Rework coarse energy coding.

This changes how the PDF used to code coarse energy.
New features:
1) The probability of 0 (p0) is now indepedent of the decay rate
 of the remaining values; this additional flexibility will allow
 us to model the actual distribution better, though that
 improvement is not part of this patch.
2) There is a guaranteed minimum number of encodable energy
    deltas.
   This ensures that even the most extreme sudden volume changes
    can be accurately represented.
3) The tail end of the distribution has an adjustable (through a
    constant in the code) minimum probability.
   This allows us to lower the worst-case bit cost of a single
    delta.
4) The codebook is interleaved as 0, -1, +1, -2, +2, ... instead
    of the 0, +1, -1, +2, -2, ... order used before (see 5).
5) There is no restriction that p0 be even.
   Any remaining, unused part of the code is assigned to an
    additional negative value (collected inter data suggests that
    very large negative deltas are more common than very large
    positive ones).
   If the minimum probability is greater than 1, then an
    additional positive delta with a smaller probablity may also
    be added.
6) Once the tail of the distribution is reached, the energy delta
    is computed directly, instead of continuing to loop through
    the codebook.
   This reduces the worst-case computational cost.

libcelt/laplace.c

@@ -35,56 +35,69 @@
 #endif
 
 #include "laplace.h"
+#include "mathops.h"
+
+/* The minimum probability of an energy delta (out of 32768). */
+#define LAPLACE_LOG_MINP (0)
+#define LAPLACE_MINP (1<<LAPLACE_LOG_MINP)
+/* The minimum number of guaranteed representable energy deltas (in one
+    direction). */
+#define LAPLACE_NMIN (16)
 
 int ec_laplace_get_start_freq(int decay)
 {
-   int fs = (((ec_uint32)32768)*(16384-decay))/(16384+decay);
-   /* Making fs even so we're sure that all the range is used for +/- values */
-   fs -= (fs&1);
-   return fs;
+   celt_uint32 ft = 32768 - LAPLACE_MINP*(2*LAPLACE_NMIN+1);
+   int fs = (ft*(16384-decay))/(16384+decay);
+   return fs+LAPLACE_MINP;
+}
+
+static int ec_laplace_get_freq1(int fs0, int decay)
+{
+   celt_int32 ft;
+   ft = 32768 - LAPLACE_MINP*(2*LAPLACE_NMIN) - fs0;
+   return ft*(16384-decay)>>15;
 }
 
 void ec_laplace_encode_start(ec_enc *enc, int *value, int decay, int fs)
 {
-   int i;
-   int fl;
-   unsigned int ft;
-   int s = 0;
+   unsigned fl;
    int val = *value;
-   if (val < 0)
-   {
-      s = 1;
-      val = -val;
-   }
-   ft = 32768;
-   fl = -fs;
-   for (i=0;i<val;i++)
+   fl = 0;
+   if (val)
    {
-      int tmp_l, tmp_s;
-      tmp_l = fl;
-      tmp_s = fs;
-      fl += fs*2;
-      fs = (fs*(ec_int32)decay)>>14;
-      if (fs == 0)
+      int s;
+      int i;
+      s = -(val<0);
+      val = val+s^s;
+      fl = fs;
+      fs = ec_laplace_get_freq1(fs, decay);
+      /* Search the decaying part of the PDF.*/
+      for (i=1; fs > 0 && i < val; i++)
+      {
+         fs *= 2;
+         fl += fs+2*LAPLACE_MINP;
+         fs = (fs*(celt_int32)decay)>>15;
+      }
+      /* Everything beyond that has probability LAPLACE_MINP. */
+      if (fs <= 0)
+      {
+         int di;
+         int di_max;
+         di_max = (32768-fl+LAPLACE_MINP-1)>>LAPLACE_LOG_MINP;
+         di_max = (di_max-s)>>1;
+         di = IMIN(val - i, di_max);
+         fl += (2*di+1+s)*LAPLACE_MINP;
+         fs = IMIN(LAPLACE_MINP, 32768-fl);
+         *value = i+di+s^s;
+      }
+      else
       {
-         if (fl+2 <= ft)
-         {
-            fs = 1;
-         } else {
-            fs = tmp_s;
-            fl = tmp_l;
-            if (s)
-               *value = -i;
-            else
-               *value = i;
-            break;
-         }
+         fs += LAPLACE_MINP;
+         fl += fs&~s;
       }
+      celt_assert(fl+fs<=32768);
+      celt_assert(fs>0);
    }
-   if (fl < 0)
-      fl = 0;
-   if (s)
-      fl += fs;
    ec_encode_bin(enc, fl, fl+fs, 15);
 }
 
@@ -99,37 +112,42 @@ void ec_laplace_encode(ec_enc *enc, int *value, int decay)
 int ec_laplace_decode_start(ec_dec *dec, int decay, int fs)
 {
    int val=0;
-   int fl, fh, fm;
-   unsigned int ft;
-   fl = 0;
-   ft = 32768;
-   fh = fs;
+   unsigned fl;
+   int fm;
    fm = ec_decode_bin(dec, 15);
-   while (fm >= fh && fs != 0)
+   fl = 0;
+   if (fm >= fs)
    {
-      fl = fh;
-      fs = (fs*(ec_int32)decay)>>14;
-      if (fs == 0 && fh+2 <= ft)
+      val++;
+      fl = fs;
+      fs = ec_laplace_get_freq1(fs, decay)+LAPLACE_MINP;
+      /* Search the decaying part of the PDF.*/
+      while(fs > LAPLACE_MINP && fm >= fl+2*fs)
       {
-         fs = 1;
+         fs *= 2;
+         fl += fs;
+         fs = ((fs-2*LAPLACE_MINP)*(celt_int32)decay)>>15;
+         fs += LAPLACE_MINP;
+         val++;
       }
-      fh += fs*2;
-      val++;
-   }
-   if (fl>0)
-   {
-      if (fm >= fl+fs)
+      /* Everything beyond that has probability LAPLACE_MINP. */
+      if (fs <= LAPLACE_MINP)
       {
+         int di;
+         di = (fm-fl)>>(LAPLACE_LOG_MINP+1);
+         val += di;
+         fl += 2*di*LAPLACE_MINP;
+      }
+      if (fm < fl+fs)
          val = -val;
+      else
          fl += fs;
-      } else {
-         fh -= fs;
-      }
    }
-   /* Preventing an infinite loop in case something screws up in the decoding */
-   if (fl==fh)
-      fl--;
-   ec_dec_update(dec, fl, fh, ft);
+   celt_assert(fl<32768);
+   celt_assert(fs>0);
+   celt_assert(fl<=fm);
+   celt_assert(fm<IMIN(fl+fs,32768));
+   ec_dec_update(dec, fl, IMIN(fl+fs,32768), 32768);
    return val;
 }