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Accuracy improvements to the fixed-point celt_rsqrt().

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Timothy B. Terriberry 2009-10-20 02:39:45 -04:00 • committed by Jean-Marc Valin
parent a3bba38b49
commit 4a275d4d8f
1 changed files with 22 additions and 5 deletions
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27
libcelt/mathops.h
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@ -190,15 +190,32 @@ static inline celt_word32 celt_rsqrt(celt_word32 x)
{
int k;
celt_word16 n;
celt_word16 r;
celt_word16 r2;
celt_word16 y;
celt_word32 rt;
const celt_word16 C[5] = {23126, -11496, 9812, -9097, 4100};
k = celt_ilog2(x)>>1;
x = VSHR32(x, (k-7)<<1);
/* Range of n is [-16384,32767] */
/* Range of n is [-16384,32767] ([-0.5,1) in Q15). */
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,k);
/* Get a rough initial guess for the root.
The optimal minimax quadratic approximation is
r = 1.4288615575712422-n*(0.8452316405039975+n*0.4519141640876117).
Coefficients here, and the final result r, are Q14.*/
r = ADD16(23410, MULT16_16_Q15(n, ADD16(-13848, MULT16_16_Q15(n, 7405))));
/* 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 [-2014,2362]. */
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*(1+y*(-0.5+y*0.375)). */
rt = ADD16(r, MULT16_16_Q15(r, MULT16_16_Q15(y,
ADD16(-16384, MULT16_16_Q15(y, 12288)))));
/* rt is now the Q14 reciprocal square root of the Q16 x, with a maximum
error of 2.70970/16384 and a MSE of 0.587003/16384^2. */
/* Most of the error in this approximation comes from the following shift: */
rt = PSHR32(rt,k);
return rt;
}