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/* Copyright (c) 2002-2008 Jean-Marc Valin
Copyright (c) 2007-2008 CSIRO
Copyright (c) 2007-2009 Xiph.Org Foundation
Written by Jean-Marc Valin */
/**
@file mathops.h
@brief Various math functions
*/
/*
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
- Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#ifdef HAVE_CONFIG_H
#include "config.h"
#endif
#include "mathops.h"
/*Compute floor(sqrt(_val)) with exact arithmetic.
_val must be greater than 0.
This has been tested on all possible 32-bit inputs greater than 0.*/
unsigned isqrt32(opus_uint32 _val){
unsigned b;
unsigned g;
int bshift;
/*Uses the second method from
http://www.azillionmonkeys.com/qed/sqroot.html
The main idea is to search for the largest binary digit b such that
(g+b)*(g+b) <= _val, and add it to the solution g.*/
g=0;
bshift=(EC_ILOG(_val)-1)>>1;
b=1U<<bshift;
do{
opus_uint32 t;
t=(((opus_uint32)g<<1)+b)<<bshift;
if(t<=_val){
g+=b;
_val-=t;
}
b>>=1;
bshift--;
}
while(bshift>=0);
return g;
}
#ifdef FIXED_POINT
opus_val32 frac_div32(opus_val32 a, opus_val32 b)
{
opus_val16 rcp;
opus_val32 result, rem;
int shift = celt_ilog2(b)-29;
a = VSHR32(a,shift);
b = VSHR32(b,shift);
/* 16-bit reciprocal */
rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
result = MULT16_32_Q15(rcp, a);
rem = PSHR32(a,2)-MULT32_32_Q31(result, b);
result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2));
if (result >= 536870912) /* 2^29 */
return 2147483647; /* 2^31 - 1 */
else if (result <= -536870912) /* -2^29 */
return -2147483647; /* -2^31 */
else
return SHL32(result, 2);
}
/** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
opus_val16 celt_rsqrt_norm(opus_val32 x)
{
opus_val16 n;
opus_val16 r;
opus_val16 r2;
opus_val16 y;
/* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
n = x-32768;
/* Get a rough initial guess for the root.
The optimal minimax quadratic approximation (using relative error) is
r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
Coefficients here, and the final result r, are Q14.*/
r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
/* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
We can compute the result from n and r using Q15 multiplies with some
adjustment, carefully done to avoid overflow.
Range of y is [-1564,1594]. */
r2 = MULT16_16_Q15(r, r);
y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
/* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
This yields the Q14 reciprocal square root of the Q16 x, with a maximum
relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
peak absolute error of 2.26591/16384. */
return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
SUB16(MULT16_16_Q15(y, 12288), 16384))));
}
/** Sqrt approximation (QX input, QX/2 output) */
opus_val32 celt_sqrt(opus_val32 x)
{
int k;
opus_val16 n;
opus_val32 rt;
static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664};
if (x==0)
return 0;
else if (x>=1073741824)
return 32767;
k = (celt_ilog2(x)>>1)-7;
x = VSHR32(x, 2*k);
n = x-32768;
rt = 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])))))))));
rt = VSHR32(rt,7-k);
return rt;
}
#define L1 32767
#define L2 -7651
#define L3 8277
#define L4 -626
static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x)
{
opus_val16 x2;
x2 = MULT16_16_P15(x,x);
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
))))))));
}
#undef L1
#undef L2
#undef L3
#undef L4
opus_val16 celt_cos_norm(opus_val32 x)
{
x = x&0x0001ffff;
if (x>SHL32(EXTEND32(1), 16))
x = SUB32(SHL32(EXTEND32(1), 17),x);
if (x&0x00007fff)
{
if (x<SHL32(EXTEND32(1), 15))
{
return _celt_cos_pi_2(EXTRACT16(x));
} else {
return NEG16(_celt_cos_pi_2(EXTRACT16(65536-x)));
}
} else {
if (x&0x0000ffff)
return 0;
else if (x&0x0001ffff)
return -32767;
else
return 32767;
}
}
/** Reciprocal approximation (Q15 input, Q16 output) */
opus_val32 celt_rcp(opus_val32 x)
{
int i;
opus_val16 n;
opus_val16 r;
celt_sig_assert(x>0);
i = celt_ilog2(x);
/* n is Q15 with range [0,1). */
n = VSHR32(x,i-15)-32768;
/* Start with a linear approximation:
r = 1.8823529411764706-0.9411764705882353*n.
The coefficients and the result are Q14 in the range [15420,30840].*/
r = ADD16(30840, MULT16_16_Q15(-15420, n));
/* Perform two Newton iterations:
r -= r*((r*n)-1.Q15)
= r*((r*n)+(r-1.Q15)). */
r = SUB16(r, MULT16_16_Q15(r,
ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
/* We subtract an extra 1 in the second iteration to avoid overflow; it also
neatly compensates for truncation error in the rest of the process. */
r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
/* r is now the Q15 solution to 2/(n+1), with a maximum relative error
of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
error of 1.24665/32768. */
return VSHR32(EXTEND32(r),i-16);
}
#endif
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