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src/share/vm/runtime/sharedRuntimeTrig.cpp@a61af66fc99e (annotated)

src/share/vm/runtime/sharedRuntimeTrig.cpp

Sat, 01 Dec 2007 00:00:00 +0000

author

duke

date

Sat, 01 Dec 2007 00:00:00 +0000

changeset 435

a61af66fc99e

child 1840

fb57d4cf76c2

permissions

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duke@435	1	/*
duke@435	2	* Copyright 2001-2005 Sun Microsystems, Inc. All Rights Reserved.
duke@435	3	* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
duke@435	4	*
duke@435	5	* This code is free software; you can redistribute it and/or modify it
duke@435	6	* under the terms of the GNU General Public License version 2 only, as
duke@435	7	* published by the Free Software Foundation.
duke@435	8	*
duke@435	9	* This code is distributed in the hope that it will be useful, but WITHOUT
duke@435	10	* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
duke@435	11	* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
duke@435	12	* version 2 for more details (a copy is included in the LICENSE file that
duke@435	13	* accompanied this code).
duke@435	14	*
duke@435	15	* You should have received a copy of the GNU General Public License version
duke@435	16	* 2 along with this work; if not, write to the Free Software Foundation,
duke@435	17	* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
duke@435	18	*
duke@435	19	* Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara,
duke@435	20	* CA 95054 USA or visit www.sun.com if you need additional information or
duke@435	21	* have any questions.
duke@435	22	*
duke@435	23	*/
duke@435	24
duke@435	25	#include "incls/_precompiled.incl"
duke@435	26	#include "incls/_sharedRuntimeTrig.cpp.incl"
duke@435	27
duke@435	28	// This file contains copies of the fdlibm routines used by
duke@435	29	// StrictMath. It turns out that it is almost always required to use
duke@435	30	// these runtime routines; the Intel CPU doesn't meet the Java
duke@435	31	// specification for sin/cos outside a certain limited argument range,
duke@435	32	// and the SPARC CPU doesn't appear to have sin/cos instructions. It
duke@435	33	// also turns out that avoiding the indirect call through function
duke@435	34	// pointer out to libjava.so in SharedRuntime speeds these routines up
duke@435	35	// by roughly 15% on both Win32/x86 and Solaris/SPARC.
duke@435	36
duke@435	37	// Enabling optimizations in this file causes incorrect code to be
duke@435	38	// generated; can not figure out how to turn down optimization for one
duke@435	39	// file in the IDE on Windows
duke@435	40	#ifdef WIN32
duke@435	41	# pragma optimize ( "", off )
duke@435	42	#endif
duke@435	43
duke@435	44	#include <math.h>
duke@435	45
duke@435	46	// VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles
duke@435	47	// [jk] this is not 100% correct because the float word order may different
duke@435	48	// from the byte order (e.g. on ARM)
duke@435	49	#ifdef VM_LITTLE_ENDIAN
duke@435	50	# define __HI(x) *(1+(int*)&x)
duke@435	51	# define __LO(x) *(int*)&x
duke@435	52	#else
duke@435	53	# define __HI(x) *(int*)&x
duke@435	54	# define __LO(x) *(1+(int*)&x)
duke@435	55	#endif
duke@435	56
duke@435	57	static double copysignA(double x, double y) {
duke@435	58	__HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
duke@435	59	return x;
duke@435	60	}
duke@435	61
duke@435	62	/*
duke@435	63	* scalbn (double x, int n)
duke@435	64	* scalbn(x,n) returns x* 2**n computed by exponent
duke@435	65	* manipulation rather than by actually performing an
duke@435	66	* exponentiation or a multiplication.
duke@435	67	*/
duke@435	68
duke@435	69	static const double
duke@435	70	two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
duke@435	71	twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */
duke@435	72	hugeX = 1.0e+300,
duke@435	73	tiny = 1.0e-300;
duke@435	74
duke@435	75	static double scalbnA (double x, int n) {
duke@435	76	int k,hx,lx;
duke@435	77	hx = __HI(x);
duke@435	78	lx = __LO(x);
duke@435	79	k = (hx&0x7ff00000)>>20; /* extract exponent */
duke@435	80	if (k==0) { /* 0 or subnormal x */
duke@435	81	if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
duke@435	82	x *= two54;
duke@435	83	hx = __HI(x);
duke@435	84	k = ((hx&0x7ff00000)>>20) - 54;
duke@435	85	if (n< -50000) return tiny*x; /*underflow*/
duke@435	86	}
duke@435	87	if (k==0x7ff) return x+x; /* NaN or Inf */
duke@435	88	k = k+n;
duke@435	89	if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */
duke@435	90	if (k > 0) /* normal result */
duke@435	91	{__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
duke@435	92	if (k <= -54) {
duke@435	93	if (n > 50000) /* in case integer overflow in n+k */
duke@435	94	return hugeX*copysignA(hugeX,x); /*overflow*/
duke@435	95	else return tiny*copysignA(tiny,x); /*underflow*/
duke@435	96	}
duke@435	97	k += 54; /* subnormal result */
duke@435	98	__HI(x) = (hx&0x800fffff)|(k<<20);
duke@435	99	return x*twom54;
duke@435	100	}
duke@435	101
duke@435	102	/*
duke@435	103	* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
duke@435	104	* double x[],y[]; int e0,nx,prec; int ipio2[];
duke@435	105	*
duke@435	106	* __kernel_rem_pio2 return the last three digits of N with
duke@435	107	* y = x - N*pi/2
duke@435	108	* so that |y| < pi/2.
duke@435	109	*
duke@435	110	* The method is to compute the integer (mod 8) and fraction parts of
duke@435	111	* (2/pi)*x without doing the full multiplication. In general we
duke@435	112	* skip the part of the product that are known to be a huge integer (
duke@435	113	* more accurately, = 0 mod 8 ). Thus the number of operations are
duke@435	114	* independent of the exponent of the input.
duke@435	115	*
duke@435	116	* (2/pi) is represented by an array of 24-bit integers in ipio2[].
duke@435	117	*
duke@435	118	* Input parameters:
duke@435	119	* x[] The input value (must be positive) is broken into nx
duke@435	120	* pieces of 24-bit integers in double precision format.
duke@435	121	* x[i] will be the i-th 24 bit of x. The scaled exponent
duke@435	122	* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
duke@435	123	* match x's up to 24 bits.
duke@435	124	*
duke@435	125	* Example of breaking a double positive z into x[0]+x[1]+x[2]:
duke@435	126	* e0 = ilogb(z)-23
duke@435	127	* z = scalbn(z,-e0)
duke@435	128	* for i = 0,1,2
duke@435	129	* x[i] = floor(z)
duke@435	130	* z = (z-x[i])*2**24
duke@435	131	*
duke@435	132	*
duke@435	133	* y[] ouput result in an array of double precision numbers.
duke@435	134	* The dimension of y[] is:
duke@435	135	* 24-bit precision 1
duke@435	136	* 53-bit precision 2
duke@435	137	* 64-bit precision 2
duke@435	138	* 113-bit precision 3
duke@435	139	* The actual value is the sum of them. Thus for 113-bit
duke@435	140	* precsion, one may have to do something like:
duke@435	141	*
duke@435	142	* long double t,w,r_head, r_tail;
duke@435	143	* t = (long double)y[2] + (long double)y[1];
duke@435	144	* w = (long double)y[0];
duke@435	145	* r_head = t+w;
duke@435	146	* r_tail = w - (r_head - t);
duke@435	147	*
duke@435	148	* e0 The exponent of x[0]
duke@435	149	*
duke@435	150	* nx dimension of x[]
duke@435	151	*
duke@435	152	* prec an interger indicating the precision:
duke@435	153	* 0 24 bits (single)
duke@435	154	* 1 53 bits (double)
duke@435	155	* 2 64 bits (extended)
duke@435	156	* 3 113 bits (quad)
duke@435	157	*
duke@435	158	* ipio2[]
duke@435	159	* integer array, contains the (24*i)-th to (24*i+23)-th
duke@435	160	* bit of 2/pi after binary point. The corresponding
duke@435	161	* floating value is
duke@435	162	*
duke@435	163	* ipio2[i] * 2^(-24(i+1)).
duke@435	164	*
duke@435	165	* External function:
duke@435	166	* double scalbn(), floor();
duke@435	167	*
duke@435	168	*
duke@435	169	* Here is the description of some local variables:
duke@435	170	*
duke@435	171	* jk jk+1 is the initial number of terms of ipio2[] needed
duke@435	172	* in the computation. The recommended value is 2,3,4,
duke@435	173	* 6 for single, double, extended,and quad.
duke@435	174	*
duke@435	175	* jz local integer variable indicating the number of
duke@435	176	* terms of ipio2[] used.
duke@435	177	*
duke@435	178	* jx nx - 1
duke@435	179	*
duke@435	180	* jv index for pointing to the suitable ipio2[] for the
duke@435	181	* computation. In general, we want
duke@435	182	* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
duke@435	183	* is an integer. Thus
duke@435	184	* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
duke@435	185	* Hence jv = max(0,(e0-3)/24).
duke@435	186	*
duke@435	187	* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
duke@435	188	*
duke@435	189	* q[] double array with integral value, representing the
duke@435	190	* 24-bits chunk of the product of x and 2/pi.
duke@435	191	*
duke@435	192	* q0 the corresponding exponent of q[0]. Note that the
duke@435	193	* exponent for q[i] would be q0-24*i.
duke@435	194	*
duke@435	195	* PIo2[] double precision array, obtained by cutting pi/2
duke@435	196	* into 24 bits chunks.
duke@435	197	*
duke@435	198	* f[] ipio2[] in floating point
duke@435	199	*
duke@435	200	* iq[] integer array by breaking up q[] in 24-bits chunk.
duke@435	201	*
duke@435	202	* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
duke@435	203	*
duke@435	204	* ih integer. If >0 it indicats q[] is >= 0.5, hence
duke@435	205	* it also indicates the *sign* of the result.
duke@435	206	*
duke@435	207	*/
duke@435	208
duke@435	209
duke@435	210	/*
duke@435	211	* Constants:
duke@435	212	* The hexadecimal values are the intended ones for the following
duke@435	213	* constants. The decimal values may be used, provided that the
duke@435	214	* compiler will convert from decimal to binary accurately enough
duke@435	215	* to produce the hexadecimal values shown.
duke@435	216	*/
duke@435	217
duke@435	218
duke@435	219	static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
duke@435	220
duke@435	221	static const double PIo2[] = {
duke@435	222	1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
duke@435	223	7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
duke@435	224	5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
duke@435	225	3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
duke@435	226	1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
duke@435	227	1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
duke@435	228	2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
duke@435	229	2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
duke@435	230	};
duke@435	231
duke@435	232	static const double
duke@435	233	zeroB = 0.0,
duke@435	234	one = 1.0,
duke@435	235	two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
duke@435	236	twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
duke@435	237
duke@435	238	static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) {
duke@435	239	int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
duke@435	240	double z,fw,f[20],fq[20],q[20];
duke@435	241
duke@435	242	/* initialize jk*/
duke@435	243	jk = init_jk[prec];
duke@435	244	jp = jk;
duke@435	245
duke@435	246	/* determine jx,jv,q0, note that 3>q0 */
duke@435	247	jx = nx-1;
duke@435	248	jv = (e0-3)/24; if(jv<0) jv=0;
duke@435	249	q0 = e0-24*(jv+1);
duke@435	250
duke@435	251	/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
duke@435	252	j = jv-jx; m = jx+jk;
duke@435	253	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j];
duke@435	254
duke@435	255	/* compute q[0],q[1],...q[jk] */
duke@435	256	for (i=0;i<=jk;i++) {
duke@435	257	for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
duke@435	258	}
duke@435	259
duke@435	260	jz = jk;
duke@435	261	recompute:
duke@435	262	/* distill q[] into iq[] reversingly */
duke@435	263	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
duke@435	264	fw = (double)((int)(twon24* z));
duke@435	265	iq[i] = (int)(z-two24B*fw);
duke@435	266	z = q[j-1]+fw;
duke@435	267	}
duke@435	268
duke@435	269	/* compute n */
duke@435	270	z = scalbnA(z,q0); /* actual value of z */
duke@435	271	z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
duke@435	272	n = (int) z;
duke@435	273	z -= (double)n;
duke@435	274	ih = 0;
duke@435	275	if(q0>0) { /* need iq[jz-1] to determine n */
duke@435	276	i = (iq[jz-1]>>(24-q0)); n += i;
duke@435	277	iq[jz-1] -= i<<(24-q0);
duke@435	278	ih = iq[jz-1]>>(23-q0);
duke@435	279	}
duke@435	280	else if(q0==0) ih = iq[jz-1]>>23;
duke@435	281	else if(z>=0.5) ih=2;
duke@435	282
duke@435	283	if(ih>0) { /* q > 0.5 */
duke@435	284	n += 1; carry = 0;
duke@435	285	for(i=0;i<jz ;i++) { /* compute 1-q */
duke@435	286	j = iq[i];
duke@435	287	if(carry==0) {
duke@435	288	if(j!=0) {
duke@435	289	carry = 1; iq[i] = 0x1000000- j;
duke@435	290	}
duke@435	291	} else iq[i] = 0xffffff - j;
duke@435	292	}
duke@435	293	if(q0>0) { /* rare case: chance is 1 in 12 */
duke@435	294	switch(q0) {
duke@435	295	case 1:
duke@435	296	iq[jz-1] &= 0x7fffff; break;
duke@435	297	case 2:
duke@435	298	iq[jz-1] &= 0x3fffff; break;
duke@435	299	}
duke@435	300	}
duke@435	301	if(ih==2) {
duke@435	302	z = one - z;
duke@435	303	if(carry!=0) z -= scalbnA(one,q0);
duke@435	304	}
duke@435	305	}
duke@435	306
duke@435	307	/* check if recomputation is needed */
duke@435	308	if(z==zeroB) {
duke@435	309	j = 0;
duke@435	310	for (i=jz-1;i>=jk;i--) j |= iq[i];
duke@435	311	if(j==0) { /* need recomputation */
duke@435	312	for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
duke@435	313
duke@435	314	for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
duke@435	315	f[jx+i] = (double) ipio2[jv+i];
duke@435	316	for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
duke@435	317	q[i] = fw;
duke@435	318	}
duke@435	319	jz += k;
duke@435	320	goto recompute;
duke@435	321	}
duke@435	322	}
duke@435	323
duke@435	324	/* chop off zero terms */
duke@435	325	if(z==0.0) {
duke@435	326	jz -= 1; q0 -= 24;
duke@435	327	while(iq[jz]==0) { jz--; q0-=24;}
duke@435	328	} else { /* break z into 24-bit if neccessary */
duke@435	329	z = scalbnA(z,-q0);
duke@435	330	if(z>=two24B) {
duke@435	331	fw = (double)((int)(twon24*z));
duke@435	332	iq[jz] = (int)(z-two24B*fw);
duke@435	333	jz += 1; q0 += 24;
duke@435	334	iq[jz] = (int) fw;
duke@435	335	} else iq[jz] = (int) z ;
duke@435	336	}
duke@435	337
duke@435	338	/* convert integer "bit" chunk to floating-point value */
duke@435	339	fw = scalbnA(one,q0);
duke@435	340	for(i=jz;i>=0;i--) {
duke@435	341	q[i] = fw*(double)iq[i]; fw*=twon24;
duke@435	342	}
duke@435	343
duke@435	344	/* compute PIo2[0,...,jp]*q[jz,...,0] */
duke@435	345	for(i=jz;i>=0;i--) {
duke@435	346	for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
duke@435	347	fq[jz-i] = fw;
duke@435	348	}
duke@435	349
duke@435	350	/* compress fq[] into y[] */
duke@435	351	switch(prec) {
duke@435	352	case 0:
duke@435	353	fw = 0.0;
duke@435	354	for (i=jz;i>=0;i--) fw += fq[i];
duke@435	355	y[0] = (ih==0)? fw: -fw;
duke@435	356	break;
duke@435	357	case 1:
duke@435	358	case 2:
duke@435	359	fw = 0.0;
duke@435	360	for (i=jz;i>=0;i--) fw += fq[i];
duke@435	361	y[0] = (ih==0)? fw: -fw;
duke@435	362	fw = fq[0]-fw;
duke@435	363	for (i=1;i<=jz;i++) fw += fq[i];
duke@435	364	y[1] = (ih==0)? fw: -fw;
duke@435	365	break;
duke@435	366	case 3: /* painful */
duke@435	367	for (i=jz;i>0;i--) {
duke@435	368	fw = fq[i-1]+fq[i];
duke@435	369	fq[i] += fq[i-1]-fw;
duke@435	370	fq[i-1] = fw;
duke@435	371	}
duke@435	372	for (i=jz;i>1;i--) {
duke@435	373	fw = fq[i-1]+fq[i];
duke@435	374	fq[i] += fq[i-1]-fw;
duke@435	375	fq[i-1] = fw;
duke@435	376	}
duke@435	377	for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
duke@435	378	if(ih==0) {
duke@435	379	y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
duke@435	380	} else {
duke@435	381	y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
duke@435	382	}
duke@435	383	}
duke@435	384	return n&7;
duke@435	385	}
duke@435	386
duke@435	387
duke@435	388	/*
duke@435	389	* ====================================================
duke@435	390	* Copyright 13 Dec 1993 Sun Microsystems, Inc. All Rights Reserved.
duke@435	391	*
duke@435	392	* Developed at SunPro, a Sun Microsystems, Inc. business.
duke@435	393	* Permission to use, copy, modify, and distribute this
duke@435	394	* software is freely granted, provided that this notice
duke@435	395	* is preserved.
duke@435	396	* ====================================================
duke@435	397	*
duke@435	398	*/
duke@435	399
duke@435	400	/* __ieee754_rem_pio2(x,y)
duke@435	401	*
duke@435	402	* return the remainder of x rem pi/2 in y[0]+y[1]
duke@435	403	* use __kernel_rem_pio2()
duke@435	404	*/
duke@435	405
duke@435	406	/*
duke@435	407	* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
duke@435	408	*/
duke@435	409	static const int two_over_pi[] = {
duke@435	410	0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
duke@435	411	0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
duke@435	412	0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
duke@435	413	0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
duke@435	414	0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
duke@435	415	0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
duke@435	416	0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
duke@435	417	0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
duke@435	418	0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
duke@435	419	0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
duke@435	420	0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
duke@435	421	};
duke@435	422
duke@435	423	static const int npio2_hw[] = {
duke@435	424	0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
duke@435	425	0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
duke@435	426	0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
duke@435	427	0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
duke@435	428	0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
duke@435	429	0x404858EB, 0x404921FB,
duke@435	430	};
duke@435	431
duke@435	432	/*
duke@435	433	* invpio2: 53 bits of 2/pi
duke@435	434	* pio2_1: first 33 bit of pi/2
duke@435	435	* pio2_1t: pi/2 - pio2_1
duke@435	436	* pio2_2: second 33 bit of pi/2
duke@435	437	* pio2_2t: pi/2 - (pio2_1+pio2_2)
duke@435	438	* pio2_3: third 33 bit of pi/2
duke@435	439	* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
duke@435	440	*/
duke@435	441
duke@435	442	static const double
duke@435	443	zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
duke@435	444	half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
duke@435	445	two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
duke@435	446	invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
duke@435	447	pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
duke@435	448	pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
duke@435	449	pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
duke@435	450	pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
duke@435	451	pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
duke@435	452	pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
duke@435	453
duke@435	454	static int __ieee754_rem_pio2(double x, double *y) {
duke@435	455	double z,w,t,r,fn;
duke@435	456	double tx[3];
duke@435	457	int e0,i,j,nx,n,ix,hx,i0;
duke@435	458
duke@435	459	i0 = ((*(int*)&two24A)>>30)^1; /* high word index */
duke@435	460	hx = *(i0+(int*)&x); /* high word of x */
duke@435	461	ix = hx&0x7fffffff;
duke@435	462	if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
duke@435	463	{y[0] = x; y[1] = 0; return 0;}
duke@435	464	if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
duke@435	465	if(hx>0) {
duke@435	466	z = x - pio2_1;
duke@435	467	if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
duke@435	468	y[0] = z - pio2_1t;
duke@435	469	y[1] = (z-y[0])-pio2_1t;
duke@435	470	} else { /* near pi/2, use 33+33+53 bit pi */
duke@435	471	z -= pio2_2;
duke@435	472	y[0] = z - pio2_2t;
duke@435	473	y[1] = (z-y[0])-pio2_2t;
duke@435	474	}
duke@435	475	return 1;
duke@435	476	} else { /* negative x */
duke@435	477	z = x + pio2_1;
duke@435	478	if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
duke@435	479	y[0] = z + pio2_1t;
duke@435	480	y[1] = (z-y[0])+pio2_1t;
duke@435	481	} else { /* near pi/2, use 33+33+53 bit pi */
duke@435	482	z += pio2_2;
duke@435	483	y[0] = z + pio2_2t;
duke@435	484	y[1] = (z-y[0])+pio2_2t;
duke@435	485	}
duke@435	486	return -1;
duke@435	487	}
duke@435	488	}
duke@435	489	if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
duke@435	490	t = fabsd(x);
duke@435	491	n = (int) (t*invpio2+half);
duke@435	492	fn = (double)n;
duke@435	493	r = t-fn*pio2_1;
duke@435	494	w = fn*pio2_1t; /* 1st round good to 85 bit */
duke@435	495	if(n<32&&ix!=npio2_hw[n-1]) {
duke@435	496	y[0] = r-w; /* quick check no cancellation */
duke@435	497	} else {
duke@435	498	j = ix>>20;
duke@435	499	y[0] = r-w;
duke@435	500	i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
duke@435	501	if(i>16) { /* 2nd iteration needed, good to 118 */
duke@435	502	t = r;
duke@435	503	w = fn*pio2_2;
duke@435	504	r = t-w;
duke@435	505	w = fn*pio2_2t-((t-r)-w);
duke@435	506	y[0] = r-w;
duke@435	507	i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
duke@435	508	if(i>49) { /* 3rd iteration need, 151 bits acc */
duke@435	509	t = r; /* will cover all possible cases */
duke@435	510	w = fn*pio2_3;
duke@435	511	r = t-w;
duke@435	512	w = fn*pio2_3t-((t-r)-w);
duke@435	513	y[0] = r-w;
duke@435	514	}
duke@435	515	}
duke@435	516	}
duke@435	517	y[1] = (r-y[0])-w;
duke@435	518	if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
duke@435	519	else return n;
duke@435	520	}
duke@435	521	/*
duke@435	522	* all other (large) arguments
duke@435	523	*/
duke@435	524	if(ix>=0x7ff00000) { /* x is inf or NaN */
duke@435	525	y[0]=y[1]=x-x; return 0;
duke@435	526	}
duke@435	527	/* set z = scalbn(|x|,ilogb(x)-23) */
duke@435	528	*(1-i0+(int*)&z) = *(1-i0+(int*)&x);
duke@435	529	e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
duke@435	530	*(i0+(int*)&z) = ix - (e0<<20);
duke@435	531	for(i=0;i<2;i++) {
duke@435	532	tx[i] = (double)((int)(z));
duke@435	533	z = (z-tx[i])*two24A;
duke@435	534	}
duke@435	535	tx[2] = z;
duke@435	536	nx = 3;
duke@435	537	while(tx[nx-1]==zeroA) nx--; /* skip zero term */
duke@435	538	n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
duke@435	539	if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
duke@435	540	return n;
duke@435	541	}
duke@435	542
duke@435	543
duke@435	544	/* __kernel_sin( x, y, iy)
duke@435	545	* kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
duke@435	546	* Input x is assumed to be bounded by ~pi/4 in magnitude.
duke@435	547	* Input y is the tail of x.
duke@435	548	* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
duke@435	549	*
duke@435	550	* Algorithm
duke@435	551	* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
duke@435	552	* 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
duke@435	553	* 3. sin(x) is approximated by a polynomial of degree 13 on
duke@435	554	* [0,pi/4]
duke@435	555	* 3 13
duke@435	556	* sin(x) ~ x + S1*x + ... + S6*x
duke@435	557	* where
duke@435	558	*
duke@435	559	* |sin(x) 2 4 6 8 10 12 | -58
duke@435	560	* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
duke@435	561	* | x |
duke@435	562	*
duke@435	563	* 4. sin(x+y) = sin(x) + sin'(x')*y
duke@435	564	* ~ sin(x) + (1-x*x/2)*y
duke@435	565	* For better accuracy, let
duke@435	566	* 3 2 2 2 2
duke@435	567	* r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
duke@435	568	* then 3 2
duke@435	569	* sin(x) = x + (S1*x + (x *(r-y/2)+y))
duke@435	570	*/
duke@435	571
duke@435	572	static const double
duke@435	573	S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
duke@435	574	S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
duke@435	575	S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
duke@435	576	S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
duke@435	577	S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
duke@435	578	S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
duke@435	579
duke@435	580	static double __kernel_sin(double x, double y, int iy)
duke@435	581	{
duke@435	582	double z,r,v;
duke@435	583	int ix;
duke@435	584	ix = __HI(x)&0x7fffffff; /* high word of x */
duke@435	585	if(ix<0x3e400000) /* |x| < 2**-27 */
duke@435	586	{if((int)x==0) return x;} /* generate inexact */
duke@435	587	z = x*x;
duke@435	588	v = z*x;
duke@435	589	r = S2+z*(S3+z*(S4+z*(S5+z*S6)));
duke@435	590	if(iy==0) return x+v*(S1+z*r);
duke@435	591	else return x-((z*(half*y-v*r)-y)-v*S1);
duke@435	592	}
duke@435	593
duke@435	594	/*
duke@435	595	* __kernel_cos( x, y )
duke@435	596	* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
duke@435	597	* Input x is assumed to be bounded by ~pi/4 in magnitude.
duke@435	598	* Input y is the tail of x.
duke@435	599	*
duke@435	600	* Algorithm
duke@435	601	* 1. Since cos(-x) = cos(x), we need only to consider positive x.
duke@435	602	* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
duke@435	603	* 3. cos(x) is approximated by a polynomial of degree 14 on
duke@435	604	* [0,pi/4]
duke@435	605	* 4 14
duke@435	606	* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
duke@435	607	* where the remez error is
duke@435	608	*
duke@435	609	* | 2 4 6 8 10 12 14 | -58
duke@435	610	* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
duke@435	611	* | |
duke@435	612	*
duke@435	613	* 4 6 8 10 12 14
duke@435	614	* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
duke@435	615	* cos(x) = 1 - x*x/2 + r
duke@435	616	* since cos(x+y) ~ cos(x) - sin(x)*y
duke@435	617	* ~ cos(x) - x*y,
duke@435	618	* a correction term is necessary in cos(x) and hence
duke@435	619	* cos(x+y) = 1 - (x*x/2 - (r - x*y))
duke@435	620	* For better accuracy when x > 0.3, let qx = |x|/4 with
duke@435	621	* the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
duke@435	622	* Then
duke@435	623	* cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
duke@435	624	* Note that 1-qx and (x*x/2-qx) is EXACT here, and the
duke@435	625	* magnitude of the latter is at least a quarter of x*x/2,
duke@435	626	* thus, reducing the rounding error in the subtraction.
duke@435	627	*/
duke@435	628
duke@435	629	static const double
duke@435	630	C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
duke@435	631	C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
duke@435	632	C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
duke@435	633	C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
duke@435	634	C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
duke@435	635	C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
duke@435	636
duke@435	637	static double __kernel_cos(double x, double y)
duke@435	638	{
duke@435	639	double a,hz,z,r,qx;
duke@435	640	int ix;
duke@435	641	ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/
duke@435	642	if(ix<0x3e400000) { /* if x < 2**27 */
duke@435	643	if(((int)x)==0) return one; /* generate inexact */
duke@435	644	}
duke@435	645	z = x*x;
duke@435	646	r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
duke@435	647	if(ix < 0x3FD33333) /* if |x| < 0.3 */
duke@435	648	return one - (0.5*z - (z*r - x*y));
duke@435	649	else {
duke@435	650	if(ix > 0x3fe90000) { /* x > 0.78125 */
duke@435	651	qx = 0.28125;
duke@435	652	} else {
duke@435	653	__HI(qx) = ix-0x00200000; /* x/4 */
duke@435	654	__LO(qx) = 0;
duke@435	655	}
duke@435	656	hz = 0.5*z-qx;
duke@435	657	a = one-qx;
duke@435	658	return a - (hz - (z*r-x*y));
duke@435	659	}
duke@435	660	}
duke@435	661
duke@435	662	/* __kernel_tan( x, y, k )
duke@435	663	* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
duke@435	664	* Input x is assumed to be bounded by ~pi/4 in magnitude.
duke@435	665	* Input y is the tail of x.
duke@435	666	* Input k indicates whether tan (if k=1) or
duke@435	667	* -1/tan (if k= -1) is returned.
duke@435	668	*
duke@435	669	* Algorithm
duke@435	670	* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
duke@435	671	* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
duke@435	672	* 3. tan(x) is approximated by a odd polynomial of degree 27 on
duke@435	673	* [0,0.67434]
duke@435	674	* 3 27
duke@435	675	* tan(x) ~ x + T1*x + ... + T13*x
duke@435	676	* where
duke@435	677	*
duke@435	678	* |tan(x) 2 4 26 | -59.2
duke@435	679	* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
duke@435	680	* | x |
duke@435	681	*
duke@435	682	* Note: tan(x+y) = tan(x) + tan'(x)*y
duke@435	683	* ~ tan(x) + (1+x*x)*y
duke@435	684	* Therefore, for better accuracy in computing tan(x+y), let
duke@435	685	* 3 2 2 2 2
duke@435	686	* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
duke@435	687	* then
duke@435	688	* 3 2
duke@435	689	* tan(x+y) = x + (T1*x + (x *(r+y)+y))
duke@435	690	*
duke@435	691	* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
duke@435	692	* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
duke@435	693	* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
duke@435	694	*/
duke@435	695
duke@435	696	static const double
duke@435	697	pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
duke@435	698	pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
duke@435	699	T[] = {
duke@435	700	3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
duke@435	701	1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
duke@435	702	5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
duke@435	703	2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
duke@435	704	8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
duke@435	705	3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
duke@435	706	1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
duke@435	707	5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
duke@435	708	2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
duke@435	709	7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
duke@435	710	7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
duke@435	711	-1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
duke@435	712	2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
duke@435	713	};
duke@435	714
duke@435	715	static double __kernel_tan(double x, double y, int iy)
duke@435	716	{
duke@435	717	double z,r,v,w,s;
duke@435	718	int ix,hx;
duke@435	719	hx = __HI(x); /* high word of x */
duke@435	720	ix = hx&0x7fffffff; /* high word of |x| */
duke@435	721	if(ix<0x3e300000) { /* x < 2**-28 */
duke@435	722	if((int)x==0) { /* generate inexact */
duke@435	723	if (((ix | __LO(x)) | (iy + 1)) == 0)
duke@435	724	return one / fabsd(x);
duke@435	725	else {
duke@435	726	if (iy == 1)
duke@435	727	return x;
duke@435	728	else { /* compute -1 / (x+y) carefully */
duke@435	729	double a, t;
duke@435	730
duke@435	731	z = w = x + y;
duke@435	732	__LO(z) = 0;
duke@435	733	v = y - (z - x);
duke@435	734	t = a = -one / w;
duke@435	735	__LO(t) = 0;
duke@435	736	s = one + t * z;
duke@435	737	return t + a * (s + t * v);
duke@435	738	}
duke@435	739	}
duke@435	740	}
duke@435	741	}
duke@435	742	if(ix>=0x3FE59428) { /* |x|>=0.6744 */
duke@435	743	if(hx<0) {x = -x; y = -y;}
duke@435	744	z = pio4-x;
duke@435	745	w = pio4lo-y;
duke@435	746	x = z+w; y = 0.0;
duke@435	747	}
duke@435	748	z = x*x;
duke@435	749	w = z*z;
duke@435	750	/* Break x^5*(T[1]+x^2*T[2]+...) into
duke@435	751	* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
duke@435	752	* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
duke@435	753	*/
duke@435	754	r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
duke@435	755	v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
duke@435	756	s = z*x;
duke@435	757	r = y + z*(s*(r+v)+y);
duke@435	758	r += T[0]*s;
duke@435	759	w = x+r;
duke@435	760	if(ix>=0x3FE59428) {
duke@435	761	v = (double)iy;
duke@435	762	return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
duke@435	763	}
duke@435	764	if(iy==1) return w;
duke@435	765	else { /* if allow error up to 2 ulp,
duke@435	766	simply return -1.0/(x+r) here */
duke@435	767	/* compute -1.0/(x+r) accurately */
duke@435	768	double a,t;
duke@435	769	z = w;
duke@435	770	__LO(z) = 0;
duke@435	771	v = r-(z - x); /* z+v = r+x */
duke@435	772	t = a = -1.0/w; /* a = -1.0/w */
duke@435	773	__LO(t) = 0;
duke@435	774	s = 1.0+t*z;
duke@435	775	return t+a*(s+t*v);
duke@435	776	}
duke@435	777	}
duke@435	778
duke@435	779
duke@435	780	//----------------------------------------------------------------------
duke@435	781	//
duke@435	782	// Routines for new sin/cos implementation
duke@435	783	//
duke@435	784	//----------------------------------------------------------------------
duke@435	785
duke@435	786	/* sin(x)
duke@435	787	* Return sine function of x.
duke@435	788	*
duke@435	789	* kernel function:
duke@435	790	* __kernel_sin ... sine function on [-pi/4,pi/4]
duke@435	791	* __kernel_cos ... cose function on [-pi/4,pi/4]
duke@435	792	* __ieee754_rem_pio2 ... argument reduction routine
duke@435	793	*
duke@435	794	* Method.
duke@435	795	* Let S,C and T denote the sin, cos and tan respectively on
duke@435	796	* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
duke@435	797	* in [-pi/4 , +pi/4], and let n = k mod 4.
duke@435	798	* We have
duke@435	799	*
duke@435	800	* n sin(x) cos(x) tan(x)
duke@435	801	* ----------------------------------------------------------
duke@435	802	* 0 S C T
duke@435	803	* 1 C -S -1/T
duke@435	804	* 2 -S -C T
duke@435	805	* 3 -C S -1/T
duke@435	806	* ----------------------------------------------------------
duke@435	807	*
duke@435	808	* Special cases:
duke@435	809	* Let trig be any of sin, cos, or tan.
duke@435	810	* trig(+-INF) is NaN, with signals;
duke@435	811	* trig(NaN) is that NaN;
duke@435	812	*
duke@435	813	* Accuracy:
duke@435	814	* TRIG(x) returns trig(x) nearly rounded
duke@435	815	*/
duke@435	816
duke@435	817	JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x))
duke@435	818	double y[2],z=0.0;
duke@435	819	int n, ix;
duke@435	820
duke@435	821	/* High word of x. */
duke@435	822	ix = __HI(x);
duke@435	823
duke@435	824	/* |x| ~< pi/4 */
duke@435	825	ix &= 0x7fffffff;
duke@435	826	if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0);
duke@435	827
duke@435	828	/* sin(Inf or NaN) is NaN */
duke@435	829	else if (ix>=0x7ff00000) return x-x;
duke@435	830
duke@435	831	/* argument reduction needed */
duke@435	832	else {
duke@435	833	n = __ieee754_rem_pio2(x,y);
duke@435	834	switch(n&3) {
duke@435	835	case 0: return __kernel_sin(y[0],y[1],1);
duke@435	836	case 1: return __kernel_cos(y[0],y[1]);
duke@435	837	case 2: return -__kernel_sin(y[0],y[1],1);
duke@435	838	default:
duke@435	839	return -__kernel_cos(y[0],y[1]);
duke@435	840	}
duke@435	841	}
duke@435	842	JRT_END
duke@435	843
duke@435	844	/* cos(x)
duke@435	845	* Return cosine function of x.
duke@435	846	*
duke@435	847	* kernel function:
duke@435	848	* __kernel_sin ... sine function on [-pi/4,pi/4]
duke@435	849	* __kernel_cos ... cosine function on [-pi/4,pi/4]
duke@435	850	* __ieee754_rem_pio2 ... argument reduction routine
duke@435	851	*
duke@435	852	* Method.
duke@435	853	* Let S,C and T denote the sin, cos and tan respectively on
duke@435	854	* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
duke@435	855	* in [-pi/4 , +pi/4], and let n = k mod 4.
duke@435	856	* We have
duke@435	857	*
duke@435	858	* n sin(x) cos(x) tan(x)
duke@435	859	* ----------------------------------------------------------
duke@435	860	* 0 S C T
duke@435	861	* 1 C -S -1/T
duke@435	862	* 2 -S -C T
duke@435	863	* 3 -C S -1/T
duke@435	864	* ----------------------------------------------------------
duke@435	865	*
duke@435	866	* Special cases:
duke@435	867	* Let trig be any of sin, cos, or tan.
duke@435	868	* trig(+-INF) is NaN, with signals;
duke@435	869	* trig(NaN) is that NaN;
duke@435	870	*
duke@435	871	* Accuracy:
duke@435	872	* TRIG(x) returns trig(x) nearly rounded
duke@435	873	*/
duke@435	874
duke@435	875	JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x))
duke@435	876	double y[2],z=0.0;
duke@435	877	int n, ix;
duke@435	878
duke@435	879	/* High word of x. */
duke@435	880	ix = __HI(x);
duke@435	881
duke@435	882	/* |x| ~< pi/4 */
duke@435	883	ix &= 0x7fffffff;
duke@435	884	if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
duke@435	885
duke@435	886	/* cos(Inf or NaN) is NaN */
duke@435	887	else if (ix>=0x7ff00000) return x-x;
duke@435	888
duke@435	889	/* argument reduction needed */
duke@435	890	else {
duke@435	891	n = __ieee754_rem_pio2(x,y);
duke@435	892	switch(n&3) {
duke@435	893	case 0: return __kernel_cos(y[0],y[1]);
duke@435	894	case 1: return -__kernel_sin(y[0],y[1],1);
duke@435	895	case 2: return -__kernel_cos(y[0],y[1]);
duke@435	896	default:
duke@435	897	return __kernel_sin(y[0],y[1],1);
duke@435	898	}
duke@435	899	}
duke@435	900	JRT_END
duke@435	901
duke@435	902	/* tan(x)
duke@435	903	* Return tangent function of x.
duke@435	904	*
duke@435	905	* kernel function:
duke@435	906	* __kernel_tan ... tangent function on [-pi/4,pi/4]
duke@435	907	* __ieee754_rem_pio2 ... argument reduction routine
duke@435	908	*
duke@435	909	* Method.
duke@435	910	* Let S,C and T denote the sin, cos and tan respectively on
duke@435	911	* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
duke@435	912	* in [-pi/4 , +pi/4], and let n = k mod 4.
duke@435	913	* We have
duke@435	914	*
duke@435	915	* n sin(x) cos(x) tan(x)
duke@435	916	* ----------------------------------------------------------
duke@435	917	* 0 S C T
duke@435	918	* 1 C -S -1/T
duke@435	919	* 2 -S -C T
duke@435	920	* 3 -C S -1/T
duke@435	921	* ----------------------------------------------------------
duke@435	922	*
duke@435	923	* Special cases:
duke@435	924	* Let trig be any of sin, cos, or tan.
duke@435	925	* trig(+-INF) is NaN, with signals;
duke@435	926	* trig(NaN) is that NaN;
duke@435	927	*
duke@435	928	* Accuracy:
duke@435	929	* TRIG(x) returns trig(x) nearly rounded
duke@435	930	*/
duke@435	931
duke@435	932	JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x))
duke@435	933	double y[2],z=0.0;
duke@435	934	int n, ix;
duke@435	935
duke@435	936	/* High word of x. */
duke@435	937	ix = __HI(x);
duke@435	938
duke@435	939	/* |x| ~< pi/4 */
duke@435	940	ix &= 0x7fffffff;
duke@435	941	if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);
duke@435	942
duke@435	943	/* tan(Inf or NaN) is NaN */
duke@435	944	else if (ix>=0x7ff00000) return x-x; /* NaN */
duke@435	945
duke@435	946	/* argument reduction needed */
duke@435	947	else {
duke@435	948	n = __ieee754_rem_pio2(x,y);
duke@435	949	return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
duke@435	950	-1 -- n odd */
duke@435	951	}
duke@435	952	JRT_END
duke@435	953
duke@435	954
duke@435	955	#ifdef WIN32
duke@435	956	# pragma optimize ( "", on )
duke@435	957	#endif