# Hacker's Delight: The Basics

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## 2–14 Condition Code Result of Add, Subtract, and Multiply

Many machines provide a “condition code” that characterizes the result of integer arithmetic operations. Often there is only one add instruction, and the characterization reflects the result for both unsigned and signed interpretation of the operands and result (but not for mixed types). The characterization usually consists of the following:

• Whether or not carry occurred (unsigned overflow)
• Whether or not signed overflow occurred
• Whether the 32-bit result, interpreted as a signed two’s-complement integer and ignoring carry and overflow, is negative, 0, or positive

Some older machines give an indication of whether the infinite precision result (that is, 33-bit result for add’s and subtract’s) is positive, negative, or 0. However, this indication is not easily used by compilers of high-level languages, and so has fallen out of favor.

For addition, only nine of the 12 combinations of these events are possible. The ones that cannot occur are “no carry, overflow, result > 0,” “no carry, overflow, result = 0,” and “carry, overflow, result < 0.” Thus, four bits are, just barely, needed for the condition code. Two of the combinations are unique in the sense that only one value of inputs produces them: Adding 0 to itself is the only way to get “no carry, no overflow, result = 0,” and adding the maximum negative number to itself is the only way to get “carry, overflow, result = 0.” These remarks remain true if there is a “carry in”—that is, if we are computing x + y + 1.

For subtraction, let us assume that to compute xy the machine actually computes x + + 1, with the carry produced as for an add (in this scheme the meaning of “carry” is reversed for subtraction, in that carry = 1 signifies that the result fits in a single word, and carry = 0 signifies that the result does not fit in a single word). Then for subtraction, only seven combinations of events are possible. The ones that cannot occur are the three that cannot occur for addition, plus “no carry, no overflow, result = 0,” and “carry, overflow, result = 0.”

If a machine’s multiplier can produce a doubleword result, then two multiply instructions are desirable: one for signed and one for unsigned operands. (On a 4-bit machine, in hexadecimal, F × F = 01 signed, and F × F = E1 unsigned.) For these instructions, neither carry nor overflow can occur, in the sense that the result will always fit in a doubleword.

For a multiplication instruction that produces a one-word result (the low-order word of the doubleword result), let us take “carry” to mean that the result does not fit in a word with the operands and result interpreted as unsigned integers, and let us take “overflow” to mean that the result does not fit in a word with the operands and result interpreted as signed two’s-complement integers. Then again, there are nine possible combinations of results, with the missing ones being “no carry, overflow, result > 0,” “no carry, overflow, result = 0,” and “carry, no overflow, result = 0.” Thus, considering addition, subtraction, and multiplication together, ten combinations can occur.