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  This has been tested on all possible 32-bit inputs.*/
42unsigned isqrt32(opus_uint32 _val){
43  unsigned b;
44  unsigned g;
45  int      bshift;
46  /*Uses the second method from
47     http://www.azillionmonkeys.com/qed/sqroot.html
48    The main idea is to search for the largest binary digit b such that
49     (g+b)*(g+b) <= _val, and add it to the solution g.*/
50  g=0;
51  bshift=(EC_ILOG(_val)-1)>>1;
52  b=1U<<bshift;
53  do{
54    opus_uint32 t;
55    t=(((opus_uint32)g<<1)+b)<<bshift;
56    if(t<=_val){
57      g+=b;
58      _val-=t;
59    }
60    b>>=1;
61    bshift--;
62  }
63  while(bshift>=0);
64  return g;
65}
66
67#ifdef FIXED_POINT
68
69opus_val32 frac_div32(opus_val32 a, opus_val32 b)
70{
71   opus_val16 rcp;
72   opus_val32 result, rem;
73   int shift = celt_ilog2(b)-29;
74   a = VSHR32(a,shift);
75   b = VSHR32(b,shift);
76   /* 16-bit reciprocal */
77   rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
78   result = MULT16_32_Q15(rcp, a);
79   rem = PSHR32(a,2)-MULT32_32_Q31(result, b);
80   result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2));
81   if (result >= 536870912)       /*  2^29 */
82      return 2147483647;          /*  2^31 - 1 */
83   else if (result <= -536870912) /* -2^29 */
84      return -2147483647;         /* -2^31 */
85   else
86      return SHL32(result, 2);
87}
88
89/** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
90opus_val16 celt_rsqrt_norm(opus_val32 x)
91{
92   opus_val16 n;
93   opus_val16 r;
94   opus_val16 r2;
95   opus_val16 y;
96   /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
97   n = x-32768;
98   /* Get a rough initial guess for the root.
99      The optimal minimax quadratic approximation (using relative error) is
100       r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
101      Coefficients here, and the final result r, are Q14.*/
102   r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
103   /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
104      We can compute the result from n and r using Q15 multiplies with some
105       adjustment, carefully done to avoid overflow.
106      Range of y is [-1564,1594]. */
107   r2 = MULT16_16_Q15(r, r);
108   y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
109   /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
110      This yields the Q14 reciprocal square root of the Q16 x, with a maximum
111       relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
112       peak absolute error of 2.26591/16384. */
113   return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
114              SUB16(MULT16_16_Q15(y, 12288), 16384))));
115}
116
117/** Sqrt approximation (QX input, QX/2 output) */
118opus_val32 celt_sqrt(opus_val32 x)
119{
120   int k;
121   opus_val16 n;
122   opus_val32 rt;
123   static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664};
124   if (x==0)
125      return 0;
126   k = (celt_ilog2(x)>>1)-7;
127   x = VSHR32(x, 2*k);
128   n = x-32768;
129   rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2],
130              MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4])))))))));
131   rt = VSHR32(rt,7-k);
132   return rt;
133}
134
135#define L1 32767
136#define L2 -7651
137#define L3 8277
138#define L4 -626
139
140static inline opus_val16 _celt_cos_pi_2(opus_val16 x)
141{
142   opus_val16 x2;
143
144   x2 = MULT16_16_P15(x,x);
145   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
146                                                                                ))))))));
147}
148
149#undef L1
150#undef L2
151#undef L3
152#undef L4
153
154opus_val16 celt_cos_norm(opus_val32 x)
155{
156   x = x&0x0001ffff;
157   if (x>SHL32(EXTEND32(1), 16))
158      x = SUB32(SHL32(EXTEND32(1), 17),x);
159   if (x&0x00007fff)
160   {
161      if (x<SHL32(EXTEND32(1), 15))
162      {
163         return _celt_cos_pi_2(EXTRACT16(x));
164      } else {
165         return NEG32(_celt_cos_pi_2(EXTRACT16(65536-x)));
166      }
167   } else {
168      if (x&0x0000ffff)
169         return 0;
170      else if (x&0x0001ffff)
171         return -32767;
172      else
173         return 32767;
174   }
175}
176
177/** Reciprocal approximation (Q15 input, Q16 output) */
178opus_val32 celt_rcp(opus_val32 x)
179{
180   int i;
181   opus_val16 n;
182   opus_val16 r;
183   celt_assert2(x>0, "celt_rcp() only defined for positive values");
184   i = celt_ilog2(x);
185   /* n is Q15 with range [0,1). */
186   n = VSHR32(x,i-15)-32768;
187   /* Start with a linear approximation:
188      r = 1.8823529411764706-0.9411764705882353*n.
189      The coefficients and the result are Q14 in the range [15420,30840].*/
190   r = ADD16(30840, MULT16_16_Q15(-15420, n));
191   /* Perform two Newton iterations:
192      r -= r*((r*n)-1.Q15)
193         = r*((r*n)+(r-1.Q15)). */
194   r = SUB16(r, MULT16_16_Q15(r,
195             ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
196   /* We subtract an extra 1 in the second iteration to avoid overflow; it also
197       neatly compensates for truncation error in the rest of the process. */
198   r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
199             ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
200   /* r is now the Q15 solution to 2/(n+1), with a maximum relative error
201       of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
202       error of 1.24665/32768. */
203   return VSHR32(EXTEND32(r),i-16);
204}
205
206#endif
207