diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/atanh.c')
| -rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/atanh.c | 117 |
1 files changed, 101 insertions, 16 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/atanh.c b/Build/source/libs/mpfr/mpfr-src/src/atanh.c index ec5a6cf6f1f..5f9953c4e7c 100644 --- a/Build/source/libs/mpfr/mpfr-src/src/atanh.c +++ b/Build/source/libs/mpfr/mpfr-src/src/atanh.c @@ -23,11 +23,84 @@ http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., #define MPFR_NEED_LONGLONG_H #include "mpfr-impl.h" - /* The computation of atanh is done by - atanh= 1/2*ln(x+1)-1/2*ln(1-x) */ +/* Put in y an approximation of atanh(x) for x small. + We assume x <= 1/2, in which case: + x <= y ~ atanh(x) = x + x^3/3 + x^5/5 + x^7/7 + ... <= 2*x. + Return k such that the error is bounded by 2^k*ulp(y). +*/ +static int +mpfr_atanh_small (mpfr_ptr y, mpfr_srcptr x) +{ + mpfr_prec_t p = MPFR_PREC(y), err; + mpfr_t x2, t, u; + unsigned long i; + int k; + + MPFR_ASSERTD(MPFR_GET_EXP (x) <= -1); + + /* in the following, theta represents a value with |theta| <= 2^(1-p) + (might be a different value each time) */ + + mpfr_init2 (t, p); + mpfr_init2 (u, p); + mpfr_init2 (x2, p); + mpfr_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */ + mpfr_set (y, t, MPFR_RNDF); /* exact */ + mpfr_mul (x2, x, x, MPFR_RNDF); /* x2 = x^2 * (1 + theta) */ + for (i = 3; ; i += 2) + { + mpfr_mul (t, t, x2, MPFR_RNDF); /* t = x^i * (1 + theta)^i */ + mpfr_div_ui (u, t, i, MPFR_RNDF); /* u = x^i/i * (1 + theta)^(i+1) */ + if (MPFR_GET_EXP (u) <= MPFR_GET_EXP (y) - p) /* |u| < ulp(y) */ + break; + mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */ + } + /* We assume |(1 + theta)^(i+1)| <= 2. + The neglected part is at most |u| + |u|/4 + |u|/16 + ... <= 4/3*|u|, + which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most + 3 ulp(y). + The rounding error on y is bounded by: + * for the (i-3)/2 add/sub, each error is bounded by ulp(y_i), + where y_i is the current value of y, which is bounded by ulp(y) + for y the final value (since it increases in absolute value), + this yields (i-3)/2*ulp(y) + * from Lemma 3.1 from [Higham02] (see algorithms.tex), + the relative error on u at step i is bounded by: + (i+1)*epsilon/(1-(i+1)*epsilon) where epsilon = 2^(1-p). + If (i+1)*epsilon <= 1/2, then the relative error on u at + step i is bounded by 2*(i+1)*epsilon, and since |u| <= 1/2^(i+1) + at step i, this gives an absolute error bound of; + 2*epsilon*x*(4/2^4 + 6/2^6 + 8/2^8 + ...) = 2*2^(1-p)*x*(7/18) = + 14/9*2^(-p)*x <= 2*ulp(x). + + If (i+1)*epsilon <= 1/2, then the relative error on u at step i + is bounded by (i+1)*epsilon/(1-(i+1)*epsilon) <= 1, thus it follows + |(1 + theta)^(i+1)| <= 2. + + Finally the total error is bounded by 3*ulp(y) + (i-3)/2*ulp(y) +2*ulp(x). + Since x <= 2*y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by: + (i+7)/2*ulp(y). + */ + err = (i + 8) / 2; /* ceil((i+7)/2) */ + k = __gmpfr_int_ceil_log2 (err); + MPFR_ASSERTN(k + 2 < p); + /* if k + 2 < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-3), + thus i+7 <= 2*err <= 2^(p-2), thus (i+7)*epsilon <= 1/2, which implies + our assumption (i+1)*epsilon <= 1/2. */ + mpfr_clear (t); + mpfr_clear (u); + mpfr_clear (x2); + return k; +} + +/* The computation of atanh is done by: + atanh = ln((1+x)/(1-x)) / 2 + except when x is very small, in which case atanh = x + tiny error, + and when x is small, where we use directly the Taylor expansion. +*/ int -mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) +mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt, mpfr_rnd_t rnd_mode) { int inexact; mpfr_t x, t, te; @@ -67,7 +140,7 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) { MPFR_SET_INF (y); MPFR_SET_SAME_SIGN (y, xt); - mpfr_set_divby0 (); + MPFR_SET_DIVBY0 (); MPFR_RET (0); } MPFR_SET_NAN (y); @@ -85,29 +158,42 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) 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 */ + /* initialize of intermediary variable */ mpfr_init2 (t, Nt); mpfr_init2 (te, Nt); - /* First computation of cosh */ MPFR_ZIV_INIT (loop, Nt); for (;;) { + int k; + + /* small case: assuming the AGM algorithm used by mpfr_log uses + log2(p) steps for a precision of p bits, we try the special + variant whenever EXP(x) <= -p/log2(p). */ + k = 1 + __gmpfr_int_ceil_log2 (Ny); /* the +1 avoids a division by 0 + when Ny=1 */ + if (MPFR_GET_EXP (x) <= - 1 - (mpfr_exp_t) (Ny / k)) + /* this implies EXP(x) <= -1 thus x < 1/2 */ + { + err = Nt - mpfr_atanh_small (t, x); + goto round; + } + /* 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))*/ + mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-x) with x = |xt| */ + mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (1+x) */ + mpfr_div (t, t, te, MPFR_RNDN); /* (1+x)/(1-x) */ + mpfr_log (t, t, MPFR_RNDN); /* ln((1+x)/(1-x)) */ + mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* ln((1+x)/(1-x)) / 2 */ /* 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 - __gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t))); */ err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1); + round: if (MPFR_LIKELY (MPFR_IS_ZERO (t) || MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) break; @@ -121,10 +207,9 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt)); - mpfr_clear(t); - mpfr_clear(te); + mpfr_clear (t); + mpfr_clear (te); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); } - |