Add an "Autosync Tempo" feature. This uses a linear Least Squares

Regression to figure out a line through the user's step errors to minimize
that error.  It then uses the slope of that line to fix the tempo and the
intercept to fix the offset.  Stops are adjusted as if they were originally
calculated as a number of beats, ie the slope is also used to change
the stops.

Also fixed is a bug where the user doesn't get a chance to keep or
reject the sync changes when ESCing (ZZing) from a song.

R=glenn

stepmania/src/RageUtil.cpp

@@ -947,6 +947,57 @@ float calc_stddev( const float *pStart, const float *pEnd, bool bSample )
 	return fDev;
 }
 
+bool CalcLeastSquares( const vector< pair<float, float> > &vCoordinates,
+                       float *pfSlope, float *pfIntercept, float *pfError )
+{
+	ASSERT( pfSlope != NULL );
+	ASSERT( pfIntercept != NULL );
+	int iNumSamples = vCoordinates.size();
+	float fNumSamples = static_cast<float>(iNumSamples);
+	if( iNumSamples == 0 ) 
+	{
+		return false;
+	}
+	float fSumXX = 0.0f, fSumXY = 0.0f, fSumX = 0.0f, fSumY = 0.0f;
+	for( int i = 0; i < iNumSamples; ++i ) 
+	{
+		fSumXX += vCoordinates[i].first * vCoordinates[i].first;
+		fSumXY += vCoordinates[i].first * vCoordinates[i].second;
+		fSumX += vCoordinates[i].first;
+		fSumY += vCoordinates[i].second;
+	}
+	float fDenominator = fNumSamples * fSumXX - fSumX * fSumX;
+	*pfSlope = (fNumSamples * fSumXY - fSumX * fSumY) / fDenominator;
+	*pfIntercept = (fSumXX * fSumY - fSumX * fSumXY) / fDenominator;
+
+	*pfError = 0.0f;
+	for( int i = 0; i < iNumSamples; ++i ) 
+	{
+		float fOneError = (vCoordinates[i].second - (*pfIntercept + *pfSlope * vCoordinates[i].first));
+		*pfError += fOneError * fOneError;
+	}
+	*pfError /= fNumSamples;
+	*pfError = sqrtf( *pfError );
+	return true;
+}
+
+void FilterHighErrorPoints( vector< pair<float, float> > *vCoordinates,
+                            float fSlope, float fIntercept, float fCutoff )
+{
+	unsigned int iOut = 0;
+	for( unsigned int iIn = 0; iIn < vCoordinates->size(); ++iIn )
+	{
+		float fError = ((*vCoordinates)[iIn].second - 
+		                (fIntercept + fSlope * (*vCoordinates)[iIn].first));
+		if( fabsf(fError) < fCutoff )
+		{
+			(*vCoordinates)[iOut] = (*vCoordinates)[iIn];
+			++iOut;
+		}
+	}
+	vCoordinates->resize(iOut);
+}
+
 void TrimLeft( RString &sStr, const char *s )
 {
 	int n = 0;