 43 44 /* Deal with special values */ 45 if (MPFR_ARE_SINGULAR (op1, op2)) 46 { 47 /* If a or b is NaN, the result is NaN */ 48 if (MPFR_IS_NAN(op1) || MPFR_IS_NAN(op2)) 49 { 50 MPFR_SET_NAN(r); 51 MPFR_RET_NAN; 52 } 53 /* now one of a or b is Inf or 0 */ 54 /* If a and b is +Inf, the result is +Inf. 55 Otherwise if a or b is -Inf or 0, the result is NaN */ 56 else if (MPFR_IS_INF(op1) || MPFR_IS_INF(op2)) 57 { 58 if (MPFR_IS_STRICTPOS(op1) && MPFR_IS_STRICTPOS(op2)) 59 { 60 MPFR_SET_INF(r); 61 MPFR_SET_SAME_SIGN(r, op1); 62 MPFR_RET(0); /* exact */ 63 } 64 else 65 { 66 MPFR_SET_NAN(r); 67 MPFR_RET_NAN; 68 } 69 } 70 else /* a and b are neither NaN nor Inf, and one is zero */ 71 { /* If a or b is 0, the result is +0 since a sqrt is positive */ 72 MPFR_ASSERTD (MPFR_IS_ZERO (op1) || MPFR_IS_ZERO (op2)); 73 MPFR_SET_POS (r); 74 MPFR_SET_ZERO (r); 75 MPFR_RET (0); /* exact */ 76 } 77 } 78 79 /* If a or b is negative (excluding -Infinity), the result is NaN */ 80 if (MPFR_UNLIKELY(MPFR_IS_NEG(op1) || MPFR_IS_NEG(op2))) 81 { 82 MPFR_SET_NAN(r); 83 MPFR_RET_NAN; 84 } 85 86 /* Precision of the following calculus */ 87 q = MPFR_PREC(r); 88 p = q + MPFR_INT_CEIL_LOG2(q) + 15; 89 MPFR_ASSERTD (p >= 7); /* see algorithms.tex */ 90 s = (p - 1) / GMP_NUMB_BITS + 1; 91 92 /* b (op2) and a (op1) are the 2 operands but we want b >= a */ 93 compare = mpfr_cmp (op1, op2); 94 if (MPFR_UNLIKELY( compare == 0 )) 95 { 96 mpfr_set (r, op1, rnd_mode); 97 MPFR_RET (0); /* exact */ 98 } 99 else if (compare > 0) 100 { 101 mpfr_srcptr t = op1; 102 op1 = op2; 103 op2 = t; 104 } 105 106 /* Now b (=op2) > a (=op1) */ 107 108 MPFR_SAVE_EXPO_MARK (expo); 109 110 MPFR_TMP_MARK(marker); 111 112 /* Main loop */ 113 MPFR_ZIV_INIT (loop, p); 114 for (;;) 115 { 116 mpfr_prec_t eq; 117 unsigned long err = 0; /* must be set to 0 at each Ziv iteration */ 118 MPFR_BLOCK_DECL (flags); 119 120 /* Init temporary vars */ 121 MPFR_TMP_INIT (up, u, p, s); 122 MPFR_TMP_INIT (vp, v, p, s); 123 MPFR_TMP_INIT (ufp, uf, p, s); 124 MPFR_TMP_INIT (vfp, vf, p, s); 125 126 /* Calculus of un and vn */ 127 retry: 128 MPFR_BLOCK (flags, 129 mpfr_mul (u, op1, op2, MPFR_RNDN); 130 /* mpfr_mul(...): faster since PREC(op) < PREC(u) */ 131 mpfr_add (v, op1, op2, MPFR_RNDN); 132 /* mpfr_add with !=prec is still good */); 133 if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags))) 134 { 135 mpfr_exp_t e1 , e2; 136 137 MPFR_ASSERTN (scaleop == 0); 138 e1 = MPFR_GET_EXP (op1); 139 e2 = MPFR_GET_EXP (op2); 140 141 /* Let's determine scaleop to avoid an overflow/underflow. */ 142 if (MPFR_OVERFLOW (flags)) 143 { 144 /* Let's recall that emin <= e1 <= e2 <= emax. 145 There has been an overflow. Thus e2 >= emax/2. 146 If the mpfr_mul overflowed, then e1 + e2 > emax. 147 If the mpfr_add overflowed, then e2 = emax. 148 We want: (e1 + scale) + (e2 + scale) <= emax, 149 i.e. scale <= (emax - e1 - e2) / 2. Let's take 150 scale = min(floor((emax - e1 - e2) / 2), -1). 151 This is OK, as: 152 1. emin <= scale <= -1. 153 2. e1 + scale >= emin. Indeed: 154 * If e1 + e2 > emax, then 155 e1 + scale >= e1 + (emax - e1 - e2) / 2 - 1 156 >= (emax + e1 - emax) / 2 - 1 157 >= e1 / 2 - 1 >= emin. 158 * Otherwise, mpfr_mul didn't overflow, therefore 159 mpfr_add overflowed and e2 = emax, so that 160 e1 > emin (see restriction below). 161 e1 + scale > emin - 1, thus e1 + scale >= emin. 162 3. e2 + scale <= emax, since scale < 0. */ 163 if (e1 + e2 > __gmpfr_emax) 164 { 165 scaleop = - (((e1 + e2) - __gmpfr_emax + 1) / 2); 166 MPFR_ASSERTN (scaleop < 0); 167 } 168 else 169 { 170 /* The addition necessarily overflowed. */ 171 MPFR_ASSERTN (e2 == __gmpfr_emax); 172 /* The case where e1 = emin and e2 = emax is not supported 173 here. This would mean that the precision of e2 would be 174 huge (and possibly not supported in practice anyway). */ 175 MPFR_ASSERTN (e1 > __gmpfr_emin); 176 scaleop = -1; 177 } 178 179 } 180 else /* underflow only (in the multiplication) */ 181 { 182 /* We have e1 + e2 <= emin (so, e1 <= e2 <= 0). 183 We want: (e1 + scale) + (e2 + scale) >= emin + 1, 184 i.e. scale >= (emin + 1 - e1 - e2) / 2. let's take 185 scale = ceil((emin + 1 - e1 - e2) / 2). This is OK, as: 186 1. 1 <= scale <= emax. 187 2. e1 + scale >= emin + 1 >= emin. 188 3. e2 + scale <= scale <= emax. */ 189 MPFR_ASSERTN (e1 <= e2 && e2 <= 0); 190 scaleop = (__gmpfr_emin + 2 - e1 - e2) / 2; 191 MPFR_ASSERTN (scaleop > 0); 192 } 193 194 MPFR_ALIAS (sc1, op1, MPFR_SIGN (op1), e1 + scaleop); 195 MPFR_ALIAS (sc2, op2, MPFR_SIGN (op2), e2 + scaleop); 196 op1 = sc1; 197 op2 = sc2; 198 MPFR_LOG_MSG (("Exception in pre-iteration, scale = %" 199 MPFR_EXP_FSPEC "d\n", scaleop)); 200 goto retry; 201 } 202 203 mpfr_clear_flags (); 204 mpfr_sqrt (u, u, MPFR_RNDN); 205 mpfr_div_2ui (v, v, 1, MPFR_RNDN); 206 207 scaleit = 0; 208 n = 1; 209 while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2) 210 { 211 MPFR_BLOCK_DECL (flags2); 212 213 MPFR_LOG_MSG (("Iteration n = %lu\n", n)); 214 215 retry2: 216 mpfr_add (vf, u, v, MPFR_RNDN); /* No overflow? */ 217 mpfr_div_2ui (vf, vf, 1, MPFR_RNDN); 218 /* See proof in algorithms.tex */ 219 if (4*eq > p) 220 { 221 mpfr_t w; 222 MPFR_BLOCK_DECL (flags3); 223 224 MPFR_LOG_MSG (("4*eq > p\n", 0)); 225 226 /* vf = V(k) */ 227 mpfr_init2 (w, (p + 1) / 2); 228 MPFR_BLOCK 229 (flags3, 230 mpfr_sub (w, v, u, MPFR_RNDN); /* e = V(k-1)-U(k-1) */ 231 mpfr_sqr (w, w, MPFR_RNDN); /* e = e^2 */ 232 mpfr_div_2ui (w, w, 4, MPFR_RNDN); /* e*= (1/2)^2*1/4 */ 233 mpfr_div (w, w, vf, MPFR_RNDN); /* 1/4*e^2/V(k) */ 234 ); 235 if (MPFR_LIKELY (! MPFR_UNDERFLOW (flags3))) 236 { 237 mpfr_sub (v, vf, w, MPFR_RNDN); 238 err = MPFR_GET_EXP (vf) - MPFR_GET_EXP (v); /* 0 or 1 */ 239 mpfr_clear (w); 240 break; 241 } 242 /* There has been an underflow because of the cancellation 243 between V(k-1) and U(k-1). Let's use the conventional 244 method. */ 245 MPFR_LOG_MSG (("4*eq > p -> underflow\n", 0)); 246 mpfr_clear (w); 247 mpfr_clear_underflow (); 248 } 249 /* U(k) increases, so that U.V can overflow (but not underflow). */ 250 MPFR_BLOCK (flags2, mpfr_mul (uf, u, v, MPFR_RNDN);); 251 if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags2))) 252 { 253 mpfr_exp_t scale2; 254 255 scale2 = - (((MPFR_GET_EXP (u) + MPFR_GET_EXP (v)) 256 - __gmpfr_emax + 1) / 2); 257 MPFR_EXP (u) += scale2; 258 MPFR_EXP (v) += scale2; 259 scaleit += scale2; 260 MPFR_LOG_MSG (("Overflow in iteration n = %lu, scaleit = %" 261 MPFR_EXP_FSPEC "d (%" MPFR_EXP_FSPEC "d)\n", 262 n, scaleit, scale2)); 263 mpfr_clear_overflow (); 264 goto retry2; 265 } 266 mpfr_sqrt (u, uf, MPFR_RNDN); 267 mpfr_swap (v, vf); 268 n ++; 269 } 270 271 MPFR_LOG_MSG (("End of iterations (n = %lu)\n", n)); 272 273 /* the error on v is bounded by (18n+51) ulps, or twice if there 274 was an exponent loss in the final subtraction */ 275 err += MPFR_INT_CEIL_LOG2(18 * n + 51); /* 18n+51 should not overflow 276 since n is about log(p) */ 277 /* we should have n+2 <= 2^(p/4) [see algorithms.tex] */ 278 if (MPFR_LIKELY (MPFR_INT_CEIL_LOG2(n + 2) <= p / 4 && 279 MPFR_CAN_ROUND (v, p - err, q, rnd_mode))) 280 break; /* Stop the loop */ 281 282 /* Next iteration */ 283 MPFR_ZIV_NEXT (loop, p); 284 s = (p - 1) / GMP_NUMB_BITS + 1; 285 } 286 MPFR_ZIV_FREE (loop); 287 288 if (MPFR_UNLIKELY ((__gmpfr_flags & (MPFR_FLAGS_ALL ^ MPFR_FLAGS_INEXACT)) 289 != 0)) 290 { 291 MPFR_ASSERTN (! mpfr_overflow_p ()); /* since mpfr_clear_flags */ 292 MPFR_ASSERTN (! mpfr_underflow_p ()); /* since mpfr_clear_flags */ 293 MPFR_ASSERTN (! mpfr_nanflag_p ()); /* since mpfr_clear_flags */ 294 } 295 296 /* Setting of the result */ 297 inexact = mpfr_set (r, v, rnd_mode); 298 MPFR_EXP (r) -= scaleop + scaleit; 299 300 /* Let's clean */ 301 MPFR_TMP_FREE(marker); 302 303 MPFR_SAVE_EXPO_FREE (expo); 304 /* From the definition of the AGM, underflow and overflow 305 are not possible. */ 306 return mpfr_check_range (r, inexact, rnd_mode); 307 /* agm(u,v) can be exact for u, v rational only for u=v. 308 Proof (due to Nicolas Brisebarre): it suffices to consider 309 u=1 and v<1. Then 1/AGM(1,v) = 2F1(1/2,1/2,1;1-v^2), 310 and a theorem due to G.V. Chudnovsky states that for x a 311 non-zero algebraic number with |x|<1, then 312 2F1(1/2,1/2,1;x) and 2F1(-1/2,1/2,1;x) are algebraically 313 independent over Q. */ 314} 315