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__FBSDID("$FreeBSD$");
18
19/* ld128 version of __ieee754_rem_pio2l(x,y)
20 *
21 * return the remainder of x rem pi/2 in y[0]+y[1]
22 * use __kernel_rem_pio2()
23 */
24
25#include <float.h>
26
27#include "math.h"
28#include "math_private.h"
29#include "fpmath.h"
30
31#define	BIAS	(LDBL_MAX_EXP - 1)
32
33/*
34 * XXX need to verify that nonzero integer multiples of pi/2 within the
35 * range get no closer to a long double than 2**-140, or that
36 * ilogb(x) + ilogb(min_delta) < 45 - -140.
37 */
38/*
39 * invpio2:  113 bits of 2/pi
40 * pio2_1:   first  68 bits of pi/2
41 * pio2_1t:  pi/2 - pio2_1
42 * pio2_2:   second 68 bits of pi/2
43 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
44 * pio2_3:   third  68 bits of pi/2
45 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
46 */
47
48static const double
49zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
50two24 =  1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
51
52static const long double
53invpio2 =  6.3661977236758134307553505349005747e-01L,	/*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
54pio2_1  =  1.5707963267948966192292994253909555e+00L,	/*  0x1921fb54442d18469800000000000.0p-112 */
55pio2_1t =  2.0222662487959507323996846200947577e-21L,	/*  0x13198a2e03707344a4093822299f3.0p-181 */
56pio2_2  =  2.0222662487959507323994779168837751e-21L,	/*  0x13198a2e03707344a400000000000.0p-181 */
57pio2_2t =  2.0670321098263988236496903051604844e-43L,	/*  0x127044533e63a0105df531d89cd91.0p-254 */
58pio2_3  =  2.0670321098263988236499468110329591e-43L,	/*  0x127044533e63a0105e00000000000.0p-254 */
59pio2_3t = -2.5650587247459238361625433492959285e-65L;	/* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
60
61static inline __always_inline int
62__ieee754_rem_pio2l(long double x, long double *y)
63{
64	union IEEEl2bits u,u1;
65	long double z,w,t,r,fn;
66	double tx[5],ty[3];
67	int64_t n;
68	int e0,ex,i,j,nx;
69	int16_t expsign;
70
71	u.e = x;
72	expsign = u.xbits.expsign;
73	ex = expsign & 0x7fff;
74	if (ex < BIAS + 45 || ex == BIAS + 45 &&
75	    u.bits.manh < 0x921fb54442d1LL) {
76	    /* |x| ~< 2^45*(pi/2), medium size */
77	    /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
78	    fn = x*invpio2+0x1.8p112;
79	    fn = fn-0x1.8p112;
80#ifdef HAVE_EFFICIENT_I64RINT
81	    n  = i64rint(fn);
82#else
83	    n  = fn;
84#endif
85	    r  = x-fn*pio2_1;
86	    w  = fn*pio2_1t;	/* 1st round good to 180 bit */
87	    {
88		union IEEEl2bits u2;
89	        int ex1;
90	        j  = ex;
91	        y[0] = r-w;
92		u2.e = y[0];
93		ex1 = u2.xbits.expsign & 0x7fff;
94	        i = j-ex1;
95	        if(i>51) {  /* 2nd iteration needed, good to 248 */
96		    t  = r;
97		    w  = fn*pio2_2;
98		    r  = t-w;
99		    w  = fn*pio2_2t-((t-r)-w);
100		    y[0] = r-w;
101		    u2.e = y[0];
102		    ex1 = u2.xbits.expsign & 0x7fff;
103		    i = j-ex1;
104		    if(i>119) {	/* 3rd iteration need, 316 bits acc */
105		    	t  = r;	/* will cover all possible cases */
106		    	w  = fn*pio2_3;
107		    	r  = t-w;
108		    	w  = fn*pio2_3t-((t-r)-w);
109		    	y[0] = r-w;
110		    }
111		}
112	    }
113	    y[1] = (r-y[0])-w;
114	    return n;
115	}
116    /*
117     * all other (large) arguments
118     */
119	if(ex==0x7fff) {		/* x is inf or NaN */
120	    y[0]=y[1]=x-x; return 0;
121	}
122    /* set z = scalbn(|x|,ilogb(x)-23) */
123	u1.e = x;
124	e0 = ex - BIAS - 23;		/* e0 = ilogb(|x|)-23; */
125	u1.xbits.expsign = ex - e0;
126	z = u1.e;
127	for(i=0;i<4;i++) {
128		tx[i] = (double)((int32_t)(z));
129		z     = (z-tx[i])*two24;
130	}
131	tx[4] = z;
132	nx = 5;
133	while(tx[nx-1]==zero) nx--;	/* skip zero term */
134	n  =  __kernel_rem_pio2(tx,ty,e0,nx,3);
135	t = (long double)ty[2] + ty[1];
136	r = t + ty[0];
137	w = ty[0] - (r - t);
138	if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
139	y[0] = r; y[1] = w; return n;
140}
141