avfilter/af_afade: improve accuracy and speed of gain computation

Gain computation for various curves was being done in a needlessly
inaccurate fashion. Of course these are all subjective curves, but when
a curve is advertised to the user, it should be matched as closely as
possible within the limitations of libm. In particular, the constants
kept here were pretty inaccurate for double precision.

Speed improvements are mainly due to the avoidance of pow, the most
notorious of the libm functions in terms of performance. To be fair, it
is the GNU libm that is among the worst, but it is not really GNU libm's fault
since others simply yield a higher error as measured in ULP.

"Magic" constants are also accordingly documented, since they take at
least a minute of thought for a casual reader.
Reviewed-by: NPaul B Mahol <onemda@gmail.com>
Signed-off-by: NGanesh Ajjanagadde <gajjanagadde@gmail.com>

libavfilter/af_afade.c

@@ -92,6 +92,7 @@ static int query_formats(AVFilterContext *ctx)
 
 static double fade_gain(int curve, int64_t index, int range)
 {
+#define CUBE(a) ((a)*(a)*(a))
     double gain;
 
     gain = av_clipd(1.0 * index / range, 0, 1.0);
@@ -101,22 +102,25 @@ static double fade_gain(int curve, int64_t index, int range)
         gain = sin(gain * M_PI / 2.0);
         break;
     case IQSIN:
-        gain = 0.636943 * asin(gain);
+        /* 0.6... = 2 / M_PI */
+        gain = 0.6366197723675814 * asin(gain);
         break;
     case ESIN:
-        gain = 1.0 - cos(M_PI / 4.0 * (pow(2.0*gain - 1, 3) + 1));
+        gain = 1.0 - cos(M_PI / 4.0 * (CUBE(2.0*gain - 1) + 1));
         break;
     case HSIN:
         gain = (1.0 - cos(gain * M_PI)) / 2.0;
         break;
     case IHSIN:
-        gain = 0.318471 * acos(1 - 2 * gain);
+        /* 0.3... = 1 / M_PI */
+        gain = 0.3183098861837907 * acos(1 - 2 * gain);
         break;
     case EXP:
-        gain = pow(0.1, (1 - gain) * 5.0);
+        /* -11.5... = 5*ln(0.1) */
+        gain = exp(-11.512925464970227 * (1 - gain));
         break;
     case LOG:
-        gain = av_clipd(0.0868589 * log(100000 * gain), 0, 1.0);
+        gain = av_clipd(1 + 0.2 * log10(gain), 0, 1.0);
         break;
     case PAR:
         gain = 1 - sqrt(1 - gain);
@@ -128,7 +132,7 @@ static double fade_gain(int curve, int64_t index, int range)
         gain *= gain;
         break;
     case CUB:
-        gain = gain * gain * gain;
+        gain = CUBE(gain);
         break;
     case SQU:
         gain = sqrt(gain);
@@ -137,10 +141,10 @@ static double fade_gain(int curve, int64_t index, int range)
         gain = cbrt(gain);
         break;
     case DESE:
-        gain = gain <= 0.5 ? pow(2 * gain, 1/3.) / 2: 1 - pow(2 * (1 - gain), 1/3.) / 2;
+        gain = gain <= 0.5 ? cbrt(2 * gain) / 2: 1 - cbrt(2 * (1 - gain)) / 2;
         break;
     case DESI:
-        gain = gain <= 0.5 ? pow(2 * gain, 3) / 2: 1 - pow(2 * (1 - gain), 3) / 2;
+        gain = gain <= 0.5 ? CUBE(2 * gain) / 2: 1 - CUBE(2 * (1 - gain)) / 2;
         break;
     }