1. "M" Standard Extension for Integer Multiplication and Division, Version 2.0

This chapter describes the standard integer multiplication and division instruction extension, which is named "M" and contains instructions that multiply or divide values held in two integer registers.

We separate integer multiply and divide out from the base to simplify low-end implementations, or for applications where integer multiply and divide operations are either infrequent or better handled in attached accelerators.

1.1. Multiplication Operations

MUL performs an XLEN-bit×XLEN-bit multiplication of rs1 by rs2 and places the lower XLEN bits in the destination register. MULH, MULHU, and MULHSU perform the same multiplication but return the upper XLEN bits of the full 2×XLEN-bit product, for signed×signed, unsigned×unsigned, and rs1×unsigned rs2 multiplication, respectively. If both the high and low bits of the same product are required, then the recommended code sequence is: MULH[[S]U] rdh, rs1, rs2; MUL rdl, rs1, rs2 (source register specifiers must be in same order and rdh cannot be the same as rs1 or rs2). Microarchitectures can then fuse these into a single multiply operation instead of performing two separate multiplies.

MULHSU is used in multi-word signed multiplication to multiply the most-significant word of the multiplicand (which contains the sign bit) with the less-significant words of the multiplier (which are unsigned).

MULW is an RV64 instruction that multiplies the lower 32 bits of the source registers, placing the sign-extension of the lower 32 bits of the result into the destination register.

In RV64, MUL can be used to obtain the upper 32 bits of the 64-bit product, but signed arguments must be proper 32-bit signed values, whereas unsigned arguments must have their upper 32 bits clear. If the arguments are not known to be sign- or zero-extended, an alternative is to shift both arguments left by 32 bits, then use MULH[[S]U].

1.2. Division Operations

DIV and DIVU perform an XLEN bits by XLEN bits signed and unsigned integer division of rs1 by rs2, rounding towards zero. REM and REMU provide the remainder of the corresponding division operation. For REM, the sign of a nonzero result equals the sign of the dividend.

For both signed and unsigned division, except in the case of overflow, it holds that \(\textrm{dividend} = \textrm{divisor} \times \textrm{quotient} + \textrm{remainder}\).

If both the quotient and remainder are required from the same division, the recommended code sequence is: DIV[U] rdq, rs1, rs2; REM[U] rdr, rs1, rs2 (rdq cannot be the same as rs1 or rs2). Microarchitectures can then fuse these into a single divide operation instead of performing two separate divides.

DIVW and DIVUW are RV64 instructions that divide the lower 32 bits of rs1 by the lower 32 bits of rs2, treating them as signed and unsigned integers respectively, placing the 32-bit quotient in rd, sign-extended to 64 bits. REMW and REMUW are RV64 instructions that provide the corresponding signed and unsigned remainder operations respectively. Both REMW and REMUW always sign-extend the 32-bit result to 64 bits, including on a divide by zero.

The semantics for division by zero and division overflow are summarized in Table 1. The quotient of division by zero has all bits set, and the remainder of division by zero equals the dividend. Signed division overflow occurs only when the most-negative integer is divided by \(-1\). The quotient of a signed division with overflow is equal to the dividend, and the remainder is zero. Unsigned division overflow cannot occur.

Table 1. Semantics for division by zero and division overflow. L is the width of the operation in bits: XLEN for DIV[U] and REM[U], or 32 for DIV[U]W and REM[U]W.

Condition	Dividend	Divisor	DIVU[W]	REMU[W]	DIV[W]	REM[W]
Division by zero Overflow (signed only)	\(x\) \(-2^{L-1}\)	0 \(-1\)	\(2^{L}-1\) -	\(x\) -	\(-1\) \(-2^{L-1}\)	\(x\) 0

We considered raising exceptions on integer divide by zero, with these exceptions causing a trap in most execution environments. However, this would be the only arithmetic trap in the standard ISA (floating-point exceptions set flags and write default values, but do not cause traps) and would require language implementors to interact with the execution environment’s trap handlers for this case. Further, where language standards mandate that a divide-by-zero exception must cause an immediate control flow change, only a single branch instruction needs to be added to each divide operation, and this branch instruction can be inserted after the divide and should normally be very predictably not taken, adding little runtime overhead. The value of all bits set is returned for both unsigned and signed divide by zero to simplify the divider circuitry. The value of all 1s is both the natural value to return for unsigned divide, representing the largest unsigned number, and also the natural result for simple unsigned divider implementations. Signed division is often implemented using an unsigned division circuit and specifying the same overflow result simplifies the hardware.

1.3. Zmmul Extension, Version 1.0

The Zmmul extension implements the multiplication subset of the M extension. It adds all of the instructions defined in Section 1.1, namely: MUL, MULH, MULHU, MULHSU, and (for RV64 only) MULW. The encodings are identical to those of the corresponding M-extension instructions. M implies Zmmul.

The Zmmul extension enables low-cost implementations that require multiplication operations but not division. For many microcontroller applications, division operations are too infrequent to justify the cost of divider hardware. By contrast, multiplication operations are more frequent, making the cost of multiplier hardware more justifiable. Simple FPGA soft cores particularly benefit from eliminating division but retaining multiplication, since many FPGAs provide hardwired multipliers but require dividers be implemented in soft logic.