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1cadc85813
In 84ba214c, I removed some redundant sign computations and in the
process, I incorrectly got rid of a temporary variable, thus passing
the absolute value of the input to bsloww1. This caused #16623.
This fix undoes the incorrect change.
1245 lines
34 KiB
C
1245 lines
34 KiB
C
/*
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* IBM Accurate Mathematical Library
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* written by International Business Machines Corp.
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* Copyright (C) 2001-2014 Free Software Foundation, Inc.
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU Lesser General Public License as published by
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* the Free Software Foundation; either version 2.1 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public License
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* along with this program; if not, see <http://www.gnu.org/licenses/>.
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*/
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/****************************************************************************/
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/* */
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/* MODULE_NAME:usncs.c */
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/* */
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/* FUNCTIONS: usin */
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/* ucos */
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/* slow */
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/* slow1 */
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/* slow2 */
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/* sloww */
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/* sloww1 */
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/* sloww2 */
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/* bsloww */
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/* bsloww1 */
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/* bsloww2 */
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/* cslow2 */
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/* csloww */
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/* csloww1 */
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/* csloww2 */
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/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h usncs.h */
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/* branred.c sincos32.c dosincos.c mpa.c */
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/* sincos.tbl */
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/* */
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/* An ultimate sin and routine. Given an IEEE double machine number x */
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/* it computes the correctly rounded (to nearest) value of sin(x) or cos(x) */
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/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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/* */
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/****************************************************************************/
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#include <errno.h>
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#include "endian.h"
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#include "mydefs.h"
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#include "usncs.h"
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#include "MathLib.h"
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#include <math_private.h>
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#include <fenv.h>
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/* Helper macros to compute sin of the input values. */
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#define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
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#define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
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/* The computed polynomial is a variation of the Taylor series expansion for
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sin(a):
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a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2
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The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
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on. The result is returned to LHS and correction in COR. */
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#define TAYLOR_SIN(xx, a, da, cor) \
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({ \
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double t = ((POLYNOMIAL (xx) * (a) - 0.5 * (da)) * (xx) + (da)); \
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double res = (a) + t; \
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(cor) = ((a) - res) + t; \
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res; \
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})
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/* This is again a variation of the Taylor series expansion with the term
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x^3/3! expanded into the following for better accuracy:
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bb * x ^ 3 + 3 * aa * x * x1 * x2 + aa * x1 ^ 3 + aa * x2 ^ 3
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The correction term is dx and bb + aa = -1/3!
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*/
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#define TAYLOR_SLOW(x0, dx, cor) \
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({ \
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static const double th2_36 = 206158430208.0; /* 1.5*2**37 */ \
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double xx = (x0) * (x0); \
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double x1 = ((x0) + th2_36) - th2_36; \
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double y = aa * x1 * x1 * x1; \
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double r = (x0) + y; \
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double x2 = ((x0) - x1) + (dx); \
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double t = (((POLYNOMIAL2 (xx) + bb) * xx + 3.0 * aa * x1 * x2) \
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* (x0) + aa * x2 * x2 * x2 + (dx)); \
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t = (((x0) - r) + y) + t; \
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double res = r + t; \
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(cor) = (r - res) + t; \
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res; \
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})
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#define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
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({ \
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int4 k = u.i[LOW_HALF] << 2; \
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sn = __sincostab.x[k]; \
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ssn = __sincostab.x[k + 1]; \
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cs = __sincostab.x[k + 2]; \
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ccs = __sincostab.x[k + 3]; \
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})
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#ifndef SECTION
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# define SECTION
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#endif
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extern const union
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{
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int4 i[880];
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double x[440];
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} __sincostab attribute_hidden;
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static const double
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sn3 = -1.66666666666664880952546298448555E-01,
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sn5 = 8.33333214285722277379541354343671E-03,
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cs2 = 4.99999999999999999999950396842453E-01,
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cs4 = -4.16666666666664434524222570944589E-02,
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cs6 = 1.38888874007937613028114285595617E-03;
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static const double t22 = 0x1.8p22;
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void __dubsin (double x, double dx, double w[]);
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void __docos (double x, double dx, double w[]);
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double __mpsin (double x, double dx, bool reduce_range);
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double __mpcos (double x, double dx, bool reduce_range);
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static double slow (double x);
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static double slow1 (double x);
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static double slow2 (double x);
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static double sloww (double x, double dx, double orig);
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static double sloww1 (double x, double dx, double orig, int m);
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static double sloww2 (double x, double dx, double orig, int n);
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static double bsloww (double x, double dx, double orig, int n);
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static double bsloww1 (double x, double dx, double orig, int n);
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static double bsloww2 (double x, double dx, double orig, int n);
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int __branred (double x, double *a, double *aa);
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static double cslow2 (double x);
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static double csloww (double x, double dx, double orig);
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static double csloww1 (double x, double dx, double orig, int m);
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static double csloww2 (double x, double dx, double orig, int n);
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/* Given a number partitioned into U and X such that U is an index into the
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sin/cos table, this macro computes the cosine of the number by combining
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the sin and cos of X (as computed by a variation of the Taylor series) with
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the values looked up from the sin/cos table to get the result in RES and a
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correction value in COR. */
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static double
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do_cos (mynumber u, double x, double *corp)
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{
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double xx, s, sn, ssn, c, cs, ccs, res, cor;
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xx = x * x;
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s = x + x * xx * (sn3 + xx * sn5);
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c = xx * (cs2 + xx * (cs4 + xx * cs6));
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SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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cor = (ccs - s * ssn - cs * c) - sn * s;
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res = cs + cor;
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cor = (cs - res) + cor;
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*corp = cor;
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return res;
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}
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/* A more precise variant of DO_COS where the number is partitioned into U, X
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and DX. EPS is the adjustment to the correction COR. */
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static double
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do_cos_slow (mynumber u, double x, double dx, double eps, double *corp)
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{
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double xx, y, x1, x2, e1, e2, res, cor;
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double s, sn, ssn, c, cs, ccs;
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xx = x * x;
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s = x * xx * (sn3 + xx * sn5);
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c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
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SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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x1 = (x + t22) - t22;
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x2 = (x - x1) + dx;
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e1 = (sn + t22) - t22;
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e2 = (sn - e1) + ssn;
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cor = (ccs - cs * c - e1 * x2 - e2 * x) - sn * s;
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y = cs - e1 * x1;
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cor = cor + ((cs - y) - e1 * x1);
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res = y + cor;
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cor = (y - res) + cor;
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if (cor > 0)
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cor = 1.0005 * cor + eps;
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else
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cor = 1.0005 * cor - eps;
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*corp = cor;
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return res;
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}
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/* Given a number partitioned into U and X and DX such that U is an index into
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the sin/cos table, this macro computes the sine of the number by combining
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the sin and cos of X (as computed by a variation of the Taylor series) with
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the values looked up from the sin/cos table to get the result in RES and a
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correction value in COR. */
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static double
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do_sin (mynumber u, double x, double dx, double *corp)
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{
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double xx, s, sn, ssn, c, cs, ccs, cor, res;
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xx = x * x;
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s = x + (dx + x * xx * (sn3 + xx * sn5));
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c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
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SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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cor = (ssn + s * ccs - sn * c) + cs * s;
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res = sn + cor;
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cor = (sn - res) + cor;
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*corp = cor;
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return res;
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}
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/* A more precise variant of res = do_sin where the number is partitioned into U, X
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and DX. EPS is the adjustment to the correction COR. */
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static double
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do_sin_slow (mynumber u, double x, double dx, double eps, double *corp)
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{
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double xx, y, x1, x2, c1, c2, res, cor;
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double s, sn, ssn, c, cs, ccs;
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xx = x * x;
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s = x * xx * (sn3 + xx * sn5);
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c = xx * (cs2 + xx * (cs4 + xx * cs6));
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SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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x1 = (x + t22) - t22;
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x2 = (x - x1) + dx;
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c1 = (cs + t22) - t22;
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c2 = (cs - c1) + ccs;
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cor = (ssn + s * ccs + cs * s + c2 * x + c1 * x2 - sn * x * dx) - sn * c;
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y = sn + c1 * x1;
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cor = cor + ((sn - y) + c1 * x1);
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res = y + cor;
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cor = (y - res) + cor;
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if (cor > 0)
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cor = 1.0005 * cor + eps;
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else
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cor = 1.0005 * cor - eps;
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*corp = cor;
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return res;
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}
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/* Reduce range of X and compute sin of a + da. K is the amount by which to
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rotate the quadrants. This allows us to use the same routine to compute cos
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by simply rotating the quadrants by 1. */
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static inline double
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__always_inline
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reduce_and_compute (double x, unsigned int k)
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{
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double retval = 0, a, da;
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unsigned int n = __branred (x, &a, &da);
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k = (n + k) % 4;
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switch (k)
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{
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case 0:
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if (a * a < 0.01588)
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retval = bsloww (a, da, x, n);
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else
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retval = bsloww1 (a, da, x, n);
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break;
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case 2:
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if (a * a < 0.01588)
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retval = bsloww (-a, -da, x, n);
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else
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retval = bsloww1 (-a, -da, x, n);
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break;
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case 1:
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case 3:
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retval = bsloww2 (a, da, x, n);
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break;
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}
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return retval;
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}
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/*******************************************************************/
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/* An ultimate sin routine. Given an IEEE double machine number x */
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/* it computes the correctly rounded (to nearest) value of sin(x) */
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/*******************************************************************/
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double
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SECTION
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__sin (double x)
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{
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double xx, res, t, cor, y, s, c, sn, ssn, cs, ccs, xn, a, da, db, eps, xn1,
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xn2;
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mynumber u, v;
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int4 k, m, n;
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double retval = 0;
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SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
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u.x = x;
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m = u.i[HIGH_HALF];
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k = 0x7fffffff & m; /* no sign */
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if (k < 0x3e500000) /* if x->0 =>sin(x)=x */
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retval = x;
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/*---------------------------- 2^-26 < |x|< 0.25 ----------------------*/
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else if (k < 0x3fd00000)
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{
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xx = x * x;
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/* Taylor series. */
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t = POLYNOMIAL (xx) * (xx * x);
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res = x + t;
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cor = (x - res) + t;
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retval = (res == res + 1.07 * cor) ? res : slow (x);
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} /* else if (k < 0x3fd00000) */
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/*---------------------------- 0.25<|x|< 0.855469---------------------- */
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else if (k < 0x3feb6000)
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{
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u.x = (m > 0) ? big + x : big - x;
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y = (m > 0) ? x - (u.x - big) : x + (u.x - big);
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xx = y * y;
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s = y + y * xx * (sn3 + xx * sn5);
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c = xx * (cs2 + xx * (cs4 + xx * cs6));
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SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
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if (m <= 0)
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{
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sn = -sn;
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ssn = -ssn;
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}
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cor = (ssn + s * ccs - sn * c) + cs * s;
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res = sn + cor;
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cor = (sn - res) + cor;
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retval = (res == res + 1.096 * cor) ? res : slow1 (x);
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} /* else if (k < 0x3feb6000) */
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/*----------------------- 0.855469 <|x|<2.426265 ----------------------*/
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else if (k < 0x400368fd)
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{
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y = (m > 0) ? hp0 - x : hp0 + x;
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if (y >= 0)
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{
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u.x = big + y;
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y = (y - (u.x - big)) + hp1;
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}
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else
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{
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u.x = big - y;
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y = (-hp1) - (y + (u.x - big));
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}
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res = do_cos (u, y, &cor);
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retval = (res == res + 1.020 * cor) ? ((m > 0) ? res : -res) : slow2 (x);
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} /* else if (k < 0x400368fd) */
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/*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
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else if (k < 0x419921FB)
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{
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t = (x * hpinv + toint);
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xn = t - toint;
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v.x = t;
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y = (x - xn * mp1) - xn * mp2;
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n = v.i[LOW_HALF] & 3;
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da = xn * mp3;
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a = y - da;
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da = (y - a) - da;
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eps = ABS (x) * 1.2e-30;
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switch (n)
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{ /* quarter of unit circle */
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case 0:
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case 2:
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xx = a * a;
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if (n)
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{
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a = -a;
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da = -da;
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}
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if (xx < 0.01588)
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{
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/* Taylor series. */
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res = TAYLOR_SIN (xx, a, da, cor);
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cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
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retval = (res == res + cor) ? res : sloww (a, da, x);
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}
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else
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{
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if (a > 0)
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m = 1;
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else
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{
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m = 0;
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a = -a;
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da = -da;
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}
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u.x = big + a;
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y = a - (u.x - big);
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res = do_sin (u, y, da, &cor);
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cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
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retval = ((res == res + cor) ? ((m) ? res : -res)
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: sloww1 (a, da, x, m));
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}
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break;
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case 1:
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case 3:
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if (a < 0)
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{
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a = -a;
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da = -da;
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}
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u.x = big + a;
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y = a - (u.x - big) + da;
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res = do_cos (u, y, &cor);
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cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
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retval = ((res == res + cor) ? ((n & 2) ? -res : res)
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: sloww2 (a, da, x, n));
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break;
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}
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} /* else if (k < 0x419921FB ) */
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/*---------------------105414350 <|x|< 281474976710656 --------------------*/
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else if (k < 0x42F00000)
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{
|
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t = (x * hpinv + toint);
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xn = t - toint;
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v.x = t;
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xn1 = (xn + 8.0e22) - 8.0e22;
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xn2 = xn - xn1;
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y = ((((x - xn1 * mp1) - xn1 * mp2) - xn2 * mp1) - xn2 * mp2);
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n = v.i[LOW_HALF] & 3;
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da = xn1 * pp3;
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t = y - da;
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da = (y - t) - da;
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da = (da - xn2 * pp3) - xn * pp4;
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a = t + da;
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da = (t - a) + da;
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eps = 1.0e-24;
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switch (n)
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{
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case 0:
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case 2:
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xx = a * a;
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if (n)
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{
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a = -a;
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da = -da;
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}
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if (xx < 0.01588)
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{
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/* Taylor series. */
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res = TAYLOR_SIN (xx, a, da, cor);
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cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
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retval = (res == res + cor) ? res : bsloww (a, da, x, n);
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}
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else
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{
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double t;
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if (a > 0)
|
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{
|
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m = 1;
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t = a;
|
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db = da;
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}
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else
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{
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m = 0;
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t = -a;
|
|
db = -da;
|
|
}
|
|
u.x = big + t;
|
|
y = t - (u.x - big);
|
|
res = do_sin (u, y, db, &cor);
|
|
cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
|
|
retval = ((res == res + cor) ? ((m) ? res : -res)
|
|
: bsloww1 (a, da, x, n));
|
|
}
|
|
break;
|
|
|
|
case 1:
|
|
case 3:
|
|
if (a < 0)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
u.x = big + a;
|
|
y = a - (u.x - big) + da;
|
|
res = do_cos (u, y, &cor);
|
|
cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
|
|
retval = ((res == res + cor) ? ((n & 2) ? -res : res)
|
|
: bsloww2 (a, da, x, n));
|
|
break;
|
|
}
|
|
} /* else if (k < 0x42F00000 ) */
|
|
|
|
/* -----------------281474976710656 <|x| <2^1024----------------------------*/
|
|
else if (k < 0x7ff00000)
|
|
retval = reduce_and_compute (x, 0);
|
|
|
|
/*--------------------- |x| > 2^1024 ----------------------------------*/
|
|
else
|
|
{
|
|
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
|
|
__set_errno (EDOM);
|
|
retval = x / x;
|
|
}
|
|
|
|
return retval;
|
|
}
|
|
|
|
|
|
/*******************************************************************/
|
|
/* An ultimate cos routine. Given an IEEE double machine number x */
|
|
/* it computes the correctly rounded (to nearest) value of cos(x) */
|
|
/*******************************************************************/
|
|
|
|
double
|
|
SECTION
|
|
__cos (double x)
|
|
{
|
|
double y, xx, res, t, cor, xn, a, da, db, eps, xn1,
|
|
xn2;
|
|
mynumber u, v;
|
|
int4 k, m, n;
|
|
|
|
double retval = 0;
|
|
|
|
SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
|
|
|
|
u.x = x;
|
|
m = u.i[HIGH_HALF];
|
|
k = 0x7fffffff & m;
|
|
|
|
/* |x|<2^-27 => cos(x)=1 */
|
|
if (k < 0x3e400000)
|
|
retval = 1.0;
|
|
|
|
else if (k < 0x3feb6000)
|
|
{ /* 2^-27 < |x| < 0.855469 */
|
|
y = ABS (x);
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
res = do_cos (u, y, &cor);
|
|
retval = (res == res + 1.020 * cor) ? res : cslow2 (x);
|
|
} /* else if (k < 0x3feb6000) */
|
|
|
|
else if (k < 0x400368fd)
|
|
{ /* 0.855469 <|x|<2.426265 */ ;
|
|
y = hp0 - ABS (x);
|
|
a = y + hp1;
|
|
da = (y - a) + hp1;
|
|
xx = a * a;
|
|
if (xx < 0.01588)
|
|
{
|
|
res = TAYLOR_SIN (xx, a, da, cor);
|
|
cor = (cor > 0) ? 1.02 * cor + 1.0e-31 : 1.02 * cor - 1.0e-31;
|
|
retval = (res == res + cor) ? res : csloww (a, da, x);
|
|
}
|
|
else
|
|
{
|
|
if (a > 0)
|
|
{
|
|
m = 1;
|
|
}
|
|
else
|
|
{
|
|
m = 0;
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
u.x = big + a;
|
|
y = a - (u.x - big);
|
|
res = do_sin (u, y, da, &cor);
|
|
cor = (cor > 0) ? 1.035 * cor + 1.0e-31 : 1.035 * cor - 1.0e-31;
|
|
retval = ((res == res + cor) ? ((m) ? res : -res)
|
|
: csloww1 (a, da, x, m));
|
|
}
|
|
|
|
} /* else if (k < 0x400368fd) */
|
|
|
|
|
|
else if (k < 0x419921FB)
|
|
{ /* 2.426265<|x|< 105414350 */
|
|
t = (x * hpinv + toint);
|
|
xn = t - toint;
|
|
v.x = t;
|
|
y = (x - xn * mp1) - xn * mp2;
|
|
n = v.i[LOW_HALF] & 3;
|
|
da = xn * mp3;
|
|
a = y - da;
|
|
da = (y - a) - da;
|
|
eps = ABS (x) * 1.2e-30;
|
|
|
|
switch (n)
|
|
{
|
|
case 1:
|
|
case 3:
|
|
xx = a * a;
|
|
if (n == 1)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
if (xx < 0.01588)
|
|
{
|
|
res = TAYLOR_SIN (xx, a, da, cor);
|
|
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
|
|
retval = (res == res + cor) ? res : csloww (a, da, x);
|
|
}
|
|
else
|
|
{
|
|
if (a > 0)
|
|
{
|
|
m = 1;
|
|
}
|
|
else
|
|
{
|
|
m = 0;
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
u.x = big + a;
|
|
y = a - (u.x - big);
|
|
res = do_sin (u, y, da, &cor);
|
|
cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
|
|
retval = ((res == res + cor) ? ((m) ? res : -res)
|
|
: csloww1 (a, da, x, m));
|
|
}
|
|
break;
|
|
|
|
case 0:
|
|
case 2:
|
|
if (a < 0)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
u.x = big + a;
|
|
y = a - (u.x - big) + da;
|
|
res = do_cos (u, y, &cor);
|
|
cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
|
|
retval = ((res == res + cor) ? ((n) ? -res : res)
|
|
: csloww2 (a, da, x, n));
|
|
break;
|
|
}
|
|
} /* else if (k < 0x419921FB ) */
|
|
|
|
else if (k < 0x42F00000)
|
|
{
|
|
t = (x * hpinv + toint);
|
|
xn = t - toint;
|
|
v.x = t;
|
|
xn1 = (xn + 8.0e22) - 8.0e22;
|
|
xn2 = xn - xn1;
|
|
y = ((((x - xn1 * mp1) - xn1 * mp2) - xn2 * mp1) - xn2 * mp2);
|
|
n = v.i[LOW_HALF] & 3;
|
|
da = xn1 * pp3;
|
|
t = y - da;
|
|
da = (y - t) - da;
|
|
da = (da - xn2 * pp3) - xn * pp4;
|
|
a = t + da;
|
|
da = (t - a) + da;
|
|
eps = 1.0e-24;
|
|
|
|
switch (n)
|
|
{
|
|
case 1:
|
|
case 3:
|
|
xx = a * a;
|
|
if (n == 1)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
if (xx < 0.01588)
|
|
{
|
|
res = TAYLOR_SIN (xx, a, da, cor);
|
|
cor = (cor > 0) ? 1.02 * cor + eps : 1.02 * cor - eps;
|
|
retval = (res == res + cor) ? res : bsloww (a, da, x, n);
|
|
}
|
|
else
|
|
{
|
|
double t;
|
|
if (a > 0)
|
|
{
|
|
m = 1;
|
|
t = a;
|
|
db = da;
|
|
}
|
|
else
|
|
{
|
|
m = 0;
|
|
t = -a;
|
|
db = -da;
|
|
}
|
|
u.x = big + t;
|
|
y = t - (u.x - big);
|
|
res = do_sin (u, y, db, &cor);
|
|
cor = (cor > 0) ? 1.035 * cor + eps : 1.035 * cor - eps;
|
|
retval = ((res == res + cor) ? ((m) ? res : -res)
|
|
: bsloww1 (a, da, x, n));
|
|
}
|
|
break;
|
|
|
|
case 0:
|
|
case 2:
|
|
if (a < 0)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
u.x = big + a;
|
|
y = a - (u.x - big) + da;
|
|
res = do_cos (u, y, &cor);
|
|
cor = (cor > 0) ? 1.025 * cor + eps : 1.025 * cor - eps;
|
|
retval = ((res == res + cor) ? ((n) ? -res : res)
|
|
: bsloww2 (a, da, x, n));
|
|
break;
|
|
}
|
|
} /* else if (k < 0x42F00000 ) */
|
|
|
|
/* 281474976710656 <|x| <2^1024 */
|
|
else if (k < 0x7ff00000)
|
|
retval = reduce_and_compute (x, 1);
|
|
|
|
else
|
|
{
|
|
if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
|
|
__set_errno (EDOM);
|
|
retval = x / x; /* |x| > 2^1024 */
|
|
}
|
|
|
|
return retval;
|
|
}
|
|
|
|
/************************************************************************/
|
|
/* Routine compute sin(x) for 2^-26 < |x|< 0.25 by Taylor with more */
|
|
/* precision and if still doesn't accurate enough by mpsin or dubsin */
|
|
/************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
slow (double x)
|
|
{
|
|
double res, cor, w[2];
|
|
res = TAYLOR_SLOW (x, 0, cor);
|
|
if (res == res + 1.0007 * cor)
|
|
return res;
|
|
else
|
|
{
|
|
__dubsin (ABS (x), 0, w);
|
|
if (w[0] == w[0] + 1.000000001 * w[1])
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
|
|
}
|
|
}
|
|
|
|
/*******************************************************************************/
|
|
/* Routine compute sin(x) for 0.25<|x|< 0.855469 by __sincostab.tbl and Taylor */
|
|
/* and if result still doesn't accurate enough by mpsin or dubsin */
|
|
/*******************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
slow1 (double x)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
y = ABS (x);
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
res = do_sin_slow (u, y, 0, 0, &cor);
|
|
if (res == res + cor)
|
|
return (x > 0) ? res : -res;
|
|
else
|
|
{
|
|
__dubsin (ABS (x), 0, w);
|
|
if (w[0] == w[0] + 1.000000005 * w[1])
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
|
|
}
|
|
}
|
|
|
|
/**************************************************************************/
|
|
/* Routine compute sin(x) for 0.855469 <|x|<2.426265 by __sincostab.tbl */
|
|
/* and if result still doesn't accurate enough by mpsin or dubsin */
|
|
/**************************************************************************/
|
|
static double
|
|
SECTION
|
|
slow2 (double x)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, y1, y2, cor, res, del;
|
|
|
|
y = ABS (x);
|
|
y = hp0 - y;
|
|
if (y >= 0)
|
|
{
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
del = hp1;
|
|
}
|
|
else
|
|
{
|
|
u.x = big - y;
|
|
y = -(y + (u.x - big));
|
|
del = -hp1;
|
|
}
|
|
res = do_cos_slow (u, y, del, 0, &cor);
|
|
if (res == res + cor)
|
|
return (x > 0) ? res : -res;
|
|
else
|
|
{
|
|
y = ABS (x) - hp0;
|
|
y1 = y - hp1;
|
|
y2 = (y - y1) - hp1;
|
|
__docos (y1, y2, w);
|
|
if (w[0] == w[0] + 1.000000005 * w[1])
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
return (x > 0) ? __mpsin (x, 0, false) : -__mpsin (-x, 0, false);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) (Double-Length number) where x is small enough*/
|
|
/* to use Taylor series around zero and (x+dx) */
|
|
/* in first or third quarter of unit circle.Routine receive also */
|
|
/* (right argument) the original value of x for computing error of */
|
|
/* result.And if result not accurate enough routine calls mpsin1 or dubsin */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
sloww (double x, double dx, double orig)
|
|
{
|
|
double y, t, res, cor, w[2], a, da, xn;
|
|
mynumber v;
|
|
int4 n;
|
|
res = TAYLOR_SLOW (x, dx, cor);
|
|
if (cor > 0)
|
|
cor = 1.0005 * cor + ABS (orig) * 3.1e-30;
|
|
else
|
|
cor = 1.0005 * cor - ABS (orig) * 3.1e-30;
|
|
|
|
if (res == res + cor)
|
|
return res;
|
|
else
|
|
{
|
|
(x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
|
|
if (w[1] > 0)
|
|
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-30;
|
|
else
|
|
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-30;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
{
|
|
t = (orig * hpinv + toint);
|
|
xn = t - toint;
|
|
v.x = t;
|
|
y = (orig - xn * mp1) - xn * mp2;
|
|
n = v.i[LOW_HALF] & 3;
|
|
da = xn * pp3;
|
|
t = y - da;
|
|
da = (y - t) - da;
|
|
y = xn * pp4;
|
|
a = t - y;
|
|
da = ((t - a) - y) + da;
|
|
if (n & 2)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
(a > 0) ? __dubsin (a, da, w) : __dubsin (-a, -da, w);
|
|
if (w[1] > 0)
|
|
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-40;
|
|
else
|
|
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-40;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (a > 0) ? w[0] : -w[0];
|
|
else
|
|
return __mpsin (orig, 0, true);
|
|
}
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) (Double-Length number) where x in first or */
|
|
/* third quarter of unit circle.Routine receive also (right argument) the */
|
|
/* original value of x for computing error of result.And if result not */
|
|
/* accurate enough routine calls mpsin1 or dubsin */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
sloww1 (double x, double dx, double orig, int m)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
u.x = big + x;
|
|
y = x - (u.x - big);
|
|
res = do_sin_slow (u, y, dx, 3.1e-30 * ABS (orig), &cor);
|
|
|
|
if (res == res + cor)
|
|
return (m > 0) ? res : -res;
|
|
else
|
|
{
|
|
__dubsin (x, dx, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (m > 0) ? w[0] : -w[0];
|
|
else
|
|
return __mpsin (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) (Double-Length number) where x in second or */
|
|
/* fourth quarter of unit circle.Routine receive also the original value */
|
|
/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
|
|
/* accurate enough routine calls mpsin1 or dubsin */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
sloww2 (double x, double dx, double orig, int n)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
u.x = big + x;
|
|
y = x - (u.x - big);
|
|
res = do_cos_slow (u, y, dx, 3.1e-30 * ABS (orig), &cor);
|
|
|
|
if (res == res + cor)
|
|
return (n & 2) ? -res : res;
|
|
else
|
|
{
|
|
__docos (x, dx, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (n & 2) ? -w[0] : w[0];
|
|
else
|
|
return __mpsin (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
|
|
/* is small enough to use Taylor series around zero and (x+dx) */
|
|
/* in first or third quarter of unit circle.Routine receive also */
|
|
/* (right argument) the original value of x for computing error of */
|
|
/* result.And if result not accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
bsloww (double x, double dx, double orig, int n)
|
|
{
|
|
double res, cor, w[2];
|
|
|
|
res = TAYLOR_SLOW (x, dx, cor);
|
|
cor = (cor > 0) ? 1.0005 * cor + 1.1e-24 : 1.0005 * cor - 1.1e-24;
|
|
if (res == res + cor)
|
|
return res;
|
|
else
|
|
{
|
|
(x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
|
|
if (w[1] > 0)
|
|
cor = 1.000000001 * w[1] + 1.1e-24;
|
|
else
|
|
cor = 1.000000001 * w[1] - 1.1e-24;
|
|
if (w[0] == w[0] + cor)
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
|
|
/* in first or third quarter of unit circle.Routine receive also */
|
|
/* (right argument) the original value of x for computing error of result.*/
|
|
/* And if result not accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
bsloww1 (double x, double dx, double orig, int n)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
y = ABS (x);
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
dx = (x > 0) ? dx : -dx;
|
|
res = do_sin_slow (u, y, dx, 1.1e-24, &cor);
|
|
if (res == res + cor)
|
|
return (x > 0) ? res : -res;
|
|
else
|
|
{
|
|
__dubsin (ABS (x), dx, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-24;
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-24;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
return (n & 1) ? __mpcos (orig, 0, true) : __mpsin (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) or cos(x+dx) (Double-Length number) where x */
|
|
/* in second or fourth quarter of unit circle.Routine receive also the */
|
|
/* original value and quarter(n= 1or 3)of x for computing error of result. */
|
|
/* And if result not accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
bsloww2 (double x, double dx, double orig, int n)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
y = ABS (x);
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
dx = (x > 0) ? dx : -dx;
|
|
res = do_cos_slow (u, y, dx, 1.1e-24, &cor);
|
|
if (res == res + cor)
|
|
return (n & 2) ? -res : res;
|
|
else
|
|
{
|
|
__docos (ABS (x), dx, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-24;
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-24;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (n & 2) ? -w[0] : w[0];
|
|
else
|
|
return (n & 1) ? __mpsin (orig, 0, true) : __mpcos (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
/************************************************************************/
|
|
/* Routine compute cos(x) for 2^-27 < |x|< 0.25 by Taylor with more */
|
|
/* precision and if still doesn't accurate enough by mpcos or docos */
|
|
/************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
cslow2 (double x)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
y = ABS (x);
|
|
u.x = big + y;
|
|
y = y - (u.x - big);
|
|
res = do_cos_slow (u, y, 0, 0, &cor);
|
|
if (res == res + cor)
|
|
return res;
|
|
else
|
|
{
|
|
y = ABS (x);
|
|
__docos (y, 0, w);
|
|
if (w[0] == w[0] + 1.000000005 * w[1])
|
|
return w[0];
|
|
else
|
|
return __mpcos (x, 0, false);
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute cos(x+dx) (Double-Length number) where x is small enough*/
|
|
/* to use Taylor series around zero and (x+dx) .Routine receive also */
|
|
/* (right argument) the original value of x for computing error of */
|
|
/* result.And if result not accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
|
|
static double
|
|
SECTION
|
|
csloww (double x, double dx, double orig)
|
|
{
|
|
double y, t, res, cor, w[2], a, da, xn;
|
|
mynumber v;
|
|
int4 n;
|
|
|
|
/* Taylor series */
|
|
res = TAYLOR_SLOW (x, dx, cor);
|
|
|
|
if (cor > 0)
|
|
cor = 1.0005 * cor + ABS (orig) * 3.1e-30;
|
|
else
|
|
cor = 1.0005 * cor - ABS (orig) * 3.1e-30;
|
|
|
|
if (res == res + cor)
|
|
return res;
|
|
else
|
|
{
|
|
(x > 0) ? __dubsin (x, dx, w) : __dubsin (-x, -dx, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-30;
|
|
else
|
|
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-30;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (x > 0) ? w[0] : -w[0];
|
|
else
|
|
{
|
|
t = (orig * hpinv + toint);
|
|
xn = t - toint;
|
|
v.x = t;
|
|
y = (orig - xn * mp1) - xn * mp2;
|
|
n = v.i[LOW_HALF] & 3;
|
|
da = xn * pp3;
|
|
t = y - da;
|
|
da = (y - t) - da;
|
|
y = xn * pp4;
|
|
a = t - y;
|
|
da = ((t - a) - y) + da;
|
|
if (n == 1)
|
|
{
|
|
a = -a;
|
|
da = -da;
|
|
}
|
|
(a > 0) ? __dubsin (a, da, w) : __dubsin (-a, -da, w);
|
|
|
|
if (w[1] > 0)
|
|
cor = 1.000000001 * w[1] + ABS (orig) * 1.1e-40;
|
|
else
|
|
cor = 1.000000001 * w[1] - ABS (orig) * 1.1e-40;
|
|
|
|
if (w[0] == w[0] + cor)
|
|
return (a > 0) ? w[0] : -w[0];
|
|
else
|
|
return __mpcos (orig, 0, true);
|
|
}
|
|
}
|
|
}
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) (Double-Length number) where x in first or */
|
|
/* third quarter of unit circle.Routine receive also (right argument) the */
|
|
/* original value of x for computing error of result.And if result not */
|
|
/* accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
csloww1 (double x, double dx, double orig, int m)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
u.x = big + x;
|
|
y = x - (u.x - big);
|
|
res = do_sin_slow (u, y, dx, 3.1e-30 * ABS (orig), &cor);
|
|
|
|
if (res == res + cor)
|
|
return (m > 0) ? res : -res;
|
|
else
|
|
{
|
|
__dubsin (x, dx, w);
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
|
|
if (w[0] == w[0] + cor)
|
|
return (m > 0) ? w[0] : -w[0];
|
|
else
|
|
return __mpcos (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
|
|
/***************************************************************************/
|
|
/* Routine compute sin(x+dx) (Double-Length number) where x in second or */
|
|
/* fourth quarter of unit circle.Routine receive also the original value */
|
|
/* and quarter(n= 1or 3)of x for computing error of result.And if result not*/
|
|
/* accurate enough routine calls other routines */
|
|
/***************************************************************************/
|
|
|
|
static double
|
|
SECTION
|
|
csloww2 (double x, double dx, double orig, int n)
|
|
{
|
|
mynumber u;
|
|
double w[2], y, cor, res;
|
|
|
|
u.x = big + x;
|
|
y = x - (u.x - big);
|
|
res = do_cos_slow (u, y, dx, 3.1e-30 * ABS (orig), &cor);
|
|
|
|
if (res == res + cor)
|
|
return (n) ? -res : res;
|
|
else
|
|
{
|
|
__docos (x, dx, w);
|
|
if (w[1] > 0)
|
|
cor = 1.000000005 * w[1] + 1.1e-30 * ABS (orig);
|
|
else
|
|
cor = 1.000000005 * w[1] - 1.1e-30 * ABS (orig);
|
|
if (w[0] == w[0] + cor)
|
|
return (n) ? -w[0] : w[0];
|
|
else
|
|
return __mpcos (orig, 0, true);
|
|
}
|
|
}
|
|
|
|
#ifndef __cos
|
|
weak_alias (__cos, cos)
|
|
# ifdef NO_LONG_DOUBLE
|
|
strong_alias (__cos, __cosl)
|
|
weak_alias (__cos, cosl)
|
|
# endif
|
|
#endif
|
|
#ifndef __sin
|
|
weak_alias (__sin, sin)
|
|
# ifdef NO_LONG_DOUBLE
|
|
strong_alias (__sin, __sinl)
|
|
weak_alias (__sin, sinl)
|
|
# endif
|
|
#endif
|