438 lines
12 KiB
C
438 lines
12 KiB
C
/* 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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- Neither the name of the Xiph.org Foundation nor the names of its
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contributors may be used to endorse or promote products derived from
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this software without specific prior written permission.
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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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#ifndef MATHOPS_H
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#define MATHOPS_H
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#include "arch.h"
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#include "entcode.h"
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#include "os_support.h"
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#ifndef OVERRIDE_FIND_MAX16
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static inline int find_max16(celt_word16 *x, int len)
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{
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celt_word16 max_corr=-VERY_LARGE16;
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int i, id = 0;
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for (i=0;i<len;i++)
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{
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if (x[i] > max_corr)
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{
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id = i;
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max_corr = x[i];
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}
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}
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return id;
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}
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#endif
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#ifndef OVERRIDE_FIND_MAX32
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static inline int find_max32(celt_word32 *x, int len)
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{
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celt_word32 max_corr=-VERY_LARGE32;
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int i, id = 0;
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for (i=0;i<len;i++)
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{
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if (x[i] > max_corr)
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{
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id = i;
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max_corr = x[i];
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}
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}
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return id;
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}
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#endif
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/* Multiplies two 16-bit fractional values. Bit-exactness of this macro is important */
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#define FRAC_MUL16(a,b) ((16384+((celt_int32)(celt_int16)(a)*(celt_int16)(b)))>>15)
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/* This is a cos() approximation designed to be bit-exact on any platform. Bit exactness
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with this approximation is important because it has an impact on the bit allocation */
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static inline celt_int16 bitexact_cos(celt_int16 x)
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{
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celt_int32 tmp;
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celt_int16 x2;
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tmp = (4096+((celt_int32)(x)*(x)))>>13;
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if (tmp > 32767)
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tmp = 32767;
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x2 = tmp;
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x2 = (32767-x2) + FRAC_MUL16(x2, (-7651 + FRAC_MUL16(x2, (8277 + FRAC_MUL16(-626, x2)))));
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if (x2 > 32766)
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x2 = 32766;
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return 1+x2;
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}
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#ifndef FIXED_POINT
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#define celt_sqrt(x) ((float)sqrt(x))
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#define celt_psqrt(x) ((float)sqrt(x))
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#define celt_rsqrt(x) (1.f/celt_sqrt(x))
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#define celt_rsqrt_norm(x) (celt_rsqrt(x))
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#define celt_acos acos
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#define celt_exp exp
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#define celt_cos_norm(x) (cos((.5f*M_PI)*(x)))
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#define celt_atan atan
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#define celt_rcp(x) (1.f/(x))
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#define celt_div(a,b) ((a)/(b))
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#define frac_div32(a,b) ((float)(a)/(b))
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#ifdef FLOAT_APPROX
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/* Note: This assumes radix-2 floating point with the exponent at bits 23..30 and an offset of 127
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denorm, +/- inf and NaN are *not* handled */
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/** Base-2 log approximation (log2(x)). */
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static inline float celt_log2(float x)
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{
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int integer;
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float frac;
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union {
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float f;
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celt_uint32 i;
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} in;
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in.f = x;
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integer = (in.i>>23)-127;
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in.i -= integer<<23;
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frac = in.f - 1.5f;
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frac = -0.41445418f + frac*(0.95909232f
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+ frac*(-0.33951290f + frac*0.16541097f));
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return 1+integer+frac;
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}
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/** Base-2 exponential approximation (2^x). */
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static inline float celt_exp2(float x)
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{
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int integer;
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float frac;
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union {
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float f;
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celt_uint32 i;
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} res;
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integer = floor(x);
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if (integer < -50)
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return 0;
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frac = x-integer;
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/* K0 = 1, K1 = log(2), K2 = 3-4*log(2), K3 = 3*log(2) - 2 */
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res.f = 0.99992522f + frac * (0.69583354f
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+ frac * (0.22606716f + 0.078024523f*frac));
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res.i = (res.i + (integer<<23)) & 0x7fffffff;
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return res.f;
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}
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#else
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#define celt_log2(x) (1.442695040888963387*log(x))
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#define celt_exp2(x) (exp(0.6931471805599453094*(x)))
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#endif
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#endif
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#ifdef FIXED_POINT
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#include "os_support.h"
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#ifndef OVERRIDE_CELT_ILOG2
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/** Integer log in base2. Undefined for zero and negative numbers */
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static inline celt_int16 celt_ilog2(celt_int32 x)
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{
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celt_assert2(x>0, "celt_ilog2() only defined for strictly positive numbers");
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return EC_ILOG(x)-1;
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}
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#endif
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#ifndef OVERRIDE_CELT_MAXABS16
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static inline celt_word16 celt_maxabs16(celt_word16 *x, int len)
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{
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int i;
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celt_word16 maxval = 0;
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for (i=0;i<len;i++)
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maxval = MAX16(maxval, ABS16(x[i]));
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return maxval;
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}
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#endif
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/** Integer log in base2. Defined for zero, but not for negative numbers */
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static inline celt_int16 celt_zlog2(celt_word32 x)
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{
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return x <= 0 ? 0 : celt_ilog2(x);
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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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static inline 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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/** Reciprocal sqrt approximation (Q30 input, Q0 output or equivalent) */
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static inline celt_word32 celt_rsqrt(celt_word32 x)
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{
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int k;
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k = celt_ilog2(x)>>1;
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x = VSHR32(x, (k-7)<<1);
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return PSHR32(celt_rsqrt_norm(x), k);
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}
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/** Sqrt approximation (QX input, QX/2 output) */
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static inline 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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/** Sqrt approximation (QX input, QX/2 output) that assumes that the input is
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strictly positive */
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static inline celt_word32 celt_psqrt(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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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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static inline 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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static inline celt_word16 celt_log2(celt_word32 x)
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{
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int i;
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celt_word16 n, frac;
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/* -0.41509302963303146, 0.9609890551383969, -0.31836011537636605,
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0.15530808010959576, -0.08556153059057618 */
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static const celt_word16 C[5] = {-6801+(1<<13-DB_SHIFT), 15746, -5217, 2545, -1401};
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if (x==0)
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return -32767;
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i = celt_ilog2(x);
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n = VSHR32(x,i-15)-32768-16384;
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frac = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2], MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, C[4]))))))));
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return SHL16(i-13,DB_SHIFT)+SHR16(frac,14-DB_SHIFT);
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}
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/*
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K0 = 1
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K1 = log(2)
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K2 = 3-4*log(2)
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K3 = 3*log(2) - 2
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*/
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#define D0 16383
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#define D1 22804
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#define D2 14819
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#define D3 10204
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/** Base-2 exponential approximation (2^x). (Q11 input, Q16 output) */
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static inline celt_word32 celt_exp2(celt_word16 x)
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{
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int integer;
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celt_word16 frac;
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integer = SHR16(x,11);
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if (integer>14)
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return 0x7f000000;
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else if (integer < -15)
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return 0;
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frac = SHL16(x-SHL16(integer,11),3);
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frac = ADD16(D0, MULT16_16_Q15(frac, ADD16(D1, MULT16_16_Q15(frac, ADD16(D2 , MULT16_16_Q15(D3,frac))))));
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return VSHR32(EXTEND32(frac), -integer-2);
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}
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/** Reciprocal approximation (Q15 input, Q16 output) */
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static inline 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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#define celt_div(a,b) MULT32_32_Q31((celt_word32)(a),celt_rcp(b))
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static inline 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 = 30-celt_ilog2(b);
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a = SHL32(a,shift);
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b = SHL32(b,shift);
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/* 16-bit reciprocal */
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rcp = ROUND16(celt_rcp(ROUND16(b,16)),2);
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result = SHL32(MULT16_32_Q15(rcp, a),1);
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rem = a-MULT32_32_Q31(result, b);
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result += SHL32(MULT16_32_Q15(rcp, rem),1);
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return result;
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}
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#define M1 32767
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#define M2 -21
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#define M3 -11943
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#define M4 4936
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/* Atan approximation using a 4th order polynomial. Input is in Q15 format
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and normalized by pi/4. Output is in Q15 format */
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static inline celt_word16 celt_atan01(celt_word16 x)
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{
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return MULT16_16_P15(x, ADD32(M1, MULT16_16_P15(x, ADD32(M2, MULT16_16_P15(x, ADD32(M3, MULT16_16_P15(M4, x)))))));
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}
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#undef M1
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#undef M2
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#undef M3
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#undef M4
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/* atan2() approximation valid for positive input values */
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static inline celt_word16 celt_atan2p(celt_word16 y, celt_word16 x)
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{
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if (y < x)
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{
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celt_word32 arg;
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arg = celt_div(SHL32(EXTEND32(y),15),x);
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if (arg >= 32767)
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arg = 32767;
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return SHR16(celt_atan01(EXTRACT16(arg)),1);
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} else {
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celt_word32 arg;
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arg = celt_div(SHL32(EXTEND32(x),15),y);
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if (arg >= 32767)
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arg = 32767;
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return 25736-SHR16(celt_atan01(EXTRACT16(arg)),1);
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}
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}
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#endif /* FIXED_POINT */
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#endif /* MATHOPS_H */
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