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In JS, I can use a function String.prototype.replace() when replacing a submatch in a regular expression. For example:

var x = 'a1b2c3'.replace(/(d+)/g, (num) => 
 return num*num+1
)
console.log(x)
// 'a2b5c10'

I've tried using sed but it seems that invoking an operator $(()) inside of the replacement is not possible.

$ echo "a1b2c3" | sed 's/([^0-9]*)([0-9])([^0-9]*)/1$((2*2+1))3/g'
# Output: a$((1*1+1))b$((2*2+1))c$((3*3+1))

Is there a similar tool or function in bash that has the functionality similar to JS's String.replace()?

The bash shell does support a native regex operator which you can enable with the ~ flag. All you need to do is define a regex, take the captured groups and replace them with the modified values

str='a1b2c3'
re='^.*([0-9]+).*([0-9]+).*([0-9]+).*$'
if [[ $str =~ $re ]]; then
 for match in "$BASH_REMATCH[@]"; do
 final="$str/$match/$((match*match+1))"
 done
fi
printf '%sn' "$final"

The [[ $str =~ $re ]] does the regex match and updates the captured group array $BASH_REMATCH[@]. So for each of the element in the order of their appearance, we do the string substitution operator $str/old/new. The replacement value in your case is the number multiplied with itself and added by 1.

Add more capturing groups to the regex .*([0-9]+) for subsequent matches.

If not for a pure bash solution above, using an external utility like perl, one could do it as

perl -pe 's/d+/$&*$&+1/ge' <<<"$str"

The $& refers to the captured digit in the string and the e flag allows do arithmetic operations over the captured groups.

You can implement this in gawk using match() and substr().

echo "a1b2c3" | awk '
 head = ""
 tail = $0

 while (match(tail, /[0-9]+/)) 
 num = substr(tail, RSTART, RLENGTH)
 num = num * num + 1
 head = head substr(tail, 1, RSTART-1) num
 tail = substr(tail, RSTART + RLENGTH)
 

 print head tail
'

Output

a2b5c10

I am not sure this is my favorite answer, but just to let you know GNU sed does have external command capabilities:

echo "a1b2c3" | sed 's/([^0-9]*)([0-9])([^0-9]*)/echo 1$((2*2+1))3/ge' | sed 's/echo //g'

e is the trick to pass the result externally.

The most annoying thing - echo is appended when the g and e flags are combined to the substitution groups following the first one, so the second sed gets rid of them. If someone knows if there is something built in that would be great.

Unfortunately

echo "a1b2c3" | sed 's/([^0-9]*)([0-9])([^0-9]*)/1$((2*2+1))3/ge'

will get a working substitution but throw an error as a2b3c4 is not a command.