1 
2 /*******************************************************************************
3 MIT License
4 -----------
5 
6 Copyright (c) 2002-2019 Advanced Micro Devices, Inc.
7 
8 Permission is hereby granted, free of charge, to any person obtaining a copy
9 of this Software and associated documentaon files (the "Software"), to deal
10 in the Software without restriction, including without limitation the rights
11 to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
12 copies of the Software, and to permit persons to whom the Software is
13 furnished to do so, subject to the following conditions:
14 
15 The above copyright notice and this permission notice shall be included in
16 all copies or substantial portions of the Software.
17 
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
24 THE SOFTWARE.
25 *******************************************************************************/
26 
27 #include "libm.h"
28 #include "libm_util.h"
29 
30 
31 /* Given positive argument x, reduce it to the range [-pi/4,pi/4] using
32    extra precision, and return the result in r.
33    Return value "region" tells how many lots of pi/2 were subtracted
34    from x to put it in the range [-pi/4,pi/4], mod 4. */
35 void __remainder_piby2f(unsigned long long ux, double *r, int *region)
36 {
37 
38 
39       /* This method simulates multi-precision floating-point
40          arithmetic and is accurate for all 1 <= x < infinity */
41 #define bitsper 36
42       unsigned long long res[10];
43       unsigned long long u, carry, mask, mant, nextbits;
44       int first, last, i, rexp, xexp, resexp, ltb, determ, bc;
45       double dx;
46       static const double
47         piby2 = 1.57079632679489655800e+00; /* 0x3ff921fb54442d18 */
48       static unsigned long long pibits[] =
49       {
50         0LL,
51         5215LL, 13000023176LL, 11362338026LL, 67174558139LL,
52         34819822259LL, 10612056195LL, 67816420731LL, 57840157550LL,
53         19558516809LL, 50025467026LL, 25186875954LL, 18152700886LL
54       };
55 
56 
57       xexp = (int)(((ux & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64);
58       ux = ((ux & MANTBITS_DP64) | IMPBIT_DP64) >> 29;
59 
60 
61       /* Now ux is the mantissa bit pattern of x as a long integer */
62       mask = 1;
63       mask = (mask << bitsper) - 1;
64 
65       /* Set first and last to the positions of the first
66          and last chunks of 2/pi that we need */
67       first = xexp / bitsper;
68       resexp = xexp - first * bitsper;
69       /* 120 is the theoretical maximum number of bits (actually
70          115 for IEEE single precision) that we need to extract
71          from the middle of 2/pi to compute the reduced argument
72          accurately enough for our purposes */
73       last = first + 120 / bitsper;
74 
75 
76       /* Do a long multiplication of the bits of 2/pi by the
77          integer mantissa */
78 #if 0
79       for (i = last; i >= first; i--)
80         {
81           u = pibits[i] * ux + carry;
82           res[i - first] = u & mask;
83           carry = u >> bitsper;
84         }
85       res[last - first + 1] = 0;
86 #else
87       /* Unroll the loop. This is only correct because we know
88          that bitsper is fixed as 36. */
89       res[4] = 0;
90       u = pibits[last] * ux;
91       res[3] = u & mask;
92       carry = u >> bitsper;
93       u = pibits[last - 1] * ux + carry;
94       res[2] = u & mask;
95       carry = u >> bitsper;
96       u = pibits[last - 2] * ux + carry;
97       res[1] = u & mask;
98       carry = u >> bitsper;
99       u = pibits[first] * ux + carry;
100       res[0] = u & mask;
101 #endif
102 
103 
104       /* Reconstruct the result */
105       ltb = (int)((((res[0] << bitsper) | res[1])
106                    >> (bitsper - 1 - resexp)) & 7);
107 
108       /* determ says whether the fractional part is >= 0.5 */
109       determ = ltb & 1;
110 
111       i = 1;
112       if (determ)
113         {
114           /* The mantissa is >= 0.5. We want to subtract it
115              from 1.0 by negating all the bits */
116           *region = ((ltb >> 1) + 1) & 3;
117           mant = 1;
118           mant = ~(res[1]) & ((mant << (bitsper - resexp)) - 1);
119           while (mant < 0x0000000000010000)
120             {
121               i++;
122               mant = (mant << bitsper) | (~(res[i]) & mask);
123             }
124           nextbits = (~(res[i+1]) & mask);
125         }
126       else
127         {
128           *region = (ltb >> 1);
129           mant = 1;
130           mant = res[1] & ((mant << (bitsper - resexp)) - 1);
131           while (mant < 0x0000000000010000)
132             {
133               i++;
134               mant = (mant << bitsper) | res[i];
135             }
136           nextbits = res[i+1];
137         }
138 
139 
140       /* Normalize the mantissa. The shift value 6 here, determined by
141          trial and error, seems to give optimal speed. */
142       bc = 0;
143       while (mant < 0x0000400000000000)
144         {
145           bc += 6;
146           mant <<= 6;
147         }
148       while (mant < 0x0010000000000000)
149         {
150           bc++;
151           mant <<= 1;
152         }
153       mant |= nextbits >> (bitsper - bc);
154 
155       rexp = 52 + resexp - bc - i * bitsper;
156 
157 
158       /* Put the result exponent rexp onto the mantissa pattern */
159       u = ((unsigned long long)rexp + EXPBIAS_DP64) << EXPSHIFTBITS_DP64;
160       ux = (mant & MANTBITS_DP64) | u;
161       if (determ)
162         /* If we negated the mantissa we negate x too */
163         ux |= SIGNBIT_DP64;
164       PUT_BITS_DP64(ux, dx);
165 
166 
167       /* x is a double precision version of the fractional part of
168          x * 2 / pi. Multiply x by pi/2 in double precision
169          to get the reduced argument r. */
170       *r = dx * piby2;
171   return;
172 
173 }
174