Computer Organization and Design 笔记  Arithmetic for Computers
Addition and Subtraction

Add (
add
), and immediate (addi
), and subtract (sub
) cause exceptions on overflow. MIPS detects overflow with an exception (or interrupt ), which is an unscheduled procedure call. The address of current instruction is saved and the computer jumps to predefined address to invoke the appropriate routine for that exception.MIPS uses exception program counter (EPC) to contain the address of the instruction that causes the exception. The instruction move from system control (
mfc0
) is used to copy EPC into a generalpurpose register. 
Add unsigned (
addu
), add immediate unsigned (addiu
), and subtract unsigned (subu
) do not cause exceptions on overflow. Programmers can trap overflow anyway: when overflow occurs, the sign bit of the result is not properly set. Compairing with sign bits of operands, the sign bit of the result can be determined.
SIMD (single instruction, multiple data): By partitioning the carry chains within a 64bit adder, a processor could perform simultaneous operations on a short vecters of eight 8bit operands, four 16bit operands, etc. Vectors and 8bit data often appears in multimedia routine.
Multiplication
multiplicand * multiplier = product
Sequential Version of the Multiplication
Refined version:
 Init: put multiplier to the left 32bit of the product register.
 Cycle:
 if the last bit of product register is 1, add the left 32bit with the multiplicand
 shift right the product register
 Final: the product register contains the 64bit product
Faster Multiplication
A way to organize these 32 addtions is in a parallel tree:
Multiply in MIPS
The registers Hi
and Lo
contains the 64bit product. Call mflo
to fetch the 32bit product, mfhi
can be used to get Hi
to test for overflow.
Division
Dividend = Quotient * Divisor + Remainder
Division Algorithm
Improved version:
 Init: put the dividend in the right 32bit of remainder register.
 Cycle:
 subtract the left 32bit of remainder by the divisor
 shift left the remaider register
 set the last bit as new quotient bit
 Final: the left 32bit contains the remainder, right 32bit contains the quotient.
Faster Division
SRT division: try to guess several quotient bits per step, using a table lookup based on the upper bits of the dividend and remainder. The key is guessing the value to subtract.
Divide in MIPS
Hi
contains the remainder, and Lo
contains the quotient after the divide instruction complete.
MIPS divide instructions ignore overflow. MIPS software must check the divisor to discover division by 0 as well as overflow.
Floating Point
scientific notation A notation that renders numbers with a single digit to the left of the decimal point.
normalized A number in floatingpoint notation that has no leading 0s.
fraction The value, generally between 0 and 1, placed in the fraction field.
exponent In the numerical representation system of floatingpoint arithmetic, the value that is placed in the exponent field.
overflow the exponent is too large to be represented in the exponent field.
floating point Computer arithmetic that represents numbers in which the binary point is not fixed.
In general, floatingpoint numbers are of the form: $(1)^S \times F \times 2^E$
MIPS float: sign(1 bit) + exponent(8 bit) + fraction(23 bit) MIPS double: s(1 bit) + exponent(11 bit) + fraction(52 bit)
IEEE 754 uses a bias of 127 for single precesion, and makes the leading 1 implicit. Since 0 has no leading 1, it’s given the reserved exponent 0 so that hardware won’t attach a leading 1.
Thus 00…00 represents 0; the representation of the rest are in the following form:
$(1)^S \times (1 + Fraction)\times 2^(Exponent  Bias)$
The exponent is located left and the bias is for comparison convenience.
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