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 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 -626 139 140 static 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 153 154 opus_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) */ 178 opus_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
