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bfdae48 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 | """Utility module of reusable algebra routines."""
import torch
import math
EPSILON = 1e-6
COS_TAYLOR_COEFFS = [
1.0,
-1.0 / math.factorial(2),
+1.0 / math.factorial(4),
-1.0 / math.factorial(6),
+1.0 / math.factorial(8),
]
SINC_TAYLOR_COEFFS = [
1.0,
-1.0 / math.factorial(3),
+1.0 / math.factorial(5),
-1.0 / math.factorial(7),
+1.0 / math.factorial(9),
]
INV_SINC_TAYLOR_COEFFS = [1, 1.0 / 6.0, 7.0 / 360.0, 31.0 / 15120.0, 127.0 / 604800.0]
INV_TANC_TAYLOR_COEFFS = [1.0, -1.0 / 3.0, -1.0 / 45.0, -2.0 / 945.0, -1.0 / 4725.0]
COSC_TAYLOR_COEFFS = [
1.0 / 2.0,
-1.0 / math.factorial(4),
+1.0 / math.factorial(6),
-1.0 / math.factorial(8),
+1.0 / math.factorial(10),
]
VAR_INV_TAN_TAYLOR_COEFFS = [1.0 / 12.0, 1.0 / 720.0, 1.0 / 30240.0, 1.0 / 1209600.0]
SINHC_TAYLOR_COEFFS = [
1.0,
1 / math.factorial(3),
1 / math.factorial(5),
1 / math.factorial(7),
1 / math.factorial(9),
]
LOG_SINHC_TAYLOR_COEFFS = [
0.0,
1 / 6,
1 / 180,
1 / 2835,
1 / 37800,
]
COSH_TAYLOR_COEFFS = [
1.0,
1 / math.factorial(2),
1 / math.factorial(4),
1 / math.factorial(6),
1 / math.factorial(8),
]
INV_SINHC_TAYLOR_COEFFS = [
1.0,
-1.0 / 6.0,
7.0 / 360.0,
-31.0 / 15120.0,
127.0 / 604800.0,
]
INV_TANH_TAYLOR_COEFFS = [1.0, 1.0 / 3.0, -1.0 / 45.0, 2.0 / 945.0, -1.0 / 4725.0]
TANH_TAYLOR_COEFFS = [1.0, -1.0 / 3.0, 2.0 / 15.0, -17.0 / 315.0, 62.0 / 2835.0]
LOG_TANH_TAYLOR_COEFFS = [0.0, -1.0 / 3.0, 7.0 / 90.0, -62.0 / 2835.0, 127.0 / 18900.0]
LOG1P_M_TANH_SQ_TAYLOR_COEFFS = [0.0, -1.0, 1.0 / 6.0, -2.0 / 45.0, 17.0 / 1260.0]
ARCTANH_CARD_TAYLOR_COEFFS = [1.0, 1.0 / 3.0, 1.0 / 5.0, 1 / 7.0, 1.0 / 9]
cos_close_0 = {"function": torch.cos, "coefficients": COS_TAYLOR_COEFFS}
sinc_close_0 = {"function": lambda x: torch.sin(x) / x, "coefficients": SINC_TAYLOR_COEFFS}
inv_sinc_close_0 = {
"function": lambda x: x / torch.sin(x),
"coefficients": INV_SINC_TAYLOR_COEFFS,
}
inv_tanc_close_0 = {
"function": lambda x: x / torch.tan(x),
"coefficients": INV_TANC_TAYLOR_COEFFS,
}
cosc_close_0 = {
"function": lambda x: (1 - torch.cos(x)) / x**2,
"coefficients": COSC_TAYLOR_COEFFS,
}
var_sinc_close_0 = {
"function": lambda x: (x - torch.sin(x)) / x**3,
"coefficients": [-k for k in SINC_TAYLOR_COEFFS[1:]],
}
var_inv_tanc_close_0 = {
"function": lambda x: (1 - (x / torch.tan(x))) / x**2,
"coefficients": VAR_INV_TAN_TAYLOR_COEFFS,
}
sinch_close_0 = {
"function": lambda x: torch.sinh(x) / x,
"coefficients": SINHC_TAYLOR_COEFFS,
}
log_sinch_close_0 = {
"function": lambda x: torch.log(torch.sinh(x) / x),
"coefficients": LOG_SINHC_TAYLOR_COEFFS,
}
cosh_close_0 = {"function": torch.cosh, "coefficients": COSH_TAYLOR_COEFFS}
inv_sinch_close_0 = {
"function": lambda x: x / torch.sinh(x),
"coefficients": INV_SINHC_TAYLOR_COEFFS,
}
inv_tanh_close_0 = {
"function": lambda x: x / torch.tanh(x),
"coefficients": INV_TANH_TAYLOR_COEFFS,
}
tanh_close_0 = {
"function": lambda x: torch.tanh(x) / x,
"coefficients": TANH_TAYLOR_COEFFS,
}
log_tanh_close_0 = {
"function": lambda x: torch.log(torch.tanh(x) / x),
"coefficients": LOG_TANH_TAYLOR_COEFFS,
}
log1p_m_tanh_sq_close_0 = {
"function": lambda x: torch.log1p(-torch.tanh(x) ** 2),
"coefficients": LOG1P_M_TANH_SQ_TAYLOR_COEFFS,
}
arctanh_card_close_0 = {
"function": lambda x: torch.arctanh(x) / x,
"coefficients": ARCTANH_CARD_TAYLOR_COEFFS,
}
taylor_coeffs = None
def taylor_exp_even_func(point, taylor_function, order=5, tol=EPSILON):
"""Taylor Approximation of an even function around zero.
Parameters
----------
point : array-like
Argument of the function to approximate.
taylor_function : dict with following keys
function : callable
Even function to approximate around zero.
coefficients : list
Taylor coefficients of even order at zero.
order : int
Order of the Taylor approximation.
Optional, Default: 5.
tol : float
Threshold to use the approximation instead of the function's value.
Where `abs(point) <= tol`, the approximation is returned.
Returns
-------
function_value: array-like
Value of the function at point.
"""
global taylor_coeffs
if taylor_coeffs is None:
taylor_coeffs = torch.stack(
[torch.tensor(v, dtype=point.dtype) for v in taylor_function["coefficients"][:5]]
).to(point.device)
taylor_coeffs_ = taylor_coeffs[:order]
approx = torch.einsum(
"k,k...->...",
taylor_coeffs_,
torch.stack([point**k for k in range(order)]),
)
point_ = torch.where(torch.abs(point) <= tol, tol * torch.ones_like(point), point)
exact = taylor_function["function"](torch.sqrt(point_))
result = torch.where(torch.abs(point) < tol, approx, exact)
return result |