opus/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.

   - Neither the name of the Xiph.org Foundation nor the names of its
   contributors may be used to endorse or promote products derived from
   this software without specific prior written permission.

   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 = 29-celt_ilog2(b);
   a = SHL32(a,shift);
   b = SHL32(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