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      1 /* Copyright (c) 2002-2008 Jean-Marc Valin
      2    Copyright (c) 2007-2008 CSIRO
      3    Copyright (c) 2007-2009 Xiph.Org Foundation
      4    Written by Jean-Marc Valin */
      5 /**
      6    @file mathops.h
      7    @brief Various math functions
      8 */
      9 /*
     10    Redistribution and use in source and binary forms, with or without
     11    modification, are permitted provided that the following conditions
     12    are met:
     13 
     14    - Redistributions of source code must retain the above copyright
     15    notice, this list of conditions and the following disclaimer.
     16 
     17    - Redistributions in binary form must reproduce the above copyright
     18    notice, this list of conditions and the following disclaimer in the
     19    documentation and/or other materials provided with the distribution.
     20 
     21    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
     22    ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
     23    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
     24    A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
     25    OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
     26    EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
     27    PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
     28    PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
     29    LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
     30    NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
     31    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
     32 */
     33 
     34 #ifdef HAVE_CONFIG_H
     35 #include "config.h"
     36 #endif
     37 
     38 #include "mathops.h"
     39 
     40 /*Compute floor(sqrt(_val)) with exact arithmetic.
     41   This has been tested on all possible 32-bit inputs.*/
     42 unsigned isqrt32(opus_uint32 _val){
     43   unsigned b;
     44   unsigned g;
     45   int      bshift;
     46   /*Uses the second method from
     47      http://www.azillionmonkeys.com/qed/sqroot.html
     48     The main idea is to search for the largest binary digit b such that
     49      (g+b)*(g+b) <= _val, and add it to the solution g.*/
     50   g=0;
     51   bshift=(EC_ILOG(_val)-1)>>1;
     52   b=1U<<bshift;
     53   do{
     54     opus_uint32 t;
     55     t=(((opus_uint32)g<<1)+b)<<bshift;
     56     if(t<=_val){
     57       g+=b;
     58       _val-=t;
     59     }
     60     b>>=1;
     61     bshift--;
     62   }
     63   while(bshift>=0);
     64   return g;
     65 }
     66 
     67 #ifdef FIXED_POINT
     68 
     69 opus_val32 frac_div32(opus_val32 a, opus_val32 b)
     70 {
     71    opus_val16 rcp;
     72    opus_val32 result, rem;
     73    int shift = celt_ilog2(b)-29;
     74    a = VSHR32(a,shift);
     75    b = VSHR32(b,shift);
     76    /* 16-bit reciprocal */
     77    rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
     78    result = MULT16_32_Q15(rcp, a);
     79    rem = PSHR32(a,2)-MULT32_32_Q31(result, b);
     80    result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2));
     81    if (result >= 536870912)       /*  2^29 */
     82       return 2147483647;          /*  2^31 - 1 */
     83    else if (result <= -536870912) /* -2^29 */
     84       return -2147483647;         /* -2^31 */
     85    else
     86       return SHL32(result, 2);
     87 }
     88 
     89 /** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
     90 opus_val16 celt_rsqrt_norm(opus_val32 x)
     91 {
     92    opus_val16 n;
     93    opus_val16 r;
     94    opus_val16 r2;
     95    opus_val16 y;
     96    /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
     97    n = x-32768;
     98    /* Get a rough initial guess for the root.
     99       The optimal minimax quadratic approximation (using relative error) is
    100        r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
    101       Coefficients here, and the final result r, are Q14.*/
    102    r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
    103    /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
    104       We can compute the result from n and r using Q15 multiplies with some
    105        adjustment, carefully done to avoid overflow.
    106       Range of y is [-1564,1594]. */
    107    r2 = MULT16_16_Q15(r, r);
    108    y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
    109    /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
    110       This yields the Q14 reciprocal square root of the Q16 x, with a maximum
    111        relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
    112        peak absolute error of 2.26591/16384. */
    113    return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
    114               SUB16(MULT16_16_Q15(y, 12288), 16384))));
    115 }
    116 
    117 /** Sqrt approximation (QX input, QX/2 output) */
    118 opus_val32 celt_sqrt(opus_val32 x)
    119 {
    120    int k;
    121    opus_val16 n;
    122    opus_val32 rt;
    123    static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664};
    124    if (x==0)
    125       return 0;
    126    else if (x>=1073741824)
    127       return 32767;
    128    k = (celt_ilog2(x)>>1)-7;
    129    x = VSHR32(x, 2*k);
    130    n = x-32768;
    131    rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2],
    132               MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4])))))))));
    133    rt = VSHR32(rt,7-k);
    134    return rt;
    135 }
    136 
    137 #define L1 32767
    138 #define L2 -7651
    139 #define L3 8277
    140 #define L4 -626
    141 
    142 static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x)
    143 {
    144    opus_val16 x2;
    145 
    146    x2 = MULT16_16_P15(x,x);
    147    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
    148                                                                                 ))))))));
    149 }
    150 
    151 #undef L1
    152 #undef L2
    153 #undef L3
    154 #undef L4
    155 
    156 opus_val16 celt_cos_norm(opus_val32 x)
    157 {
    158    x = x&0x0001ffff;
    159    if (x>SHL32(EXTEND32(1), 16))
    160       x = SUB32(SHL32(EXTEND32(1), 17),x);
    161    if (x&0x00007fff)
    162    {
    163       if (x<SHL32(EXTEND32(1), 15))
    164       {
    165          return _celt_cos_pi_2(EXTRACT16(x));
    166       } else {
    167          return NEG32(_celt_cos_pi_2(EXTRACT16(65536-x)));
    168       }
    169    } else {
    170       if (x&0x0000ffff)
    171          return 0;
    172       else if (x&0x0001ffff)
    173          return -32767;
    174       else
    175          return 32767;
    176    }
    177 }
    178 
    179 /** Reciprocal approximation (Q15 input, Q16 output) */
    180 opus_val32 celt_rcp(opus_val32 x)
    181 {
    182    int i;
    183    opus_val16 n;
    184    opus_val16 r;
    185    celt_assert2(x>0, "celt_rcp() only defined for positive values");
    186    i = celt_ilog2(x);
    187    /* n is Q15 with range [0,1). */
    188    n = VSHR32(x,i-15)-32768;
    189    /* Start with a linear approximation:
    190       r = 1.8823529411764706-0.9411764705882353*n.
    191       The coefficients and the result are Q14 in the range [15420,30840].*/
    192    r = ADD16(30840, MULT16_16_Q15(-15420, n));
    193    /* Perform two Newton iterations:
    194       r -= r*((r*n)-1.Q15)
    195          = r*((r*n)+(r-1.Q15)). */
    196    r = SUB16(r, MULT16_16_Q15(r,
    197              ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
    198    /* We subtract an extra 1 in the second iteration to avoid overflow; it also
    199        neatly compensates for truncation error in the rest of the process. */
    200    r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
    201              ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
    202    /* r is now the Q15 solution to 2/(n+1), with a maximum relative error
    203        of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
    204        error of 1.24665/32768. */
    205    return VSHR32(EXTEND32(r),i-16);
    206 }
    207 
    208 #endif
    209