amath  1.8.5
Simple command line calculator
remp2.c
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1 /*-
2  * Copyright (c) 2014-2018 Carsten Sonne Larsen <cs@innolan.net>
3  * All rights reserved.
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25  * Project homepage:
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27  *
28  * The original source code can be obtained from:
29  * http://www.netlib.org/fdlibm/e_rem_pio2.c
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39  */
40 
41 #include "prim.h"
42 
43 /**
44  * @file remp2.c
45  * @brief
46  */
47 
48 /* rem_pio2(x,y)
49  *
50  * return the remainder of x rem pi/2 in y[0]+y[1]
51  * use __kernel_rem_pio2()
52  */
53 
54 
55 /*
56  * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
57  */
58 
59 static const int two_over_pi[] = {
60  0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
61  0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
62  0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
63  0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
64  0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
65  0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
66  0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
67  0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
68  0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
69  0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
70  0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
71 };
72 
73 static const int npio2_hw[] = {
74  0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
75  0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
76  0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
77  0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
78  0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
79  0x404858EB, 0x404921FB,
80 };
81 
82 /*
83  * invpio2: 53 bits of 2/pi
84  * pio2_1: first 33 bit of pi/2
85  * pio2_1t: pi/2 - pio2_1
86  * pio2_2: second 33 bit of pi/2
87  * pio2_2t: pi/2 - (pio2_1+pio2_2)
88  * pio2_3: third 33 bit of pi/2
89  * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
90  */
91 
92 static const double
93 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
94 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
95 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
96 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
97 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
98 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
99 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
100 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
101 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
102 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
103 
104 int32_t rempio2(double x, double *y)
105 {
106  double z = 0.,w,t,r,fn;
107  double tx[3];
108  int32_t i,j,n,ix,hx;
109  int e0,nx;
110  uint32_t low;
111 
112  GET_HIGH_WORD(hx,x); /* high word of x */
113  ix = hx&0x7fffffff;
114  if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
115  {
116  y[0] = x;
117  y[1] = 0;
118  return 0;
119  }
120  if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
121  if(hx>0) {
122  z = x - pio2_1;
123  if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
124  y[0] = z - pio2_1t;
125  y[1] = (z-y[0])-pio2_1t;
126  } else { /* near pi/2, use 33+33+53 bit pi */
127  z -= pio2_2;
128  y[0] = z - pio2_2t;
129  y[1] = (z-y[0])-pio2_2t;
130  }
131  return 1;
132  } else { /* negative x */
133  z = x + pio2_1;
134  if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
135  y[0] = z + pio2_1t;
136  y[1] = (z-y[0])+pio2_1t;
137  } else { /* near pi/2, use 33+33+53 bit pi */
138  z += pio2_2;
139  y[0] = z + pio2_2t;
140  y[1] = (z-y[0])+pio2_2t;
141  }
142  return -1;
143  }
144  }
145  if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
146  t = fabs(x);
147  n = (int32_t) (t*invpio2+half);
148  fn = (double)n;
149  r = t-fn*pio2_1;
150  w = fn*pio2_1t; /* 1st round good to 85 bit */
151  if(n<32&&ix!=npio2_hw[n-1]) {
152  y[0] = r-w; /* quick check no cancellation */
153  } else {
154  uint32_t high;
155 
156  j = ix>>20;
157  y[0] = r-w;
158  GET_HIGH_WORD(high, y[0]);
159  i = j-((high>>20)&0x7ff);
160  if(i>16) { /* 2nd iteration needed, good to 118 */
161  t = r;
162  w = fn*pio2_2;
163  r = t-w;
164  w = fn*pio2_2t-((t-r)-w);
165  y[0] = r-w;
166  GET_HIGH_WORD(high,y[0]);
167  i = j-((high>>20)&0x7ff);
168  if(i>49) { /* 3rd iteration need, 151 bits acc */
169  t = r; /* will cover all possible cases */
170  w = fn*pio2_3;
171  r = t-w;
172  w = fn*pio2_3t-((t-r)-w);
173  y[0] = r-w;
174  }
175  }
176  }
177  y[1] = (r-y[0])-w;
178  if(hx<0) {
179  y[0] = -y[0];
180  y[1] = -y[1];
181  return -n;
182  }
183  else return n;
184  }
185  /*
186  * all other (large) arguments
187  */
188  if(ix>=0x7ff00000) { /* x is inf or NaN */
189  y[0]=y[1]=x-x;
190  return 0;
191  }
192  /* set z = scalbn(|x|,ilogb(x)-23) */
193  GET_LOW_WORD(low,x);
194  SET_LOW_WORD(z,low);
195  e0 = (int32_t)(ix>>20)-1046; /* e0 = ilogb(z)-23; */
196  SET_HIGH_WORD(z,ix - (e0<<20));
197  for(i=0; i<2; i++) {
198  tx[i] = (double)((int32_t)(z));
199  z = (z-tx[i])*two24;
200  }
201  tx[2] = z;
202  nx = 3;
203  while(tx[nx-1]==zero) nx--; /* skip zero term */
205  if(hx<0) {
206  y[0] = -y[0];
207  y[1] = -y[1];
208  return -n;
209  }
210  return n;
211 }
static const double pio2_3
Definition: remp2.c:101
#define GET_HIGH_WORD(i, d)
Get the more significant 32 bit int from a double.
Definition: prim.h:167
static const double zero
Definition: remp2.c:93
static const double half
Definition: remp2.c:94
static const double pio2_2t
Definition: remp2.c:100
#define GET_LOW_WORD(i, d)
Get the less significant 32 bit int from a double.
Definition: prim.h:177
static const double pio2_1t
Definition: remp2.c:98
static const double pio2_1
Definition: remp2.c:97
double fabs(double x)
Returns the absolute value of x.
Definition: fabs.c:51
static const double invpio2
Definition: remp2.c:96
static const int two_over_pi[]
Definition: remp2.c:59
static const double two24
Definition: remp2.c:95
static const double pio2_2
Definition: remp2.c:99
static const int npio2_hw[]
Definition: remp2.c:73
#define SET_LOW_WORD(d, v)
Set the less significant 32 bits of a double from an int.
Definition: prim.h:209
int32_t rempio2(double x, double *y)
Definition: remp2.c:104
#define SET_HIGH_WORD(d, v)
Set the more significant 32 bits of a double from an int.
Definition: prim.h:198
static const double pio2_3t
Definition: remp2.c:102
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2)
Kernel reduction function.
Definition: kremp2.c:184