diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/log1p.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/log1p.c | 117 |
1 files changed, 101 insertions, 16 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/log1p.c b/Build/source/libs/mpfr/mpfr-src/src/log1p.c index 4ef43c2ae68..96c2f6831ed 100644 --- a/Build/source/libs/mpfr/mpfr-src/src/log1p.c +++ b/Build/source/libs/mpfr/mpfr-src/src/log1p.c @@ -23,8 +23,80 @@ 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 log1p is done by - log1p(x)=log(1+x) */ +/* Put in y an approximation of log(1+x) for x small. + We assume |x| < 1, in which case: + |x/2| <= |log(1+x)| = |x - x^2/2 + x^3/3 - x^4/4 + ...| <= |x|. + Return k such that the error is bounded by 2^k*ulp(y). +*/ +static int +mpfr_log1p_small (mpfr_ptr y, mpfr_srcptr x) +{ + mpfr_prec_t p = MPFR_PREC(y), err; + mpfr_t t, u; + unsigned long i; + int k; + + MPFR_ASSERTD(MPFR_GET_EXP (x) <= 0); /* ensures |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_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */ + mpfr_set (y, t, MPFR_RNDF); /* exact */ + for (i = 2; ; i++) + { + mpfr_mul (t, t, x, 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; + if (i & 1) + mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */ + else + mpfr_sub (y, y, u, MPFR_RNDF); /* error <= ulp(y) */ + } + /* We assume |(1 + theta)^(i+1)| <= 2. + The neglected part is at most |u| + |u|/2 + ... <= 2|u| < 2 ulp(y) + which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most + 4 ulp(y). + The rounding error on y is bounded by: + * for the (i-2) add/sub, each error is bounded by ulp(y), + and since |y| <= |x|, this yields (i-2)*ulp(x) + * 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*(3/2^3 + 4/2^4 + 5/2^5 + ...) <= 2*2^(1-p)*x = + 4*2^(-p)*x <= 4*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 4*ulp(y) + (i-2)*ulp(x) + 4*ulp(x) + = 4*ulp(y) + (i+2)*ulp(x). + Since x/2 <= y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by: + (2*i+8)*ulp(y). + */ + err = 2 * i + 8; + k = __gmpfr_int_ceil_log2 (err); + MPFR_ASSERTN(k < p); + /* if k < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-1), + thus i+4 = err/2 <= 2^(p-2), thus (i+4)*epsilon <= 1/2, which implies + our assumption (i+1)*epsilon <= 1/2. */ + mpfr_clear (t); + mpfr_clear (u); + return k; +} + +/* The computation of log1p is done by + log1p(x) = log(1+x) + except when x is very small, in which case log1p(x) = x + tiny error, + or when x is small, where we use directly the Taylor expansion. +*/ int mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) @@ -89,7 +161,7 @@ mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { MPFR_SET_INF (y); MPFR_SET_NEG (y); - mpfr_set_divby0 (); + MPFR_SET_DIVBY0 (); MPFR_RET (0); } MPFR_SET_NAN (y); @@ -117,27 +189,40 @@ mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) if (MPFR_EXP(x) < 0) Nt += -MPFR_EXP(x); - /* initialise of intermediary variable */ + /* initialize 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, MPFR_RNDN); /* 1+x */ - /* if inexact = 0, then t = x+1, and the result is simply log(t) */ - if (inexact == 0) + 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) <= - (mpfr_exp_t) (Ny / k)) + /* this implies EXP(x) <= 0 thus x < 1 */ + err = Nt - mpfr_log1p_small (t, x); + else { - inexact = mpfr_log (y, t, rnd_mode); - goto end; + /* compute log1p */ + inexact = mpfr_add_ui (t, x, 1, MPFR_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, MPFR_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)); } - mpfr_log (t, t, MPFR_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; |