ref: 730fa4c5de29be500f4fbf041386b473d4319a3b
dir: /libcelt/mathops.c/
/* 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 FOUNDATION 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.
This has been tested on all possible 32-bit inputs.*/
unsigned isqrt32(celt_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{
celt_uint32 t;
t=((celt_uint32)g<<1)+b<<bshift;
if(t<=_val){
g+=b;
_val-=t;
}
b>>=1;
bshift--;
}
while(bshift>=0);
return g;
}
#ifdef FIXED_POINT
celt_word32 frac_div32(celt_word32 a, celt_word32 b)
{
celt_word16 rcp;
celt_word32 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 = SHL32(MULT16_32_Q15(rcp, a),2);
rem = a-MULT32_32_Q31(result, b);
result += SHL32(MULT16_32_Q15(rcp, rem),2);
return result;
}
/** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
celt_word16 celt_rsqrt_norm(celt_word32 x)
{
celt_word16 n;
celt_word16 r;
celt_word16 r2;
celt_word16 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) */
celt_word32 celt_sqrt(celt_word32 x)
{
int k;
celt_word16 n;
celt_word32 rt;
static const celt_word16 C[5] = {23175, 11561, -3011, 1699, -664};
if (x==0)
return 0;
k = (celt_ilog2(x)>>1)-7;
x = VSHR32(x, (k<<1));
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 inline celt_word16 _celt_cos_pi_2(celt_word16 x)
{
celt_word16 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
celt_word16 celt_cos_norm(celt_word32 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 NEG32(_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) */
celt_word32 celt_rcp(celt_word32 x)
{
int i;
celt_word16 n;
celt_word16 r;
celt_assert2(x>0, "celt_rcp() only defined for positive values");
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