Correct way to add numbers to get lots of 8's

Bash + Common Linux utilities, 80

s=~0my3[_p^?{}s
h()(tr 0-9 $s<<<$1|xxd -p)
dc -e$[0x`h $1`|0x`h $2`]P|tr $s 0-9aQ

Note the ^? in the source should be replaced with an ASCII 0x7f character.

The string s is each 7 segment digit 0-9, a, Q encoded with each segment corresponding to a bit of an ASCII char.

The h() function transliterates the input number from decimal to the encoding specified by s, then outputs the result as a raw hex string.

The two resulting raw hex strings are ORed together using regular bash arithmetic, then output by dc's P command as a bytestream. This bytestream is then transliterated back to decimal + a + Q and output.

Note also that when using the <<< bash herestring construct in function h() a newline is implicitly appended to the redirected string. This doesn't matter - it is simply translated to 0x0a at the end of each hex string; when the two hex numbers are ORed together, the result is still 0x0a in the last char which doesn't get transliterated and thus simply translates back to a newline which is output after the result.

Test output:

$ for testcase in \
> "12345 123" \
> "88888 42" \
> "0 23" \
> "1234 56789" \
> "4 7"; do 
> ./7segadd.sh $testcase
> done
12389
88888
28
58a89
Q
$ 

Python 2, 155 bytes

def f(a,b):exec"a=[ord('?(u|j^_,♥~'[int(c)])for c in a];a=max(len(b)-len(a),0)*[0]+a;a,b=b,a;"*2;print`['214567q3a980'[(c|d)%13]for c,d in zip(a,b)]`[2::5]

Replace the ♥ with a DEL character (0x7F).

Calling f("12345", "123") prints 12389.

JavaScript (ES6), 158 144 bytes

f=(s,t)=>t[s.length]?f(t,s):s[t.length]?f(s,' '+t):s.replace(/./g,(c,i)=>"540q9361278a"[(a[c]|a[t[i]])%13],a=[119,20,47,31,92,91,123,22,127,95])

Saved 14 bytes by shamelessly stealing @Lynn's %13 trick.

f=(s,t)=>t[s.length]?f(t,s):s[t.length]?f(s,' '+t):s.replace(/./g,(c,i)=>"540q9361278a"[(a[c]|a[t[i]])%13],a=[119,20,47,31,92,91,123,22,127,95])
;o.textContent=[...s="0123456789"].map(c=>f(c.repeat(10),s)).join`
`;

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