# 
# Fast discrete cosine transform algorithms (Python)
# 
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# https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms
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import cmath, fft


# DCT type II, unscaled
def transform(vector: list[float]) -> list[float]:
	temp0: list[float] = vector[ : : 2] + vector[-1 - len(vector) % 2 : : -2]
	temp1: list[complex] = fft.transform(temp0, False)
	factor: complex = -1j * cmath.pi / (len(vector) * 2)
	return [(val * cmath.exp(i * factor)).real for (i, val) in enumerate(temp1)]


# DCT type III, unscaled
def inverse_transform(vector: list[float]) -> list[float]:
	n: int = len(vector)
	factor: complex = -1j * cmath.pi / (len(vector) * 2)
	temp0: list[complex] = [(val if i > 0 else val / 2) * cmath.exp(i * factor)
		for (i, val) in enumerate(vector)]
	temp0 = fft.transform(temp0, False)
	
	temp1: list[float] = [val.real for val in temp0]
	result: list[float] = [cmath.nan] * n
	result[ : : 2] = temp1[ : (n + 1) // 2]
	result[-1 - len(vector) % 2 : : -2] = temp1[(n + 1) // 2 : ]
	return result