Carry flag
Encyclopedia
In computer processors the carry flag (usually indicated as the C flag) is a single bit
in a system status (flag) register
used to indicate when an arithmetic
carry or borrow has been generated out of the most significant
ALU
bit position. The carry flag enables numbers larger than a single ALU width to be added/subtracted by carrying (adding) a binary digit from a partial addition/subtraction to the least significant bit
position of a more significant word. It is also used to extend bit shifts and rotates in a similar manner on many processors (sometimes done via a dedicated X flag). For subtractive operations, two (opposite) conventions are employed as most machines sets the carry flag on borrow while some machines (such as the 6502
and the PIC
) instead resets the carry flag on borrow (and vice versa).
s these instructions are called by names such as ADD/SUB, ADC/SBC (ADD/SUB including carry), SHL/SHR (bit shifts), ROL/ROR (bit rotates), RCR/RCL (rotate through carry), and so on. The use of the carry flag in this manner enables multi-word add, subtract, shift, and rotate operations.
An example is what happens if you were to add 255 and 255 using 8-bit
registers. The result should be 510 which is 1_1111_1110 in binary, requiring 9 bits. The 8 least significant bits always stored in the register would be 1111_1110 binary (254 decimal) but since there is carry out of bit 7 (the eight bit), the carry is set, indicating that the result needs 9-bits. The valid 9-bit result is the concatenation of the carry flag with the result. Note that in an 8-bit two's complement
interpretation, this operation is -1 + -1 and yields the correct result of -2, with no overflow
, even if the carry is ignored.
Another example may be an 8-bit register
with the bit pattern 0101_0101 and the carry flag set; if we execute a rotate left through carry instruction, the result would be 1010_1011 with the carry flag cleared because the most significant bit (bit 7) was rotated into the carry while the carry was rotated into the least significant bit (bit 0).
While the first micro processors Intel 4004
(CLC) and Intel 8008
(FC) had specific operations to set the carry bit to 0, the later Intel 8080
and it's compatible processors (such as the Z80) only had operations to set the carry bit to 1. For setting it to zero they use the logic instructions (AND, OR, XOR) which are actual does not use the carry bit at all for their functions.
One uses the bit as a borrow flag, setting it if a<b when computing a−b, and a borrow must be performed. A subtract with borrow (SBB) instruction will compute a−b−C = a−(b+C), while a subtract without borrow (SUB) acts as if the borrow bit were clear. The 8080, Z80, x86 and 68k
families (among others) use a borrow bit.
The other takes advantage of the identity that −x = not(x)+1 and computes a−b as a+not(b)+1. The carry flag is set according to this addition, and subtract with carry computes a+not(b)+C, while subtract without carry acts as if the carry bit were set. The 6502
and PowerPC
processors use this convention. The 6502 is a particularly well-known example because it does not have a subtract without carry operation, so software must ensure that the carry flag is set before every subtract operation where a borrow is not required.
The modern convention is to refer to the first alternative as a "borrow bit", while the second is called a "carry bit". However, there are exceptions in both directions; the VAX
and NS320xx
architectures use the borrow bit convention, but call their a−b−C operation "subtract with carry" (
uses a carry bit convention, but calls its a+not(b)+C operation "subtract with borrow" (
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
in a system status (flag) register
Status register
A status register or flag register is a collection of flag bits for a processor. An example is the FLAGS register of the x86 architecture....
used to indicate when an arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
carry or borrow has been generated out of the most significant
Most significant bit
In computing, the most significant bit is the bit position in a binary number having the greatest value...
ALU
Arithmetic logic unit
In computing, an arithmetic logic unit is a digital circuit that performs arithmetic and logical operations.The ALU is a fundamental building block of the central processing unit of a computer, and even the simplest microprocessors contain one for purposes such as maintaining timers...
bit position. The carry flag enables numbers larger than a single ALU width to be added/subtracted by carrying (adding) a binary digit from a partial addition/subtraction to the least significant bit
Least significant bit
In computing, the least significant bit is the bit position in a binary integer giving the units value, that is, determining whether the number is even or odd. The lsb is sometimes referred to as the right-most bit, due to the convention in positional notation of writing less significant digits...
position of a more significant word. It is also used to extend bit shifts and rotates in a similar manner on many processors (sometimes done via a dedicated X flag). For subtractive operations, two (opposite) conventions are employed as most machines sets the carry flag on borrow while some machines (such as the 6502
MOS Technology 6502
The MOS Technology 6502 is an 8-bit microprocessor that was designed by Chuck Peddle and Bill Mensch for MOS Technology in 1975. When it was introduced, it was the least expensive full-featured microprocessor on the market by a considerable margin, costing less than one-sixth the price of...
and the PIC
PIC
-Government and politics:* Partido Independiente de Color, a former Cuban political party* Peace Implementation Council, an international body charged with implementing the Dayton Peace Agreement for Bosnia and Herzegovina...
) instead resets the carry flag on borrow (and vice versa).
Uses
The carry flag is affected by the result of most arithmetic (and typically several bitwise) instructions and is also used as an input to many of them. Several of these instructions has two forms which either reads or ignores the carry. In assembly languageAssembly language
An assembly language is a low-level programming language for computers, microprocessors, microcontrollers, and other programmable devices. It implements a symbolic representation of the machine codes and other constants needed to program a given CPU architecture...
s these instructions are called by names such as ADD/SUB, ADC/SBC (ADD/SUB including carry), SHL/SHR (bit shifts), ROL/ROR (bit rotates), RCR/RCL (rotate through carry), and so on. The use of the carry flag in this manner enables multi-word add, subtract, shift, and rotate operations.
An example is what happens if you were to add 255 and 255 using 8-bit
8-bit
The first widely adopted 8-bit microprocessor was the Intel 8080, being used in many hobbyist computers of the late 1970s and early 1980s, often running the CP/M operating system. The Zilog Z80 and the Motorola 6800 were also used in similar computers...
registers. The result should be 510 which is 1_1111_1110 in binary, requiring 9 bits. The 8 least significant bits always stored in the register would be 1111_1110 binary (254 decimal) but since there is carry out of bit 7 (the eight bit), the carry is set, indicating that the result needs 9-bits. The valid 9-bit result is the concatenation of the carry flag with the result. Note that in an 8-bit two's complement
Two's complement
The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two...
interpretation, this operation is -1 + -1 and yields the correct result of -2, with no overflow
Overflow flag
In computer processors, the overflow flag is usually a single bit in a system status register used to indicate when an arithmetic overflow has occurred in an operation, indicating that the signed two's-complement result would not fit in the number of bits used for the operation...
, even if the carry is ignored.
Another example may be an 8-bit register
Processor register
In computer architecture, a processor register is a small amount of storage available as part of a CPU or other digital processor. Such registers are addressed by mechanisms other than main memory and can be accessed more quickly...
with the bit pattern 0101_0101 and the carry flag set; if we execute a rotate left through carry instruction, the result would be 1010_1011 with the carry flag cleared because the most significant bit (bit 7) was rotated into the carry while the carry was rotated into the least significant bit (bit 0).
While the first micro processors Intel 4004
Intel 4004
The Intel 4004 was a 4-bit central processing unit released by Intel Corporation in 1971. It was the first complete CPU on one chip, and also the first commercially available microprocessor...
(CLC) and Intel 8008
Intel 8008
The Intel 8008 was an early byte-oriented microprocessor designed and manufactured by Intel and introduced in April 1972. It was an 8-bit CPU with an external 14-bit address bus that could address 16KB of memory...
(FC) had specific operations to set the carry bit to 0, the later Intel 8080
Intel 8080
The Intel 8080 was the second 8-bit microprocessor designed and manufactured by Intel and was released in April 1974. It was an extended and enhanced variant of the earlier 8008 design, although without binary compatibility...
and it's compatible processors (such as the Z80) only had operations to set the carry bit to 1. For setting it to zero they use the logic instructions (AND, OR, XOR) which are actual does not use the carry bit at all for their functions.
Carry flag vs. Borrow flag
While the carry flag is well-defined for addition, there are two possible ways to use the carry flag for subtractive operations.One uses the bit as a borrow flag, setting it if a<b when computing a−b, and a borrow must be performed. A subtract with borrow (SBB) instruction will compute a−b−C = a−(b+C), while a subtract without borrow (SUB) acts as if the borrow bit were clear. The 8080, Z80, x86 and 68k
68k
The Motorola 680x0/m68000/68000 is a family of 32-bit CISC microprocessors. During the 1980s and early 1990s, they were popular in personal computers and workstations and were the primary competitors of Intel's x86 microprocessors...
families (among others) use a borrow bit.
The other takes advantage of the identity that −x = not(x)+1 and computes a−b as a+not(b)+1. The carry flag is set according to this addition, and subtract with carry computes a+not(b)+C, while subtract without carry acts as if the carry bit were set. The 6502
MOS Technology 6502
The MOS Technology 6502 is an 8-bit microprocessor that was designed by Chuck Peddle and Bill Mensch for MOS Technology in 1975. When it was introduced, it was the least expensive full-featured microprocessor on the market by a considerable margin, costing less than one-sixth the price of...
and PowerPC
PowerPC
PowerPC is a RISC architecture created by the 1991 Apple–IBM–Motorola alliance, known as AIM...
processors use this convention. The 6502 is a particularly well-known example because it does not have a subtract without carry operation, so software must ensure that the carry flag is set before every subtract operation where a borrow is not required.
The modern convention is to refer to the first alternative as a "borrow bit", while the second is called a "carry bit". However, there are exceptions in both directions; the VAX
VAX
VAX was an instruction set architecture developed by Digital Equipment Corporation in the mid-1970s. A 32-bit complex instruction set computer ISA, it was designed to extend or replace DEC's various Programmed Data Processor ISAs...
and NS320xx
NS320xx
The 320xx or NS32000 was a series of microprocessors from National Semiconductor . They were likely the first 32-bit general-purpose microprocessors on the market, but due to a number of factors never managed to become a major player...
architectures use the borrow bit convention, but call their a−b−C operation "subtract with carry" (
SBWC). PA-RISCPA-RISC
PA-RISC is an instruction set architecture developed by Hewlett-Packard. As the name implies, it is a reduced instruction set computer architecture, where the PA stands for Precision Architecture...
uses a carry bit convention, but calls its a+not(b)+C operation "subtract with borrow" (
SUBB).