michael@0: /* Copyright (c) 2002-2008 Jean-Marc Valin
michael@0:    Copyright (c) 2007-2008 CSIRO
michael@0:    Copyright (c) 2007-2009 Xiph.Org Foundation
michael@0:    Written by Jean-Marc Valin */
michael@0: /**
michael@0:    @file mathops.h
michael@0:    @brief Various math functions
michael@0: */
michael@0: /*
michael@0:    Redistribution and use in source and binary forms, with or without
michael@0:    modification, are permitted provided that the following conditions
michael@0:    are met:
michael@0: 
michael@0:    - Redistributions of source code must retain the above copyright
michael@0:    notice, this list of conditions and the following disclaimer.
michael@0: 
michael@0:    - Redistributions in binary form must reproduce the above copyright
michael@0:    notice, this list of conditions and the following disclaimer in the
michael@0:    documentation and/or other materials provided with the distribution.
michael@0: 
michael@0:    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
michael@0:    ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
michael@0:    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
michael@0:    A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
michael@0:    OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
michael@0:    EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
michael@0:    PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
michael@0:    PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
michael@0:    LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
michael@0:    NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
michael@0:    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
michael@0: */
michael@0: 
michael@0: #ifdef HAVE_CONFIG_H
michael@0: #include "config.h"
michael@0: #endif
michael@0: 
michael@0: #include "mathops.h"
michael@0: 
michael@0: /*Compute floor(sqrt(_val)) with exact arithmetic.
michael@0:   This has been tested on all possible 32-bit inputs.*/
michael@0: unsigned isqrt32(opus_uint32 _val){
michael@0:   unsigned b;
michael@0:   unsigned g;
michael@0:   int      bshift;
michael@0:   /*Uses the second method from
michael@0:      http://www.azillionmonkeys.com/qed/sqroot.html
michael@0:     The main idea is to search for the largest binary digit b such that
michael@0:      (g+b)*(g+b) <= _val, and add it to the solution g.*/
michael@0:   g=0;
michael@0:   bshift=(EC_ILOG(_val)-1)>>1;
michael@0:   b=1U<<bshift;
michael@0:   do{
michael@0:     opus_uint32 t;
michael@0:     t=(((opus_uint32)g<<1)+b)<<bshift;
michael@0:     if(t<=_val){
michael@0:       g+=b;
michael@0:       _val-=t;
michael@0:     }
michael@0:     b>>=1;
michael@0:     bshift--;
michael@0:   }
michael@0:   while(bshift>=0);
michael@0:   return g;
michael@0: }
michael@0: 
michael@0: #ifdef FIXED_POINT
michael@0: 
michael@0: opus_val32 frac_div32(opus_val32 a, opus_val32 b)
michael@0: {
michael@0:    opus_val16 rcp;
michael@0:    opus_val32 result, rem;
michael@0:    int shift = celt_ilog2(b)-29;
michael@0:    a = VSHR32(a,shift);
michael@0:    b = VSHR32(b,shift);
michael@0:    /* 16-bit reciprocal */
michael@0:    rcp = ROUND16(celt_rcp(ROUND16(b,16)),3);
michael@0:    result = MULT16_32_Q15(rcp, a);
michael@0:    rem = PSHR32(a,2)-MULT32_32_Q31(result, b);
michael@0:    result = ADD32(result, SHL32(MULT16_32_Q15(rcp, rem),2));
michael@0:    if (result >= 536870912)       /*  2^29 */
michael@0:       return 2147483647;          /*  2^31 - 1 */
michael@0:    else if (result <= -536870912) /* -2^29 */
michael@0:       return -2147483647;         /* -2^31 */
michael@0:    else
michael@0:       return SHL32(result, 2);
michael@0: }
michael@0: 
michael@0: /** Reciprocal sqrt approximation in the range [0.25,1) (Q16 in, Q14 out) */
michael@0: opus_val16 celt_rsqrt_norm(opus_val32 x)
michael@0: {
michael@0:    opus_val16 n;
michael@0:    opus_val16 r;
michael@0:    opus_val16 r2;
michael@0:    opus_val16 y;
michael@0:    /* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
michael@0:    n = x-32768;
michael@0:    /* Get a rough initial guess for the root.
michael@0:       The optimal minimax quadratic approximation (using relative error) is
michael@0:        r = 1.437799046117536+n*(-0.823394375837328+n*0.4096419668459485).
michael@0:       Coefficients here, and the final result r, are Q14.*/
michael@0:    r = ADD16(23557, MULT16_16_Q15(n, ADD16(-13490, MULT16_16_Q15(n, 6713))));
michael@0:    /* We want y = x*r*r-1 in Q15, but x is 32-bit Q16 and r is Q14.
michael@0:       We can compute the result from n and r using Q15 multiplies with some
michael@0:        adjustment, carefully done to avoid overflow.
michael@0:       Range of y is [-1564,1594]. */
michael@0:    r2 = MULT16_16_Q15(r, r);
michael@0:    y = SHL16(SUB16(ADD16(MULT16_16_Q15(r2, n), r2), 16384), 1);
michael@0:    /* Apply a 2nd-order Householder iteration: r += r*y*(y*0.375-0.5).
michael@0:       This yields the Q14 reciprocal square root of the Q16 x, with a maximum
michael@0:        relative error of 1.04956E-4, a (relative) RMSE of 2.80979E-5, and a
michael@0:        peak absolute error of 2.26591/16384. */
michael@0:    return ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
michael@0:               SUB16(MULT16_16_Q15(y, 12288), 16384))));
michael@0: }
michael@0: 
michael@0: /** Sqrt approximation (QX input, QX/2 output) */
michael@0: opus_val32 celt_sqrt(opus_val32 x)
michael@0: {
michael@0:    int k;
michael@0:    opus_val16 n;
michael@0:    opus_val32 rt;
michael@0:    static const opus_val16 C[5] = {23175, 11561, -3011, 1699, -664};
michael@0:    if (x==0)
michael@0:       return 0;
michael@0:    else if (x>=1073741824)
michael@0:       return 32767;
michael@0:    k = (celt_ilog2(x)>>1)-7;
michael@0:    x = VSHR32(x, 2*k);
michael@0:    n = x-32768;
michael@0:    rt = ADD16(C[0], MULT16_16_Q15(n, ADD16(C[1], MULT16_16_Q15(n, ADD16(C[2],
michael@0:               MULT16_16_Q15(n, ADD16(C[3], MULT16_16_Q15(n, (C[4])))))))));
michael@0:    rt = VSHR32(rt,7-k);
michael@0:    return rt;
michael@0: }
michael@0: 
michael@0: #define L1 32767
michael@0: #define L2 -7651
michael@0: #define L3 8277
michael@0: #define L4 -626
michael@0: 
michael@0: static OPUS_INLINE opus_val16 _celt_cos_pi_2(opus_val16 x)
michael@0: {
michael@0:    opus_val16 x2;
michael@0: 
michael@0:    x2 = MULT16_16_P15(x,x);
michael@0:    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
michael@0:                                                                                 ))))))));
michael@0: }
michael@0: 
michael@0: #undef L1
michael@0: #undef L2
michael@0: #undef L3
michael@0: #undef L4
michael@0: 
michael@0: opus_val16 celt_cos_norm(opus_val32 x)
michael@0: {
michael@0:    x = x&0x0001ffff;
michael@0:    if (x>SHL32(EXTEND32(1), 16))
michael@0:       x = SUB32(SHL32(EXTEND32(1), 17),x);
michael@0:    if (x&0x00007fff)
michael@0:    {
michael@0:       if (x<SHL32(EXTEND32(1), 15))
michael@0:       {
michael@0:          return _celt_cos_pi_2(EXTRACT16(x));
michael@0:       } else {
michael@0:          return NEG32(_celt_cos_pi_2(EXTRACT16(65536-x)));
michael@0:       }
michael@0:    } else {
michael@0:       if (x&0x0000ffff)
michael@0:          return 0;
michael@0:       else if (x&0x0001ffff)
michael@0:          return -32767;
michael@0:       else
michael@0:          return 32767;
michael@0:    }
michael@0: }
michael@0: 
michael@0: /** Reciprocal approximation (Q15 input, Q16 output) */
michael@0: opus_val32 celt_rcp(opus_val32 x)
michael@0: {
michael@0:    int i;
michael@0:    opus_val16 n;
michael@0:    opus_val16 r;
michael@0:    celt_assert2(x>0, "celt_rcp() only defined for positive values");
michael@0:    i = celt_ilog2(x);
michael@0:    /* n is Q15 with range [0,1). */
michael@0:    n = VSHR32(x,i-15)-32768;
michael@0:    /* Start with a linear approximation:
michael@0:       r = 1.8823529411764706-0.9411764705882353*n.
michael@0:       The coefficients and the result are Q14 in the range [15420,30840].*/
michael@0:    r = ADD16(30840, MULT16_16_Q15(-15420, n));
michael@0:    /* Perform two Newton iterations:
michael@0:       r -= r*((r*n)-1.Q15)
michael@0:          = r*((r*n)+(r-1.Q15)). */
michael@0:    r = SUB16(r, MULT16_16_Q15(r,
michael@0:              ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768))));
michael@0:    /* We subtract an extra 1 in the second iteration to avoid overflow; it also
michael@0:        neatly compensates for truncation error in the rest of the process. */
michael@0:    r = SUB16(r, ADD16(1, MULT16_16_Q15(r,
michael@0:              ADD16(MULT16_16_Q15(r, n), ADD16(r, -32768)))));
michael@0:    /* r is now the Q15 solution to 2/(n+1), with a maximum relative error
michael@0:        of 7.05346E-5, a (relative) RMSE of 2.14418E-5, and a peak absolute
michael@0:        error of 1.24665/32768. */
michael@0:    return VSHR32(EXTEND32(r),i-16);
michael@0: }
michael@0: 
michael@0: #endif