sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated decimal integers on standard input; prints their sum to standard output

Explanation

We represent each input in unary-coded decimal, using x as our digit, and 0 as separator. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.

sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated non-negative decimal integers on standard input; prints their sum to standard output

Explanation

We represent each input in unary-coded decimal, using x as our digit, and 0 as separator. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.

#sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated decimal integers on standard input; prints their sum to standard output

##Explanation We represent each input in unary-coded decimal, using x as our digit, and 0 as separator. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.

sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated decimal integers on standard input; prints their sum to standard output

Explanation

We represent each input in unary-coded decimal, using x as our digit, and 0 as separator. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.

#sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated decimal integers on standard input; prints their sum to standard output

##Explanation We represent each input in unary-coded decimal, using x as our digit, and 0 as separater. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.

#sed, 275 bytes

s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
s/$/ /
:a
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
ta
s/ *//g
:c
s/0xxxxxxxxxx/x0/g
tc
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

Takes 2 space-separated decimal integers on standard input; prints their sum to standard output

##Explanation We represent each input in unary-coded decimal, using x as our digit, and 0 as separator. For example, 42 is written 0xxxx0xx.

#!/bin/sed -f
# Convert to unary decimal
s/[1-9]/0&/g
s/[5-9]/4&/g
s/[89]/4&/g
s/[2367]/xx&/g
s/[13579]/x/g
s/4/xxxx/g
s/[1-8]//g
# Append space to separate second input from output
s/$/ /
# Consume the smallest of each digit from input, and push
# the sum to output
:add
s/\(.*\)0\(x*\) \(.*\)0\(x*\) /1円 3円 02円4円/
tadd
# Everything is now accumulated in output; remove the spaces
s/ *//g
# Add carry to the next position
:carry
s/0xxxxxxxxxx/x0/g
tcarry
# Back to decimal
s/0x/-x/g
s/xx/2/g
y/x/1/
s/22/4/g
s/44/8/g
s/81/9/g
s/21/3/g
s/42/6/g
s/43/7/g
s/41/5/g
s/-//g

The extension to support decimal fractions is left as an exercise for the reader.