/* mpfr_compound_si --- compound(x,n) = (1+x)^n Copyright 2021-2023 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. */ #define MPFR_NEED_LONGLONG_H /* needed for MPFR_INT_CEIL_LOG2 */ #include "mpfr-impl.h" /* assuming |(1+x)^n - 1| < 1/4*ulp(1), return correct rounding, where s is the sign of n*log2(1+x) */ static int mpfr_compound_near_one (mpfr_ptr y, int s, mpfr_rnd_t rnd_mode) { mpfr_set_ui (y, 1, rnd_mode); /* exact */ if (rnd_mode == MPFR_RNDN || rnd_mode == MPFR_RNDF || (s > 0 && (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD)) || (s < 0 && (rnd_mode == MPFR_RNDA || rnd_mode == MPFR_RNDU))) { /* round toward 1 */ return -s; } else if (s > 0) /* necessarily RNDA or RNDU */ { /* round toward +Inf */ mpfr_nextabove (y); return +1; } else /* necessarily s < 0 and RNDZ or RNDD */ { /* round toward 0 */ mpfr_nextbelow (y); return -1; } } /* put in y the correctly rounded value of (1+x)^n */ int mpfr_compound_si (mpfr_ptr y, mpfr_srcptr x, long n, mpfr_rnd_t rnd_mode) { int inexact, compared, k, nloop; mpfr_t t, u; mpfr_prec_t py, prec, extra; mpfr_rnd_t rnd1; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pd]=%.*Rg n=%ld rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, n, rnd_mode), ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); /* Special cases */ if (MPFR_IS_SINGULAR (x)) { if (MPFR_IS_INF (x) && MPFR_IS_NEG (x)) { /* compound(-Inf,n) is NaN */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (n == 0 || MPFR_IS_ZERO (x)) { /* compound(x,0) = 1 for x >= -1 or NaN (the only special value of x that is not concerned is -Inf, already handled); compound(0,n) = 1 */ return mpfr_set_ui (y, 1, rnd_mode); } else if (MPFR_IS_NAN (x)) { /* compound(NaN,n) is NaN, except for n = 0, already handled. */ MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_INF (x)) /* x = +Inf */ { MPFR_ASSERTD (MPFR_IS_POS (x)); if (n < 0) /* (1+Inf)^n = +0 for n < 0 */ MPFR_SET_ZERO (y); else /* n > 0: (1+Inf)^n = +Inf */ MPFR_SET_INF (y); MPFR_SET_POS (y); MPFR_RET (0); /* exact 0 or infinity */ } } /* (1+x)^n = NaN for x < -1 */ compared = mpfr_cmp_si (x, -1); if (compared < 0) { MPFR_SET_NAN (y); MPFR_RET_NAN; } /* compound(x,0) gives 1 for x >= 1 */ if (n == 0) return mpfr_set_ui (y, 1, rnd_mode); if (compared == 0) { if (n < 0) { /* compound(-1,n) = +Inf with divide-by-zero exception */ MPFR_SET_INF (y); MPFR_SET_POS (y); MPFR_SET_DIVBY0 (); MPFR_RET (0); } else { /* compound(-1,n) = +0 */ MPFR_SET_ZERO (y); MPFR_SET_POS (y); MPFR_RET (0); } } if (n == 1) return mpfr_add_ui (y, x, 1, rnd_mode); MPFR_SAVE_EXPO_MARK (expo); py = MPFR_GET_PREC (y); prec = py + MPFR_INT_CEIL_LOG2 (py) + 6; mpfr_init2 (t, prec); mpfr_init2 (u, prec); k = MPFR_INT_CEIL_LOG2(SAFE_ABS (unsigned long, n)); /* thus |n| <= 2^k */ /* We compute u=log2p1(x) with prec+extra bits, since we lose some bits in 2^u. */ extra = 0; rnd1 = VSIGN (n) == MPFR_SIGN (x) ? MPFR_RNDD : MPFR_RNDU; MPFR_ZIV_INIT (loop, prec); for (nloop = 0; ; nloop++) { unsigned int inex; mpfr_exp_t e, e2, ex; mpfr_prec_t precu = MPFR_ADD_PREC (prec, extra); mpfr_prec_t new_extra; mpfr_rnd_t rnd2; /* We compute (1+x)^n as 2^(n*log2p1(x)), and we round toward 1, thus we round n*log2p1(x) toward 0, thus for x*n > 0 we round log2p1(x) toward -Inf, and for x*n < 0 we round log2p1(x) toward +Inf. */ inex = mpfr_log2p1 (u, x, rnd1) != 0; e = MPFR_GET_EXP (u); /* |u - log2(1+x)| <= ulp(t) = 2^(e-precu) */ inex |= mpfr_mul_si (u, u, n, MPFR_RNDZ) != 0; e2 = MPFR_GET_EXP (u); /* |u - n*log2(1+x)| <= 2^(e2-precu) + |n|*2^(e-precu) <= 2^(e2-precu) + 2^(e+k-precu) <= 2^(e+k+1-precu) where |n| <= 2^k, and e2 is the new exponent of u. */ MPFR_ASSERTD (e2 <= e + k); e += k + 1; MPFR_ASSERTN (e2 <= MPFR_PREC_MAX); new_extra = e2 > 0 ? e2 : 0; /* |u - n*log2(1+x)| <= 2^(e-precu) */ /* detect overflow: since we rounded n*log2p1(x) toward 0, if n*log2p1(x) >= __gmpfr_emax, we are sure there is overflow. */ if (mpfr_cmp_si (u, __gmpfr_emax) >= 0) { MPFR_ZIV_FREE (loop); mpfr_clear (t); mpfr_clear (u); MPFR_SAVE_EXPO_FREE (expo); return mpfr_overflow (y, rnd_mode, 1); } /* detect underflow: similarly, since we rounded n*log2p1(x) toward 0, if n*log2p1(x) < __gmpfr_emin-1, we are sure there is underflow. */ if (mpfr_cmp_si (u, __gmpfr_emin - 1) < 0) { MPFR_ZIV_FREE (loop); mpfr_clear (t); mpfr_clear (u); MPFR_SAVE_EXPO_FREE (expo); return mpfr_underflow (y, rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode, 1); } /* Detect cases where result is 1 or 1+ulp(1) or 1-1/2*ulp(1): |2^u - 1| = |exp(u*log(2)) - 1| <= |u|*log(2) < |u| */ if (nloop == 0 && MPFR_GET_EXP(u) < - py) { /* since ulp(1) = 2^(1-py), we have |u| < 1/4*ulp(1) */ /* mpfr_compound_near_one must be called in the extended exponent range, so that 1 is representable. */ inexact = mpfr_compound_near_one (y, MPFR_SIGN (u), rnd_mode); goto end; } /* round 2^u toward 1 */ rnd2 = MPFR_IS_POS (u) ? MPFR_RNDD : MPFR_RNDU; inex |= mpfr_exp2 (t, u, rnd2) != 0; /* we had |u - n*log2(1+x)| < 2^(e-precu) thus u = n*log2(1+x) + delta with |delta| < 2^(e-precu) then 2^u = (1+x)^n * 2^delta with |delta| < 2^(e-precu). For |delta| < 0.5, |2^delta - 1| <= |delta| thus |t - (1+x)^n| <= ulp(t) + |t|*2^(e-precu) < 2^(EXP(t)-prec) + 2^(EXP(t)+e-precu) */ e = (precu - prec >= e) ? 1 : e + 1 - (precu - prec); /* now |t - (1+x)^n| < 2^(EXP(t)+e-prec) */ if (MPFR_LIKELY (!inex || MPFR_CAN_ROUND (t, prec - e, py, rnd_mode))) break; /* If t fits in the target precision (or with 1 more bit), then we can round, assuming the working precision is large enough, but the above MPFR_CAN_ROUND() will fail because we cannot determine the ternary value. However since we rounded t toward 1, we can determine it. Since the error in the approximation t is at most 2^e ulp(t), this error should be less than 1/2 ulp(y), thus we should have prec - py >= e + 1. */ if (mpfr_min_prec (t) <= py + 1 && prec - py >= e + 1) { /* we add/subtract one ulp to get the correct rounding */ if (rnd2 == MPFR_RNDD) /* t was rounded downwards */ mpfr_nextabove (t); else mpfr_nextbelow (t); break; } /* Detect particular cases where Ziv's strategy may take too much memory and be too long, i.e. when x^n fits in the target precision (+ 1 additional bit for rounding to nearest) and the exact result (1+x)^n is very close to x^n. Necessarily, x is a large even integer and n > 0 (thus n > 1). Since this does not depend on the working precision, we only check this at the first iteration (nloop == 0). Hence the first "if" below and the kx < ex test of the second "if" (x is an even integer iff its least bit 1 has exponent >= 1). The second test of the second "if" corresponds to another simple condition that implies that x^n fits in the target precision. Here are the details: Let k be the minimum length of the significand of x, and x' the odd (integer) significand of x. This means that 2^(k-1) <= x' < 2^k. Thus 2^(n*(k-1)) <= (x')^n < 2^(k*n), and x^n has between n*(k-1)+1 and k*n bits. So x^n can fit into p bits only if p >= n*(k-1)+1, i.e. n*(k-1) <= p-1. Note that x >= 2^k, so that x^n >= 2^(k*n). Since raw overflow has already been detected, k*n cannot overflow if computed with the mpfr_exp_t type. Hence the second test of the second "if", which cannot overflow. */ MPFR_ASSERTD (n < 0 || n > 1); if (nloop == 0 && n > 1 && (ex = MPFR_GET_EXP (x)) >= 17) { mpfr_prec_t kx = mpfr_min_prec (x); mpfr_prec_t p = py + (rnd_mode == MPFR_RNDN); MPFR_LOG_MSG (("Check if x^n fits... n=%ld kx=%Pd p=%Pd\n", n, kx, p)); if (kx < ex && n * (mpfr_exp_t) (kx - 1) <= p - 1) { mpfr_t v; /* Check whether x^n really fits into p bits. */ mpfr_init2 (v, p); inexact = mpfr_pow_ui (v, x, n, MPFR_RNDZ); if (inexact == 0) { MPFR_LOG_MSG (("x^n fits into p bits\n", 0)); /* (x+1)^n = x^n * (1 + 1/x)^n For directed rounding, we can round when (1 + 1/x)^n < 1 + 2^-p, and then the result is x^n, except for rounding up. Indeed, if (1 + 1/x)^n < 1 + 2^-p, 1 <= (x+1)^n < x^n * (1 + 2^-p) = x^n + x^n/2^p < x^n + ulp(x^n). For rounding to nearest, we can round when (1 + 1/x)^n < 1 + 2^-p, and then the result is x^n when x^n fits into p-1 bits, and nextabove(x^n) otherwise. */ mpfr_ui_div (t, 1, x, MPFR_RNDU); mpfr_add_ui (t, t, 1, MPFR_RNDU); mpfr_pow_ui (t, t, n, MPFR_RNDU); mpfr_sub_ui (t, t, 1, MPFR_RNDU); /* t cannot be zero */ if (MPFR_GET_EXP(t) < - py) { mpfr_set (y, v, MPFR_RNDZ); if ((rnd_mode == MPFR_RNDN && mpfr_min_prec (v) == p) || rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA) { /* round up */ mpfr_nextabove (y); inexact = 1; } else inexact = -1; mpfr_clear (v); goto end; } } mpfr_clear (v); } } /* Exact cases like compound(0.5,2) = 9/4 must be detected, since except for 1+x power of 2, the log2p1 above will be inexact, so that in the Ziv test, inexact != 0 and MPFR_CAN_ROUND will fail (even for RNDN, as the ternary value cannot be determined), yielding an infinite loop. For an exact case in precision prec(y), 1+x will necessarily be exact in precision prec(y), thus also in prec(t), where prec(t) >= prec(y), and we can use mpfr_pow_si under this condition (which will also evaluate some non-exact cases). */ if (mpfr_add_ui (t, x, 1, MPFR_RNDZ) == 0) { inexact = mpfr_pow_si (y, t, n, rnd_mode); goto end; } MPFR_ZIV_NEXT (loop, prec); mpfr_set_prec (t, prec); extra = new_extra; mpfr_set_prec (u, MPFR_ADD_PREC (prec, extra)); } inexact = mpfr_set (y, t, rnd_mode); end: MPFR_ZIV_FREE (loop); mpfr_clear (t); mpfr_clear (u); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }