Playing Darts: become an amateur with the most efficient prioritizing

Question

Introduction:

I think most people will know how darts work. But since some might not, here are the rules of playing Darts in a nutshell:

• Amount of players: 2
• Starting points: 501
• Goal: Get your points to 0 first
• Darts per player: 3
• Available numbers on the board: 1 through 20, and Bulls-eye

Here is the layout of a Dartboard:

• Each number is worth an equal amount of points
• Each triple (the inner red-green circle) is 3x the number
• Each double (the outer red-green circle) is 2x the number
• The red center is called the Bulls-eye or Double-Bull, which is worth 50 points
• And the outlined part of the center around the Bulls-eye (also known as Single-Bull) is worth 25 points

Some other rules:

• You'll have to finish to 0 exactly
• The very last dart that puts your points to 0 must be a double or the Bulls-eye
• If a dart would leave your remaining total on 1 or less than 0, the entire set of three darts of which it forms a part is discarded (since you aren't able to finish 1 or less than 0 with a double)
• Once you've reached (exactly) 0, you won't have to throw any remaining darts of the three you throw in a single turn

Relevant information regarding this Challenge:

Although every player has of course preferred doubles to finish with (Double-20 is preferred by some of the pros for example), in general - and for the sake of this challenge - this will be the order of prioritization on finishing doubles:

16, 8, 20, 12, 4, 18, 14, 10, 6, 2, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, Bulls-Eye

Why? Let's compare a double-16 and double-17 finish. If you want to throw the double-16, but throw in the 16 instead by accident, you can continue to double-8. If you want to throw the double-17, but throw in the 17 instead, you must first throw an odd number (like 1 or 3), before you can try your best on a finishing double again. If we look at the following, you can see the next move in case you accidentally throw a single instead of a double:

Bulls-eye
20-10-5
19
18-9
17
16-8-4-2-1
15
14-7
13
12-6-3
11

Clearly 16 is your best bet, hence the given order of prioritizing above. (And since the Bulls-eye is the smallest, that has the lowest preference for the sake of this challenge.)

Challenge:

So, those are the rules of playing Darts. As for this challenge:

Given an input integer in the range of 2-170 (excluding 169, 168, 166, 165, 163, 162, 159; these cannot be finished within 3 darts), output the steps required to finish it following these rules in this exact order of precedence:

1. You must try to finish it in as few darts as possible
2. You must keep the prioritizing in mind when doing so for your finishing darts
3. When multiple options are available with the same priority, we pick the option with the highest number. I.e. 1.) For 131 the expected result is T20, T13, D16 instead of T19, T14, D16 or T18, T15, D16 or T17, T14, D16 etc., because T20 is the highest number. 2.) For 109 the expected result is 17, T20, D16 instead of 20, T19, D16, because T20 is the highest number.
4. If exactly three darts must be thrown to finish, sort the first two throws by Single, Triple, Double, Bulls-Eye, Single-Bull

Keeping the rules above in mind, these will be your strict input-output pairs (format of both input and output is flexible, though):

Legend: D = Double; T = Triple; BE = Bulls-eye; SB = Single-Bull

Points left:   Darts to throw (in exact order)
2              D1
3              1, D1
4              D2
5              1, D2
6              D3
7              3, D2
8              D4
9              1, D4
10             D5
11             3, D4
12             D6
13             5, D4
14             D7
15             7, D4
16             D8
17             1, D8
18             D9
19             3, D8
20             D10
21             5, D8
22             D11
23             7, D8
24             D12
25             9, D8
26             D13
27             11, D8
28             D14
29             13, D8
30             D15
31             15, D8
32             D16
33             1, D16
34             D17
35             3, D16
36             D18
37             5, D16
38             D19
39             7, D16
40             D20
41             9, D16
42             10, D16
43             11, D16
44             12, D16
45             13, D16
46             14, D16
47             15, D16
48             16, D16
49             17, D16
50             BE
51             19, D16
52             20, D16
53             T7, D16
54             D11, D16
55             T13, D8
56             T8, D16
57             SB, D16
58             D13, D16
59             T9, D16
60             D14, D16
61             T15, D8
62             T10, D16
63             T13, D12
64             D16, D16
65             T11, D16
66             D17, D16
67             T17, D8
68             T12, D16
69             T15, D12
70             D19, D16
71             T13, D16
72             D20, D16
73             T19, D8
74             T14, D16
75             T17, D12
76             T20, D8
77             T15, D16
78             D19, D20
79             T13, D20
80             T16, D16
81             T19, D12
82             BE, D16
83             T17, D16
84             T20, D12
85             T15, D20
86             T18, D16
87             T17, D18
88             T16, D20
89             T19, D16
90             BE, D20
91             T17, D20
91             T17, D20
92             T20, D16
93             T19, D18
94             T18, D20
95             T19, D19
96             T20, D18
97             T19, D20
98             T20, D19
99             7, T20, D16
100            T20, D20
101            T17, BE
102            10, T20, D16
103            11, T20, D16
104            T18, BE
105            13, T20, D16
106            14, T20, D16
107            T19, BE
108            16, T20, D16
109            17, T20, D16
110            T20, BE
111            19, T20, D16
112            20, T20, D16
113            T20, T7, D16
114            T20, D11, D16
115            T19, D13, D16
116            T20, T8, D16
117            T20, SB, D16
118            T20, D13, D16
119            T20, T9, D16
120            T20, D14, D16
121            T19, D16, D16
122            T20, T10, D16
123            T19, D17, D16
124            T20, D16, D16
125            T20, T11, D16
126            T20, D17, D16
127            T19, D19, D16
128            T20, T12, D16
129            T19, D20, D16
130            T20, D19, D16
131            T20, T13, D16
132            T20, D20, D16
133            T17, BE, D16
134            T20, T14, D16
135            T19, D19, D20
136            T18, BE, D16
137            T20, T15, D16
138            T20, D19, D20
139            T19, BE, D16
140            T20, T16, D16
141            T17, BE, D20
142            T20, BE, D16
143            T20, T17, D16
144            T18, BE, D20
145            T20, T15, D20
146            T20, T18, D16
147            T19, BE, D20
148            T20, T16, D20
149            T20, T19, D16
150            T20, BE, D20
151            T20, T17, D20
152            T20, T20, D16
153            T20, T19, D18
154            T20, T18, D20
155            T20, T19, D19
156            T20, T20, D18
157            T20, T19, D20
158            T20, T20, D19
160            T20, T20, D20
161            T20, T17, BE
164            T20, T18, BE
167            T20, T19, BE
170            T20, T20, BE

NOTE: Test cases are done by hand, so if you see any mistakes, please comment down below.
Multiple fixed thanks to @JohanKarlsson, @JonathanAllan, and @Sebastian

Challenge rules:

• Input and output format are flexible. Can be a list of Strings, array of Objects, single STDOUT-printline, your call. You can also choose your own formatting regarding the Triples, Doubles, Bulls-Eye and Single-Bull. Please state what you've used in your answer.

General rules:

• This is code-golf, so shortest answer in bytes wins.
  Don't let code-golf languages discourage you from posting answers with non-codegolfing languages. Try to come up with an as short as possible answer for 'any' programming language.
• Standard rules apply for your answer, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters and return-type, full programs. Your call.
• Default Loopholes are forbidden.
• If possible, please add a link with a test for your code.
• Also, please add an explanation if necessary.

Answer 1

Python 3, 987 1,003 586 bytes

Edit: Fixed problem with T20's using some hacky code.

Edit: Added in golfed version.

from itertools import*
def d(i):
 f={v*2:"D"+str(v) for v in [16,8,20,12,4,18,14,10,6,2,19,17,15,13,11,9,7,5,3,1]}
 f[50]="BE"
 t={3*i:"T"+str(i) for i in range(20,6,-1)}
 t.update({i:str(i) for i in range(20,0,-1)})
 t.update({2*i:"D"+str(i) for i in range(20,10,-1) if i%3!=0})
 t.update({50:"BE", 25:"SB"})
 if i in f:
  return f.get(i)
 else:
  for n,x in product(f,t):
   r=i-n
   if r in t:
    return ' '.join((t.get(r),f.get(n)))
  for n,x,y in product(f,t,t):
   r=i-n
   if x+y==r:
    b=x!=60 or y>20
    return ' '.join((t.get(x if b else y),t.get(y if b else x), f.get(n)))

Try it online! Ungolfed version:

from itertools import product
def darts(inp):
    list_finishes = [16, 8, 20, 12, 4, 18, 14, 10, 6, 2, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1]
    finishes = {finish*2:"D"+str(finish) for finish in list_finishes}
    finishes.update({50:"BE"})
    throws = {3*i:"T"+str(i) for i in range(20,6,-1)}
    throws.update({i:str(i) for i in range(20,0,-1)})
    throws.update({2*i:"D"+str(i) for i in range(20,10,-1) if i%3!=0})
    throws.update({50:"BE", 25:"SB"})

    if inp in finishes:
        return finishes.get(inp)
    else:
        for finish, x in product(finishes, throws):
            remaining = inp - finish
            if remaining in throws:
                return ', '.join((throws.get(remaining), finishes.get(finish)))
        for finish, x, y in product(finishes, throws, throws):
            remaining = inp - finish
            if x + y == remaining:
                    return ', '.join((throws.get(x if x!= 60 or y>20 else y), throws.get(y if x!= 60 or y>20 else x), finishes.get(finish)))

Little effort has gone into golfing this code, as I intended it to be a reference solution more than anything. Anyone who wants to improve this code is free to post it as their own solution or edit my answer. I have also created a function to test your function against the given values (Try it online with my function):

def test_func(func, has_comma=True):
    answers  =  """
2              D1
3              1, D1
4              D2
5              1, D2
6              D3
7              3, D2
8              D4
9              1, D4
10             D5
11             3, D4
12             D6
13             5, D4
14             D7
15             7, D4
16             D8
17             1, D8
18             D9
19             3, D8
20             D10
21             5, D8
22             D11
23             7, D8
24             D12
25             9, D8
26             D13
27             11, D8
28             D14
29             13, D8
30             D15
31             15, D8
32             D16
33             1, D16
34             D17
35             3, D16
36             D18
37             5, D16
38             D19
39             7, D16
40             D20
41             9, D16
42             10, D16
43             11, D16
44             12, D16
45             13, D16
46             14, D16
47             15, D16
48             16, D16
49             17, D16
50             BE
51             19, D16
52             20, D16
53             T7, D16
54             D11, D16
55             T13, D8
56             T8, D16
57             SB, D16
58             D13, D16
59             T9, D16
60             D14, D16
61             T15, D8
62             T10, D16
63             T13, D12
64             D16, D16
65             T11, D16
66             D17, D16
67             T17, D8
68             T12, D16
69             T15, D12
70             D19, D16
71             T13, D16
72             D20, D16
73             T19, D8
74             T14, D16
75             T17, D12
76             T20, D8
77             T15, D16
78             D19, D20
79             T13, D20
80             T16, D16
81             T19, D12
82             BE, D16
83             T17, D16
84             T20, D12
85             T15, D20
86             T18, D16
87             T17, D18
88             T16, D20
89             T19, D16
90             BE, D20
91             T17, D20
91             T17, D20
92             T20, D16
93             T19, D18
94             T18, D20
95             T19, D19
96             T20, D18
97             T19, D20
98             T20, D19
99             7, T20, D16
100            T20, D20
101            T17, BE
102            10, T20, D16
103            11, T20, D16
104            T18, BE
105            13, T20, D16
106            14, T20, D16
107            T19, BE
108            16, T20, D16
109            17, T20, D16
110            T20, BE
111            19, T20, D16
112            20, T20, D16
113            T20, T7, D16
114            T20, D11, D16
115            T19, D13, D16
116            T20, T8, D16
117            T20, SB, D16
118            T20, D13, D16
119            T20, T9, D16
120            T20, D14, D16
121            T19, D16, D16
122            T20, T10, D16
123            T19, D17, D16
124            T20, D16, D16
125            T20, T11, D16
126            T20, D17, D16
127            T19, D19, D16
128            T20, T12, D16
129            T19, D20, D16
130            T20, D19, D16
131            T20, T13, D16
132            T20, D20, D16
133            T17, BE, D16
134            T20, T14, D16
135            T19, D19, D20
136            T18, BE, D16
137            T20, T15, D16
138            T20, D19, D20
139            T19, BE, D16
140            T20, T16, D16
141            T17, BE, D20
142            T20, BE, D16
143            T20, T17, D16
144            T18, BE, D20
145            T20, T15, D20
146            T20, T18, D16
147            T19, BE, D20
148            T20, T16, D20
149            T20, T19, D16
150            T20, BE, D20
151            T20, T17, D20
152            T20, T20, D16
153            T20, T19, D18
154            T20, T18, D20
155            T20, T19, D19
156            T20, T20, D18
157            T20, T19, D20
158            T20, T20, D19
160            T20, T20, D20
161            T20, T17, BE
164            T20, T18, BE
167            T20, T19, BE
170            T20, T20, BE
"""
    if not has_comma:
        answers = answers.replace(',','')
    answers_dict = {int(answer.split('  ')[0]):answer.split('  ', 1)[1].lstrip() for answer in answers.split('\n')[1:-1]}
    all_valid = True
    for key in answers_dict.keys():
        try:
           assert func(key)==answers_dict.get(key)
        except:
            print("Failure on {}: Your answer '{}' != Given answer '{}'".format(key, func(key), answers_dict.get(key)))
            all_valid = False
    if all_valid:
        print("All tests completed successfully.")