1/* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */
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#include "libm.h"
16#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
17/* ld80 and ld128 version of __rem_pio2(x,y)
18 *
19 * return the remainder of x rem pi/2 in y[0]+y[1]
20 * use __rem_pio2_large() for large x
21 */
22
23static const long double toint = 1.5/LDBL_EPSILON;
24
25#if LDBL_MANT_DIG == 64
26/* u ~< 0x1p25*pi/2 */
27#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
28#define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
29#define ROUND1 22
30#define ROUND2 61
31#define NX 3
32#define NY 2
33/*
34 * invpio2:  64 bits of 2/pi
35 * pio2_1:   first  39 bits of pi/2
36 * pio2_1t:  pi/2 - pio2_1
37 * pio2_2:   second 39 bits of pi/2
38 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
39 * pio2_3:   third  39 bits of pi/2
40 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
41 */
42static const double
43pio2_1 =  1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
44pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
45pio2_3 =  6.36831716351370313614e-25; /*  0x18a2e037074000.0p-133 */
46static const long double
47invpio2 =  6.36619772367581343076e-01L, /*  0xa2f9836e4e44152a.0p-64 */
48pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
49pio2_2t =  6.36831716351095013979e-25L, /*  0xc51701b839a25205.0p-144 */
50pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
51#elif LDBL_MANT_DIG == 113
52/* u ~< 0x1p45*pi/2 */
53#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
54#define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
55#define ROUND1 51
56#define ROUND2 119
57#define NX 5
58#define NY 3
59static const long double
60invpio2 =  6.3661977236758134307553505349005747e-01L,	/*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
61pio2_1  =  1.5707963267948966192292994253909555e+00L,	/*  0x1921fb54442d18469800000000000.0p-112 */
62pio2_1t =  2.0222662487959507323996846200947577e-21L,	/*  0x13198a2e03707344a4093822299f3.0p-181 */
63pio2_2  =  2.0222662487959507323994779168837751e-21L,	/*  0x13198a2e03707344a400000000000.0p-181 */
64pio2_2t =  2.0670321098263988236496903051604844e-43L,	/*  0x127044533e63a0105df531d89cd91.0p-254 */
65pio2_3  =  2.0670321098263988236499468110329591e-43L,	/*  0x127044533e63a0105e00000000000.0p-254 */
66pio2_3t = -2.5650587247459238361625433492959285e-65L;	/* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
67#endif
68
69int __rem_pio2l(long double x, long double *y)
70{
71	union ldshape u,uz;
72	long double z,w,t,r,fn;
73	double tx[NX],ty[NY];
74	int ex,ey,n,i;
75
76	u.f = x;
77	ex = u.i.se & 0x7fff;
78	if (SMALL(u)) {
79		/* rint(x/(pi/2)), Assume round-to-nearest. */
80		fn = x*invpio2 + toint - toint;
81		n = QUOBITS(fn);
82		r = x-fn*pio2_1;
83		w = fn*pio2_1t;  /* 1st round good to 102/180 bits (ld80/ld128) */
84		y[0] = r-w;
85		u.f = y[0];
86		ey = u.i.se & 0x7fff;
87		if (ex - ey > ROUND1) {  /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
88			t = r;
89			w = fn*pio2_2;
90			r = t-w;
91			w = fn*pio2_2t-((t-r)-w);
92			y[0] = r-w;
93			u.f = y[0];
94			ey = u.i.se & 0x7fff;
95			if (ex - ey > ROUND2) {  /* 3rd iteration, good to 180/316 bits */
96				t = r; /* will cover all possible cases (not verified for ld128) */
97				w = fn*pio2_3;
98				r = t-w;
99				w = fn*pio2_3t-((t-r)-w);
100				y[0] = r-w;
101			}
102		}
103		y[1] = (r - y[0]) - w;
104		return n;
105	}
106	/*
107	 * all other (large) arguments
108	 */
109	if (ex == 0x7fff) {                /* x is inf or NaN */
110		y[0] = y[1] = x - x;
111		return 0;
112	}
113	/* set z = scalbn(|x|,-ilogb(x)+23) */
114	uz.f = x;
115	uz.i.se = 0x3fff + 23;
116	z = uz.f;
117	for (i=0; i < NX - 1; i++) {
118		tx[i] = (double)(int32_t)z;
119		z     = (z-tx[i])*0x1p24;
120	}
121	tx[i] = z;
122	while (tx[i] == 0)
123		i--;
124	n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
125	w = ty[1];
126	if (NY == 3)
127		w += ty[2];
128	r = ty[0] + w;
129	/* TODO: for ld128 this does not follow the recommendation of the
130	comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
131	w -= r - ty[0];
132	if (u.i.se >> 15) {
133		y[0] = -r;
134		y[1] = -w;
135		return -n;
136	}
137	y[0] = r;
138	y[1] = w;
139	return n;
140}
141#endif
142