# 
# Fast discrete cosine transform algorithms (Python)
# 
# Copyright (c) 2020 Project Nayuki. (MIT License)
# https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms
# 
# Permission is hereby granted, free of charge, to any person obtaining a copy of
# this software and associated documentation files (the "Software"), to deal in
# the Software without restriction, including without limitation the rights to
# use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
# the Software, and to permit persons to whom the Software is furnished to do so,
# subject to the following conditions:
# - The above copyright notice and this permission notice shall be included in
#   all copies or substantial portions of the Software.
# - The Software is provided "as is", without warranty of any kind, express or
#   implied, including but not limited to the warranties of merchantability,
#   fitness for a particular purpose and noninfringement. In no event shall the
#   authors or copyright holders be liable for any claim, damages or other
#   liability, whether in an action of contract, tort or otherwise, arising from,
#   out of or in connection with the Software or the use or other dealings in the
#   Software.
# 

import math


# DCT type II, scaled. Algorithm by Arai, Agui, Nakajima, 1988.
# See: https://web.stanford.edu/class/ee398a/handouts/lectures/07-TransformCoding.pdf#page=30
def transform(vector):
	v0 = vector[0] + vector[7]
	v1 = vector[1] + vector[6]
	v2 = vector[2] + vector[5]
	v3 = vector[3] + vector[4]
	v4 = vector[3] - vector[4]
	v5 = vector[2] - vector[5]
	v6 = vector[1] - vector[6]
	v7 = vector[0] - vector[7]
	
	v8 = v0 + v3
	v9 = v1 + v2
	v10 = v1 - v2
	v11 = v0 - v3
	v12 = -v4 - v5
	v13 = (v5 + v6) * A[3]
	v14 = v6 + v7
	
	v15 = v8 + v9
	v16 = v8 - v9
	v17 = (v10 + v11) * A[1]
	v18 = (v12 + v14) * A[5]
	
	v19 = -v12 * A[2] - v18
	v20 = v14 * A[4] - v18
	
	v21 = v17 + v11
	v22 = v11 - v17
	v23 = v13 + v7
	v24 = v7 - v13
	
	v25 = v19 + v24
	v26 = v23 + v20
	v27 = v23 - v20
	v28 = v24 - v19
	
	return [
		S[0] * v15,
		S[1] * v26,
		S[2] * v21,
		S[3] * v28,
		S[4] * v16,
		S[5] * v25,
		S[6] * v22,
		S[7] * v27,
	]


# DCT type III, scaled. A straightforward inverse of the forward algorithm.
def inverse_transform(vector):
	v15 = vector[0] / S[0]
	v26 = vector[1] / S[1]
	v21 = vector[2] / S[2]
	v28 = vector[3] / S[3]
	v16 = vector[4] / S[4]
	v25 = vector[5] / S[5]
	v22 = vector[6] / S[6]
	v27 = vector[7] / S[7]
	
	v19 = (v25 - v28) / 2
	v20 = (v26 - v27) / 2
	v23 = (v26 + v27) / 2
	v24 = (v25 + v28) / 2
	
	v7  = (v23 + v24) / 2
	v11 = (v21 + v22) / 2
	v13 = (v23 - v24) / 2
	v17 = (v21 - v22) / 2
	
	v8 = (v15 + v16) / 2
	v9 = (v15 - v16) / 2
	
	v18 = (v19 - v20) * A[5]  # Different from original
	v12 = (v19 * A[4] - v18) / (A[2] * A[5] - A[2] * A[4] - A[4] * A[5])
	v14 = (v18 - v20 * A[2]) / (A[2] * A[5] - A[2] * A[4] - A[4] * A[5])
	
	v6 = v14 - v7
	v5 = v13 / A[3] - v6
	v4 = -v5 - v12
	v10 = v17 / A[1] - v11
	
	v0 = (v8 + v11) / 2
	v1 = (v9 + v10) / 2
	v2 = (v9 - v10) / 2
	v3 = (v8 - v11) / 2
	
	return [
		(v0 + v7) / 2,
		(v1 + v6) / 2,
		(v2 + v5) / 2,
		(v3 + v4) / 2,
		(v3 - v4) / 2,
		(v2 - v5) / 2,
		(v1 - v6) / 2,
		(v0 - v7) / 2,
	]


# ---- Tables of constants ----

C = [math.cos(math.pi / 16 * i) for i in range(8)]
S = [1 / (4 * val) for val in C]
S[0] = 1 / (2 * math.sqrt(2))
A = [
	None,
	C[4],
	C[2] - C[6],
	C[4],
	C[6] + C[2],
	C[6],
]