# Binary coded decimal

Binary-Coded Decimal (or BCD for short) is a method of representing decimal numbers by storing what appears to be a decimal number but is actually stored as hexadecimal. Many CISC CPUs (e.g. X86 Assembly have special hardware routines for displaying these kinds of numbers.) On low-level hardware, such as 7-segment displays, binary-coded decimal is very important for outputting data in a format the end user can understand.

Binary coded decimal is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Use your language's built-in BCD functions, OR create your own conversion function, that converts an addition of hexadecimal numbers to binary-coded decimal. You should get the following results with these test cases:

•   0x19 + 1 = 0x20
•   0x30 - 1 = 0x29
•   0x99 + 1 = 0x100
Bonus Points

Demonstrate the above test cases in both "packed BCD" (two digits per byte) and "unpacked BCD" (one digit per byte).

## 6502 Assembly

Doesn't work with: Ricoh 2A03

The 6502 is a bit different in that it has a special operating mode where all addition and subtraction is handled as binary-coded decimal. Like the 68000, this must be invoked ahead of time, rather than using the Intel method of doing the math normally and then correcting it after the fact. (This special operating mode won't work on the aforementioned Ricoh 2A03, which performs math in "normal" mode even if the decimal flag is set.)

```sed ;set decimal flag; now all math is BCD
lda #\$19
clc
cld           ;chances are, PrintHex won't work properly when in decimal mode.
JSR PrintHex  ;unimplemented print routine
JSR NewLine

sed
lda #\$30
sec
sbc #1
cld
jsr PrintHex
JSR NewLine

sed
lda #\$99
clc
pha
lda #0
cld
jsr PrintHex
pla
jsr PrintHex
jsr NewLine
Output:
```20
29
0100```

## 68000 Assembly

The 68000 has special mathematics commands for binary-coded decimal. However, they only work at byte length, and cannot use immediate operands. Even adding by 1 this way requires you to load 1 into a register first.

```	MOVEQ #\$19,D0
MOVEQ #1,D1
MOVEQ #0,D2

ABCD D1,D0
JSR PrintHex
JSR NewLine

MOVEQ #\$30,D0
SBCD D1,D0
JSR PrintHex
JSR NewLine

MOVE.B #\$99,D0
ABCD D1,D0		;D0 has rolled over to 00 and set both the extend and carry flags.
ADDX D2,D2		;add the extend flag which was set by the above operation
;this can't use immediate operands either so we're using D2 which we set to zero at the start.

MOVE.L D0,D3	;back up the output since PrintHex takes D0 as its argument.
MOVE.L D2,D0	;print the 01
JSR PrintHex
MOVE.L D3,D0	;then the 00
JSR PrintHex

jmp *
```
Output:
```20
29
0100```

## ALGOL 68

Algol 68 does not have BCD as standard. This sample implements 2-digit unsigned packed decimal numbers, similar to the PL/M sample. The 2-digit numbers are then used to provide addition/subtraction of larger numbers.

```BEGIN # implements packed BCD arithmetic                                     #
INT x99 = ( 9 * 16 ) + 9;           # maximum unsigned 2-digit BCD value #
# structure to hold BCD values                                           #
MODE BCD = STRUCT( INT  value           # BCD value - signed -x99 to x99 #
, BOOL carry           # TRUE if the value overflowed,  #
);                     # FALSE otherwise                #

# constructs a BCD value from a, assuming it is in the correct format    #
# if the value has overflowed, it is truncated to a valid value and      #
# carry is set                                                           #
OP ASBCD    = ( INT a )BCD:
BEGIN
INT  v    := ABS a;
BOOL carry = v > x99;
IF carry THEN
v := ( ( ( v OVER 16 ) MOD 10 ) * 16 ) + ( v MOD 16 )
FI;
BCD( v * SIGN a, carry )
END # ASBCD # ;
# returns a converted to BCD format, truncating and setting carry        #
#         if necessary                                                   #
OP TOBCD    = ( INT a )BCD:
IF a < 0
THEN - TOBCD ABS a
ELSE BCD( ( ( ( a OVER 10 ) MOD 10 ) * 16 ) + ( a MOD 10 ), a > x99 )
FI # TOBCD # ;

BCD  bcd 99 = TOBCD 99;
BCD  bcd 1  = TOBCD  1;
BCD  bcd 0  = TOBCD  0;

# returns a two-digit string representation of the BCD value a           #
OP TOSTRING = ( BCD a )STRING: IF value OF a < 0 THEN "-" ELSE "" FI
+ whole( ABS value OF a OVER 16, 0 )
+ whole( ABS value OF a MOD  16, 0 )
;
# returns a string representation of the row of BCD values in a          #
#         assumes the most significant digits are in a[ LWB a ]          #
OP TOSTRING = ( []BCD a )STRING:
BEGIN
STRING result := "";
FOR b pos FROM LWB a TO UPB a DO result +:= TOSTRING a[ b pos ] OD;
result
END # TOSTRING # ;
# returns the sum of a and b, a and b can be positive or negative        #
#         the result is always positive, if it would be negative, it is  #
#         tens complemented                                              #
OP +        = ( BCD a, b )BCD:
BEGIN
INT  av = ABS value OF a,      bv = ABS value OF b;
BOOL ap =     value OF a >= 0, bp =     value OF b >= 0;
INT  a2 = av MOD 16,           b2 = bv MOD 16;
INT bcd value =
IF   ap = bp
THEN # both positive or both negative                        #
INT result := av + bv;
IF a2 + b2 > 9 THEN result +:= 6 FI;
IF ap THEN result ELSE - result FI
ELIF av >= bv
THEN # different signs, magnitude of a at least that of b    #
INT result := av - bv;
IF a2 < b2 THEN result -:= 6 FI;
IF ap THEN result ELSE - result FI
ELSE # different signs, magnitude of a less than that of b   #
INT result := bv - av;
IF b2 < a2 THEN result -:= 6 FI;
IF ap THEN - result ELSE - result FI
FI;
IF bcd value >= 0 THEN # result is positive                      #
ASBCD bcd value
ELSE                   # result is negative - tens complement    #
BCD result := ( bcd 99 + ASBCD bcd value ) + bcd 1;
carry OF result := TRUE;
result
FI
END # + # ;
# returns the value of b negated, carry is preserved                     #
OP -        = ( BCD a )BCD: BCD( - value OF a, carry OF a );
# returns the difference of a and b, a and b can be positive or negative #
OP -        = ( BCD a, b )BCD: a + - b;
# adds b to a and resurns a                                              #
OP +:=      = ( REF BCD a, BCD b )REF BCD: a := a + b;
# subtracts b from a and resurns a                                       #
OP -:=      = ( REF BCD a, BCD b )REF BCD: a := a - b;

print( ( TOSTRING ( TOBCD 19 + bcd 1 ), newline ) );
print( ( TOSTRING ( TOBCD 30 - bcd 1 ), newline ) );
BCD r = TOBCD 99 + bcd 1;
print( ( IF carry OF r THEN "1" ELSE "" FI, TOSTRING r, newline ) );
print( ( newline ) );

# use the 2-digit BCD to add/subtract larger numbers                     #
[ 1 : 6 ]BCD d12 :=
( TOBCD  1, TOBCD 23, TOBCD 45, TOBCD 67, TOBCD 89, TOBCD 01 );
[]BCD        a12  =
( TOBCD  1, TOBCD 11, TOBCD 11, TOBCD 11, TOBCD 11, TOBCD 11 );
TO 10 DO                                     # repeatedly add s12 to d12 #
print( ( TOSTRING d12, " + ", TOSTRING a12, " = " ) );
BOOL carry := FALSE;
FOR b pos FROM UPB d12 BY -1 TO LWB d12 DO
d12[ b pos ] +:= a12[ b pos ];
BOOL need carry = carry OF d12[ b pos ];
IF carry THEN d12[ b pos ] +:= bcd 1 FI;
carry := need carry OR carry OF d12[ b pos ]
OD;
print( ( TOSTRING d12, newline ) )
OD;
TO 10 DO                              # repeatedly subtract a12 from d12 #
print( ( TOSTRING d12, " - ", TOSTRING a12, " = " ) );
BOOL carry := FALSE;
FOR b pos FROM UPB d12 BY -1 TO LWB d12 DO
d12[ b pos ] -:= a12[ b pos ];
BOOL need carry = carry OF d12[ b pos ];
IF carry THEN d12[ b pos ] -:= bcd 1 FI;
carry := need carry OR carry OF d12[ b pos ]
OD;
print( ( TOSTRING d12, newline ) )
OD

END```
Output:
```20
29
100

012345678901 + 011111111111 = 023456790012
023456790012 + 011111111111 = 034567901123
034567901123 + 011111111111 = 045679012234
045679012234 + 011111111111 = 056790123345
056790123345 + 011111111111 = 067901234456
067901234456 + 011111111111 = 079012345567
079012345567 + 011111111111 = 090123456678
090123456678 + 011111111111 = 101234567789
101234567789 + 011111111111 = 112345678900
112345678900 + 011111111111 = 123456790011
123456790011 - 011111111111 = 112345678900
112345678900 - 011111111111 = 101234567789
101234567789 - 011111111111 = 090123456678
090123456678 - 011111111111 = 079012345567
079012345567 - 011111111111 = 067901234456
067901234456 - 011111111111 = 056790123345
056790123345 - 011111111111 = 045679012234
045679012234 - 011111111111 = 034567901123
034567901123 - 011111111111 = 023456790012
023456790012 - 011111111111 = 012345678901
```

## ALGOL W

Translation of: ALGOL 68
```begin % implements packed BCD arithmetic                                     %
integer X99;                        % maximum unsigned 2-digit BCD value %
% structure to hold BCD values                                           %
record BCD ( integer dValue             % signed BCD value:  -x99 to x99 %
; logical dCarry             % TRUE if the value overflowed,  %
);                           % FALSE otherwise                %
reference(BCD) bcd99, bcd1, bcd0;
% constructs a BCD value from a, assuming it is in the correct format    %
% if the value has overflowed, it is truncated to a valid value and      %
% carry is set                                                           %
reference(BCD) procedure asBcd ( integer value a ) ;
begin
integer v;
logical carry;
v     := abs a;
carry := v > X99;
if carry then v := ( ( ( v div 16 ) rem 10 ) * 16 ) + ( v rem 16 );
BCD( if a < 0 then - v else v, carry )
end asBcd ;
% returns a converted to BCD format, truncating and setting carry        %
%         if necessary                                                   %
reference(BCD) procedure toBcd ( integer value a ) ;
if   a < 0
then negateBcd( toBcd( abs a ) )
else BCD( ( ( ( a div 10 ) rem 10 ) * 16 ) + ( a rem 10 ), a > X99 )
;
% returns the value of b negated, carry is preserved                     %
reference(BCD) procedure negateBcd ( reference(BCD) value a ) ; BCD( - dValue(a), dCarry(a) );
% writes a two-digit string representation of the BCD value a            %
procedure writeOnBcd ( reference(BCD) value a ) ;
begin
if dValue(a) < 0 then writeon( s_w := 0, "-" );
writeon( i_w := 1, s_w := 0
, abs dValue(a) div 16
, abs dValue(a) rem 16
)
end writeOnBcd;
% writes a BCD value with a preceeding newline                           %
procedure writeBcd ( reference(BCD) value a ) ; begin write(); writeOnBcd( a ) end;
% writes an array of BCD values - the bounds should be 1 :: ub           %
procedure showBcd ( reference(BCD) array a ( * ); integer value ub ) ;
for i := 1 until ub do writeOnBcd( a( i ) );

% returns the sum of a and b, a and b can be positive or negative        %
reference(BCD) procedure addBcd ( reference(BCD) value a, b ) ;
begin
integer av, bv, a2, b2, bcdResult;
logical ap, bp;
av := abs dValue(a);      bv := abs dValue(b);
ap :=     dValue(a) >= 0; bp :=     dValue(b) >= 0;
a2 := av rem 16;          b2 := bv rem 16;
if    ap = bp then begin
bcdResult := av + bv;
if a2 + b2 > 9 then bcdResult :=   bcdResult + 6;
if not ap      then bcdResult := - bcdResult
end
else if av >= bv then begin
bcdResult := av - bv;
if a2 < b2 then bcdResult :=   bcdResult - 6;
if not ap  then bcdResult := - bcdResult
end
else begin
bcdResult := bv - av;
if b2 < a2 then bcdResult :=   bcdResult - 6;
if ap      then bcdResult := - bcdResult
end if_ap_eq_bp__av_ge_bv__;
if bcdResult >= 0 then begin                    % result is positive %
asBcd( bcdResult )
end
else begin                       % negative result - tens complement %
reference(BCD) sum;
dCarry(sum) := true;
sum
end if_bcdResult_ge_0__
% returns the difference of a and b, a and b can be positive or negative %
reference(BCD) procedure subtractBcd ( reference(BCD) value a, b ) ; addBcd( a, negateBcd( b ) );

X99   := ( 9 * 16 ) + 9;
bcd99 := toBcd( 99 );
bcd1  := toBcd(  1 );
bcd0  := toBcd(  0 );

begin % task test cases                                                  %
reference(BCD) r;
writeBcd( addBcd(      toBcd( 19 ), toBcd( 1 ) ) );
writeBcd( subtractBcd( toBcd( 30 ), toBcd( 1 ) ) );
r := addBcd(           toBcd( 99 ), toBcd( 1 ) );
if dCarry(r) then write( s_w := 0, "1" );
writeOnBcd( r );
end;

begin % use the 2-digit BCD to add/subtract larger numbers               %
reference(BCD) array d12, a12 ( 1 :: 6 );
integer dPos;
write();
dPos := 0;
for v := 1, 23, 45, 67, 89, 01 do begin
dPos := dPos + 1;
d12( dPos ) := toBcd( v )
end for_v ;
dPos := 0;
for v := 1, 11, 11, 11, 11, 11 do begin
dPos := dPos + 1;
a12( dPos ) := toBcd( v )
end for_v ;
for i := 1 until 10 do begin             % repeatedly add a12 to d12 %
logical carry;
write();showBcd( d12, 6 );writeon( " + " );showBcd( a12, 6 );writeon( " = " );
carry := false;
for bPos := 6 step -1 until 1 do begin
logical needCarry;
d12( bPos ) := addBcd( d12( bPos ), a12( bPos ) );
needCarry := dCarry(d12( bPos ));
if carry then d12( bPos ) := addBcd( d12( bPOs ), bcd1 );
carry := needCarry or dCarry(d12( bPos ))
end for_bPos ;
showBcd( d12, 6 )
end for_i;
for i := 1 until 10 do begin      % repeatedly subtract a12 from d12 %
logical carry;
write();showBcd( d12, 6 );writeon( " - " );showBcd( a12, 6 );writeon( " = " );
carry := false;
for bPos := 6 step -1 until 1 do begin
logical needCarry;
d12( bPos ) := subtractBcd( d12( bPos ), a12( bPos ) );
needCarry := dCarry(d12( bPos ));
if carry then d12( bPos ) := subtractBcd( d12( bPOs ), bcd1 );
carry := needCarry or dCarry(d12( bPos ))
end for_bPos ;
showBcd( d12, 6 )
end for_i;
end

end.
```
Output:
```20
29
100

012345678901 + 011111111111 = 023456790012
023456790012 + 011111111111 = 034567901123
034567901123 + 011111111111 = 045679012234
045679012234 + 011111111111 = 056790123345
056790123345 + 011111111111 = 067901234456
067901234456 + 011111111111 = 079012345567
079012345567 + 011111111111 = 090123456678
090123456678 + 011111111111 = 101234567789
101234567789 + 011111111111 = 112345678900
112345678900 + 011111111111 = 123456790011
123456790011 - 011111111111 = 112345678900
112345678900 - 011111111111 = 101234567789
101234567789 - 011111111111 = 090123456678
090123456678 - 011111111111 = 079012345567
079012345567 - 011111111111 = 067901234456
067901234456 - 011111111111 = 056790123345
056790123345 - 011111111111 = 045679012234
045679012234 - 011111111111 = 034567901123
034567901123 - 011111111111 = 023456790012
023456790012 - 011111111111 = 012345678901
```

## C++

Translation of: Rust
```#include <cassert>
#include <cstdint>
#include <iostream>

class bcd64 {
public:
constexpr explicit bcd64(uint64_t bits = 0) : bits_(bits) {}
constexpr bcd64& operator+=(bcd64 other) {
uint64_t t1 = bits_ + 0x0666666666666666;
uint64_t t2 = t1 + other.bits_;
uint64_t t3 = t1 ^ other.bits_;
uint64_t t4 = ~(t2 ^ t3) & 0x1111111111111110;
uint64_t t5 = (t4 >> 2) | (t4 >> 3);
bits_ = t2 - t5;
return *this;
}
constexpr bcd64 operator-() const {
uint64_t t1 = static_cast<uint64_t>(-static_cast<int64_t>(bits_));
uint64_t t2 = t1 + 0xFFFFFFFFFFFFFFFF;
uint64_t t3 = t2 ^ 1;
uint64_t t4 = ~(t2 ^ t3) & 0x1111111111111110;
uint64_t t5 = (t4 >> 2) | (t4 >> 3);
return bcd64(t1 - t5);
}
friend constexpr bool operator==(bcd64 a, bcd64 b);
friend std::ostream& operator<<(std::ostream& os, bcd64 a);

private:
uint64_t bits_;
};

constexpr bool operator==(bcd64 a, bcd64 b) { return a.bits_ == b.bits_; }

constexpr bool operator!=(bcd64 a, bcd64 b) { return !(a == b); }

constexpr bcd64 operator+(bcd64 a, bcd64 b) {
bcd64 sum(a);
sum += b;
return sum;
}

constexpr bcd64 operator-(bcd64 a, bcd64 b) { return a + -b; }

std::ostream& operator<<(std::ostream& os, bcd64 a) {
auto f = os.flags();
os << std::showbase << std::hex << a.bits_;
os.flags(f);
return os;
}

int main() {
constexpr bcd64 one(0x01);
assert(bcd64(0x19) + one == bcd64(0x20));
std::cout << bcd64(0x19) + one << '\n';
assert(bcd64(0x30) - one == bcd64(0x29));
std::cout << bcd64(0x30) - one << '\n';
assert(bcd64(0x99) + one == bcd64(0x100));
std::cout << bcd64(0x99) + one << '\n';
}
```
Output:
```0x20
0x29
0x100
```

## Forth

This code implements direct BCD arithmetic using notes from Douglas Jones from the University of Iowa: https://homepage.cs.uiowa.edu/~jones/bcd/bcd.html#packed

```\ add two 15 digit bcd numbers
\
: bcd+ ( n1 n2 -- n3 )
0x0666666666666666 +    \ offset the digits in n2
2dup xor                \ add, discounting carry
-rot + swap             \ add with carry (only carries have correct digit)
over xor                \ bitmask of where carries occurred.
invert 0x1111111111111110 and   \ invert then change digit to 6
dup 2 rshift swap 3 rshift or   \ in each non-carry position

: bcdneg ( n -- n )    \ reduction of 9999...9999 swap - 1 bcd+
negate 0x0FFFFFFFFFFFFFFF and dup 1-
1 xor over xor invert 0x1111111111111110 and
dup 2 rshift swap 3 rshift or - ;

: bcd-  bcdneg bcd+ ;
```
Output:
```Gforth 0.7.3, Copyright (C) 1995-2008 Free Software Foundation, Inc.
Gforth comes with ABSOLUTELY NO WARRANTY; for details type `license'
Type `bye' to exit
hex  ok
19 1 bcd+ . 20  ok
30 1 bcd- . 29  ok
99 1 bcd+ . 100  ok
```

## J

Here, we represent hexadecimal numbers using J's constant notation, and to demonstrate bcd we generate results in that representation:

```   bcd=: &.((10 #. 16 #.inv ". ::]) :. ('16b',16 hfd@#. 10 #.inv ]))
16b19 +bcd 1
16b20
16b30 -bcd 1
16b29
16b99 +bcd 1
16b100
(16b99 +bcd 1) -bcd 1
16b99
```

Note that we're actually using a hex representation as an intermediate result here. Technically, though, sticking with built in arithmetic and formatting as decimal, but gluing the '16b' prefix onto the formatted result would have been more efficient. And that says a lot about bcd representation. (The value of bcd is not efficiency, but how it handles edge cases. Consider the decimal IEEE 754 format as an example where this might be considered significant. There are other ways to achieve those edge cases -- bcd happens to be relevant when building the mechanisms into hardware.)

For reference, here are decimal and binary representations of the above numbers:

```   (":,_16{.' '-.~'2b',":@#:) 16b19
25         2b11001
(":,_16{.' '-.~'2b',":@#:) 16b20
32        2b100000
(":,_16{.' '-.~'2b',":@#:) 16b29
41        2b101001
(":,_16{.' '-.~'2b',":@#:) 16b30
48        2b110000
(":,_16{.' '-.~'2b',":@#:) 16b99
153      2b10011001
(":,_16{.' '-.~'2b',":@#:) 16b100
256     2b100000000
2b11001
25
NB. ...
```

## Julia

Handles negative and floating point numbers (but avoid BigFloats due to very long decimal places from binary to decimal conversion).

```const nibs = [0b0, 0b1, 0b10, 0b11, 0b100, 0b101, 0b110, 0b111, 0b1000, 0b1001]

"""
function bcd_decode(data::Vector{codeunit}, sgn, decimalplaces; table = nibs)

Decode BCD number
bcd: packed BCD data as vector of bytes
sgn: sign(positive 1, negative -1, zero 0)
decimalplaces: decimal places from end for placing decimal point (-1 if none)
table: translation table, defaults to same as nibble (nibs table)
"""
function bcd_decode(bcd::Vector{UInt8}, sgn, decimalplaces = 0; table = nibs)
decoded = 0
for (i, byt) in enumerate(bcd)
decoded = decoded * 10 + table[byt >> 4 + 1]
decoded = decoded * 10 + table[byt & 0b1111 + 1]
end
return decimalplaces == 0 ? sgn * decoded : sgn * decoded / 10^decimalplaces
end

"""
function bcd_encode(number::Real; table::Vector{UInt8} = nibs)

Encode real number as BCD.
`number`` is in native binary formats
`table`` is the table used for encoding the nibbles of the decimal digits, default `nibs`
Returns: BCD encoding vector of UInt8, number's sign (1, 0 -1), and position of decimal point
"""
function bcd_encode(number::Real; table::Vector{UInt8} = nibs)
if (sgn = sign(number)) < 0
number = -number
end
s = string(number)
if (exponentfound = findlast(ch -> ch in ['e', 'E'], s)) != nothing
expplace = parse(Int, s[exponentfound+1:end])
s = s[begin:exponentfound-1]
else
expplace = 0
end
if (decimalplaces = findfirst(==('.'), s)) != nothing
s = s[begin:decimalplaces-1] * s[decimalplaces+1:end]
decimalplaces = length(s) - decimalplaces + 1
decimalplaces -= expplace
else
decimalplaces = -expplace
end
len = length(s)
if isodd(len)
s = "0" * s
len += 1
end
return [table[s[i+1]-'0'+1] | (table[s[i]-'0'+1] << 4) for i in 1:2:len-1], sgn, decimalplaces
end

"""
function bcd_encode(number::Integer; table::Vector{UInt8} = nibs)

Encode integer as BCD.
`number`` is in native binary formats
`table`` is the table used for encoding the nibbles of the decimal digits, default `nibs`
Returns: Tuple containg two values: a BCD encoded vector of UInt8 and the number's sign (1, 0 -1)
"""
function bcd_encode(number::Integer; table::Vector{UInt8} = nibs)
if (sgn = sign(number)) < 0
number = -number
end
s = string(number)
len = length(s)
if isodd(len)
s = "0" * s
len += 1
end
return [table[s[i+1]-'0'+1] | (table[s[i]-'0'+1] << 4) for i in 1:2:len-1], sgn
end

for test in [1, 2, 3, -9876, 10, 12342436]
enc = bcd_encode(test, table = nibs)
dec = bcd_decode(enc..., table = nibs)
println("\$test encoded is \$enc, decoded is \$dec")
end

for test in [-987654.321, -10.0, 9.9999, 123424367.0089]
enc = bcd_encode(test, table = nibs)
dec = bcd_decode(enc..., table = nibs)
println("\$test encoded is \$enc, decoded is \$dec")
end

println("BCD 19 (\$(bcd_encode(19))) + BCD 1 (\$(bcd_encode(1))) = BCD 20 " *
"(\$(bcd_encode(bcd_decode(bcd_encode(19)...) + bcd_decode(bcd_encode(1)...))))")
println("BCD 30 (\$(bcd_encode(30))) - BCD 1 (\$(bcd_encode(1))) = BCD 29 " *
"(\$(bcd_encode(bcd_decode(bcd_encode(30)...) - bcd_decode(bcd_encode(1)...))))")
println("BCD 99 (\$(bcd_encode(99))) + BCD 1 (\$(bcd_encode(1))) = BCD 100 " *
"(\$(bcd_encode(bcd_decode(bcd_encode(99)...) + bcd_decode(bcd_encode(1)...))))")
```
Output:
```1 encoded is (UInt8[0x01], 1), decoded is 1
2 encoded is (UInt8[0x02], 1), decoded is 2
3 encoded is (UInt8[0x03], 1), decoded is 3
-9876 encoded is (UInt8[0x98, 0x76], -1), decoded is -9876
10 encoded is (UInt8[0x10], 1), decoded is 10
12342436 encoded is (UInt8[0x12, 0x34, 0x24, 0x36], 1), decoded is 12342436
-987654.321 encoded is (UInt8[0x09, 0x87, 0x65, 0x43, 0x21], -1.0, 3), decoded is -987654.321
-10.0 encoded is (UInt8[0x01, 0x00], -1.0, 1), decoded is -10.0
9.9999 encoded is (UInt8[0x09, 0x99, 0x99], 1.0, 4), decoded is 9.9999
1.234243670089e8 encoded is (UInt8[0x01, 0x23, 0x42, 0x43, 0x67, 0x00, 0x89], 1.0, 4), decoded is 1.234243670089e8
BCD 19 (UInt8[0x19]) + BCD 1 ((UInt8[0x01], 1)) = BCD 20 ((UInt8[0x20], 1))
BCD 30 (UInt8[0x30]) - BCD 1 ((UInt8[0x01], 1)) = BCD 29 ((UInt8[0x29], 1))
BCD 99 (UInt8[0x99]) + BCD 1 ((UInt8[0x01], 1)) = BCD 100 ((UInt8[0x01, 0x00], 1))
```

## Pascal

### Free Pascal

There exist a special unit for BCD, even with fractions.Obvious for Delphi compatibility.

```program CheckBCD;
// See https://wiki.freepascal.org/BcdUnit
{\$IFDEF FPC}  {\$MODE objFPC}{\$ELSE} {\$APPTYPE CONSOLE} {\$ENDIF}
uses
sysutils,fmtBCD {\$IFDEF WINDOWS},Windows{\$ENDIF}  ;

{type
TBcd  = packed record
Precision: Byte;
SignSpecialPlaces: Byte;
Fraction: packed array [0..31] of Byte;
end;}
var
Bcd0,Bcd1,BcdOut : tBCD;
Begin
Bcd1 := IntegerToBcd(1);
//         0x19 + 1 = 0x20
Bcd0 := IntegerToBcd(19);
writeln(BcdToStr(Bcd0),'+',BcdToStr(Bcd1),' =',BcdToStr(BcdOut));
//      0x30 - 1 = 0x29
Bcd0 := IntegerToBcd(29);
writeln(BcdToStr(Bcd0),'+',BcdToStr(Bcd1),' =',BcdToStr(BcdOut));
//      0x99 + 1 = 0x100
Bcd0 := IntegerToBcd(99);
writeln(BcdToStr(Bcd0),'+',BcdToStr(Bcd1),' =',BcdToStr(BcdOut));
BcdMultiply(Bcd0,Bcd0,BcdOut);
writeln(BcdToStr(Bcd0),'*',BcdToStr(Bcd0),' =',BcdToStr(BcdOut));
end.
```
Output:
```19+1 =20
29+1 =30
99+1 =100
99*99 =9801
```

## Phix

### using fbld and fbstp

The FPU maths is all as normal (decimal), it is only the load and store that convert from/to BCD.
While I supply everything in decimal, you could easily return and pass around the likes of acc and res.

```without javascript_semantics -- (not a chance!)

function h(string s)
-- convert the 10 bytes BCD, as held in
-- a binary string, to a decimal string.
for i=length(s) to 1 by -1 do
if s[i]!='\0' or i=1 then
string res = sprintf("%x",s[i])
for j=i-1 to 1 by -1 do
res &= sprintf("%02x",s[j])
end for
return res
end if
end for
end function

procedure test(integer a, b)
-- Some (binary) strings to hold 10 byte BCDs:
string acc = repeat('\0',10),
res = repeat('\0',10)
#ilASM{
mov eax,[a]
mov edx,[b]
mov esi,[acc]
mov edi,[res]
push eax
fild dword[esp]
fbstp tbyte[ebx+esi*4]  -- save as 10 byte BCD
fbld tbyte[ebx+esi*4]   -- reload proves we can
mov [esp],edx
fild dword[esp]
fbstp tbyte[ebx+edi*4]
pop eax     -- (discard temp workspace)
}
integer pm = iff(b>=0?'+':'-')
printf(1,"%s %c %d = %s\n",{h(acc),pm,abs(b),h(res)})
end procedure
test(19,+1)
test(30,-1)
test(99,+1)
```
Output:
```19 + 1 = 20
30 - 1 = 29
99 + 1 = 100
```

### using daa and das

This time we'll supply the arguments in hex/BCD. Note the result is limited to 16 bits plus one carry bit here.
The aaa, aas, aam, and aad instructions are also available. Same output as above, of course

```without javascript_semantics -- (not a chance!)
requires(32)        -- aaa etc not valid on 64 bit

procedure test2(integer bcd, op)
integer res
#ilASM{
mov eax,[bcd]
mov ecx, 1
cmp [op],'+'
jne :sub1
daa
jmp @f
::sub1
sub al,cl
das
@@:
mov[res],eax
}
printf(1,"%x %c 1 = %x\n",{bcd,op,res})
end procedure
test2(#19,'+')
test2(#30,'-')
test2(#99,'+')
```

### hll bit fiddling

With routines to convert between decimal and bcd, same output as above, of course. No attempt has been made to support fractions or negative numbers...

```with javascript_semantics -- (no requires() needed here)
function bcd_decode(integer bcd)
assert(bcd>=0)
integer res = 0, dec = 1
while bcd do
res += and_bits(bcd,#F)*dec
bcd = bcd >> 4
dec *= 10
end while
return res
end function

function bcd_encode(integer dec)
assert(dec>=0)
integer res = 0, shift = 0
while dec do
res += remainder(dec,10) << shift
dec = trunc(dec/10)
shift += 4
end while
return res
end function

procedure test3(integer dec, op)
integer bcd = bcd_encode(dec),
work = bcd, res = 0, shift = 0,
carry = 1
while work or carry do
integer digit = (work && #F)
if op='+' then
digit += carry
if digit>9 then
digit -= 10
carry = 1
else
carry = 0
end if
else
digit -= carry
if digit<0 then
digit += 10
carry = 1
else
carry = 0
end if
end if
res += digit<<shift
work = work>>4
shift += 4
end while
printf(1,"%d %c 1 = %d\n",{bcd_decode(bcd),op,bcd_decode(res)})
end procedure
test3(19,'+')
test3(30,'-')
test3(99,'+')
```

## PL/M

Works with: 8080 PL/M Compiler
... under CP/M (or an emulator)

The 8080 PL/M compiler supports packed BCD by wrapping the 8080/Z80 DAA instruction with the DEC built in function, demonstrated here. Unfortunately, I couldn't get the first use of DEC to yeild the correct result without first doing a shift operation. Not sure if this is a bug in the program, the compiler or the 8080 emulator or that I'm misunderstanding something...

This is basically
Translation of: Z80 Assembly
```100H: /* DEMONSTRATE PL/M'S BCD HANDLING                                     */

BDOS: PROCEDURE( FN, ARG ); /* CP/M BDOS SYSTEM CALL                      */
GOTO 5;
END BDOS;
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C ); END;
PR\$NL:     PROCEDURE; CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH );  END;

PR\$BCD:    PROCEDURE( V );                  /* PRINT A 2-DIGIT BCD NUMBER */
DECLARE V BYTE;
DECLARE D BYTE;
D = SHR( V AND 0F0H, 4 );
CALL PR\$CHAR( D + '0' );
D = V AND 0FH;
CALL PR\$CHAR( D + '0' );
END PR\$BCD ;

DECLARE ( A, B, I ) BYTE;

A = SHL( 1, 4 );      /* WORKS AROUND A POSSIBLE BUG IN THE 8080 EMULATOR */
/* OR MY UNDERSTANDING OF THE DEC() FUNCTION...     */
A = 19H;
CALL PR\$BCD( DEC( A + 1 ) ); CALL PR\$NL;
A = 30H;
CALL PR\$BCD( DEC( A - 1 ) ); CALL PR\$NL;
B = 00H;
A = 99H;
A = DEC( A  +   1 );           /*       ADD 1 TO 99 - THIS WILL SET CARRY */
B = DEC( B PLUS 0 );           /* ADD THE CARRY TO GET THE LEADING DIGITS */
CALL PR\$BCD( B ); CALL PR\$BCD( A ); CALL PR\$NL;

EOF```
Output:
```20
29
0100
```

A more complex example, showing how the DEC function can be used to perform unsigned BCD addition and subtraction on arbitrary length BCD numbers.

```100H: /* DEMONSTRATE PL/M'S BCD HANDLING                                     */

BDOS: PROCEDURE( FN, ARG ); /* CP/M BDOS SYSTEM CALL                      */
GOTO 5;
END BDOS;
PR\$CHAR:   PROCEDURE( C ); DECLARE C BYTE;    CALL BDOS( 2, C ); END;
PR\$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;
PR\$NL:     PROCEDURE; CALL PR\$CHAR( 0DH ); CALL PR\$CHAR( 0AH );  END;

PR\$BCD:    PROCEDURE( V );                  /* PRINT A 2-DIGIT BCD NUMBER */
DECLARE V BYTE;
DECLARE D BYTE;
D = SHR( V AND 0F0H, 4 );
CALL PR\$CHAR( D + '0' );
D = V AND 0FH;
CALL PR\$CHAR( D + '0' );
END PR\$BCD ;

DECLARE ( A, B, C, D, E, F, I ) BYTE;

F =  1H;                /* CONSTRUCT 12345678901 AS A 12 DIGIT BCD NUMBER */
E = 23H;                                           /* IN F, E, D, C, B. A */
D = 45H;
C = 67H;
B = 89H;
A = 01H;

DO I = 1 TO 10;               /* REPEATEDLY ADD 11111111111 TO THE NUMBER */
CALL PR\$BCD( F );
CALL PR\$BCD( E );
CALL PR\$BCD( D );
CALL PR\$BCD( C );
CALL PR\$BCD( B );
CALL PR\$BCD( A );
CALL PR\$STRING( .' + 011111111111 = \$' );
A = DEC( A  +   11H );    /* THE PARAMETER TO THE DEC BUILTIN FUNCTION */
B = DEC( B PLUS 11H );    /* MUST BE A CONSTANT OR UNSCRIPTED VARIABLE */
C = DEC( C PLUS 11H );    /* +/-/PLUS/MINUS ANOTHER CONSTANT OR        */
D = DEC( D PLUS 11H );    /* UNSUBSCRIPTED VARIABLE                    */
E = DEC( E PLUS 11H );    /* ( WHICH MUST CONTAIN 2-DIGIT BCD VALUES ).*/
F = DEC( F PLUS  1  );    /* PLUS/MINUS PERFORM ADDITION/SUBTRACTION   */
CALL PR\$BCD( F );         /* INCLUDING THE CARRY FROM THE PREVIOUS     */
CALL PR\$BCD( E );         /* OPERATION, +/- IGNORE THE CARRY.          */
CALL PR\$BCD( D );         /* THE RESULT IS ADJUSTED TO BE A 2-DIGIT    */
CALL PR\$BCD( C );         /* BCD VALUE AND THE CARRY FLAG IS SET       */
CALL PR\$BCD( B );         /* ACCORDINGLY                               */
CALL PR\$BCD( A );
CALL PR\$NL;
END;

DO I = 1 TO 10;        /* REPEATEDLY SUBTRACT 11111111111 FROM THE NUMBER */
CALL PR\$BCD( F );
CALL PR\$BCD( E );
CALL PR\$BCD( D );
CALL PR\$BCD( C );
CALL PR\$BCD( B );
CALL PR\$BCD( A );
CALL PR\$STRING( .' - 011111111111 = \$' );
A = DEC( A   -   11H );
B = DEC( B MINUS 11H );
C = DEC( C MINUS 11H );
D = DEC( D MINUS 11H );
E = DEC( E MINUS 11H );
F = DEC( F MINUS  1  );
CALL PR\$BCD( F );
CALL PR\$BCD( E );
CALL PR\$BCD( D );
CALL PR\$BCD( C );
CALL PR\$BCD( B );
CALL PR\$BCD( A );
CALL PR\$NL;
END;

EOF```
Output:
```012345678901 + 011111111111 = 023456790012
023456790012 + 011111111111 = 034567901123
034567901123 + 011111111111 = 045679012234
045679012234 + 011111111111 = 056790123345
056790123345 + 011111111111 = 067901234456
067901234456 + 011111111111 = 079012345567
079012345567 + 011111111111 = 090123456678
090123456678 + 011111111111 = 101234567789
101234567789 + 011111111111 = 112345678900
112345678900 + 011111111111 = 123456790011
123456790011 - 011111111111 = 112345678900
112345678900 - 011111111111 = 101234567789
101234567789 - 011111111111 = 090123456678
090123456678 - 011111111111 = 079012345567
079012345567 - 011111111111 = 067901234456
067901234456 - 011111111111 = 056790123345
056790123345 - 011111111111 = 045679012234
045679012234 - 011111111111 = 034567901123
034567901123 - 011111111111 = 023456790012
023456790012 - 011111111111 = 012345678901
```

## Raku

Translation of: Rust
```# 20220930 Raku programming solution

class Bcd64 { has uint64 \$.bits }

multi infix:<⊞> (Bcd64 \p, Bcd64 \q) {
my \$t1 = p.bits + 0x0666_6666_6666_6666;
my \$t2 = ( \$t1 + q.bits ) % uint64.Range.max ;
my \$t3 = \$t1 +^ q.bits;
my \$t4 = +^(\$t2 +^ \$t3) +& 0x1111_1111_1111_1110;
my \$t5 = (\$t4 +> 2) +| (\$t4 +> 3);
Bcd64.new: bits => (\$t2 - \$t5)
}

multi prefix:<⊟> (Bcd64 \p) {
my \$t1 = uint64.Range.max + 1 - p.bits ;
my \$t2 = ( \$t1 + 0xFFFF_FFFF_FFFF_FFFF ) % uint64.Range.max;
my \$t3 = \$t2 +^ 1;
my \$t4 = +^(\$t2 +^ \$t3) +& 0x1111_1111_1111_1110;
my \$t5 = (\$t4 +> 2) +| (\$t4 +> 3);
Bcd64.new: bits => (\$t1 - \$t5)
}

multi infix:<⊟> (Bcd64 \p, Bcd64 \q) { p ⊞ ( ⊟q ) }

my (\$one,\$n19,\$n30,\$n99) = (0x01,0x19,0x30,0x99).map: { Bcd64.new: bits=>\$_ };

{ .bits.base(16).say } for (\$n19 ⊞ \$one,\$n30 ⊟ \$one,\$n99 ⊞ \$one);
```
Output:
```20
29
100
```

## RPL

Translation of: Forth
Works with: Halcyon Calc version 4.2.7
```≪ #666666666666666h + DUP2 XOR ROT ROT + SWAP OVER XOR
NOT #1111111111111110h AND
DUP SR SR SWAP SR SR SR OR -
#FFFFFFFFFFFFFFFh AND

≪ NOT 1 + #FFFFFFFFFFFFFFFh AND DUP 1 - 1 XOR OVER XOR
NOT #1111111111111110h AND
DUP SR SR SWAP SR SR SR OR -
≫ 'NGBCD' STO

'SUBCD' STO

64 STWS HEX
#30 #1 SUBCD
```
Output:
```3: #20h
2: #100h
1: #29h
```

## Rust

Based on the Forth implementation re: how to implement BCD arithmetic in software. Uses operator overloading for new BCD type.

```#[derive(Copy, Clone)]
pub struct Bcd64 {
bits: u64
}

use std::ops::*;

type Output = Self;
fn add(self, other: Self) -> Self {
let t1 = self.bits + 0x0666_6666_6666_6666;
let t3 = t1 ^ other.bits;
let t4 = !(t2 ^ t3) & 0x1111_1111_1111_1110;
let t5 = (t4 >> 2) | (t4 >> 3);
return Bcd64{ bits: t2 - t5 };
}
}

impl Neg for Bcd64 {
type Output = Self;
fn neg(self) -> Self { // return 10's complement
let t1 = -(self.bits as i64) as u64;
let t3 = t2 ^ 1;
let t4 = !(t2 ^ t3) & 0x1111_1111_1111_1110;
let t5 = (t4 >> 2) | (t4 >> 3);
return Bcd64{ bits: t1 - t5 };
}
}

impl Sub for Bcd64 {
type Output = Self;
fn sub(self, other: Self) -> Self {
return self + -other;
}
}

#[test]
let one = Bcd64{ bits: 0x01 };
assert_eq!((Bcd64{ bits: 0x19 } + one).bits, 0x20);
assert_eq!((Bcd64{ bits: 0x30 } - one).bits, 0x29);
assert_eq!((Bcd64{ bits: 0x99 } + one).bits, 0x100);
}
```
Output:

For the output, use "cargo test" to run the unit test for this module.

```running 1 test

test result: ok. 1 passed; 0 failed; 0 ignored; 0 measured; 0 filtered out; finished in 0.00s
```

## Wren

Library: Wren-check
Library: Wren-math
Library: Wren-str
Library: Wren-fmt

In Wren all numbers are represented by 64 bit floats and the language has no real concept of bytes, nibbles or even integers.

The following is therefore a simulation of BCD arithmetic using packed binary strings to represent decimal digits. It only works for non-negative integral numbers.

We can change to 'unpacked' notation simply by prepending '0000' to each 'digit' of the 'packed' notation.

In what follows, the hex prefix '0x' is simply a way of representing BCD literals and has nothing to do with hexadecimal as such.

```import "./check" for Check
import "./math" for Int
import "./str" for Str
import "./fmt" for Fmt

class BCD {
static init_() {
__bcd = [
"0000", "0001", "0010", "0011", "0100",
"0101", "0110", "0111", "1000", "1001"
]
__dec = {
"0000": "0", "0001": "1", "0010": "2", "0011": "3", "0100": "4",
"0101": "5", "0110": "6", "0111": "7", "1000": "8", "1001": "9"
}
}

construct new(n) {
if (n is String) {
if (n.startsWith("0x")) n = n[2..-1]
n = Num.fromString(n)
}
Check.nonNegInt("n", n)
if (!__bcd) BCD.init_()
_b = ""
for (digit in Int.digits(n)) _b = _b + __bcd[digit]
}

toInt {
var ns = ""
for (nibble in Str.chunks(_b, 4)) ns = ns + __dec[nibble]
return Num.fromString(ns)
}

+(other) {
if (!(other is BCD)) other = BCD.new(other)
return BCD.new(this.toInt + other.toInt)
}

-(other) {
if (!(other is BCD)) other = BCD.new(other)
return BCD.new(this.toInt - other.toInt)
}

toString {
var ret = _b.trimStart("0")
if (ret == "") ret = "0"
return ret
}

toUnpacked {
var ret = ""
for (nibble in Str.chunks(_b, 4)) ret = ret + "0000" + nibble
ret = ret.trimStart("0")
if (ret == "") ret = "0"
return ret
}

toHex { "0x" + this.toInt.toString }
}

var hexs = ["0x19", "0x30", "0x99"]
var ops  = ["+", "-", "+"]
for (packed in [true, false]) {
for (i in 0...hexs.count) {
var op = ops[i]
var bcd = BCD.new(hexs[i])
var bcd2 = (op == "+") ? bcd + 1 : bcd - 1
var str = packed ? bcd.toString : bcd.toUnpacked
var str2 = packed ? bcd2.toString : bcd2.toUnpacked
var hex = bcd.toHex
var hex2 = bcd2.toHex
var un = packed ? "" : "un"
var w = packed ? 8 : 12
var args = [hex, op, hex2, un, w, str, op, str2]
Fmt.lprint("\$s \$s 1 = \$-5s or, in \$0spacked BCD, \$*s \$s 1 = \$s", args)
}
if (packed) System.print()
}
```
Output:
```0x19 + 1 = 0x20  or, in packed BCD,    11001 + 1 = 100000
0x30 - 1 = 0x29  or, in packed BCD,   110000 - 1 = 101001
0x99 + 1 = 0x100 or, in packed BCD, 10011001 + 1 = 100000000

0x19 + 1 = 0x20  or, in unpacked BCD,    100001001 + 1 = 1000000000
0x30 - 1 = 0x29  or, in unpacked BCD,   1100000000 - 1 = 1000001001
0x99 + 1 = 0x100 or, in unpacked BCD, 100100001001 + 1 = 10000000000000000
```

## Z80 Assembly

The `DAA` function will convert an 8-bit hexadecimal value to BCD after an addition or subtraction is performed. The algorithm used is actually quite complex, but the Z80's dedicated hardware for it makes it all happen in 4 clock cycles, tied with the fastest instructions the CPU can perform.

```PrintChar equ &BB5A  ;Amstrad CPC kernel's print routine
org &1000

ld a,&19
daa
call ShowHex
call NewLine

ld a,&30
sub 1
daa
call ShowHex
call NewLine

ld a,&99
daa
;this rolls over to 00 since DAA only works with the accumulator.
;But the carry is set by this operation, so we can work accordingly.

jr nc,continue  ;this branch is never taken, it exists to demonstrate the concept of how DAA affects the carry flag.
push af
ld a,1
call ShowHex
pop af
continue:
call ShowHex
call NewLine

ShowHex:
push af
and %11110000
rrca
rrca
rrca
rrca
call PrintHexChar
pop af
and %00001111
;call PrintHexChar
;execution flows into it naturally.
PrintHexChar:
;this little trick converts hexadecimal or BCD to ASCII.
or a	;Clear Carry Flag
daa
```20