int mpfr_sub_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int u, mp_rnd_t rnd_mode) { if (MPFR_LIKELY (u != 0)) /* if u=0, do nothing */ { mpfr_t uu; mp_limb_t up[1]; unsigned long cnt; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_TMP_INIT1 (up, uu, BITS_PER_MP_LIMB); MPFR_ASSERTN (u == (mp_limb_t) u); count_leading_zeros (cnt, (mp_limb_t) u); *up = (mp_limb_t) u << cnt; /* Optimization note: Exponent save/restore operations may be removed if mpfr_sub works even when uu is out-of-range. */ MPFR_SAVE_EXPO_MARK (expo); MPFR_SET_EXP (uu, BITS_PER_MP_LIMB - cnt); inex = mpfr_sub (y, x, uu, rnd_mode); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inex, rnd_mode); } else return mpfr_set (y, x, rnd_mode); }
/* This function does not change the flags. */ static void mpfr_get_zexp (mpz_ptr ez, mpfr_srcptr x) { mpz_init (ez); if (MPFR_IS_UBF (x)) mpz_set (ez, MPFR_ZEXP (x)); else { mp_limb_t e_limb[MPFR_EXP_LIMB_SIZE]; mpfr_t e; int inex; MPFR_SAVE_EXPO_DECL (expo); /* TODO: Once this has been tested, optimize based on whether _MPFR_EXP_FORMAT <= 3. */ MPFR_TMP_INIT1 (e_limb, e, sizeof (mpfr_exp_t) * CHAR_BIT); MPFR_SAVE_EXPO_MARK (expo); MPFR_DBGRES (inex = mpfr_set_exp_t (e, MPFR_GET_EXP (x), MPFR_RNDN)); MPFR_ASSERTD (inex == 0); MPFR_DBGRES (inex = mpfr_get_z (ez, e, MPFR_RNDN)); MPFR_ASSERTD (inex == 0); MPFR_SAVE_EXPO_FREE (expo); } }
int mpfr_sqrt_ui (mpfr_ptr r, unsigned long u, mpfr_rnd_t rnd_mode) { if (u) { mpfr_t uu; mp_limb_t up[1]; unsigned long cnt; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_TMP_INIT1 (up, uu, GMP_NUMB_BITS); MPFR_ASSERTN (u == (mp_limb_t) u); count_leading_zeros (cnt, (mp_limb_t) u); *up = (mp_limb_t) u << cnt; MPFR_SAVE_EXPO_MARK (expo); MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt); inex = mpfr_sqrt(r, uu, rnd_mode); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range(r, inex, rnd_mode); } else /* sqrt(0) = 0 */ { MPFR_SET_ZERO(r); MPFR_SET_POS(r); MPFR_RET(0); } }
int mpfr_add_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long int u, mpfr_rnd_t rnd_mode) { MPFR_LOG_FUNC (("x[%Pu]=%.*Rg u=%lu rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, u, rnd_mode), ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y)); if (MPFR_LIKELY(u != 0) ) /* if u=0, do nothing */ { mpfr_t uu; mp_limb_t up[1]; unsigned long cnt; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_TMP_INIT1 (up, uu, GMP_NUMB_BITS); MPFR_ASSERTD (u == (mp_limb_t) u); count_leading_zeros(cnt, (mp_limb_t) u); up[0] = (mp_limb_t) u << cnt; /* Optimization note: Exponent save/restore operations may be removed if mpfr_add works even when uu is out-of-range. */ MPFR_SAVE_EXPO_MARK (expo); MPFR_SET_EXP (uu, GMP_NUMB_BITS - cnt); inex = mpfr_add(y, x, uu, rnd_mode); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range(y, inex, rnd_mode); } else /* (unsigned long) 0 is assumed to be a real 0 (unsigned) */ return mpfr_set (y, x, rnd_mode); }
int mpfr_mul_d (mpfr_ptr a, mpfr_srcptr b, double c, mpfr_rnd_t rnd_mode) { int inexact; mpfr_t d; mp_limb_t tmp_man[MPFR_LIMBS_PER_DOUBLE]; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("b[%Pu]=%.*Rg c=%.20g rnd=%d", mpfr_get_prec(b), mpfr_log_prec, b, c, rnd_mode), ("a[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (a), mpfr_log_prec, a, inexact)); MPFR_SAVE_EXPO_MARK (expo); MPFR_TMP_INIT1(tmp_man, d, IEEE_DBL_MANT_DIG); inexact = mpfr_set_d (d, c, rnd_mode); MPFR_ASSERTD (inexact == 0); MPFR_CLEAR_FLAGS (); inexact = mpfr_mul (a, b, d, rnd_mode); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (a, inexact, rnd_mode); }
int mpfr_d_div (mpfr_ptr a, double b, mpfr_srcptr c, mpfr_rnd_t rnd_mode) { int inexact; mpfr_t d; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC ( ("b=%.20g c[%Pu]=%*.Rg rnd=%d", b, mpfr_get_prec (c), mpfr_log_prec, c, rnd_mode), ("a[%Pu]=%*.Rg", mpfr_get_prec (a), mpfr_log_prec, a)); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (d, IEEE_DBL_MANT_DIG); inexact = mpfr_set_d (d, b, rnd_mode); MPFR_ASSERTN (inexact == 0); mpfr_clear_flags (); inexact = mpfr_div (a, d, c, rnd_mode); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); mpfr_clear(d); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (a, inexact, rnd_mode); }
int mpfr_mul_d (mpfr_ptr a, mpfr_srcptr b, double c, mpfr_rnd_t rnd_mode) { int inexact; mpfr_t d; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("b[%Pu]=%.*Rg c=%.20g rnd=%d", mpfr_get_prec(b), mpfr_log_prec, b, c, rnd_mode), ("a[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (a), mpfr_get_prec, a, inexact)); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (d, IEEE_DBL_MANT_DIG); inexact = mpfr_set_d (d, c, rnd_mode); MPFR_ASSERTN (inexact == 0); mpfr_clear_flags (); inexact = mpfr_mul (a, b, d, rnd_mode); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); mpfr_clear(d); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (a, inexact, rnd_mode); }
/* Convert an mpz_t to an mpfr_exp_t, restricted to the interval [MPFR_EXP_MIN,MPFR_EXP_MAX]. */ mpfr_exp_t mpfr_ubf_zexp2exp (mpz_ptr ez) { mp_size_t n; mpfr_eexp_t e; mpfr_t d; int inex; MPFR_SAVE_EXPO_DECL (expo); n = ABSIZ (ez); /* limb size of ez */ if (n == 0) return 0; MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (d, n * GMP_NUMB_BITS); MPFR_DBGRES (inex = mpfr_set_z (d, ez, MPFR_RNDN)); MPFR_ASSERTD (inex == 0); e = mpfr_get_exp_t (d, MPFR_RNDZ); mpfr_clear (d); MPFR_SAVE_EXPO_FREE (expo); if (MPFR_UNLIKELY (e < MPFR_EXP_MIN)) return MPFR_EXP_MIN; if (MPFR_UNLIKELY (e > MPFR_EXP_MAX)) return MPFR_EXP_MAX; return e; }
int mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, 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_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg k=%lu rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); if (MPFR_UNLIKELY (k <= 1)) { if (k < 1) /* k==0 => y=x^(1/0)=x^(+Inf) */ #if 0 /* For 0 <= x < 1 => +0. For x = 1 => 1. For x > 1, => +Inf. For x < 0 => NaN. */ { if (MPFR_IS_NEG (x) && !MPFR_IS_ZERO (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } inexact = mpfr_cmp (x, __gmpfr_one); if (inexact == 0) return mpfr_set_ui (y, 1, rnd_mode); /* 1 may be Out of Range */ else if (inexact < 0) return mpfr_set_ui (y, 0, rnd_mode); /* 0+ */ else { mpfr_set_inf (y, 1); return 0; } } #endif { MPFR_SET_NAN (y); MPFR_RET_NAN; } else /* y =x^(1/1)=x */ return mpfr_set (y, x, rnd_mode); }
/* special code for IEEE 754 little-endian extended format */ long double mpfr_get_ld (mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_long_double_t ld; mpfr_t tmp; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (tmp, MPFR_LDBL_MANT_DIG); inex = mpfr_set (tmp, x, rnd_mode); mpfr_set_emin (-16382-63); mpfr_set_emax (16384); mpfr_subnormalize (tmp, mpfr_check_range (tmp, inex, rnd_mode), rnd_mode); mpfr_prec_round (tmp, 64, MPFR_RNDZ); /* exact */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (tmp))) ld.ld = (long double) mpfr_get_d (tmp, rnd_mode); else { mp_limb_t *tmpmant; mpfr_exp_t e, denorm; tmpmant = MPFR_MANT (tmp); e = MPFR_GET_EXP (tmp); /* The smallest positive normal number is 2^(-16382), which is 0.5*2^(-16381) in MPFR, thus any exponent <= -16382 corresponds to a subnormal number. The smallest positive subnormal number is 2^(-16445) which is 0.5*2^(-16444) in MPFR thus 0 <= denorm <= 63. */ denorm = MPFR_UNLIKELY (e <= -16382) ? - e - 16382 + 1 : 0; MPFR_ASSERTD (0 <= denorm && denorm < 64); #if GMP_NUMB_BITS >= 64 ld.s.manl = (tmpmant[0] >> denorm); ld.s.manh = (tmpmant[0] >> denorm) >> 32; #elif GMP_NUMB_BITS == 32 if (MPFR_LIKELY (denorm == 0)) { ld.s.manl = tmpmant[0]; ld.s.manh = tmpmant[1]; } else if (denorm < 32) { ld.s.manl = (tmpmant[0] >> denorm) | (tmpmant[1] << (32 - denorm)); ld.s.manh = tmpmant[1] >> denorm; } else /* 32 <= denorm < 64 */ {
int mpfr_ui_pow (mpfr_ptr y, unsigned long int n, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_t t; int inexact; mp_limb_t tmp_mant[(sizeof (n) - 1) / sizeof (mp_limb_t) + 1]; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); MPFR_TMP_INIT1(tmp_mant, t, sizeof(n) * CHAR_BIT); inexact = mpfr_set_ui (t, n, MPFR_RNDN); MPFR_ASSERTD (inexact == 0); inexact = mpfr_pow (y, t, x, rnd_mode); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }
int mpfr_get_z (mpz_ptr z, mpfr_srcptr f, mpfr_rnd_t rnd) { int inex; mpfr_t r; mpfr_exp_t exp; MPFR_SAVE_EXPO_DECL (expo); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (f))) { if (MPFR_UNLIKELY (MPFR_NOTZERO (f))) MPFR_SET_ERANGEFLAG (); mpz_set_ui (z, 0); /* The ternary value is 0 even for infinity. Giving the rounding direction in this case would not make much sense anyway, and the direction would not necessarily match rnd. */ return 0; } MPFR_SAVE_EXPO_MARK (expo); exp = MPFR_GET_EXP (f); /* if exp <= 0, then |f|<1, thus |o(f)|<=1 */ MPFR_ASSERTN (exp < 0 || exp <= MPFR_PREC_MAX); mpfr_init2 (r, (exp < (mpfr_exp_t) MPFR_PREC_MIN ? MPFR_PREC_MIN : (mpfr_prec_t) exp)); inex = mpfr_rint (r, f, rnd); MPFR_ASSERTN (inex != 1 && inex != -1); /* integral part of f is representable in r */ MPFR_ASSERTN (MPFR_IS_FP (r)); /* The flags from mpfr_rint are the wanted ones. In particular, it sets the inexact flag when necessary. */ MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); exp = mpfr_get_z_2exp (z, r); if (exp >= 0) mpz_mul_2exp (z, z, exp); else mpz_fdiv_q_2exp (z, z, -exp); mpfr_clear (r); MPFR_SAVE_EXPO_FREE (expo); return inex; }
int mpfr_erfc (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) { int inex; mpfr_t tmp; mpfr_exp_t te, err; mpfr_prec_t prec; mpfr_exp_t emin = mpfr_get_emin (); MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex)); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } /* erfc(+inf) = 0+, erfc(-inf) = 2 erfc (0) = 1 */ else if (MPFR_IS_INF (x)) return mpfr_set_ui (y, MPFR_IS_POS (x) ? 0 : 2, rnd); else return mpfr_set_ui (y, 1, rnd); } if (MPFR_SIGN (x) > 0) { /* by default, emin = 1-2^30, thus the smallest representable number is 1/2*2^emin = 2^(-2^30): for x >= 27282, erfc(x) < 2^(-2^30-1), and for x >= 1787897414, erfc(x) < 2^(-2^62-1). */ if ((emin >= -1073741823 && mpfr_cmp_ui (x, 27282) >= 0) || mpfr_cmp_ui (x, 1787897414) >= 0) { /* May be incorrect if MPFR_EMAX_MAX >= 2^62. */ MPFR_ASSERTN ((MPFR_EMAX_MAX >> 31) >> 31 == 0); return mpfr_underflow (y, (rnd == MPFR_RNDN) ? MPFR_RNDZ : rnd, 1); } }
static int pi_div_2ui (mpfr_ptr dest, int i, int neg, mpfr_rnd_t rnd_mode) { int inexact; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); if (neg) /* -PI/2^i */ { inexact = - mpfr_const_pi (dest, MPFR_INVERT_RND (rnd_mode)); MPFR_CHANGE_SIGN (dest); } else /* PI/2^i */ { inexact = mpfr_const_pi (dest, rnd_mode); } mpfr_div_2ui (dest, dest, i, rnd_mode); /* exact */ MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (dest, inexact, rnd_mode); }
int mpfr_cmp_d (mpfr_srcptr b, double d) { mpfr_t tmp; int res; mp_limb_t tmp_man[MPFR_LIMBS_PER_DOUBLE]; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); MPFR_TMP_INIT1(tmp_man, tmp, IEEE_DBL_MANT_DIG); res = mpfr_set_d (tmp, d, MPFR_RNDN); MPFR_ASSERTD (res == 0); MPFR_CLEAR_FLAGS (); res = mpfr_cmp (b, tmp); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); MPFR_SAVE_EXPO_FREE (expo); return res; }
int mpfr_rint_trunc (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode) { if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(u) ) || mpfr_integer_p (u)) return mpfr_set (r, u, rnd_mode); else { mpfr_t tmp; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (tmp, MPFR_PREC (u)); /* trunc(u) is always representable in tmp */ mpfr_trunc (tmp, u); inex = mpfr_set (r, tmp, rnd_mode); mpfr_clear (tmp); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (r, inex, rnd_mode); } }
long double mpfr_get_ld (mpfr_srcptr x, mp_rnd_t rnd_mode) { mpfr_long_double_t ld; mpfr_t tmp; MPFR_SAVE_EXPO_DECL (expo); MPFR_SAVE_EXPO_MARK (expo); mpfr_set_emin (-16382-63); mpfr_set_emax (16383); mpfr_init2 (tmp, MPFR_LDBL_MANT_DIG); mpfr_subnormalize(tmp, mpfr_set (tmp, x, rnd_mode), rnd_mode); mpfr_prec_round (tmp, 64, GMP_RNDZ); /* exact */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (tmp))) ld.ld = (long double) mpfr_get_d (tmp, rnd_mode); else { mp_limb_t *tmpmant; mp_exp_t e, denorm; tmpmant = MPFR_MANT (tmp); e = MPFR_GET_EXP (tmp); denorm = MPFR_UNLIKELY (e < -16382) ? - e - 16382 + 1 : 0; #if BITS_PER_MP_LIMB >= 64 ld.s.manl = (tmpmant[0] >> denorm); ld.s.manh = (tmpmant[0] >> denorm) >> 32; #elif BITS_PER_MP_LIMB == 32 if (MPFR_LIKELY (denorm == 0)) { ld.s.manl = tmpmant[0]; ld.s.manh = tmpmant[1]; } else if (denorm < 32) { ld.s.manl = (tmpmant[0] >> denorm) | (tmpmant[1] << (32 - denorm)); ld.s.manh = tmpmant[1] >> denorm; } else /* 32 <= denorm <= 64 */ {
int mpfr_div_d (mpfr_ptr a, mpfr_srcptr b, double c, mp_rnd_t rnd_mode) { int inexact; mpfr_t d; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("b[%#R]=%R c%.20g rnd=%d", b, b, c, rnd_mode), ("a[%#R]=%R", a, a)); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (d, IEEE_DBL_MANT_DIG); inexact = mpfr_set_d (d, c, rnd_mode); MPFR_ASSERTN (inexact == 0); mpfr_clear_flags (); inexact = mpfr_div (a, b, d, rnd_mode); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); mpfr_clear(d); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (a, inexact, rnd_mode); }
int mpfr_rint_floor (mpfr_ptr r, mpfr_srcptr u, mpfr_rnd_t rnd_mode) { if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(u) ) || mpfr_integer_p (u)) return mpfr_set (r, u, rnd_mode); else { mpfr_t tmp; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_BLOCK_DECL (flags); MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (tmp, MPFR_PREC (u)); /* floor(u) is representable in tmp unless an overflow occurs */ MPFR_BLOCK (flags, mpfr_floor (tmp, u)); inex = (MPFR_OVERFLOW (flags) ? mpfr_overflow (r, rnd_mode, MPFR_SIGN_NEG) : mpfr_set (r, tmp, rnd_mode)); mpfr_clear (tmp); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (r, inex, rnd_mode); } }
/* (y, z) <- (sin(x), cos(x)), return value is 0 iff both results are exact ie, iff x = 0 */ int mpfr_sin_cos (mpfr_ptr y, mpfr_ptr z, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_prec_t prec, m; int neg, reduce; mpfr_t c, xr; mpfr_srcptr xx; mpfr_exp_t err, expx; int inexy, inexz; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_ASSERTN (y != z); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN(x) || MPFR_IS_INF(x)) { MPFR_SET_NAN (y); MPFR_SET_NAN (z); MPFR_RET_NAN; } else /* x is zero */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); /* y = 0, thus exact, but z is inexact in case of underflow or overflow */ inexy = 0; /* y is exact */ inexz = mpfr_set_ui (z, 1, rnd_mode); return INEX(inexy,inexz); } } MPFR_LOG_FUNC (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), ("sin[%Pu]=%.*Rg cos[%Pu]=%.*Rg", mpfr_get_prec(y), mpfr_log_prec, y, mpfr_get_prec (z), mpfr_log_prec, z)); MPFR_SAVE_EXPO_MARK (expo); prec = MAX (MPFR_PREC (y), MPFR_PREC (z)); m = prec + MPFR_INT_CEIL_LOG2 (prec) + 13; expx = MPFR_GET_EXP (x); /* When x is close to 0, say 2^(-k), then there is a cancellation of about 2k bits in 1-cos(x)^2. FIXME: in that case, it would be more efficient to compute sin(x) directly. VL: This is partly done by using MPFR_FAST_COMPUTE_IF_SMALL_INPUT from the mpfr_sin and mpfr_cos functions. Moreover, any overflow on m is avoided. */ if (expx < 0) { /* Warning: in case y = x, and the first call to MPFR_FAST_COMPUTE_IF_SMALL_INPUT succeeds but the second fails, we will have clobbered the original value of x. The workaround is to first compute z = cos(x) in that case, since y and z are different. */ if (y != x) /* y and x differ, thus we can safely try to compute y first */ { MPFR_FAST_COMPUTE_IF_SMALL_INPUT ( y, x, -2 * expx, 2, 0, rnd_mode, { inexy = _inexact; goto small_input; });
/* We use the reflection formula Gamma(1+t) Gamma(1-t) = - Pi t / sin(Pi (1 + t)) in order to treat the case x <= 1, i.e. with x = 1-t, then Gamma(x) = -Pi*(1-x)/sin(Pi*(2-x))/GAMMA(2-x) */ int mpfr_gamma (mpfr_ptr gamma, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_t xp, GammaTrial, tmp, tmp2; mpz_t fact; mpfr_prec_t realprec; int compared, is_integer; int inex = 0; /* 0 means: result gamma not set yet */ MPFR_GROUP_DECL (group); MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), ("gamma[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (gamma), mpfr_log_prec, gamma, inex)); /* Trivial cases */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (gamma); MPFR_RET_NAN; } else if (MPFR_IS_INF (x)) { if (MPFR_IS_NEG (x)) { MPFR_SET_NAN (gamma); MPFR_RET_NAN; } else { MPFR_SET_INF (gamma); MPFR_SET_POS (gamma); MPFR_RET (0); /* exact */ } } else /* x is zero */ { MPFR_ASSERTD(MPFR_IS_ZERO(x)); MPFR_SET_INF(gamma); MPFR_SET_SAME_SIGN(gamma, x); MPFR_SET_DIVBY0 (); MPFR_RET (0); /* exact */ } } /* Check for tiny arguments, where gamma(x) ~ 1/x - euler + .... We know from "Bound on Runs of Zeros and Ones for Algebraic Functions", Proceedings of Arith15, T. Lang and J.-M. Muller, 2001, that the maximal number of consecutive zeroes or ones after the round bit is n-1 for an input of n bits. But we need a more precise lower bound. Assume x has n bits, and 1/x is near a floating-point number y of n+1 bits. We can write x = X*2^e, y = Y/2^f with X, Y integers of n and n+1 bits. Thus X*Y^2^(e-f) is near from 1, i.e., X*Y is near from 2^(f-e). Two cases can happen: (i) either X*Y is exactly 2^(f-e), but this can happen only if X and Y are themselves powers of two, i.e., x is a power of two; (ii) or X*Y is at distance at least one from 2^(f-e), thus |xy-1| >= 2^(e-f), or |y-1/x| >= 2^(e-f)/x = 2^(-f)/X >= 2^(-f-n). Since ufp(y) = 2^(n-f) [ufp = unit in first place], this means that the distance |y-1/x| >= 2^(-2n) ufp(y). Now assuming |gamma(x)-1/x| <= 1, which is true for x <= 1, if 2^(-2n) ufp(y) >= 2, the error is at most 2^(-2n-1) ufp(y), and round(1/x) with precision >= 2n+2 gives the correct result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). A sufficient condition is thus EXP(x) + 2 <= -2 MAX(PREC(x),PREC(Y)). */ if (MPFR_GET_EXP (x) + 2 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(gamma))) { int sign = MPFR_SIGN (x); /* retrieve sign before possible override */ int special; MPFR_BLOCK_DECL (flags); MPFR_SAVE_EXPO_MARK (expo); /* for overflow cases, see below; this needs to be done before x possibly gets overridden. */ special = MPFR_GET_EXP (x) == 1 - MPFR_EMAX_MAX && MPFR_IS_POS_SIGN (sign) && MPFR_IS_LIKE_RNDD (rnd_mode, sign) && mpfr_powerof2_raw (x); MPFR_BLOCK (flags, inex = mpfr_ui_div (gamma, 1, x, rnd_mode)); if (inex == 0) /* x is a power of two */ { /* return RND(1/x - euler) = RND(+/- 2^k - eps) with eps > 0 */ if (rnd_mode == MPFR_RNDN || MPFR_IS_LIKE_RNDU (rnd_mode, sign)) inex = 1; else { mpfr_nextbelow (gamma); inex = -1; } } else if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags))) { /* Overflow in the division 1/x. This is a real overflow, except in RNDZ or RNDD when 1/x = 2^emax, i.e. x = 2^(-emax): due to the "- euler", the rounded value in unbounded exponent range is 0.111...11 * 2^emax (not an overflow). */ if (!special) MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, flags); } MPFR_SAVE_EXPO_FREE (expo); /* Note: an overflow is possible with an infinite result; in this case, the overflow flag will automatically be restored by mpfr_check_range. */ return mpfr_check_range (gamma, inex, rnd_mode); } is_integer = mpfr_integer_p (x); /* gamma(x) for x a negative integer gives NaN */ if (is_integer && MPFR_IS_NEG(x)) { MPFR_SET_NAN (gamma); MPFR_RET_NAN; } compared = mpfr_cmp_ui (x, 1); if (compared == 0) return mpfr_set_ui (gamma, 1, rnd_mode); /* if x is an integer that fits into an unsigned long, use mpfr_fac_ui if argument is not too large. If precision is p, fac_ui costs O(u*p), whereas gamma costs O(p*M(p)), so for u <= M(p), fac_ui should be faster. We approximate here M(p) by p*log(p)^2, which is not a bad guess. Warning: since the generic code does not handle exact cases, we want all cases where gamma(x) is exact to be treated here. */ if (is_integer && mpfr_fits_ulong_p (x, MPFR_RNDN)) { unsigned long int u; mpfr_prec_t p = MPFR_PREC(gamma); u = mpfr_get_ui (x, MPFR_RNDN); if (u < 44787929UL && bits_fac (u - 1) <= p + (rnd_mode == MPFR_RNDN)) /* bits_fac: lower bound on the number of bits of m, where gamma(x) = (u-1)! = m*2^e with m odd. */ return mpfr_fac_ui (gamma, u - 1, rnd_mode); /* if bits_fac(...) > p (resp. p+1 for rounding to nearest), then gamma(x) cannot be exact in precision p (resp. p+1). FIXME: remove the test u < 44787929UL after changing bits_fac to return a mpz_t or mpfr_t. */ } MPFR_SAVE_EXPO_MARK (expo); /* check for overflow: according to (6.1.37) in Abramowitz & Stegun, gamma(x) >= exp(-x) * x^(x-1/2) * sqrt(2*Pi) >= 2 * (x/e)^x / x for x >= 1 */ if (compared > 0) { mpfr_t yp; mpfr_exp_t expxp; MPFR_BLOCK_DECL (flags); /* quick test for the default exponent range */ if (mpfr_get_emax () >= 1073741823UL && MPFR_GET_EXP(x) <= 25) { MPFR_SAVE_EXPO_FREE (expo); return mpfr_gamma_aux (gamma, x, rnd_mode); } /* 1/e rounded down to 53 bits */ #define EXPM1_STR "0.010111100010110101011000110110001011001110111100111" mpfr_init2 (xp, 53); mpfr_init2 (yp, 53); mpfr_set_str_binary (xp, EXPM1_STR); mpfr_mul (xp, x, xp, MPFR_RNDZ); mpfr_sub_ui (yp, x, 2, MPFR_RNDZ); mpfr_pow (xp, xp, yp, MPFR_RNDZ); /* (x/e)^(x-2) */ mpfr_set_str_binary (yp, EXPM1_STR); mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^(x-1) */ mpfr_mul (xp, xp, yp, MPFR_RNDZ); /* x^(x-2) / e^x */ mpfr_mul (xp, xp, x, MPFR_RNDZ); /* lower bound on x^(x-1) / e^x */ MPFR_BLOCK (flags, mpfr_mul_2ui (xp, xp, 1, MPFR_RNDZ)); expxp = MPFR_GET_EXP (xp); mpfr_clear (xp); mpfr_clear (yp); MPFR_SAVE_EXPO_FREE (expo); return MPFR_OVERFLOW (flags) || expxp > __gmpfr_emax ? mpfr_overflow (gamma, rnd_mode, 1) : mpfr_gamma_aux (gamma, x, rnd_mode); } /* now compared < 0 */ /* check for underflow: for x < 1, gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x). Since gamma(2-x) >= 2 * ((2-x)/e)^(2-x) / (2-x), we have |gamma(x)| <= Pi*(1-x)*(2-x)/2/((2-x)/e)^(2-x) / |sin(Pi*(2-x))| <= 12 * ((2-x)/e)^x / |sin(Pi*(2-x))|. To avoid an underflow in ((2-x)/e)^x, we compute the logarithm. */ if (MPFR_IS_NEG(x)) { int underflow = 0, sgn, ck; mpfr_prec_t w; mpfr_init2 (xp, 53); mpfr_init2 (tmp, 53); mpfr_init2 (tmp2, 53); /* we want an upper bound for x * [log(2-x)-1]. since x < 0, we need a lower bound on log(2-x) */ mpfr_ui_sub (xp, 2, x, MPFR_RNDD); mpfr_log (xp, xp, MPFR_RNDD); mpfr_sub_ui (xp, xp, 1, MPFR_RNDD); mpfr_mul (xp, xp, x, MPFR_RNDU); /* we need an upper bound on 1/|sin(Pi*(2-x))|, thus a lower bound on |sin(Pi*(2-x))|. If 2-x is exact, then the error of Pi*(2-x) is (1+u)^2 with u = 2^(-p) thus the error on sin(Pi*(2-x)) is less than 1/2ulp + 3Pi(2-x)u, assuming u <= 1, thus <= u + 3Pi(2-x)u */ w = mpfr_gamma_2_minus_x_exact (x); /* 2-x is exact for prec >= w */ w += 17; /* to get tmp2 small enough */ mpfr_set_prec (tmp, w); mpfr_set_prec (tmp2, w); MPFR_DBGRES (ck = mpfr_ui_sub (tmp, 2, x, MPFR_RNDN)); MPFR_ASSERTD (ck == 0); /* tmp = 2-x exactly */ mpfr_const_pi (tmp2, MPFR_RNDN); mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); /* Pi*(2-x) */ mpfr_sin (tmp, tmp2, MPFR_RNDN); /* sin(Pi*(2-x)) */ sgn = mpfr_sgn (tmp); mpfr_abs (tmp, tmp, MPFR_RNDN); mpfr_mul_ui (tmp2, tmp2, 3, MPFR_RNDU); /* 3Pi(2-x) */ mpfr_add_ui (tmp2, tmp2, 1, MPFR_RNDU); /* 3Pi(2-x)+1 */ mpfr_div_2ui (tmp2, tmp2, mpfr_get_prec (tmp), MPFR_RNDU); /* if tmp2<|tmp|, we get a lower bound */ if (mpfr_cmp (tmp2, tmp) < 0) { mpfr_sub (tmp, tmp, tmp2, MPFR_RNDZ); /* low bnd on |sin(Pi*(2-x))| */ mpfr_ui_div (tmp, 12, tmp, MPFR_RNDU); /* upper bound */ mpfr_log2 (tmp, tmp, MPFR_RNDU); mpfr_add (xp, tmp, xp, MPFR_RNDU); /* The assert below checks that expo.saved_emin - 2 always fits in a long. FIXME if we want to allow mpfr_exp_t to be a long long, for instance. */ MPFR_ASSERTN (MPFR_EMIN_MIN - 2 >= LONG_MIN); underflow = mpfr_cmp_si (xp, expo.saved_emin - 2) <= 0; } mpfr_clear (xp); mpfr_clear (tmp); mpfr_clear (tmp2); if (underflow) /* the sign is the opposite of that of sin(Pi*(2-x)) */ { MPFR_SAVE_EXPO_FREE (expo); return mpfr_underflow (gamma, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, -sgn); } } realprec = MPFR_PREC (gamma); /* we want both 1-x and 2-x to be exact */ { mpfr_prec_t w; w = mpfr_gamma_1_minus_x_exact (x); if (realprec < w) realprec = w; w = mpfr_gamma_2_minus_x_exact (x); if (realprec < w) realprec = w; } realprec = realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20; MPFR_ASSERTD(realprec >= 5); MPFR_GROUP_INIT_4 (group, realprec + MPFR_INT_CEIL_LOG2 (realprec) + 20, xp, tmp, tmp2, GammaTrial); mpz_init (fact); MPFR_ZIV_INIT (loop, realprec); for (;;) { mpfr_exp_t err_g; int ck; MPFR_GROUP_REPREC_4 (group, realprec, xp, tmp, tmp2, GammaTrial); /* reflection formula: gamma(x) = Pi*(x-1)/sin(Pi*(2-x))/gamma(2-x) */ ck = mpfr_ui_sub (xp, 2, x, MPFR_RNDN); /* 2-x, exact */ MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */ mpfr_gamma (tmp, xp, MPFR_RNDN); /* gamma(2-x), error (1+u) */ mpfr_const_pi (tmp2, MPFR_RNDN); /* Pi, error (1+u) */ mpfr_mul (GammaTrial, tmp2, xp, MPFR_RNDN); /* Pi*(2-x), error (1+u)^2 */ err_g = MPFR_GET_EXP(GammaTrial); mpfr_sin (GammaTrial, GammaTrial, MPFR_RNDN); /* sin(Pi*(2-x)) */ /* If tmp is +Inf, we compute exp(lngamma(x)). */ if (mpfr_inf_p (tmp)) { inex = mpfr_explgamma (gamma, x, &expo, tmp, tmp2, rnd_mode); if (inex) goto end; else goto ziv_next; } err_g = err_g + 1 - MPFR_GET_EXP(GammaTrial); /* let g0 the true value of Pi*(2-x), g the computed value. We have g = g0 + h with |h| <= |(1+u^2)-1|*g. Thus sin(g) = sin(g0) + h' with |h'| <= |(1+u^2)-1|*g. The relative error is thus bounded by |(1+u^2)-1|*g/sin(g) <= |(1+u^2)-1|*2^err_g. <= 2.25*u*2^err_g for |u|<=1/4. With the rounding error, this gives (0.5 + 2.25*2^err_g)*u. */ ck = mpfr_sub_ui (xp, x, 1, MPFR_RNDN); /* x-1, exact */ MPFR_ASSERTD(ck == 0); (void) ck; /* use ck to avoid a warning */ mpfr_mul (xp, tmp2, xp, MPFR_RNDN); /* Pi*(x-1), error (1+u)^2 */ mpfr_mul (GammaTrial, GammaTrial, tmp, MPFR_RNDN); /* [1 + (0.5 + 2.25*2^err_g)*u]*(1+u)^2 = 1 + (2.5 + 2.25*2^err_g)*u + (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2. For err_g <= realprec-2, we have (0.5 + 2.25*2^err_g)*u <= 0.5*u + 2.25/4 <= 0.6875 and u^2 <= u/4, thus (0.5 + 2.25*2^err_g)*u*(2u+u^2) + u^2 <= 0.6875*(2u+u/4) + u/4 <= 1.8*u, thus the rel. error is bounded by (4.5 + 2.25*2^err_g)*u. */ mpfr_div (GammaTrial, xp, GammaTrial, MPFR_RNDN); /* the error is of the form (1+u)^3/[1 + (4.5 + 2.25*2^err_g)*u]. For realprec >= 5 and err_g <= realprec-2, [(4.5 + 2.25*2^err_g)*u]^2 <= 0.71, and for |y|<=0.71, 1/(1-y) can be written 1+a*y with a<=4. (1+u)^3 * (1+4*(4.5 + 2.25*2^err_g)*u) = 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (55+27*2^err_g)*u^3 + (18+9*2^err_g)*u^4 <= 1 + (21 + 9*2^err_g)*u + (57+27*2^err_g)*u^2 + (56+28*2^err_g)*u^3 <= 1 + (21 + 9*2^err_g)*u + (59+28*2^err_g)*u^2 <= 1 + (23 + 10*2^err_g)*u. The final error is thus bounded by (23 + 10*2^err_g) ulps, which is <= 2^6 for err_g<=2, and <= 2^(err_g+4) for err_g >= 2. */ err_g = (err_g <= 2) ? 6 : err_g + 4; if (MPFR_LIKELY (MPFR_CAN_ROUND (GammaTrial, realprec - err_g, MPFR_PREC(gamma), rnd_mode))) break; ziv_next: MPFR_ZIV_NEXT (loop, realprec); } end: MPFR_ZIV_FREE (loop); if (inex == 0) inex = mpfr_set (gamma, GammaTrial, rnd_mode); MPFR_GROUP_CLEAR (group); mpz_clear (fact); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (gamma, inex, rnd_mode); }
int mpfr_exp (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { mpfr_exp_t expx; mpfr_prec_t precy; int inexact; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode), ("y[%#R]=%R inexact=%d", y, y, inexact)); if (MPFR_UNLIKELY( MPFR_IS_SINGULAR(x) )) { if (MPFR_IS_NAN(x)) { MPFR_SET_NAN(y); MPFR_RET_NAN; } else if (MPFR_IS_INF(x)) { if (MPFR_IS_POS(x)) MPFR_SET_INF(y); else MPFR_SET_ZERO(y); MPFR_SET_POS(y); MPFR_RET(0); } else { MPFR_ASSERTD(MPFR_IS_ZERO(x)); return mpfr_set_ui (y, 1, rnd_mode); } } /* First, let's detect most overflow and underflow cases. */ { mpfr_t e, bound; /* We must extended the exponent range and save the flags now. */ MPFR_SAVE_EXPO_MARK (expo); mpfr_init2 (e, sizeof (mpfr_exp_t) * CHAR_BIT); mpfr_init2 (bound, 32); inexact = mpfr_set_exp_t (e, expo.saved_emax, MPFR_RNDN); MPFR_ASSERTD (inexact == 0); mpfr_const_log2 (bound, expo.saved_emax < 0 ? MPFR_RNDD : MPFR_RNDU); mpfr_mul (bound, bound, e, MPFR_RNDU); if (MPFR_UNLIKELY (mpfr_cmp (x, bound) >= 0)) { /* x > log(2^emax), thus exp(x) > 2^emax */ mpfr_clears (e, bound, (mpfr_ptr) 0); MPFR_SAVE_EXPO_FREE (expo); return mpfr_overflow (y, rnd_mode, 1); } inexact = mpfr_set_exp_t (e, expo.saved_emin, MPFR_RNDN); MPFR_ASSERTD (inexact == 0); inexact = mpfr_sub_ui (e, e, 2, MPFR_RNDN); MPFR_ASSERTD (inexact == 0); mpfr_const_log2 (bound, expo.saved_emin < 0 ? MPFR_RNDU : MPFR_RNDD); mpfr_mul (bound, bound, e, MPFR_RNDD); if (MPFR_UNLIKELY (mpfr_cmp (x, bound) <= 0)) { /* x < log(2^(emin - 2)), thus exp(x) < 2^(emin - 2) */ mpfr_clears (e, bound, (mpfr_ptr) 0); MPFR_SAVE_EXPO_FREE (expo); return mpfr_underflow (y, rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode, 1); } /* Other overflow/underflow cases must be detected by the generic routines. */ mpfr_clears (e, bound, (mpfr_ptr) 0); MPFR_SAVE_EXPO_FREE (expo); } expx = MPFR_GET_EXP (x); precy = MPFR_PREC (y); /* if x < 2^(-precy), then exp(x) i.e. gives 1 +/- 1 ulp(1) */ if (MPFR_UNLIKELY (expx < 0 && (mpfr_uexp_t) (-expx) > precy)) { mpfr_exp_t emin = __gmpfr_emin; mpfr_exp_t emax = __gmpfr_emax; int signx = MPFR_SIGN (x); MPFR_SET_POS (y); if (MPFR_IS_NEG_SIGN (signx) && (rnd_mode == MPFR_RNDD || rnd_mode == MPFR_RNDZ)) { __gmpfr_emin = 0; __gmpfr_emax = 0; mpfr_setmax (y, 0); /* y = 1 - epsilon */ inexact = -1; } else { __gmpfr_emin = 1; __gmpfr_emax = 1; mpfr_setmin (y, 1); /* y = 1 */ if (MPFR_IS_POS_SIGN (signx) && (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA)) { mp_size_t yn; int sh; yn = 1 + (MPFR_PREC(y) - 1) / GMP_NUMB_BITS; sh = (mpfr_prec_t) yn * GMP_NUMB_BITS - MPFR_PREC(y); MPFR_MANT(y)[0] += MPFR_LIMB_ONE << sh; inexact = 1; } else inexact = -MPFR_FROM_SIGN_TO_INT(signx); } __gmpfr_emin = emin; __gmpfr_emax = emax; } else /* General case */ { if (MPFR_UNLIKELY (precy >= MPFR_EXP_THRESHOLD)) /* mpfr_exp_3 saves the exponent range and flags itself, otherwise the flag changes in mpfr_exp_3 are lost */ inexact = mpfr_exp_3 (y, x, rnd_mode); /* O(M(n) log(n)^2) */ else { MPFR_SAVE_EXPO_MARK (expo); inexact = mpfr_exp_2 (y, x, rnd_mode); /* O(n^(1/3) M(n)) */ MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); MPFR_SAVE_EXPO_FREE (expo); } } return mpfr_check_range (y, inexact, rnd_mode); }
int mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) { int inexact; mpfr_t x, t, te; mpfr_prec_t Nx, Ny, Nt; mpfr_exp_t err; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); /* Special cases */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt))) { /* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result between -1 and 1 */ if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else /* necessarily xt is 0 */ { MPFR_ASSERTD (MPFR_IS_ZERO (xt)); MPFR_SET_ZERO (y); /* atanh(0) = 0 */ MPFR_SET_SAME_SIGN (y,xt); MPFR_RET (0); } } /* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */ if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0)) { if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt)) { MPFR_SET_INF (y); MPFR_SET_SAME_SIGN (y, xt); mpfr_set_divby0 (); MPFR_RET (0); } MPFR_SET_NAN (y); MPFR_RET_NAN; } /* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */ MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1, rnd_mode, {}); MPFR_SAVE_EXPO_MARK (expo); /* Compute initial precision */ Nx = MPFR_PREC (xt); MPFR_TMP_INIT_ABS (x, xt); Ny = MPFR_PREC (y); Nt = MAX (Nx, Ny); /* the optimal number of bits : see algorithms.ps */ Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4; /* initialise of intermediary variable */ mpfr_init2 (t, Nt); mpfr_init2 (te, Nt); /* First computation of cosh */ MPFR_ZIV_INIT (loop, Nt); for (;;) { /* compute atanh */ mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/ mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/ mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/ mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/ mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/ /* error estimate: see algorithms.tex */ /* FIXME: this does not correspond to the value in algorithms.tex!!! */ /* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/ err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1); if (MPFR_LIKELY (MPFR_IS_ZERO (t) || MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) break; /* reactualisation of the precision */ MPFR_ZIV_NEXT (loop, Nt); mpfr_set_prec (t, Nt); mpfr_set_prec (te, Nt); } MPFR_ZIV_FREE (loop); inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt)); mpfr_clear(t); mpfr_clear(te); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }
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_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC ( ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact)); /* special values */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_INF (x)) { MPFR_SET_INF (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } /* case 0: cbrt(+/- 0) = +/- 0 */ else /* x is necessarily 0 */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } } /* General case */ MPFR_SAVE_EXPO_MARK (expo); mpz_init (m); e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */ if ((negative = MPFR_IS_NEG(x))) mpz_neg (m, m); r = e % 3; if (r < 0) r += 3; /* x = (m*2^r) * 2^(e-r) = (m*2^r) * 2^(3*q) */ 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) { mpz_mul_2exp (m, m, sh); e = e - sh; } else if (r > 0) { mpz_mul_2exp (m, m, r); e = e - 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; 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 (inexact) { if (negative) 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); else inexact = -1; } /* 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); if (negative) { MPFR_CHANGE_SIGN (y); inexact = -inexact; } mpz_clear (m); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }
int mpfr_exp2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { int inexact; long xint; mpfr_t xfrac; MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode), ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y, inexact)); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_INF (x)) { if (MPFR_IS_POS (x)) MPFR_SET_INF (y); else MPFR_SET_ZERO (y); MPFR_SET_POS (y); MPFR_RET (0); } else /* 2^0 = 1 */ { MPFR_ASSERTD (MPFR_IS_ZERO(x)); return mpfr_set_ui (y, 1, rnd_mode); } } /* since the smallest representable non-zero float is 1/2*2^__gmpfr_emin, if x < __gmpfr_emin - 1, the result is either 1/2*2^__gmpfr_emin or 0 */ MPFR_ASSERTN (MPFR_EMIN_MIN >= LONG_MIN + 2); if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emin - 1) < 0)) { mpfr_rnd_t rnd2 = rnd_mode; /* in round to nearest mode, round to zero when x <= __gmpfr_emin-2 */ if (rnd_mode == MPFR_RNDN && mpfr_cmp_si_2exp (x, __gmpfr_emin - 2, 0) <= 0) rnd2 = MPFR_RNDZ; return mpfr_underflow (y, rnd2, 1); } MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX); if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emax) >= 0)) return mpfr_overflow (y, rnd_mode, 1); /* We now know that emin - 1 <= x < emax. */ MPFR_SAVE_EXPO_MARK (expo); /* 2^x = 1 + x*log(2) + O(x^2) for x near zero, and for |x| <= 1 we have |2^x - 1| <= x < 2^EXP(x). If x > 0 we must round away from 0 (dir=1); if x < 0 we must round toward 0 (dir=0). */ MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, - MPFR_GET_EXP (x), 0, MPFR_IS_POS (x), rnd_mode, expo, {}); xint = mpfr_get_si (x, MPFR_RNDZ); mpfr_init2 (xfrac, MPFR_PREC (x)); mpfr_sub_si (xfrac, x, xint, MPFR_RNDN); /* exact */ if (MPFR_IS_ZERO (xfrac)) { mpfr_set_ui (y, 1, MPFR_RNDN); inexact = 0; } else { /* Declaration of the intermediary variable */ mpfr_t t; /* Declaration of the size variable */ mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */ mpfr_prec_t Nt; /* working precision */ mpfr_exp_t err; /* error */ MPFR_ZIV_DECL (loop); /* compute the precision of intermediary variable */ /* the optimal number of bits : see algorithms.tex */ Nt = Ny + 5 + MPFR_INT_CEIL_LOG2 (Ny); /* initialize of intermediary variable */ mpfr_init2 (t, Nt); /* First computation */ MPFR_ZIV_INIT (loop, Nt); for (;;) { /* compute exp(x*ln(2))*/ mpfr_const_log2 (t, MPFR_RNDU); /* ln(2) */ mpfr_mul (t, xfrac, t, MPFR_RNDU); /* xfrac * ln(2) */ err = Nt - (MPFR_GET_EXP (t) + 2); /* Estimate of the error */ mpfr_exp (t, t, MPFR_RNDN); /* exp(xfrac * ln(2)) */ if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) break; /* Actualisation of the precision */ MPFR_ZIV_NEXT (loop, Nt); mpfr_set_prec (t, Nt); } MPFR_ZIV_FREE (loop); inexact = mpfr_set (y, t, rnd_mode); mpfr_clear (t); } mpfr_clear (xfrac); MPFR_CLEAR_FLAGS (); mpfr_mul_2si (y, y, xint, MPFR_RNDN); /* exact or overflow */ /* Note: We can have an overflow only when t was rounded up to 2. */ MPFR_ASSERTD (MPFR_IS_PURE_FP (y) || inexact > 0); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }
int mpfr_sinh (mpfr_ptr y, mpfr_srcptr xt, mp_rnd_t rnd_mode) { mpfr_t x; int inexact; MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", xt, xt, rnd_mode), ("y[%#R]=%R inexact=%d", y, y, inexact)); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt))) { if (MPFR_IS_NAN (xt)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_INF (xt)) { MPFR_SET_INF (y); MPFR_SET_SAME_SIGN (y, xt); MPFR_RET (0); } else /* xt is zero */ { MPFR_ASSERTD (MPFR_IS_ZERO (xt)); MPFR_SET_ZERO (y); /* sinh(0) = 0 */ MPFR_SET_SAME_SIGN (y, xt); MPFR_RET (0); } } /* sinh(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */ MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP(xt), 2, 1, rnd_mode, {}); MPFR_TMP_INIT_ABS (x, xt); { mpfr_t t, ti; mp_exp_t d; mp_prec_t Nt; /* Precision of the intermediary variable */ long int err; /* Precision of error */ MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_GROUP_DECL (group); MPFR_SAVE_EXPO_MARK (expo); /* compute the precision of intermediary variable */ Nt = MAX (MPFR_PREC (x), MPFR_PREC (y)); /* the optimal number of bits : see algorithms.ps */ Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4; /* If x is near 0, exp(x) - 1/exp(x) = 2*x+x^3/3+O(x^5) */ if (MPFR_GET_EXP (x) < 0) Nt -= 2*MPFR_GET_EXP (x); /* initialise of intermediary variables */ MPFR_GROUP_INIT_2 (group, Nt, t, ti); /* First computation of sinh */ MPFR_ZIV_INIT (loop, Nt); for (;;) { /* compute sinh */ mpfr_clear_flags (); mpfr_exp (t, x, GMP_RNDD); /* exp(x) */ /* exp(x) can overflow! */ /* BUG/TODO/FIXME: exp can overflow but sinh may be representable! */ if (MPFR_UNLIKELY (mpfr_overflow_p ())) { inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt)); MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW); break; } d = MPFR_GET_EXP (t); mpfr_ui_div (ti, 1, t, GMP_RNDU); /* 1/exp(x) */ mpfr_sub (t, t, ti, GMP_RNDN); /* exp(x) - 1/exp(x) */ mpfr_div_2ui (t, t, 1, GMP_RNDN); /* 1/2(exp(x) - 1/exp(x)) */ /* it may be that t is zero (in fact, it can only occur when te=1, and thus ti=1 too) */ if (MPFR_IS_ZERO (t)) err = Nt; /* double the precision */ else { /* calculation of the error */ d = d - MPFR_GET_EXP (t) + 2; /* error estimate: err = Nt-(__gmpfr_ceil_log2(1+pow(2,d)));*/ err = Nt - (MAX (d, 0) + 1); if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y), rnd_mode))) { inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt)); break; } } /* actualisation of the precision */ Nt += err; MPFR_ZIV_NEXT (loop, Nt); MPFR_GROUP_REPREC_2 (group, Nt, t, ti); } MPFR_ZIV_FREE (loop); MPFR_GROUP_CLEAR (group); MPFR_SAVE_EXPO_FREE (expo); } return mpfr_check_range (y, inexact, rnd_mode); }
int main (int argc, char *argv[]) { mpfr_t x, y, z, s; MPFR_SAVE_EXPO_DECL (expo); tests_start_mpfr (); mpfr_init (x); mpfr_init (s); mpfr_init (y); mpfr_init (z); /* check special cases */ mpfr_set_prec (x, 2); mpfr_set_prec (y, 2); mpfr_set_prec (z, 2); mpfr_set_prec (s, 2); mpfr_set_str (x, "-0.75", 10, GMP_RNDN); mpfr_set_str (y, "0.5", 10, GMP_RNDN); mpfr_set_str (z, "-0.375", 10, GMP_RNDN); mpfr_fms (s, x, y, z, GMP_RNDU); /* result is 0 */ if (mpfr_cmp_ui(s, 0)) { printf("Error: -0.75 * 0.5 - -0.375 should be equal to 0 for prec=2\n"); exit(1); } mpfr_set_prec (x, 27); mpfr_set_prec (y, 27); mpfr_set_prec (z, 27); mpfr_set_prec (s, 27); mpfr_set_str_binary (x, "1.11111111111111111111111111e-1"); mpfr_set (y, x, GMP_RNDN); mpfr_set_str_binary (z, "1.00011110100011001011001001e-1"); if (mpfr_fms (s, x, y, z, GMP_RNDN) >= 0) { printf ("Wrong inexact flag for x=y=1-2^(-27)\n"); exit (1); } mpfr_set_nan (x); mpfr_urandomb (y, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in x=NAN does not return NAN"); exit (1); } mpfr_set_nan (y); mpfr_urandomb (x, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p(s)) { printf ("evaluation of function in y=NAN does not return NAN"); exit (1); } mpfr_set_nan (z); mpfr_urandomb (y, RANDS); mpfr_urandomb (x, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in z=NAN does not return NAN"); exit (1); } mpfr_set_inf (x, 1); mpfr_set_inf (y, 1); mpfr_set_inf (z, -1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) < 0) { printf ("Error for (+inf) * (+inf) - (-inf)\n"); exit (1); } mpfr_set_inf (x, -1); mpfr_set_inf (y, -1); mpfr_set_inf (z, -1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) < 0) { printf ("Error for (-inf) * (-inf) - (-inf)\n"); exit (1); } mpfr_set_inf (x, 1); mpfr_set_inf (y, -1); mpfr_set_inf (z, 1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) > 0) { printf ("Error for (+inf) * (-inf) - (+inf)\n"); exit (1); } mpfr_set_inf (x, -1); mpfr_set_inf (y, 1); mpfr_set_inf (z, 1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) > 0) { printf ("Error for (-inf) * (+inf) - (+inf)\n"); exit (1); } mpfr_set_inf (x, 1); mpfr_set_ui (y, 0, GMP_RNDN); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in x=INF y=0 does not return NAN"); exit (1); } mpfr_set_inf (y, 1); mpfr_set_ui (x, 0, GMP_RNDN); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in x=0 y=INF does not return NAN"); exit (1); } mpfr_set_inf (x, 1); mpfr_urandomb (y, RANDS); /* always positive */ mpfr_set_inf (z, 1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in x=INF y>0 z=INF does not return NAN"); exit (1); } mpfr_set_inf (y, 1); mpfr_urandomb (x, RANDS); mpfr_set_inf (z, 1); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_nan_p (s)) { printf ("evaluation of function in x>0 y=INF z=INF does not return NAN"); exit (1); } mpfr_set_inf (x, 1); mpfr_urandomb (y, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) < 0) { printf ("evaluation of function in x=INF does not return INF"); exit (1); } mpfr_set_inf (y, 1); mpfr_urandomb (x, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) < 0) { printf ("evaluation of function in y=INF does not return INF"); exit (1); } mpfr_set_inf (z, -1); mpfr_urandomb (x, RANDS); mpfr_urandomb (y, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); if (!mpfr_inf_p (s) || mpfr_sgn (s) < 0) { printf ("evaluation of function in z=-INF does not return INF"); exit (1); } mpfr_set_ui (x, 0, GMP_RNDN); mpfr_urandomb (y, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); mpfr_neg (z, z, GMP_RNDN); if (mpfr_cmp (s, z)) { printf ("evaluation of function in x=0 does not return -z\n"); exit (1); } mpfr_set_ui (y, 0, GMP_RNDN); mpfr_urandomb (x, RANDS); mpfr_urandomb (z, RANDS); mpfr_fms (s, x, y, z, GMP_RNDN); mpfr_neg (z, z, GMP_RNDN); if (mpfr_cmp (s, z)) { printf ("evaluation of function in y=0 does not return -z\n"); exit (1); } { mp_prec_t prec; mpfr_t t, slong; mp_rnd_t rnd; int inexact, compare; unsigned int n; mp_prec_t p0=2, p1=200; unsigned int N=200; mpfr_init (t); mpfr_init (slong); /* generic test */ for (prec = p0; prec <= p1; prec++) { mpfr_set_prec (x, prec); mpfr_set_prec (y, prec); mpfr_set_prec (z, prec); mpfr_set_prec (s, prec); mpfr_set_prec (t, prec); for (n=0; n<N; n++) { mpfr_urandomb (x, RANDS); mpfr_urandomb (y, RANDS); mpfr_urandomb (z, RANDS); if (randlimb () % 2) mpfr_neg (x, x, GMP_RNDN); if (randlimb () % 2) mpfr_neg (y, y, GMP_RNDN); if (randlimb () % 2) mpfr_neg (z, z, GMP_RNDN); rnd = RND_RAND (); mpfr_set_prec (slong, 2 * prec); if (mpfr_mul (slong, x, y, rnd)) { printf ("x*y should be exact\n"); exit (1); } compare = mpfr_sub (t, slong, z, rnd); inexact = mpfr_fms (s, x, y, z, rnd); if (mpfr_cmp (s, t)) { printf ("results differ for x="); mpfr_out_str (stdout, 2, prec, x, GMP_RNDN); printf (" y="); mpfr_out_str (stdout, 2, prec, y, GMP_RNDN); printf (" z="); mpfr_out_str (stdout, 2, prec, z, GMP_RNDN); printf (" prec=%u rnd_mode=%s\n", (unsigned int) prec, mpfr_print_rnd_mode (rnd)); printf ("got "); mpfr_out_str (stdout, 2, prec, s, GMP_RNDN); puts (""); printf ("expected "); mpfr_out_str (stdout, 2, prec, t, GMP_RNDN); puts (""); printf ("approx "); mpfr_print_binary (slong); puts (""); exit (1); } if (((inexact == 0) && (compare != 0)) || ((inexact < 0) && (compare >= 0)) || ((inexact > 0) && (compare <= 0))) { printf ("Wrong inexact flag for rnd=%s: expected %d, got %d\n", mpfr_print_rnd_mode (rnd), compare, inexact); printf (" x="); mpfr_out_str (stdout, 2, 0, x, GMP_RNDN); printf (" y="); mpfr_out_str (stdout, 2, 0, y, GMP_RNDN); printf (" z="); mpfr_out_str (stdout, 2, 0, z, GMP_RNDN); printf (" s="); mpfr_out_str (stdout, 2, 0, s, GMP_RNDN); printf ("\n"); exit (1); } } } mpfr_clear (t); mpfr_clear (slong); } mpfr_clear (x); mpfr_clear (y); mpfr_clear (z); mpfr_clear (s); test_exact (); MPFR_SAVE_EXPO_MARK (expo); test_overflow1 (); test_overflow2 (); test_underflow1 (); test_underflow2 (); MPFR_SAVE_EXPO_FREE (expo); tests_end_mpfr (); return 0; }
int mpfr_fac_ui (mpfr_ptr y, unsigned long int x, mpfr_rnd_t rnd_mode) { mpfr_t t; /* Variable of Intermediary Calculation*/ unsigned long i; int round, inexact; mpfr_prec_t Ny; /* Precision of output variable */ mpfr_prec_t Nt; /* Precision of Intermediary Calculation variable */ mpfr_prec_t err; /* Precision of error */ mpfr_rnd_t rnd; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); /***** test x = 0 and x == 1******/ if (MPFR_UNLIKELY (x <= 1)) return mpfr_set_ui (y, 1, rnd_mode); /* 0! = 1 and 1! = 1 */ MPFR_SAVE_EXPO_MARK (expo); /* Initialisation of the Precision */ Ny = MPFR_PREC (y); /* compute the size of intermediary variable */ Nt = Ny + 2 * MPFR_INT_CEIL_LOG2 (x) + 7; mpfr_init2 (t, Nt); /* initialise of intermediary variable */ rnd = MPFR_RNDZ; MPFR_ZIV_INIT (loop, Nt); for (;;) { /* compute factorial */ inexact = mpfr_set_ui (t, 1, rnd); for (i = 2 ; i <= x ; i++) { round = mpfr_mul_ui (t, t, i, rnd); /* assume the first inexact product gives the sign of difference: is that always correct? */ if (inexact == 0) inexact = round; } err = Nt - 1 - MPFR_INT_CEIL_LOG2 (Nt); round = !inexact || mpfr_can_round (t, err, rnd, MPFR_RNDZ, Ny + (rnd_mode == MPFR_RNDN)); if (MPFR_LIKELY (round)) { /* If inexact = 0, then t is exactly x!, so round is the correct inexact flag. Otherwise, t != x! since we rounded to zero or away. */ round = mpfr_set (y, t, rnd_mode); if (inexact == 0) { inexact = round; break; } else if ((inexact < 0 && round <= 0) || (inexact > 0 && round >= 0)) break; else /* inexact and round have opposite signs: we cannot compute the inexact flag. Restart using the symmetric rounding. */ rnd = (rnd == MPFR_RNDZ) ? MPFR_RNDU : MPFR_RNDZ; } MPFR_ZIV_NEXT (loop, Nt); mpfr_set_prec (t, Nt); } MPFR_ZIV_FREE (loop); mpfr_clear (t); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }
/* set f to the rational q */ int mpfr_set_q (mpfr_ptr f, mpq_srcptr q, mp_rnd_t rnd) { mpz_srcptr num, den; mpfr_t n, d; int inexact; int cn, cd; long shift; mp_size_t sn, sd; MPFR_SAVE_EXPO_DECL (expo); num = mpq_numref (q); den = mpq_denref (q); /* NAN and INF for mpq are not really documented, but could be found */ if (MPFR_UNLIKELY (mpz_sgn (num) == 0)) { if (MPFR_UNLIKELY (mpz_sgn (den) == 0)) { MPFR_SET_NAN (f); MPFR_RET_NAN; } else { MPFR_SET_ZERO (f); MPFR_SET_POS (f); MPFR_RET (0); } } if (MPFR_UNLIKELY (mpz_sgn (den) == 0)) { MPFR_SET_INF (f); MPFR_SET_SIGN (f, mpz_sgn (num)); MPFR_RET (0); } MPFR_SAVE_EXPO_MARK (expo); cn = set_z (n, num, &sn); cd = set_z (d, den, &sd); sn -= sd; if (MPFR_UNLIKELY (sn > MPFR_EMAX_MAX / BITS_PER_MP_LIMB)) { inexact = mpfr_overflow (f, rnd, MPFR_SIGN (f)); goto end; } if (MPFR_UNLIKELY (sn < MPFR_EMIN_MIN / BITS_PER_MP_LIMB -1)) { if (rnd == GMP_RNDN) rnd = GMP_RNDZ; inexact = mpfr_underflow (f, rnd, MPFR_SIGN (f)); goto end; } inexact = mpfr_div (f, n, d, rnd); shift = BITS_PER_MP_LIMB*sn+cn-cd; MPFR_ASSERTD (shift == BITS_PER_MP_LIMB*sn+cn-cd); cd = mpfr_mul_2si (f, f, shift, rnd); MPFR_SAVE_EXPO_FREE (expo); if (MPFR_UNLIKELY (cd != 0)) inexact = cd; else inexact = mpfr_check_range (f, inexact, rnd); end: mpfr_clear (d); mpfr_clear (n); return inexact; }
int mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode) { int comp, inexact; mp_exp_t ex; MPFR_SAVE_EXPO_DECL (expo); if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } /* check for inf or -inf (result is not defined) */ else if (MPFR_IS_INF (x)) { if (MPFR_IS_POS (x)) { MPFR_SET_INF (y); MPFR_SET_POS (y); MPFR_RET (0); } else { MPFR_SET_NAN (y); MPFR_RET_NAN; } } else /* x is zero */ { MPFR_ASSERTD (MPFR_IS_ZERO (x)); MPFR_SET_ZERO (y); /* log1p(+/- 0) = +/- 0 */ MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } } ex = MPFR_GET_EXP (x); if (ex < 0) /* -0.5 < x < 0.5 */ { /* For x > 0, abs(log(1+x)-x) < x^2/2. For x > -0.5, abs(log(1+x)-x) < x^2. */ if (MPFR_IS_POS (x)) MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex - 1, 0, 0, rnd_mode, {}); else MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {}); } comp = mpfr_cmp_si (x, -1); /* log1p(x) is undefined for x < -1 */ if (MPFR_UNLIKELY(comp <= 0)) { if (comp == 0) /* x=0: log1p(-1)=-inf (division by zero) */ { MPFR_SET_INF (y); MPFR_SET_NEG (y); MPFR_RET (0); } MPFR_SET_NAN (y); MPFR_RET_NAN; } MPFR_SAVE_EXPO_MARK (expo); /* General case */ { /* Declaration of the intermediary variable */ mpfr_t t; /* Declaration of the size variable */ mp_prec_t Ny = MPFR_PREC(y); /* target precision */ mp_prec_t Nt; /* working precision */ mp_exp_t err; /* error */ MPFR_ZIV_DECL (loop); /* compute the precision of intermediary variable */ /* the optimal number of bits : see algorithms.tex */ Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6; /* if |x| is smaller than 2^(-e), we will loose about e bits in log(1+x) */ if (MPFR_EXP(x) < 0) Nt += -MPFR_EXP(x); /* initialise of intermediary variable */ mpfr_init2 (t, Nt); /* First computation of log1p */ MPFR_ZIV_INIT (loop, Nt); for (;;) { /* compute log1p */ inexact = mpfr_add_ui (t, x, 1, GMP_RNDN); /* 1+x */ /* if inexact = 0, then t = x+1, and the result is simply log(t) */ if (inexact == 0) { inexact = mpfr_log (y, t, rnd_mode); goto end; } mpfr_log (t, t, GMP_RNDN); /* log(1+x) */ /* the error is bounded by (1/2+2^(1-EXP(t))*ulp(t) (cf algorithms.tex) if EXP(t)>=2, then error <= ulp(t) if EXP(t)<=1, then error <= 2^(2-EXP(t))*ulp(t) */ err = Nt - MAX (0, 2 - MPFR_GET_EXP (t)); if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) break; /* increase the precision */ MPFR_ZIV_NEXT (loop, Nt); mpfr_set_prec (t, Nt); } inexact = mpfr_set (y, t, rnd_mode); end: MPFR_ZIV_FREE (loop); mpfr_clear (t); } MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }