28/*
29 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
30 */
31static const int32_t two_over_pi[] = {
320xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
330x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
340x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
350xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
360x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
370x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
380x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
390xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
400x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
410x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
420x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
43};
44
| |
45static const int32_t npio2_hw[] = {
460x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
470x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
480x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
490x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
500x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
510x404858EB, 0x404921FB,
52};
53
54/*
55 * invpio2: 53 bits of 2/pi
56 * pio2_1: first 33 bit of pi/2
57 * pio2_1t: pi/2 - pio2_1
58 * pio2_2: second 33 bit of pi/2
59 * pio2_2t: pi/2 - (pio2_1+pio2_2)
60 * pio2_3: third 33 bit of pi/2
61 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
62 */
63
64static const double
65zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
66half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
67two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
68invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
69pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
70pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
71pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
72pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
73pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
74pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
75
76 int32_t __ieee754_rem_pio2(double x, double *y)
77{
78 double z,w,t,r,fn;
79 double tx[3];
80 int32_t e0,i,j,nx,n,ix,hx;
81 u_int32_t low;
82
83 GET_HIGH_WORD(hx,x); /* high word of x */
84 ix = hx&0x7fffffff;
85 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
86 {y[0] = x; y[1] = 0; return 0;}
87 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
88 if(hx>0) {
89 z = x - pio2_1;
90 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
91 y[0] = z - pio2_1t;
92 y[1] = (z-y[0])-pio2_1t;
93 } else { /* near pi/2, use 33+33+53 bit pi */
94 z -= pio2_2;
95 y[0] = z - pio2_2t;
96 y[1] = (z-y[0])-pio2_2t;
97 }
98 return 1;
99 } else { /* negative x */
100 z = x + pio2_1;
101 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
102 y[0] = z + pio2_1t;
103 y[1] = (z-y[0])+pio2_1t;
104 } else { /* near pi/2, use 33+33+53 bit pi */
105 z += pio2_2;
106 y[0] = z + pio2_2t;
107 y[1] = (z-y[0])+pio2_2t;
108 }
109 return -1;
110 }
111 }
112 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
113 t = fabs(x);
114 n = (int32_t) (t*invpio2+half);
115 fn = (double)n;
116 r = t-fn*pio2_1;
117 w = fn*pio2_1t; /* 1st round good to 85 bit */
118 if(n<32&&ix!=npio2_hw[n-1]) {
119 y[0] = r-w; /* quick check no cancellation */
120 } else {
121 u_int32_t high;
122 j = ix>>20;
123 y[0] = r-w;
124 GET_HIGH_WORD(high,y[0]);
125 i = j-((high>>20)&0x7ff);
126 if(i>16) { /* 2nd iteration needed, good to 118 */
127 t = r;
128 w = fn*pio2_2;
129 r = t-w;
130 w = fn*pio2_2t-((t-r)-w);
131 y[0] = r-w;
132 GET_HIGH_WORD(high,y[0]);
133 i = j-((high>>20)&0x7ff);
134 if(i>49) { /* 3rd iteration need, 151 bits acc */
135 t = r; /* will cover all possible cases */
136 w = fn*pio2_3;
137 r = t-w;
138 w = fn*pio2_3t-((t-r)-w);
139 y[0] = r-w;
140 }
141 }
142 }
143 y[1] = (r-y[0])-w;
144 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
145 else return n;
146 }
147 /*
148 * all other (large) arguments
149 */
150 if(ix>=0x7ff00000) { /* x is inf or NaN */
151 y[0]=y[1]=x-x; return 0;
152 }
153 /* set z = scalbn(|x|,ilogb(x)-23) */
154 GET_LOW_WORD(low,x);
155 SET_LOW_WORD(z,low);
156 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
157 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
158 for(i=0;i<2;i++) {
159 tx[i] = (double)((int32_t)(z));
160 z = (z-tx[i])*two24;
161 }
162 tx[2] = z;
163 nx = 3;
164 while(tx[nx-1]==zero) nx--; /* skip zero term */
| 28static const int32_t npio2_hw[] = {
290x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
300x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
310x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
320x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
330x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
340x404858EB, 0x404921FB,
35};
36
37/*
38 * invpio2: 53 bits of 2/pi
39 * pio2_1: first 33 bit of pi/2
40 * pio2_1t: pi/2 - pio2_1
41 * pio2_2: second 33 bit of pi/2
42 * pio2_2t: pi/2 - (pio2_1+pio2_2)
43 * pio2_3: third 33 bit of pi/2
44 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
45 */
46
47static const double
48zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
49half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
50two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
51invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
52pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
53pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
54pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
55pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
56pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
57pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
58
59 int32_t __ieee754_rem_pio2(double x, double *y)
60{
61 double z,w,t,r,fn;
62 double tx[3];
63 int32_t e0,i,j,nx,n,ix,hx;
64 u_int32_t low;
65
66 GET_HIGH_WORD(hx,x); /* high word of x */
67 ix = hx&0x7fffffff;
68 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
69 {y[0] = x; y[1] = 0; return 0;}
70 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
71 if(hx>0) {
72 z = x - pio2_1;
73 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
74 y[0] = z - pio2_1t;
75 y[1] = (z-y[0])-pio2_1t;
76 } else { /* near pi/2, use 33+33+53 bit pi */
77 z -= pio2_2;
78 y[0] = z - pio2_2t;
79 y[1] = (z-y[0])-pio2_2t;
80 }
81 return 1;
82 } else { /* negative x */
83 z = x + pio2_1;
84 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
85 y[0] = z + pio2_1t;
86 y[1] = (z-y[0])+pio2_1t;
87 } else { /* near pi/2, use 33+33+53 bit pi */
88 z += pio2_2;
89 y[0] = z + pio2_2t;
90 y[1] = (z-y[0])+pio2_2t;
91 }
92 return -1;
93 }
94 }
95 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
96 t = fabs(x);
97 n = (int32_t) (t*invpio2+half);
98 fn = (double)n;
99 r = t-fn*pio2_1;
100 w = fn*pio2_1t; /* 1st round good to 85 bit */
101 if(n<32&&ix!=npio2_hw[n-1]) {
102 y[0] = r-w; /* quick check no cancellation */
103 } else {
104 u_int32_t high;
105 j = ix>>20;
106 y[0] = r-w;
107 GET_HIGH_WORD(high,y[0]);
108 i = j-((high>>20)&0x7ff);
109 if(i>16) { /* 2nd iteration needed, good to 118 */
110 t = r;
111 w = fn*pio2_2;
112 r = t-w;
113 w = fn*pio2_2t-((t-r)-w);
114 y[0] = r-w;
115 GET_HIGH_WORD(high,y[0]);
116 i = j-((high>>20)&0x7ff);
117 if(i>49) { /* 3rd iteration need, 151 bits acc */
118 t = r; /* will cover all possible cases */
119 w = fn*pio2_3;
120 r = t-w;
121 w = fn*pio2_3t-((t-r)-w);
122 y[0] = r-w;
123 }
124 }
125 }
126 y[1] = (r-y[0])-w;
127 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
128 else return n;
129 }
130 /*
131 * all other (large) arguments
132 */
133 if(ix>=0x7ff00000) { /* x is inf or NaN */
134 y[0]=y[1]=x-x; return 0;
135 }
136 /* set z = scalbn(|x|,ilogb(x)-23) */
137 GET_LOW_WORD(low,x);
138 SET_LOW_WORD(z,low);
139 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
140 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
141 for(i=0;i<2;i++) {
142 tx[i] = (double)((int32_t)(z));
143 z = (z-tx[i])*two24;
144 }
145 tx[2] = z;
146 nx = 3;
147 while(tx[nx-1]==zero) nx--; /* skip zero term */
|