1/*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 * Optimized by Bruce D. Evans.
13 */
14
15/* ld80 version of __ieee754_rem_pio2l(x,y)
16 *
17 * return the remainder of x rem pi/2 in y[0]+y[1]
18 * use __kernel_rem_pio2()
19 */
20
21#include <float.h>
22
23#include "math.h"
24#include "math_private.h"
25#include "fpmath.h"
26
27#define	BIAS	(LDBL_MAX_EXP - 1)
28
29/*
30 * invpio2:  64 bits of 2/pi
31 * pio2_1:   first  39 bits of pi/2
32 * pio2_1t:  pi/2 - pio2_1
33 * pio2_2:   second 39 bits of pi/2
34 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
35 * pio2_3:   third  39 bits of pi/2
36 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
37 */
38
39static const double
40zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
41two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
42pio2_1  =  1.57079632679597125389e+00,	/* 0x3FF921FB, 0x54444000 */
43pio2_2  = -1.07463465549783099519e-12,	/* -0x12e7b967674000.0p-92 */
44pio2_3  =  6.36831716351370313614e-25;	/*  0x18a2e037074000.0p-133 */
45
46#if defined(__amd64__) || defined(__i386__)
47/* Long double constants are slow on these arches, and broken on i386. */
48static const volatile double
49invpio2hi =  6.3661977236758138e-01,	/*  0x145f306dc9c883.0p-53 */
50invpio2lo = -3.9356538861223811e-17,	/* -0x16b00000000000.0p-107 */
51pio2_1thi = -1.0746346554971943e-12,	/* -0x12e7b9676733af.0p-92 */
52pio2_1tlo =  8.8451028997905949e-29,	/*  0x1c080000000000.0p-146 */
53pio2_2thi =  6.3683171635109499e-25,	/*  0x18a2e03707344a.0p-133 */
54pio2_2tlo =  2.3183081793789774e-41,	/*  0x10280000000000.0p-187 */
55pio2_3thi = -2.7529965190440717e-37,	/* -0x176b7ed8fbbacc.0p-174 */
56pio2_3tlo = -4.2006647512740502e-54;	/* -0x19c00000000000.0p-230 */
57#define	invpio2	((long double)invpio2hi + invpio2lo)
58#define	pio2_1t	((long double)pio2_1thi + pio2_1tlo)
59#define	pio2_2t	((long double)pio2_2thi + pio2_2tlo)
60#define	pio2_3t	((long double)pio2_3thi + pio2_3tlo)
61#else
62static const long double
63invpio2 =  6.36619772367581343076e-01L,	/*  0xa2f9836e4e44152a.0p-64 */
64pio2_1t = -1.07463465549719416346e-12L,	/* -0x973dcb3b399d747f.0p-103 */
65pio2_2t =  6.36831716351095013979e-25L,	/*  0xc51701b839a25205.0p-144 */
66pio2_3t = -2.75299651904407171810e-37L;	/* -0xbb5bf6c7ddd660ce.0p-185 */
67#endif
68
69static inline __always_inline int
70__ieee754_rem_pio2l(long double x, long double *y)
71{
72	union IEEEl2bits u,u1;
73	long double z,w,t,r,fn;
74	double tx[3],ty[2];
75	int e0,ex,i,j,nx,n;
76	int16_t expsign;
77
78	u.e = x;
79	expsign = u.xbits.expsign;
80	ex = expsign & 0x7fff;
81	if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) {
82	    /* |x| ~< 2^25*(pi/2), medium size */
83	    fn = rnintl(x*invpio2);
84	    n  = irint(fn);
85	    r  = x-fn*pio2_1;
86	    w  = fn*pio2_1t;	/* 1st round good to 102 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>22) {  /* 2nd iteration needed, good to 141 */
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>61) {	/* 3rd iteration need, 180 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<2;i++) {
128		tx[i] = (double)((int32_t)(z));
129		z     = (z-tx[i])*two24;
130	}
131	tx[2] = z;
132	nx = 3;
133	while(tx[nx-1]==zero) nx--;	/* skip zero term */
134	n  =  __kernel_rem_pio2(tx,ty,e0,nx,2);
135	r = (long double)ty[0] + ty[1];
136	w = ty[1] - (r - ty[0]);
137	if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
138	y[0] = r; y[1] = w; return n;
139}
140