diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/cbrt.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/cbrt.c | 104 |
1 files changed, 64 insertions, 40 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/cbrt.c b/Build/source/libs/mpfr/mpfr-src/src/cbrt.c index 50301761348..85f6cd2fb91 100644 --- a/Build/source/libs/mpfr/mpfr-src/src/cbrt.c +++ b/Build/source/libs/mpfr/mpfr-src/src/cbrt.c @@ -1,6 +1,6 @@ /* mpfr_cbrt -- cube root function. -Copyright 2002-2019 Free Software Foundation, Inc. +Copyright 2002-2020 Free Software Foundation, Inc. Contributed by the AriC and Caramba projects, INRIA. This file is part of the GNU MPFR Library. @@ -23,31 +23,35 @@ https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., #define MPFR_NEED_LONGLONG_H #include "mpfr-impl.h" - /* The computation of y = x^(1/3) is done as follows: +/* The computation of y = x^(1/3) is done as follows. - Let x = sign * m * 2^(3*e) where m is an integer + Let n = PREC(y), or PREC(y) + 1 if the rounding mode is MPFR_RNDN. + We seek to compute an integer cube root in precision n and the + associated inexact bit (non-zero iff the remainder is non-zero). - with 2^(3n-3) <= m < 2^(3n) where n = PREC(y) + Let us write x, possibly truncated, under the form sign * m * 2^(3*e) + where m is an integer such that 2^(3n-3) <= m < 2^(3n), i.e. m has + between 3n-2 and 3n bits. - and m = s^3 + r where 0 <= r and m < (s+1)^3 + Let s be the integer cube root of m, i.e. the maximum integer such that + m = s^3 + t with t >= 0. Thus 2^(n-1) <= s < 2^n, i.e. s has n bits. - we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(3n-3) - i.e. m must have at least 3n-2 bits + Then |x|^(1/3) = s * 2^e or (s+1) * 2^e depending on the rounding mode, + the sign, and whether s is "inexact" (i.e. t > 0 or the truncation of x + was not equal to x). - then x^(1/3) = s * 2^e if r=0 - x^(1/3) = (s+1) * 2^e if round up - x^(1/3) = (s-1) * 2^e if round down - x^(1/3) = s * 2^e if nearest and r < 3/2*s^2+3/4*s+1/8 - (s+1) * 2^e otherwise - */ + Note: The truncation of x was allowed because any breakpoint has n bits + and its cube has at most 3n bits. Thus the truncation of x cannot yield + a cube root below RNDZ(x^(1/3)) in precision n. [TODO: add details.] +*/ int mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpz_t m; - mpfr_exp_t e, r, sh; - mpfr_prec_t n, size_m, tmp; - int inexact, negative; + mpfr_exp_t e, d, sh; + mpfr_prec_t n, size_m; + int inexact, inexact2, negative, r; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC ( @@ -89,40 +93,55 @@ mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) r = e % 3; if (r < 0) r += 3; + MPFR_ASSERTD (r >= 0 && r < 3 && (e - r) % 3 == 0); + /* x = (m*2^r) * 2^(e-r) = (m*2^r) * 2^(3*q) */ + MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d r=%d\n", (mpfr_eexp_t) e, r)); + MPFR_MPZ_SIZEINBASE2 (size_m, m); n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN); - /* we want 3*n-2 <= size_m + 3*sh + r <= 3*n - i.e. 3*sh + size_m + r <= 3*n */ - sh = (3 * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / 3; - sh = 3 * sh + r; - if (sh >= 0) + /* We will need to multiply m by 2^(r'), truncated if r' < 0, and + subtract r' from e, so that m has between 3n-2 and 3n bits and + e becomes a multiple of 3. + Since r = e % 3, we write r' = 3 * sh + r. + We want 3 * n - 2 <= size_m + 3 * sh + r <= 3 * n. + Let d = 3 * n - size_m - r. Thus we want 0 <= d - 3 * sh <= 2, + i.e. sh = floor(d/3). */ + d = 3 * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r; + sh = d >= 0 ? d / 3 : - ((2 - d) / 3); /* floor(d/3) */ + r += 3 * sh; /* denoted r' above */ + + e -= r; + MPFR_ASSERTD (e % 3 == 0); + e /= 3; + + inexact = 0; + + if (r > 0) { - mpz_mul_2exp (m, m, sh); - e = e - sh; + mpz_mul_2exp (m, m, r); } - else if (r > 0) + else if (r < 0) { - mpz_mul_2exp (m, m, r); - e = e - r; + r = -r; + inexact = mpz_scan1 (m, 0) < r; + mpz_fdiv_q_2exp (m, m, r); } - /* invariant: x = m*2^e, with e divisible by 3 */ - /* we reuse the variable m to store the cube root, since it is not needed any more: we just need to know if the root is exact */ - inexact = mpz_root (m, m, 3) == 0; + inexact = ! mpz_root (m, m, 3) || inexact; - MPFR_MPZ_SIZEINBASE2 (tmp, m); - sh = tmp - n; - if (sh > 0) /* we have to flush to 0 the last sh bits from m */ - { - inexact = inexact || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh); - mpz_fdiv_q_2exp (m, m, sh); - e += 3 * sh; - } +#if MPFR_WANT_ASSERT > 0 + { + mpfr_prec_t tmp; + + MPFR_MPZ_SIZEINBASE2 (tmp, m); + MPFR_ASSERTN (tmp == n); + } +#endif if (inexact) { @@ -130,7 +149,10 @@ mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) rnd_mode = MPFR_INVERT_RND (rnd_mode); if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0))) - inexact = 1, mpz_add_ui (m, m, 1); + { + inexact = 1; + mpz_add_ui (m, m, 1); + } else inexact = -1; } @@ -138,8 +160,10 @@ mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) /* either inexact is not zero, and the conversion is exact, i.e. inexact is not changed; or inexact=0, and inexact is set only when rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */ - inexact += mpfr_set_z (y, m, MPFR_RNDN); - MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / 3); + inexact2 = mpfr_set_z (y, m, MPFR_RNDN); + MPFR_ASSERTD (inexact == 0 || inexact2 == 0); + inexact += inexact2; + MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e); if (negative) { |