xref: /freebsd/lib/msun/ld80/e_rem_pio2l.h (revision 4b9d6057)
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 #include <sys/cdefs.h>
16 /* ld80 version of __ieee754_rem_pio2l(x,y)
17  *
18  * return the remainder of x rem pi/2 in y[0]+y[1]
19  * use __kernel_rem_pio2()
20  */
21 
22 #include <float.h>
23 
24 #include "math.h"
25 #include "math_private.h"
26 #include "fpmath.h"
27 
28 #define	BIAS	(LDBL_MAX_EXP - 1)
29 
30 /*
31  * invpio2:  64 bits of 2/pi
32  * pio2_1:   first  39 bits of pi/2
33  * pio2_1t:  pi/2 - pio2_1
34  * pio2_2:   second 39 bits of pi/2
35  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
36  * pio2_3:   third  39 bits of pi/2
37  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
38  */
39 
40 static const double
41 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
42 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
43 pio2_1  =  1.57079632679597125389e+00,	/* 0x3FF921FB, 0x54444000 */
44 pio2_2  = -1.07463465549783099519e-12,	/* -0x12e7b967674000.0p-92 */
45 pio2_3  =  6.36831716351370313614e-25;	/*  0x18a2e037074000.0p-133 */
46 
47 #if defined(__amd64__) || defined(__i386__)
48 /* Long double constants are slow on these arches, and broken on i386. */
49 static const volatile double
50 invpio2hi =  6.3661977236758138e-01,	/*  0x145f306dc9c883.0p-53 */
51 invpio2lo = -3.9356538861223811e-17,	/* -0x16b00000000000.0p-107 */
52 pio2_1thi = -1.0746346554971943e-12,	/* -0x12e7b9676733af.0p-92 */
53 pio2_1tlo =  8.8451028997905949e-29,	/*  0x1c080000000000.0p-146 */
54 pio2_2thi =  6.3683171635109499e-25,	/*  0x18a2e03707344a.0p-133 */
55 pio2_2tlo =  2.3183081793789774e-41,	/*  0x10280000000000.0p-187 */
56 pio2_3thi = -2.7529965190440717e-37,	/* -0x176b7ed8fbbacc.0p-174 */
57 pio2_3tlo = -4.2006647512740502e-54;	/* -0x19c00000000000.0p-230 */
58 #define	invpio2	((long double)invpio2hi + invpio2lo)
59 #define	pio2_1t	((long double)pio2_1thi + pio2_1tlo)
60 #define	pio2_2t	((long double)pio2_2thi + pio2_2tlo)
61 #define	pio2_3t	((long double)pio2_3thi + pio2_3tlo)
62 #else
63 static const long double
64 invpio2 =  6.36619772367581343076e-01L,	/*  0xa2f9836e4e44152a.0p-64 */
65 pio2_1t = -1.07463465549719416346e-12L,	/* -0x973dcb3b399d747f.0p-103 */
66 pio2_2t =  6.36831716351095013979e-25L,	/*  0xc51701b839a25205.0p-144 */
67 pio2_3t = -2.75299651904407171810e-37L;	/* -0xbb5bf6c7ddd660ce.0p-185 */
68 #endif
69 
70 static inline __always_inline int
71 __ieee754_rem_pio2l(long double x, long double *y)
72 {
73 	union IEEEl2bits u,u1;
74 	long double z,w,t,r,fn;
75 	double tx[3],ty[2];
76 	int e0,ex,i,j,nx,n;
77 	int16_t expsign;
78 
79 	u.e = x;
80 	expsign = u.xbits.expsign;
81 	ex = expsign & 0x7fff;
82 	if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) {
83 	    /* |x| ~< 2^25*(pi/2), medium size */
84 	    fn = rnintl(x*invpio2);
85 	    n  = irint(fn);
86 	    r  = x-fn*pio2_1;
87 	    w  = fn*pio2_1t;	/* 1st round good to 102 bit */
88 	    {
89 		union IEEEl2bits u2;
90 	        int ex1;
91 	        j  = ex;
92 	        y[0] = r-w;
93 		u2.e = y[0];
94 		ex1 = u2.xbits.expsign & 0x7fff;
95 	        i = j-ex1;
96 	        if(i>22) {  /* 2nd iteration needed, good to 141 */
97 		    t  = r;
98 		    w  = fn*pio2_2;
99 		    r  = t-w;
100 		    w  = fn*pio2_2t-((t-r)-w);
101 		    y[0] = r-w;
102 		    u2.e = y[0];
103 		    ex1 = u2.xbits.expsign & 0x7fff;
104 		    i = j-ex1;
105 		    if(i>61) {	/* 3rd iteration need, 180 bits acc */
106 		    	t  = r;	/* will cover all possible cases */
107 		    	w  = fn*pio2_3;
108 		    	r  = t-w;
109 		    	w  = fn*pio2_3t-((t-r)-w);
110 		    	y[0] = r-w;
111 		    }
112 		}
113 	    }
114 	    y[1] = (r-y[0])-w;
115 	    return n;
116 	}
117     /*
118      * all other (large) arguments
119      */
120 	if(ex==0x7fff) {		/* x is inf or NaN */
121 	    y[0]=y[1]=x-x; return 0;
122 	}
123     /* set z = scalbn(|x|,ilogb(x)-23) */
124 	u1.e = x;
125 	e0 = ex - BIAS - 23;		/* e0 = ilogb(|x|)-23; */
126 	u1.xbits.expsign = ex - e0;
127 	z = u1.e;
128 	for(i=0;i<2;i++) {
129 		tx[i] = (double)((int32_t)(z));
130 		z     = (z-tx[i])*two24;
131 	}
132 	tx[2] = z;
133 	nx = 3;
134 	while(tx[nx-1]==zero) nx--;	/* skip zero term */
135 	n  =  __kernel_rem_pio2(tx,ty,e0,nx,2);
136 	r = (long double)ty[0] + ty[1];
137 	w = ty[1] - (r - ty[0]);
138 	if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
139 	y[0] = r; y[1] = w; return n;
140 }
141