chromium / native_client / nacl-gcc / f80d6b9ee7f94755c697ffb7194fb01dd0c537dd / . / mpfr-2.4.1 / cos.c

/* mpfr_cos -- cosine of a floating-point number | |

Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. | |

Contributed by the Arenaire and Cacao projects, INRIA. | |

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" | |

/* f <- 1 - r/2! + r^2/4! + ... + (-1)^l r^l/(2l)! + ... | |

Assumes |r| < 1/2, and f, r have the same precision. | |

Returns e such that the error on f is bounded by 2^e ulps. | |

*/ | |

static int | |

mpfr_cos2_aux (mpfr_ptr f, mpfr_srcptr r) | |

{ | |

mpz_t x, t, s; | |

mp_exp_t ex, l, m; | |

mp_prec_t p, q; | |

unsigned long i, maxi, imax; | |

MPFR_ASSERTD(mpfr_get_exp (r) <= -1); | |

/* compute minimal i such that i*(i+1) does not fit in an unsigned long, | |

assuming that there are no padding bits. */ | |

maxi = 1UL << (CHAR_BIT * sizeof(unsigned long) / 2); | |

if (maxi * (maxi / 2) == 0) /* test checked at compile time */ | |

{ | |

/* can occur only when there are padding bits. */ | |

/* maxi * (maxi-1) is representable iff maxi * (maxi / 2) != 0 */ | |

do | |

maxi /= 2; | |

while (maxi * (maxi / 2) == 0); | |

} | |

mpz_init (x); | |

mpz_init (s); | |

mpz_init (t); | |

ex = mpfr_get_z_exp (x, r); /* r = x*2^ex */ | |

/* remove trailing zeroes */ | |

l = mpz_scan1 (x, 0); | |

ex += l; | |

mpz_div_2exp (x, x, l); | |

/* since |r| < 1, r = x*2^ex, and x is an integer, necessarily ex < 0 */ | |

p = mpfr_get_prec (f); /* same than r */ | |

/* bound for number of iterations */ | |

imax = p / (-mpfr_get_exp (r)); | |

imax += (imax == 0); | |

q = 2 * MPFR_INT_CEIL_LOG2(imax) + 4; /* bound for (3l)^2 */ | |

mpz_set_ui (s, 1); /* initialize sum with 1 */ | |

mpz_mul_2exp (s, s, p + q); /* scale all values by 2^(p+q) */ | |

mpz_set (t, s); /* invariant: t is previous term */ | |

for (i = 1; (m = mpz_sizeinbase (t, 2)) >= q; i += 2) | |

{ | |

/* adjust precision of x to that of t */ | |

l = mpz_sizeinbase (x, 2); | |

if (l > m) | |

{ | |

l -= m; | |

mpz_div_2exp (x, x, l); | |

ex += l; | |

} | |

/* multiply t by r */ | |

mpz_mul (t, t, x); | |

mpz_div_2exp (t, t, -ex); | |

/* divide t by i*(i+1) */ | |

if (i < maxi) | |

mpz_div_ui (t, t, i * (i + 1)); | |

else | |

{ | |

mpz_div_ui (t, t, i); | |

mpz_div_ui (t, t, i + 1); | |

} | |

/* if m is the (current) number of bits of t, we can consider that | |

all operations on t so far had precision >= m, so we can prove | |

by induction that the relative error on t is of the form | |

(1+u)^(3l)-1, where |u| <= 2^(-m), and l=(i+1)/2 is the # of loops. | |

Since |(1+x^2)^(1/x) - 1| <= 4x/3 for |x| <= 1/2, | |

for |u| <= 1/(3l)^2, the absolute error is bounded by | |

4/3*(3l)*2^(-m)*t <= 4*l since |t| < 2^m. | |

Therefore the error on s is bounded by 2*l*(l+1). */ | |

/* add or subtract to s */ | |

if (i % 4 == 1) | |

mpz_sub (s, s, t); | |

else | |

mpz_add (s, s, t); | |

} | |

mpfr_set_z (f, s, GMP_RNDN); | |

mpfr_div_2ui (f, f, p + q, GMP_RNDN); | |

mpz_clear (x); | |

mpz_clear (s); | |

mpz_clear (t); | |

l = (i - 1) / 2; /* number of iterations */ | |

return 2 * MPFR_INT_CEIL_LOG2 (l + 1) + 1; /* bound is 2l(l+1) */ | |

} | |

int | |

mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode) | |

{ | |

mp_prec_t K0, K, precy, m, k, l; | |

int inexact, reduce = 0; | |

mpfr_t r, s, xr, c; | |

mp_exp_t exps, cancel = 0, expx; | |

MPFR_ZIV_DECL (loop); | |

MPFR_SAVE_EXPO_DECL (expo); | |

MPFR_GROUP_DECL (group); | |

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_IS_INF (x)) | |

{ | |

MPFR_SET_NAN (y); | |

MPFR_RET_NAN; | |

} | |

else | |

{ | |

MPFR_ASSERTD (MPFR_IS_ZERO (x)); | |

return mpfr_set_ui (y, 1, rnd_mode); | |

} | |

} | |

MPFR_SAVE_EXPO_MARK (expo); | |

/* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */ | |

expx = MPFR_GET_EXP (x); | |

MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, -2 * expx, | |

1, 0, rnd_mode, expo, {}); | |

/* Compute initial precision */ | |

precy = MPFR_PREC (y); | |

K0 = __gmpfr_isqrt (precy / 3); | |

m = precy + 2 * MPFR_INT_CEIL_LOG2 (precy) + 2 * K0; | |

if (expx >= 3) | |

{ | |

reduce = 1; | |

mpfr_init2 (xr, m); | |

mpfr_init2 (c, expx + m - 1); | |

} | |

MPFR_GROUP_INIT_2 (group, m, r, s); | |

MPFR_ZIV_INIT (loop, m); | |

for (;;) | |

{ | |

/* If |x| >= 4, first reduce x cmod (2*Pi) into xr, using mpfr_remainder: | |

let e = EXP(x) >= 3, and m the target precision: | |

(1) c <- 2*Pi [precision e+m-1, nearest] | |

(2) xr <- remainder (x, c) [precision m, nearest] | |

We have |c - 2*Pi| <= 1/2ulp(c) = 2^(3-e-m) | |

|xr - x - k c| <= 1/2ulp(xr) <= 2^(1-m) | |

|k| <= |x|/(2*Pi) <= 2^(e-2) | |

Thus |xr - x - 2kPi| <= |k| |c - 2Pi| + 2^(1-m) <= 2^(2-m). | |

It follows |cos(xr) - cos(x)| <= 2^(2-m). */ | |

if (reduce) | |

{ | |

mpfr_const_pi (c, GMP_RNDN); | |

mpfr_mul_2ui (c, c, 1, GMP_RNDN); /* 2Pi */ | |

mpfr_remainder (xr, x, c, GMP_RNDN); | |

/* now |xr| <= 4, thus r <= 16 below */ | |

mpfr_mul (r, xr, xr, GMP_RNDU); /* err <= 1 ulp */ | |

} | |

else | |

mpfr_mul (r, x, x, GMP_RNDU); /* err <= 1 ulp */ | |

/* now |x| < 4 (or xr if reduce = 1), thus |r| <= 16 */ | |

/* we need |r| < 1/2 for mpfr_cos2_aux, i.e., EXP(r) - 2K <= -1 */ | |

K = K0 + 1 + MAX(0, MPFR_EXP(r)) / 2; | |

/* since K0 >= 0, if EXP(r) < 0, then K >= 1, thus EXP(r) - 2K <= -3; | |

otherwise if EXP(r) >= 0, then K >= 1/2 + EXP(r)/2, thus | |

EXP(r) - 2K <= -1 */ | |

MPFR_SET_EXP (r, MPFR_GET_EXP (r) - 2 * K); /* Can't overflow! */ | |

/* s <- 1 - r/2! + ... + (-1)^l r^l/(2l)! */ | |

l = mpfr_cos2_aux (s, r); | |

/* l is the error bound in ulps on s */ | |

MPFR_SET_ONE (r); | |

for (k = 0; k < K; k++) | |

{ | |

mpfr_sqr (s, s, GMP_RNDU); /* err <= 2*olderr */ | |

MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); /* Can't overflow */ | |

mpfr_sub (s, s, r, GMP_RNDN); /* err <= 4*olderr */ | |

if (MPFR_IS_ZERO(s)) | |

goto ziv_next; | |

MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1); | |

} | |

/* The absolute error on s is bounded by (2l+1/3)*2^(2K-m) | |

2l+1/3 <= 2l+1. | |

If |x| >= 4, we need to add 2^(2-m) for the argument reduction | |

by 2Pi: if K = 0, this amounts to add 4 to 2l+1/3, i.e., to add | |

2 to l; if K >= 1, this amounts to add 1 to 2*l+1/3. */ | |

l = 2 * l + 1; | |

if (reduce) | |

l += (K == 0) ? 4 : 1; | |

k = MPFR_INT_CEIL_LOG2 (l) + 2*K; | |

/* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */ | |

exps = MPFR_GET_EXP (s); | |

if (MPFR_LIKELY (MPFR_CAN_ROUND (s, exps + m - k, precy, rnd_mode))) | |

break; | |

if (MPFR_UNLIKELY (exps == 1)) | |

/* s = 1 or -1, and except x=0 which was already checked above, | |

cos(x) cannot be 1 or -1, so we can round if the error is less | |

than 2^(-precy) for directed rounding, or 2^(-precy-1) for rounding | |

to nearest. */ | |

{ | |

if (m > k && (m - k >= precy + (rnd_mode == GMP_RNDN))) | |

{ | |

/* If round to nearest or away, result is s = 1 or -1, | |

otherwise it is round(nexttoward (s, 0)). However in order to | |

have the inexact flag correctly set below, we set |s| to | |

1 - 2^(-m) in all cases. */ | |

mpfr_nexttozero (s); | |

break; | |

} | |

} | |

if (exps < cancel) | |

{ | |

m += cancel - exps; | |

cancel = exps; | |

} | |

ziv_next: | |

MPFR_ZIV_NEXT (loop, m); | |

MPFR_GROUP_REPREC_2 (group, m, r, s); | |

if (reduce) | |

{ | |

mpfr_set_prec (xr, m); | |

mpfr_set_prec (c, expx + m - 1); | |

} | |

} | |

MPFR_ZIV_FREE (loop); | |

inexact = mpfr_set (y, s, rnd_mode); | |

MPFR_GROUP_CLEAR (group); | |

if (reduce) | |

{ | |

mpfr_clear (xr); | |

mpfr_clear (c); | |

} | |

MPFR_SAVE_EXPO_FREE (expo); | |

return mpfr_check_range (y, inexact, rnd_mode); | |

} |