1 /* Copyright (c) 2002-2008 Jean-Marc Valin
2 Copyright (c) 2007-2008 CSIRO
3 Copyright (c) 2007-2009 Xiph.Org Foundation
4 Written by Jean-Marc Valin */
5 /**
6 @file mathops.h
7 @brief Various math functions
8 */
9 /*
10 Redistribution and use in source and binary forms, with or without
11 modification, are permitted provided that the following conditions
12 are met:
13
14 - Redistributions of source code must retain the above copyright
15 notice, this list of conditions and the following disclaimer.
16
17 - Redistributions in binary form must reproduce the above copyright
18 notice, this list of conditions and the following disclaimer in the
19 documentation and/or other materials provided with the distribution.
20
21 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
22 ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
23 LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
24 A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
25 OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
26 EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
27 PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
28 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
29 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
30 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
31 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
32 */
33
34 #ifdef HAVE_CONFIG_H
35 #include "config.h"
36 #endif
37
38 #include "mathops.h"
39
40 /*Compute floor(sqrt(_val)) with exact arithmetic.
41 _val must be greater than 0.
42 This has been tested on all possible 32-bit inputs greater than 0.*/
isqrt32(opus_uint32 _val)43 unsigned isqrt32(opus_uint32 _val){
44 unsigned b;
45 unsigned g;
46 int bshift;
47 /*Uses the second method from
48 http://www.azillionmonkeys.com/qed/sqroot.html
49 The main idea is to search for the largest binary digit b such that
50 (g+b)*(g+b) <= _val, and add it to the solution g.*/
51 g=0;
52 bshift=(EC_ILOG(_val)-1)>>1;
53 b=1U<<bshift;
54 do{
55 opus_uint32 t;
56 t=(((opus_uint32)g<<1)+b)<<bshift;
57 if(t<=_val){
58 g+=b;
59 _val-=t;
60 }
61 b>>=1;
62 bshift--;
63 }
64 while(bshift>=0);
65 return g;
66 }
67
68 #ifdef FIXED_POINT
69
frac_div32(opus_val32 a,opus_val32 b)70 opus_val32 frac_div32(opus_val32 a, opus_val32 b)
71 {
72 opus_val16 rcp;
73 opus_val32 result, rem;
74 int shift = celt_ilog2(b)-29;
75 a = VSHR32(a,shift);
76 b = VSHR32(b,shift);
77 /* 16-bit reciprocal */
78 rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
79 result = MULT16_32_Q15(rcp, a);
80 rem = PSHR32(a,2)-MULT32_32_Q31(result, b);
81 result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2));
82 if (result >= 536870912) /* 2^29 */
83 return 2147483647; /* 2^31 - 1 */
84 else if (result <= -536870912) /* -2^29 */
85 return -2147483647; /* -2^31 */
86 else
87 return SHL32(result, 2);
88 }
89
90 /** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
celt_rsqrt_norm(opus_val32 x)91 opus_val16 celt_rsqrt_norm(opus_val32 x)
92 {
93 opus_val16 n;
94 opus_val16 r;
95 opus_val16 r2;
96 opus_val16 y;
97 /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
98 n = x-32768;
99 /* Get a rough initial guess for the root.
100 The optimal minimax quadratic approximation (using relative error) is
101 r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
102 Coefficients here, and the final result r, are Q14.*/
103 r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
104 /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
105 We can compute the result from n and r using Q15 multiplies with some
106 adjustment, carefully done to avoid overflow.
107 Range of y is [-1564,1594]. */
108 r2 = MULT16_16_Q15(r, r);
109 y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
110 /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
111 This yields the Q14 reciprocal square root of the Q16 x, with a maximum
112 relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
113 peak absolute error of 2.26591/16384. */
114 return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
115 SUB16(MULT16_16_Q15(y, 12288), 16384))));
116 }
117
118 /** Sqrt approximation (QX input, QX/2 output) */
celt_sqrt(opus_val32 x)119 opus_val32 celt_sqrt(opus_val32 x)
120 {
121 int k;
122 opus_val16 n;
123 opus_val32 rt;
124 static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664};
125 if (x==0)
126 return 0;
127 else if (x>=1073741824)
128 return 32767;
129 k = (celt_ilog2(x)>>1)-7;
130 x = VSHR32(x, 2*k);
131 n = x-32768;
132 rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2],
133 MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4])))))))));
134 rt = VSHR32(rt,7-k);
135 return rt;
136 }
137
138 #define L1 32767
139 #define L2 -7651
140 #define L3 8277
141 #define L4 -626
142
_celt_cos_pi_2(opus_val16 x)143 static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x)
144 {
145 opus_val16 x2;
146
147 x2 = MULT16_16_P15(x,x);
148 return ADD16(1,MIN16(32766,ADD32(SUB16(L1,x2), MULT16_16_P15(x2, ADD32(L2, MULT16_16_P15(x2, ADD32(L3, MULT16_16_P15(L4, x2
149 ))))))));
150 }
151
152 #undef L1
153 #undef L2
154 #undef L3
155 #undef L4
156
celt_cos_norm(opus_val32 x)157 opus_val16 celt_cos_norm(opus_val32 x)
158 {
159 x = x&0x0001ffff;
160 if (x>SHL32(EXTEND32(1), 16))
161 x = SUB32(SHL32(EXTEND32(1), 17),x);
162 if (x&0x00007fff)
163 {
164 if (x<SHL32(EXTEND32(1), 15))
165 {
166 return _celt_cos_pi_2(EXTRACT16(x));
167 } else {
168 return NEG16(_celt_cos_pi_2(EXTRACT16(65536-x)));
169 }
170 } else {
171 if (x&0x0000ffff)
172 return 0;
173 else if (x&0x0001ffff)
174 return -32767;
175 else
176 return 32767;
177 }
178 }
179
180 /** Reciprocal approximation (Q15 input, Q16 output) */
celt_rcp(opus_val32 x)181 opus_val32 celt_rcp(opus_val32 x)
182 {
183 int i;
184 opus_val16 n;
185 opus_val16 r;
186 celt_sig_assert(x>0);
187 i = celt_ilog2(x);
188 /* n is Q15 with range [0,1). */
189 n = VSHR32(x,i-15)-32768;
190 /* Start with a linear approximation:
191 r = 1.8823529411764706-0.9411764705882353*n.
192 The coefficients and the result are Q14 in the range [15420,30840].*/
193 r = ADD16(30840, MULT16_16_Q15(-15420, n));
194 /* Perform two Newton iterations:
195 r -= r*((r*n)-1.Q15)
196 = r*((r*n)+(r-1.Q15)). */
197 r = SUB16(r, MULT16_16_Q15(r,
198 ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
199 /* We subtract an extra 1 in the second iteration to avoid overflow; it also
200 neatly compensates for truncation error in the rest of the process. */
201 r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
202 ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
203 /* r is now the Q15 solution to 2/(n+1), with a maximum relative error
204 of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
205 error of 1.24665/32768. */
206 return VSHR32(EXTEND32(r),i-16);
207 }
208
209 #endif
210