Chris@69: /* Copyright (c) 2002-2008 Jean-Marc Valin Chris@69: Copyright (c) 2007-2008 CSIRO Chris@69: Copyright (c) 2007-2009 Xiph.Org Foundation Chris@69: Written by Jean-Marc Valin */ Chris@69: /** Chris@69: @file mathops.h Chris@69: @brief Various math functions Chris@69: */ Chris@69: /* Chris@69: Redistribution and use in source and binary forms, with or without Chris@69: modification, are permitted provided that the following conditions Chris@69: are met: Chris@69: Chris@69: - Redistributions of source code must retain the above copyright Chris@69: notice, this list of conditions and the following disclaimer. Chris@69: Chris@69: - Redistributions in binary form must reproduce the above copyright Chris@69: notice, this list of conditions and the following disclaimer in the Chris@69: documentation and/or other materials provided with the distribution. Chris@69: Chris@69: THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS Chris@69: ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT Chris@69: LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR Chris@69: A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER Chris@69: OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, Chris@69: EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, Chris@69: PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR Chris@69: PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF Chris@69: LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING Chris@69: NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS Chris@69: SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. Chris@69: */ Chris@69: Chris@69: #ifdef HAVE_CONFIG_H Chris@69: #include "config.h" Chris@69: #endif Chris@69: Chris@69: #include "mathops.h" Chris@69: Chris@69: /*Compute floor(sqrt(_val)) with exact arithmetic. Chris@69: _val must be greater than 0. Chris@69: This has been tested on all possible 32-bit inputs greater than 0.*/ Chris@69: unsigned isqrt32(opus_uint32 _val){ Chris@69: unsigned b; Chris@69: unsigned g; Chris@69: int bshift; Chris@69: /*Uses the second method from Chris@69: http://www.azillionmonkeys.com/qed/sqroot.html Chris@69: The main idea is to search for the largest binary digit b such that Chris@69: (g+b)*(g+b) <= _val, and add it to the solution g.*/ Chris@69: g=0; Chris@69: bshift=(EC_ILOG(_val)-1)>>1; Chris@69: b=1U<>=1; Chris@69: bshift--; Chris@69: } Chris@69: while(bshift>=0); Chris@69: return g; Chris@69: } Chris@69: Chris@69: #ifdef FIXED_POINT Chris@69: Chris@69: opus_val32 frac_div32(opus_val32 a, opus_val32 b) Chris@69: { Chris@69: opus_val16 rcp; Chris@69: opus_val32 result, rem; Chris@69: int shift = celt_ilog2(b)-29; Chris@69: a = VSHR32(a,shift); Chris@69: b = VSHR32(b,shift); Chris@69: /* 16-bit reciprocal */ Chris@69: rcp = ROUND16(celt_rcp(ROUND16(b,16)),3); Chris@69: result = MULT16_32_Q15(rcp, a); Chris@69: rem = PSHR32(a,2)-MULT32_32_Q31(result, b); Chris@69: result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2)); Chris@69: if (result >= 536870912) /* 2^29 */ Chris@69: return 2147483647; /* 2^31 - 1 */ Chris@69: else if (result <= -536870912) /* -2^29 */ Chris@69: return -2147483647; /* -2^31 */ Chris@69: else Chris@69: return SHL32(result, 2); Chris@69: } Chris@69: Chris@69: /** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */ Chris@69: opus_val16 celt_rsqrt_norm(opus_val32 x) Chris@69: { Chris@69: opus_val16 n; Chris@69: opus_val16 r; Chris@69: opus_val16 r2; Chris@69: opus_val16 y; Chris@69: /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */ Chris@69: n = x-32768; Chris@69: /* Get a rough initial guess for the root. Chris@69: The optimal minimax quadratic approximation (using relative error) is Chris@69: r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485). Chris@69: Coefficients here, and the final result r, are Q14.*/ Chris@69: r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713)))); Chris@69: /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14. Chris@69: We can compute the result from n and r using Q15 multiplies with some Chris@69: adjustment, carefully done to avoid overflow. Chris@69: Range of y is [-1564,1594]. */ Chris@69: r2 = MULT16_16_Q15(r, r); Chris@69: y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1); Chris@69: /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5). Chris@69: This yields the Q14 reciprocal square root of the Q16 x, with a maximum Chris@69: relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a Chris@69: peak absolute error of 2.26591/16384. */ Chris@69: return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y, Chris@69: SUB16(MULT16_16_Q15(y, 12288), 16384)))); Chris@69: } Chris@69: Chris@69: /** Sqrt approximation (QX input, QX/2 output) */ Chris@69: opus_val32 celt_sqrt(opus_val32 x) Chris@69: { Chris@69: int k; Chris@69: opus_val16 n; Chris@69: opus_val32 rt; Chris@69: static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664}; Chris@69: if (x==0) Chris@69: return 0; Chris@69: else if (x>=1073741824) Chris@69: return 32767; Chris@69: k = (celt_ilog2(x)>>1)-7; Chris@69: x = VSHR32(x, 2*k); Chris@69: n = x-32768; Chris@69: rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2], Chris@69: MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4]))))))))); Chris@69: rt = VSHR32(rt,7-k); Chris@69: return rt; Chris@69: } Chris@69: Chris@69: #define L1 32767 Chris@69: #define L2 -7651 Chris@69: #define L3 8277 Chris@69: #define L4 -626 Chris@69: Chris@69: static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x) Chris@69: { Chris@69: opus_val16 x2; Chris@69: Chris@69: x2 = MULT16_16_P15(x,x); Chris@69: 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 Chris@69: )))))))); Chris@69: } Chris@69: Chris@69: #undef L1 Chris@69: #undef L2 Chris@69: #undef L3 Chris@69: #undef L4 Chris@69: Chris@69: opus_val16 celt_cos_norm(opus_val32 x) Chris@69: { Chris@69: x = x&0x0001ffff; Chris@69: if (x>SHL32(EXTEND32(1), 16)) Chris@69: x = SUB32(SHL32(EXTEND32(1), 17),x); Chris@69: if (x&0x00007fff) Chris@69: { Chris@69: if (x0); Chris@69: i = celt_ilog2(x); Chris@69: /* n is Q15 with range [0,1). */ Chris@69: n = VSHR32(x,i-15)-32768; Chris@69: /* Start with a linear approximation: Chris@69: r = 1.8823529411764706-0.9411764705882353*n. Chris@69: The coefficients and the result are Q14 in the range [15420,30840].*/ Chris@69: r = ADD16(30840, MULT16_16_Q15(-15420, n)); Chris@69: /* Perform two Newton iterations: Chris@69: r -= r*((r*n)-1.Q15) Chris@69: = r*((r*n)+(r-1.Q15)). */ Chris@69: r = SUB16(r, MULT16_16_Q15(r, Chris@69: ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))); Chris@69: /* We subtract an extra 1 in the second iteration to avoid overflow; it also Chris@69: neatly compensates for truncation error in the rest of the process. */ Chris@69: r = SUB16(r, ADD16(1, MULT16_16_Q15(r, Chris@69: ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))))); Chris@69: /* r is now the Q15 solution to 2/(n+1), with a maximum relative error Chris@69: of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute Chris@69: error of 1.24665/32768. */ Chris@69: return VSHR32(EXTEND32(r),i-16); Chris@69: } Chris@69: Chris@69: #endif