e_rem_pio2l.h revision 1.2 
1/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 *
13 * Optimized by Bruce D. Evans.
14 */
15
16#include <sys/cdefs.h>
17#if 0
18__FBSDID("$FreeBSD: head/lib/msun/ld80/e_rem_pio2l.h 336545 2018-07-20 12:42:24Z bde $");
19#endif
20
21/* ld80 version of __ieee754_rem_pio2l(x,y)
22 *
23 * return the remainder of x rem pi/2 in y[0]+y[1]
24 * use __kernel_rem_pio2()
25 */
26
27#include <float.h>
28#include <machine/ieee.h>
29
30#include "math.h"
31#include "math_private.h"
32
33#define	BIAS	(LDBL_MAX_EXP - 1)
34
35/*
36 * invpio2:  64 bits of 2/pi
37 * pio2_1:   first  39 bits of pi/2
38 * pio2_1t:  pi/2 - pio2_1
39 * pio2_2:   second 39 bits of pi/2
40 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
41 * pio2_3:   third  39 bits of pi/2
42 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
43 */
44
45static const double
46zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
47two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
48pio2_1  =  1.57079632679597125389e+00,	/* 0x3FF921FB, 0x54444000 */
49pio2_2  = -1.07463465549783099519e-12,	/* -0x12e7b967674000.0p-92 */
50pio2_3  =  6.36831716351370313614e-25;	/*  0x18a2e037074000.0p-133 */
51
52#if defined(__amd64__) || defined(__i386__)
53/* Long double constants are slow on these arches, and broken on i386. */
54static const volatile double
55invpio2hi =  6.3661977236758138e-01,	/*  0x145f306dc9c883.0p-53 */
56invpio2lo = -3.9356538861223811e-17,	/* -0x16b00000000000.0p-107 */
57pio2_1thi = -1.0746346554971943e-12,	/* -0x12e7b9676733af.0p-92 */
58pio2_1tlo =  8.8451028997905949e-29,	/*  0x1c080000000000.0p-146 */
59pio2_2thi =  6.3683171635109499e-25,	/*  0x18a2e03707344a.0p-133 */
60pio2_2tlo =  2.3183081793789774e-41,	/*  0x10280000000000.0p-187 */
61pio2_3thi = -2.7529965190440717e-37,	/* -0x176b7ed8fbbacc.0p-174 */
62pio2_3tlo = -4.2006647512740502e-54;	/* -0x19c00000000000.0p-230 */
63#define	invpio2	((long double)invpio2hi + invpio2lo)
64#define	pio2_1t	((long double)pio2_1thi + pio2_1tlo)
65#define	pio2_2t	((long double)pio2_2thi + pio2_2tlo)
66#define	pio2_3t	((long double)pio2_3thi + pio2_3tlo)
67#else
68static const long double
69invpio2 =  6.36619772367581343076e-01L,	/*  0xa2f9836e4e44152a.0p-64 */
70pio2_1t = -1.07463465549719416346e-12L,	/* -0x973dcb3b399d747f.0p-103 */
71pio2_2t =  6.36831716351095013979e-25L,	/*  0xc51701b839a25205.0p-144 */
72pio2_3t = -2.75299651904407171810e-37L;	/* -0xbb5bf6c7ddd660ce.0p-185 */
73#endif
74
75static inline __always_inline int
76__ieee754_rem_pio2l(long double x, long double *y)
77{
78	union ieee_ext_u u, u1;
79	long double z,w,t,r,fn;
80	double tx[3],ty[2];
81	int e0,ex,i,j,nx,n;
82	int16_t expsign, expsign1;
83
84	u.extu_ld = x;
85	ex = u.extu_exp;
86	expsign = u.extu_exp | (u.extu_sign << EXT_EXPBITS);
87	if (ex < BIAS + 25 || (ex == BIAS + 25 && u.extu_frach < 0xc90fdaa2U)) {
88	    /* |x| ~< 2^25*(pi/2), medium size */
89	    fn = rnintl(x*invpio2);
90	    n  = irint(fn);
91	    r  = x-fn*pio2_1;
92	    w  = fn*pio2_1t;	/* 1st round good to 102 bit */
93	    {
94		union ieee_ext_u u2;
95	        int ex1;
96	        j  = ex;
97	        y[0] = r-w;
98		u2.extu_ld = y[0];
99		ex1 = u2.extu_exp;
100	        i = j-ex1;
101	        if(i>22) {  /* 2nd iteration needed, good to 141 */
102		    t  = r;
103		    w  = fn*pio2_2;
104		    r  = t-w;
105		    w  = fn*pio2_2t-((t-r)-w);
106		    y[0] = r-w;
107		    u2.extu_ld = y[0];
108		    ex1 = u2.extu_exp;
109		    i = j-ex1;
110		    if(i>61) {	/* 3rd iteration need, 180 bits acc */
111			t  = r;	/* will cover all possible cases */
112			w  = fn*pio2_3;
113			r  = t-w;
114			w  = fn*pio2_3t-((t-r)-w);
115			y[0] = r-w;
116		    }
117		}
118	    }
119	    y[1] = (r-y[0])-w;
120	    return n;
121	}
122    /*
123     * all other (large) arguments
124     */
125	if(ex==0x7fff) {		/* x is inf or NaN */
126	    y[0]=y[1]=x-x; return 0;
127	}
128    /* set z = scalbn(|x|,ilogb(x)-23) */
129	u1.extu_ld = x;
130	e0 = ex - BIAS - 23;		/* e0 = ilogb(|x|)-23; */
131	expsign1 = ex - e0;
132	u1.extu_exp = expsign1;
133	u1.extu_sign = expsign1 >> EXT_EXPBITS;
134	z = u1.extu_ld;
135	for(i=0;i<2;i++) {
136		tx[i] = (double)((int32_t)(z));
137		z     = (z-tx[i])*two24;
138	}
139	tx[2] = z;
140	nx = 3;
141	while(tx[nx-1]==zero) nx--;	/* skip zero term */
142	n  =  __kernel_rem_pio2(tx,ty,e0,nx,2);
143	r = (long double)ty[0] + ty[1];
144	w = ty[1] - (r - ty[0]);
145	if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
146	y[0] = r; y[1] = w; return n;
147}
148