mathematics - Make numbers 1-100 with only four 8s

Make numbers 1, 2, 3, 4, 5,.....100 using only four 8s. You can use multiplication, division, addition, and subtraction. No negatives. You may use exponents.

Example:

$1 = 8 \times 8 \div 8 \div 8$

Answer

It is not possible to generate all the numbers with only the give operations.
I used a computer for that.
The only ones are 1,2,3,4,6,7,8,9,10,12,15,16,17,19,24,32,48,56,63,64,65,72,80,87,88,89 (marked in bold).

So I took the liberty to use other functions like square root ($\sqrt{x}$), ceil ($\lceil x \rceil$), floor ($\lfloor x \rfloor$).

1 = $\frac{88}{88}$
2 = $\frac88 + \frac88$
3 = $\frac{88}{8} - 8$
4 = $\frac{8 \times 8}{8+8}$
5 = $\sqrt{8+8} + \frac88$
6 = $8 - \frac{8 + 8}{8}$
7 = $\frac{8 \times 8 - 8}{8}$
8 = $8 + 8 \times (8 - 8)$
9 = $\frac{8 \times 8 + 8}{8}$

10 = $\frac{88 - 8}{8}$
11 = $\lceil \sqrt8 \rceil + 8-8+8$
12 = $\frac{88 + 8}{8}$
13 = $\lfloor \sqrt8 \rfloor + \frac{88}{8}$
14 = $\lceil \sqrt8 \rceil + \frac{88}{8}$
15 = $8 + 8 - \frac88$
16 = $8 \times \frac{8+8}{8}$
17 = $8 + 8 + \frac88$
18 = $8 + 8 + \sqrt{\sqrt{8+8}}$
19 = $\frac{88}{8} + 8$

20 = $8 + 8 + \sqrt{8+8}$
21 = $8 + 8 + \lceil \sqrt{8} \rceil + \lfloor \sqrt{8} \rfloor$
22 = $\frac{88}{\sqrt{8+8}}$
23 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{8+8}$
24 = $88-8\times 8$
25 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{\sqrt{8+8}}$
26 = $8 +8 +8 + \lfloor \sqrt{8} \rfloor$
27 = $\lceil \sqrt8 \rceil + 8+8+8$
28 = $8 \times \lceil\sqrt8\rceil + \sqrt{8+8}$
29 = $8 \times \sqrt{8+8} - \lceil \sqrt8 \rceil$

30 = $8 \times \sqrt{8+8} - \lfloor \sqrt8 \rfloor$
31 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} + \sqrt{8+8}$
32 = $8 + 8 + 8 + 8$
33 = $8 \times \sqrt{8+8} + \lfloor \sqrt{\sqrt8} \rfloor$
34 = $8 \times \sqrt{8+8} + \lfloor \sqrt8 \rfloor$
35 = $8 \times \sqrt{8+8} + \lceil \sqrt8 \rceil$
36 = $(8 + \lfloor\sqrt{\sqrt8}\rfloor) \times \sqrt{8+8}$
37 = $\lfloor\frac{88}{\lceil\sqrt8\rceil}\rfloor + 8$
38 = $\lceil\frac{88}{\lceil\sqrt8\rceil}\rceil + 8$
39 = $(8+ \lceil\sqrt8\rceil + \lfloor\sqrt8\rfloor) \times \lceil\sqrt8\rceil$

40 = $8 \times \sqrt{8+8} + 8$
41 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lceil\sqrt8\rceil$
42 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lfloor\sqrt8\rfloor$
43 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lfloor\sqrt{\sqrt8}\rfloor$
44 = $\frac{88}{\sqrt{\sqrt{8+8}}}$
45 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lfloor\sqrt{\sqrt8}\rfloor$
46 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lfloor\sqrt8\rfloor$
47 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lceil\sqrt8\rceil$
48 = $8 \times 8 - 8 -8$
49 = $(8 - \lfloor \sqrt{\sqrt8} \rfloor) \times (8 - \lfloor \sqrt{\sqrt8} \rfloor)$

50 = $(\lfloor \sqrt8 \rfloor + \lceil \sqrt8 \rceil) \times (8 + \lfloor \sqrt8 \rfloor)$
51 = $(8+8) \times \lceil \sqrt8 \rceil + \lceil \sqrt8 \rceil$
52 = $\frac{88}{\lfloor\sqrt8\rfloor} + 8$
53 = $8 \times 8 - 8 - \lceil \sqrt8 \rceil$
54 = $8 \times 8 - 8 - \lfloor \sqrt8 \rfloor$
55 = $8 \times 8 - 8 - \lfloor \sqrt{\sqrt8} \rfloor$
56 = $8 \times (8 - \frac88)$
57 = $8 \times 8 - 8 + \lfloor \sqrt{\sqrt8} \rfloor$
58 = $8 \times 8 - 8 + \lfloor \sqrt8 \rfloor$
59 = $8 \times 8 - 8 + \lceil \sqrt8 \rceil$

60 = $8 \times 8 - \sqrt{8+8}$
61 = $8\times 8 - \lceil \frac{8}{\sqrt8}\rceil$
62 = $8\times 8 - \sqrt{\sqrt{8+8}}$
63 = $8 \times 8 - \frac88$
64 = $8 \times 8 \times \frac88$
65 = $8 * 8 + \frac88$
66 = $8\times 8 + \sqrt{\sqrt{8+8}}$
67 = $8\times 8 + \lceil \frac{8}{\sqrt8}\rceil$
68 = $8 \times 8 + \sqrt{8+8}$
69 = $8 + 8 \times 8 - \lceil \sqrt8 \rceil$

70 = $8 \times 8 + \sqrt{8+8}$
71 = $8 +8 \times 8 - \lfloor \sqrt{\sqrt8} \rfloor$
72 = $88 - 8 - 8$
73 = $\lfloor \sqrt{\sqrt8} \rfloor + 8 +8 \times 8$
74 = $\lfloor \sqrt8 \rfloor + 8 +8 \times 8$
75 = $\lceil \sqrt8 \rceil + 8 +8 \times 8$
76 = $\lceil \sqrt{88 \times 8 \times 8} \rceil$ - thanks to h34
77 = $88 - 8 - \lceil\sqrt8\rceil$
78 = $88 - 8 - \lfloor\sqrt8\rfloor$
79 = $\lceil\sqrt8\rceil^{\sqrt{8+8}} - \lfloor\sqrt8\rfloor$

80 = $8\times 8 + 8 + 8$
81 = $(\lfloor\sqrt8\rfloor + \lfloor\sqrt{\sqrt8}\rfloor)^{\sqrt{8+8}}$
82 = $88 - 8 + \lfloor\sqrt8\rfloor$
83 = $88 - 8 + \lceil\sqrt8\rceil$
84 = $88 - \sqrt{8+8}$
85 = $88 - \lceil\frac{8}{\sqrt8}\rceil$
86 = $88 - \lfloor\frac{8}{\sqrt8}\rfloor$
87 = $88 - \frac88$
88 = $88 - 8 + 8$
89 = $88 + \frac88$

90 = $88 + \lfloor\frac{8}{\sqrt8}\rfloor$
91 = $88 + \lceil\frac{8}{\sqrt8}\rceil$
92 = $88 + \sqrt{8+8}$
93 = $88 +8 - \lceil\sqrt8\rceil$
94 = $88 +8 - \lfloor\sqrt8\rfloor$
95 = $88 +8 - \lfloor\sqrt{\sqrt8}\rfloor$
96 = $8 \times \sqrt{8+8} \times \lceil \sqrt8 \rceil$
97 = $88 +8 + \lfloor\sqrt{\sqrt8}\rfloor$
98 = $88 +8 + \lfloor\sqrt8\rfloor$
99 = $88 +8 + \lceil\sqrt8\rceil$

100 = $(8 + \lfloor \sqrt8 \rfloor)^{\sqrt{\sqrt{8+8}}}$

Bonus:
If we allow log functions we can generate every number like this:

$x = \log_{\frac{\lfloor\sqrt{\sqrt8}\rfloor}{\lfloor\sqrt8\rfloor}}\left({\log_8\underbrace{\sqrt{\sqrt{\dots\sqrt{8\,}\,}\,}}_\text{x square roots}}\right)$

This is equivalent to

$x = \log_{\frac12}\left({\log_8{8^{\frac{1}{2^x}}}}\right)$

Going further:

$x = \log_{\frac12}\left({\frac{1}{2^x}}\right)$

Depending on he number of square roots, we can get any number.

And the snarky solution:

Turn one eight 90 degrees to get: $\infty$ and then $\frac{8-8}{8} \times \infty=0 \times \infty$ which can be equal to anything. (l know, I know, you will say it's undefined, but think... limits).