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.*/ 42 unsigned 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 69 opus_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) */ 90 opus_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) */ 118 opus_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 else if (x>=1073741824) 127 return 32767; 128 k = (celt_ilog2(x)>>1)-7; 129 x = VSHR32(x, 2*k); 130 n = x-32768; 131 rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2], 132 MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4]))))))))); 133 rt = VSHR32(rt,7-k); 134 return rt; 135 } 136 137 #define L1 32767 138 #define L2 -7651 139 #define L3 8277 140 #define L4 -626 141 142 static inline opus_val16 _celt_cos_pi_2(opus_val16 x) 143 { 144 opus_val16 x2; 145 146 x2 = MULT16_16_P15(x,x); 147 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 148 )))))))); 149 } 150 151 #undef L1 152 #undef L2 153 #undef L3 154 #undef L4 155 156 opus_val16 celt_cos_norm(opus_val32 x) 157 { 158 x = x&0x0001ffff; 159 if (x>SHL32(EXTEND32(1), 16)) 160 x = SUB32(SHL32(EXTEND32(1), 17),x); 161 if (x&0x00007fff) 162 { 163 if (x<SHL32(EXTEND32(1), 15)) 164 { 165 return _celt_cos_pi_2(EXTRACT16(x)); 166 } else { 167 return NEG32(_celt_cos_pi_2(EXTRACT16(65536-x))); 168 } 169 } else { 170 if (x&0x0000ffff) 171 return 0; 172 else if (x&0x0001ffff) 173 return -32767; 174 else 175 return 32767; 176 } 177 } 178 179 /** Reciprocal approximation (Q15 input, Q16 output) */ 180 opus_val32 celt_rcp(opus_val32 x) 181 { 182 int i; 183 opus_val16 n; 184 opus_val16 r; 185 celt_assert2(x>0, "celt_rcp() only defined for positive values"); 186 i = celt_ilog2(x); 187 /* n is Q15 with range [0,1). */ 188 n = VSHR32(x,i-15)-32768; 189 /* Start with a linear approximation: 190 r = 1.8823529411764706-0.9411764705882353*n. 191 The coefficients and the result are Q14 in the range [15420,30840].*/ 192 r = ADD16(30840, MULT16_16_Q15(-15420, n)); 193 /* Perform two Newton iterations: 194 r -= r*((r*n)-1.Q15) 195 = r*((r*n)+(r-1.Q15)). */ 196 r = SUB16(r, MULT16_16_Q15(r, 197 ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))); 198 /* We subtract an extra 1 in the second iteration to avoid overflow; it also 199 neatly compensates for truncation error in the rest of the process. */ 200 r = SUB16(r, ADD16(1, MULT16_16_Q15(r, 201 ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))))); 202 /* r is now the Q15 solution to 2/(n+1), with a maximum relative error 203 of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute 204 error of 1.24665/32768. */ 205 return VSHR32(EXTEND32(r),i-16); 206 } 207 208 #endif 209
