| /* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str |
| |
| Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao projects, INRIA. |
| Contributed by Alain Delplanque and Paul Zimmermann. |
| |
| 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 2.1 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.LIB. If not, write to |
| the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, |
| MA 02110-1301, USA. */ |
| |
| |
| #define MPFR_NEED_LONGLONG_H |
| #include "mpfr-impl.h" |
| |
| /* this function computes an approximation of b^e in {a, n}, with exponent |
| stored in exp_r. The computed value is rounded towards zero (truncated). |
| It returns an integer f such that the final error is bounded by 2^f ulps, |
| that is: |
| a*2^exp_r <= b^e <= 2^exp_r (a + 2^f), |
| where a represents {a, n}, i.e. the integer |
| a[0] + a[1]*B + ... + a[n-1]*B^(n-1) where B=2^BITS_PER_MP_LIMB |
| |
| Return -1 is the result is exact. |
| Return -2 if an overflow occurred in the computation of exp_r. |
| */ |
| |
| long |
| mpfr_mpn_exp (mp_limb_t *a, mp_exp_t *exp_r, int b, mp_exp_t e, size_t n) |
| { |
| mp_limb_t *c, B; |
| mp_exp_t f, h; |
| int i; |
| unsigned long t; /* number of bits in e */ |
| unsigned long bits; |
| size_t n1; |
| unsigned int error; /* (number - 1) of loop a^2b inexact */ |
| /* error == t means no error */ |
| int err_s_a2 = 0; |
| int err_s_ab = 0; /* number of error when shift A^2, AB */ |
| MPFR_TMP_DECL(marker); |
| |
| MPFR_ASSERTN(e > 0); |
| MPFR_ASSERTN((2 <= b) && (b <= 36)); |
| |
| MPFR_TMP_MARK(marker); |
| |
| /* initialization of a, b, f, h */ |
| |
| /* normalize the base */ |
| B = (mp_limb_t) b; |
| count_leading_zeros (h, B); |
| |
| bits = BITS_PER_MP_LIMB - h; |
| |
| B = B << h; |
| h = - h; |
| |
| /* allocate space for A and set it to B */ |
| c = (mp_limb_t*) MPFR_TMP_ALLOC(2 * n * BYTES_PER_MP_LIMB); |
| a [n - 1] = B; |
| MPN_ZERO (a, n - 1); |
| /* initial exponent for A: invariant is A = {a, n} * 2^f */ |
| f = h - (n - 1) * BITS_PER_MP_LIMB; |
| |
| /* determine number of bits in e */ |
| count_leading_zeros (t, (mp_limb_t) e); |
| |
| t = BITS_PER_MP_LIMB - t; /* number of bits of exponent e */ |
| |
| error = t; /* error <= BITS_PER_MP_LIMB */ |
| |
| MPN_ZERO (c, 2 * n); |
| |
| for (i = t - 2; i >= 0; i--) |
| { |
| |
| /* determine precision needed */ |
| bits = n * BITS_PER_MP_LIMB - mpn_scan1 (a, 0); |
| n1 = (n * BITS_PER_MP_LIMB - bits) / BITS_PER_MP_LIMB; |
| |
| /* square of A : {c+2n1, 2(n-n1)} = {a+n1, n-n1}^2 */ |
| mpn_sqr_n (c + 2 * n1, a + n1, n - n1); |
| |
| /* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */ |
| |
| /* check overflow on f */ |
| if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2)) |
| { |
| overflow: |
| MPFR_TMP_FREE(marker); |
| return -2; |
| } |
| /* FIXME: Could f = 2*f + n * BITS_PER_MP_LIMB be used? */ |
| f = 2*f; |
| MPFR_SADD_OVERFLOW (f, f, n * BITS_PER_MP_LIMB, |
| mp_exp_t, mp_exp_unsigned_t, |
| MPFR_EXP_MIN, MPFR_EXP_MAX, |
| goto overflow, goto overflow); |
| if ((c[2*n - 1] & MPFR_LIMB_HIGHBIT) == 0) |
| { |
| /* shift A by one bit to the left */ |
| mpn_lshift (a, c + n, n, 1); |
| a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1); |
| f --; |
| if (error != t) |
| err_s_a2 ++; |
| } |
| else |
| MPN_COPY (a, c + n, n); |
| |
| if ((error == t) && (2 * n1 <= n) && |
| (mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * BITS_PER_MP_LIMB)) |
| error = i; |
| |
| if (e & ((mp_exp_t) 1 << i)) |
| { |
| /* multiply A by B */ |
| c[2 * n - 1] = mpn_mul_1 (c + n - 1, a, n, B); |
| f += h + BITS_PER_MP_LIMB; |
| if ((c[2 * n - 1] & MPFR_LIMB_HIGHBIT) == 0) |
| { /* shift A by one bit to the left */ |
| mpn_lshift (a, c + n, n, 1); |
| a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1); |
| f --; |
| } |
| else |
| { |
| MPN_COPY (a, c + n, n); |
| if (error != t) |
| err_s_ab ++; |
| } |
| if ((error == t) && (c[n - 1] != 0)) |
| error = i; |
| } |
| } |
| |
| MPFR_TMP_FREE(marker); |
| |
| *exp_r = f; |
| |
| if (error == t) |
| return -1; /* result is exact */ |
| else /* error <= t-2 <= BITS_PER_MP_LIMB-2 |
| err_s_ab, err_s_a2 <= t-1 */ |
| { |
| /* if there are p loops after the first inexact result, with |
| j shifts in a^2 and l shifts in a*b, then the final error is |
| at most 2^(p+ceil((j+1)/2)+l+1)*ulp(res). |
| This is bounded by 2^(5/2*t-1/2) where t is the number of bits of e. |
| */ |
| error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */ |
| #if 0 |
| if ((error - 1) >= ((n * BITS_PER_MP_LIMB - 1) / 2)) |
| error = n * BITS_PER_MP_LIMB; /* result is completely wrong: |
| this is very unlikely since error is |
| at most 5/2*log_2(e), and |
| n * BITS_PER_MP_LIMB is at least |
| 3*log_2(e) */ |
| #endif |
| return error; |
| } |
| } |