mathops.c (11265B)
1 /* Copyright (c) 2002-2008 Jean-Marc Valin 2 Copyright (c) 2007-2008 CSIRO 3 Copyright (c) 2007-2009 Xiph.Org Foundation 4 Copyright (c) 2024 Arm Limited 5 Written by Jean-Marc Valin */ 6 /** 7 @file mathops.h 8 @brief Various math functions 9 */ 10 /* 11 Redistribution and use in source and binary forms, with or without 12 modification, are permitted provided that the following conditions 13 are met: 14 15 - Redistributions of source code must retain the above copyright 16 notice, this list of conditions and the following disclaimer. 17 18 - Redistributions in binary form must reproduce the above copyright 19 notice, this list of conditions and the following disclaimer in the 20 documentation and/or other materials provided with the distribution. 21 22 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 23 ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 24 LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR 25 A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER 26 OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, 27 EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, 28 PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR 29 PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF 30 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING 31 NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS 32 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 33 */ 34 35 #ifdef HAVE_CONFIG_H 36 #include "config.h" 37 #endif 38 39 #include "float_cast.h" 40 #include "mathops.h" 41 42 /*Compute floor(sqrt(_val)) with exact arithmetic. 43 _val must be greater than 0. 44 This has been tested on all possible 32-bit inputs greater than 0.*/ 45 unsigned isqrt32(opus_uint32 _val){ 46 unsigned b; 47 unsigned g; 48 int bshift; 49 /*Uses the second method from 50 http://www.azillionmonkeys.com/qed/sqroot.html 51 The main idea is to search for the largest binary digit b such that 52 (g+b)*(g+b) <= _val, and add it to the solution g.*/ 53 g=0; 54 bshift=(EC_ILOG(_val)-1)>>1; 55 b=1U<<bshift; 56 do{ 57 opus_uint32 t; 58 t=(((opus_uint32)g<<1)+b)<<bshift; 59 if(t<=_val){ 60 g+=b; 61 _val-=t; 62 } 63 b>>=1; 64 bshift--; 65 } 66 while(bshift>=0); 67 return g; 68 } 69 70 #ifdef FIXED_POINT 71 72 opus_val32 frac_div32_q29(opus_val32 a, opus_val32 b) 73 { 74 opus_val16 rcp; 75 opus_val32 result, rem; 76 int shift = celt_ilog2(b)-29; 77 a = VSHR32(a,shift); 78 b = VSHR32(b,shift); 79 /* 16-bit reciprocal */ 80 rcp = ROUND16(celt_rcp(ROUND16(b,16)),3); 81 result = MULT16_32_Q15(rcp, a); 82 rem = PSHR32(a,2)-MULT32_32_Q31(result, b); 83 result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2)); 84 return result; 85 } 86 87 opus_val32 frac_div32(opus_val32 a, opus_val32 b) { 88 opus_val32 result = frac_div32_q29(a,b); 89 if (result >= 536870912) /* 2^29 */ 90 return 2147483647; /* 2^31 - 1 */ 91 else if (result <= -536870912) /* -2^29 */ 92 return -2147483647; /* -2^31 */ 93 else 94 return SHL32(result, 2); 95 } 96 97 /** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */ 98 opus_val16 celt_rsqrt_norm(opus_val32 x) 99 { 100 opus_val16 n; 101 opus_val16 r; 102 opus_val16 r2; 103 opus_val16 y; 104 /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */ 105 n = x-32768; 106 /* Get a rough initial guess for the root. 107 The optimal minimax quadratic approximation (using relative error) is 108 r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485). 109 Coefficients here, and the final result r, are Q14.*/ 110 r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713)))); 111 /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14. 112 We can compute the result from n and r using Q15 multiplies with some 113 adjustment, carefully done to avoid overflow. 114 Range of y is [-1564,1594]. */ 115 r2 = MULT16_16_Q15(r, r); 116 y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1); 117 /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5). 118 This yields the Q14 reciprocal square root of the Q16 x, with a maximum 119 relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a 120 peak absolute error of 2.26591/16384. */ 121 return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y, 122 SUB16(MULT16_16_Q15(y, 12288), 16384)))); 123 } 124 125 /** Reciprocal sqrt approximation in the range [0.25,1) (Q31 in, Q29 out) */ 126 opus_val32 celt_rsqrt_norm32(opus_val32 x) 127 { 128 opus_int32 tmp; 129 /* Use the first-order Newton-Raphson method to refine the root estimate. 130 * r = r * (1.5 - 0.5*x*r*r) */ 131 opus_int32 r_q29 = SHL32(celt_rsqrt_norm(SHR32(x, 31-16)), 15); 132 /* Split evaluation in steps to avoid exploding macro expansion. */ 133 tmp = MULT32_32_Q31(r_q29, r_q29); 134 tmp = MULT32_32_Q31(1073741824 /* Q31 */, tmp); 135 tmp = MULT32_32_Q31(x, tmp); 136 return SHL32(MULT32_32_Q31(r_q29, SUB32(201326592 /* Q27 */, tmp)), 4); 137 } 138 139 /** Sqrt approximation (QX input, QX/2 output) */ 140 opus_val32 celt_sqrt(opus_val32 x) 141 { 142 int k; 143 opus_val16 n; 144 opus_val32 rt; 145 /* These coeffs are optimized in fixed-point to minimize both RMS and max error 146 of sqrt(x) over .25<x<1 without exceeding 32767. 147 The RMS error is 3.4e-5 and the max is 8.2e-5. */ 148 static const opus_val16 C[6] = {23171, 11574, -2901, 1592, -1002, 336}; 149 if (x==0) 150 return 0; 151 else if (x>=1073741824) 152 return 32767; 153 k = (celt_ilog2(x)>>1)-7; 154 x = VSHR32(x, 2*k); 155 n = x-32768; 156 rt = ADD32(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2], 157 MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, ADD16(C[4], MULT16_16_Q15(n, (C[5]))))))))))); 158 rt = VSHR32(rt,7-k); 159 return rt; 160 } 161 162 /* Perform fixed-point arithmetic to approximate the square root. When the input 163 * is in Qx format, the output will be in Q(x/2 + 16) format. */ 164 opus_val32 celt_sqrt32(opus_val32 x) 165 { 166 int k; 167 opus_int32 x_frac; 168 if (x==0) 169 return 0; 170 else if (x>=1073741824) 171 return 2147483647; /* 2^31 -1 */ 172 k = (celt_ilog2(x)>>1); 173 x_frac = VSHR32(x, 2*(k-14)-1); 174 x_frac = MULT32_32_Q31(celt_rsqrt_norm32(x_frac), x_frac); 175 if (k < 12) return PSHR32(x_frac, 12-k); 176 else return SHL32(x_frac, k-12); 177 } 178 179 #define L1 32767 180 #define L2 -7651 181 #define L3 8277 182 #define L4 -626 183 184 static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x) 185 { 186 opus_val16 x2; 187 188 x2 = MULT16_16_P15(x,x); 189 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 190 )))))))); 191 } 192 193 #undef L1 194 #undef L2 195 #undef L3 196 #undef L4 197 198 opus_val16 celt_cos_norm(opus_val32 x) 199 { 200 x = x&0x0001ffff; 201 if (x>SHL32(EXTEND32(1), 16)) 202 x = SUB32(SHL32(EXTEND32(1), 17),x); 203 if (x&0x00007fff) 204 { 205 if (x<SHL32(EXTEND32(1), 15)) 206 { 207 return _celt_cos_pi_2(EXTRACT16(x)); 208 } else { 209 return NEG16(_celt_cos_pi_2(EXTRACT16(65536-x))); 210 } 211 } else { 212 if (x&0x0000ffff) 213 return 0; 214 else if (x&0x0001ffff) 215 return -32767; 216 else 217 return 32767; 218 } 219 } 220 221 /* Calculates the cosine of (PI*0.5*x) where the input x ranges from -1 to 1 and 222 * is in Q30 format. The output will also be in Q31 format. */ 223 opus_val32 celt_cos_norm32(opus_val32 x) 224 { 225 static const opus_val32 COS_NORM_COEFF_A0 = 134217720; /* Q27 */ 226 static const opus_val32 COS_NORM_COEFF_A1 = -662336704; /* Q29 */ 227 static const opus_val32 COS_NORM_COEFF_A2 = 544710848; /* Q31 */ 228 static const opus_val32 COS_NORM_COEFF_A3 = -178761936; /* Q33 */ 229 static const opus_val32 COS_NORM_COEFF_A4 = 29487206; /* Q35 */ 230 opus_int32 x_sq_q29, tmp; 231 /* The expected x is in the range of [-1.0f, 1.0f] */ 232 celt_sig_assert((x >= -1073741824) && (x <= 1073741824)); 233 /* Make cos(+/- pi/2) exactly zero. */ 234 if (ABS32(x) == 1<<30) return 0; 235 x_sq_q29 = MULT32_32_Q31(x, x); 236 /* Split evaluation in steps to avoid exploding macro expansion. */ 237 tmp = ADD32(COS_NORM_COEFF_A3, MULT32_32_Q31(x_sq_q29, COS_NORM_COEFF_A4)); 238 tmp = ADD32(COS_NORM_COEFF_A2, MULT32_32_Q31(x_sq_q29, tmp)); 239 tmp = ADD32(COS_NORM_COEFF_A1, MULT32_32_Q31(x_sq_q29, tmp)); 240 return SHL32(ADD32(COS_NORM_COEFF_A0, MULT32_32_Q31(x_sq_q29, tmp)), 4); 241 } 242 243 /* Computes a 16 bit approximate reciprocal (1/x) for a normalized Q15 input, 244 * resulting in a Q15 output. */ 245 opus_val16 celt_rcp_norm16(opus_val16 x) 246 { 247 opus_val16 r; 248 /* Start with a linear approximation: 249 r = 1.8823529411764706-0.9411764705882353*n. 250 The coefficients and the result are Q14 in the range [15420,30840].*/ 251 r = ADD16(30840, MULT16_16_Q15(-15420, x)); 252 /* Perform two Newton iterations: 253 r -= r*((r*n)+(r-1.Q15)) 254 = r*((r*n)+(r-1.Q15)). */ 255 r = SUB16(r, MULT16_16_Q15(r, 256 ADD16(MULT16_16_Q15(r, x), ADD16(r, -32768)))); 257 /* We subtract an extra 1 in the second iteration to avoid overflow; it also 258 neatly compensates for truncation error in the rest of the process. */ 259 return SUB16(r, ADD16(1, MULT16_16_Q15(r, 260 ADD16(MULT16_16_Q15(r, x), ADD16(r, -32768))))); 261 } 262 263 /* Computes a 32 bit approximated reciprocal (1/x) for a normalized Q31 input, 264 * resulting in a Q30 output. The expected input range is [0.5f, 1.0f) in Q31 265 * and the expected output range is [1.0f, 2.0f) in Q30. */ 266 opus_val32 celt_rcp_norm32(opus_val32 x) 267 { 268 opus_val32 r_q30; 269 celt_sig_assert(x >= 1073741824); 270 r_q30 = SHL32(EXTEND32(celt_rcp_norm16(SHR32(x, 15)-32768)), 16); 271 /* Solving f(y) = a - 1/y using the Newton Method 272 * Note: f(y)' = 1/y^2 273 * r = r - f(r)/f(r)' = r - (x * r*r - r) 274 * = r - r*(r*x - 1) 275 * where 276 * - r means 1/y's approximation. 277 * - x means a, the input of function. 278 * Please note that: 279 * - It adds 1 to avoid overflow 280 * - -1.0f in Q30 is -1073741824. */ 281 return SUB32(r_q30, ADD32(SHL32( 282 MULT32_32_Q31(ADD32(MULT32_32_Q31(r_q30, x), -1073741824), 283 r_q30), 1), 1)); 284 } 285 286 /** Reciprocal approximation (Q15 input, Q16 output) */ 287 opus_val32 celt_rcp(opus_val32 x) 288 { 289 int i; 290 opus_val16 r; 291 celt_sig_assert(x>0); 292 i = celt_ilog2(x); 293 294 /* Compute the reciprocal of a Q15 number in the range [0, 1). */ 295 r = celt_rcp_norm16(VSHR32(x,i-15)-32768); 296 297 /* r is now the Q15 solution to 2/(n+1), with a maximum relative error 298 of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute 299 error of 1.24665/32768. */ 300 return VSHR32(EXTEND32(r),i-16); 301 } 302 303 #endif 304 305 #ifndef DISABLE_FLOAT_API 306 307 void celt_float2int16_c(const float * OPUS_RESTRICT in, short * OPUS_RESTRICT out, int cnt) 308 { 309 int i; 310 for (i = 0; i < cnt; i++) 311 { 312 out[i] = FLOAT2INT16(in[i]); 313 } 314 } 315 316 int opus_limit2_checkwithin1_c(float * samples, int cnt) 317 { 318 int i; 319 if (cnt <= 0) 320 { 321 return 1; 322 } 323 324 for (i = 0; i < cnt; i++) 325 { 326 float clippedVal = samples[i]; 327 clippedVal = FMAX(-2.0f, clippedVal); 328 clippedVal = FMIN(2.0f, clippedVal); 329 samples[i] = clippedVal; 330 } 331 332 /* C implementation can't provide quick hint. Assume it might exceed -1/+1. */ 333 return 0; 334 } 335 336 #endif /* DISABLE_FLOAT_API */