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 
48 static const double
49 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
50 two24 =  1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
51 
52 static const long double
53 invpio2 =  6.3661977236758134307553505349005747e-01L,	/*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
54 pio2_1  =  1.5707963267948966192292994253909555e+00L,	/*  0x1921fb54442d18469800000000000.0p-112 */
55 pio2_1t =  2.0222662487959507323996846200947577e-21L,	/*  0x13198a2e03707344a4093822299f3.0p-181 */
56 pio2_2  =  2.0222662487959507323994779168837751e-21L,	/*  0x13198a2e03707344a400000000000.0p-181 */
57 pio2_2t =  2.0670321098263988236496903051604844e-43L,	/*  0x127044533e63a0105df531d89cd91.0p-254 */
58 pio2_3  =  2.0670321098263988236499468110329591e-43L,	/*  0x127044533e63a0105e00000000000.0p-254 */
59 pio2_3t = -2.5650587247459238361625433492959285e-65L;	/* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
60 
61 static inline __always_inline int
__ieee754_rem_pio2l(long double x,long double * y)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