202 lines
6.4 KiB
C
202 lines
6.4 KiB
C
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/* Copyright (c) 2002-2008 Jean-Marc Valin
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Copyright (c) 2007-2008 CSIRO
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Copyright (c) 2007-2009 Xiph.Org Foundation
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Written by Jean-Marc Valin */
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/**
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@file mathops.h
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@brief Various math functions
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*/
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/*
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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- Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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- Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE FOUNDATION OR
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CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
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EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
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PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*/
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#ifdef HAVE_CONFIG_H
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#include "config.h"
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#endif
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#include "mathops.h"
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/*Compute floor(sqrt(_val)) with exact arithmetic.
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This has been tested on all possible 32-bit inputs.*/
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unsigned isqrt32(celt_uint32 _val){
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unsigned b;
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unsigned g;
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int bshift;
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/*Uses the second method from
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http://www.azillionmonkeys.com/qed/sqroot.html
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The main idea is to search for the largest binary digit b such that
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(g+b)*(g+b) <= _val, and add it to the solution g.*/
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g=0;
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bshift=EC_ILOG(_val)-1>>1;
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b=1U<<bshift;
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do{
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celt_uint32 t;
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t=((celt_uint32)g<<1)+b<<bshift;
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if(t<=_val){
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g+=b;
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_val-=t;
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}
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b>>=1;
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bshift--;
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}
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while(bshift>=0);
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return g;
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}
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#ifdef FIXED_POINT
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celt_word32 frac_div32(celt_word32 a, celt_word32 b)
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{
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celt_word16 rcp;
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celt_word32 result, rem;
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int shift = celt_ilog2(b)-29;
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a = VSHR32(a,shift);
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b = VSHR32(b,shift);
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/* 16-bit reciprocal */
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rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
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result = SHL32(MULT16_32_Q15(rcp, a),2);
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rem = a-MULT32_32_Q31(result, b);
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result += SHL32(MULT16_32_Q15(rcp, rem),2);
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return result;
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}
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/** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
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celt_word16 celt_rsqrt_norm(celt_word32 x)
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{
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celt_word16 n;
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celt_word16 r;
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celt_word16 r2;
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celt_word16 y;
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/* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
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n = x-32768;
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/* Get a rough initial guess for the root.
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The optimal minimax quadratic approximation (using relative error) is
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r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
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Coefficients here, and the final result r, are Q14.*/
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r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
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/* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
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We can compute the result from n and r using Q15 multiplies with some
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adjustment, carefully done to avoid overflow.
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Range of y is [-1564,1594]. */
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r2 = MULT16_16_Q15(r, r);
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y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
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/* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
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This yields the Q14 reciprocal square root of the Q16 x, with a maximum
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relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
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peak absolute error of 2.26591/16384. */
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return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
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SUB16(MULT16_16_Q15(y, 12288), 16384))));
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}
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/** Sqrt approximation (QX input, QX/2 output) */
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celt_word32 celt_sqrt(celt_word32 x)
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{
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int k;
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celt_word16 n;
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celt_word32 rt;
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static const celt_word16 C[5] = {23175, 11561, -3011, 1699, -664};
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if (x==0)
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return 0;
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k = (celt_ilog2(x)>>1)-7;
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x = VSHR32(x, (k<<1));
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n = x-32768;
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rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2],
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MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4])))))))));
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rt = VSHR32(rt,7-k);
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return rt;
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}
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#define L1 32767
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#define L2 -7651
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#define L3 8277
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#define L4 -626
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static inline celt_word16 _celt_cos_pi_2(celt_word16 x)
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{
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celt_word16 x2;
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x2 = MULT16_16_P15(x,x);
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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
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))))))));
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}
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#undef L1
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#undef L2
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#undef L3
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#undef L4
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celt_word16 celt_cos_norm(celt_word32 x)
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{
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x = x&0x0001ffff;
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if (x>SHL32(EXTEND32(1), 16))
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x = SUB32(SHL32(EXTEND32(1), 17),x);
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if (x&0x00007fff)
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{
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if (x<SHL32(EXTEND32(1), 15))
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{
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return _celt_cos_pi_2(EXTRACT16(x));
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} else {
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return NEG32(_celt_cos_pi_2(EXTRACT16(65536-x)));
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}
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} else {
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if (x&0x0000ffff)
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return 0;
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else if (x&0x0001ffff)
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return -32767;
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else
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return 32767;
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}
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}
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/** Reciprocal approximation (Q15 input, Q16 output) */
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celt_word32 celt_rcp(celt_word32 x)
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{
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int i;
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celt_word16 n;
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celt_word16 r;
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celt_assert2(x>0, "celt_rcp() only defined for positive values");
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i = celt_ilog2(x);
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/* n is Q15 with range [0,1). */
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n = VSHR32(x,i-15)-32768;
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/* Start with a linear approximation:
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r = 1.8823529411764706-0.9411764705882353*n.
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The coefficients and the result are Q14 in the range [15420,30840].*/
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r = ADD16(30840, MULT16_16_Q15(-15420, n));
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/* Perform two Newton iterations:
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r -= r*((r*n)-1.Q15)
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= r*((r*n)+(r-1.Q15)). */
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r = SUB16(r, MULT16_16_Q15(r,
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ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
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/* We subtract an extra 1 in the second iteration to avoid overflow; it also
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neatly compensates for truncation error in the rest of the process. */
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r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
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ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
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/* r is now the Q15 solution to 2/(n+1), with a maximum relative error
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of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
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error of 1.24665/32768. */
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return VSHR32(EXTEND32(r),i-16);
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}
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#endif
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