Integer Multiplication

Transcription

1 Integer Multiplication Multiplying two numbers: multiplicand 1 1 multiplier m-bits n-bits = (m + n)-bit result m-bits: 2 m 1 is the largest number (2 m 1)(2 n 1) = 2 m+n 2 m 2 n + 1

2 Integer Multiplication: First Try 64 bit multiplicand shift_left bit multiplier shift_right 64b ALU alu_control lsb bit product write control FSM 64 How do we build this?

3 Registers And Shift Registers Register with shift left: 1 D 1 1 Q D Q D FF FF FF Q 1 D FF Q Register with write: Q Q D FF FF FF D D Q D Q 1 FF

4 Control start 1 add multiplicand to product check lsb shift multiplicand left by 1 shift multiplier right by 1 done Y 32nd iteration? N

5 Integer Multiplication Observations: 32 iterations for multiplication 32 cycles How long does 1 iteration take? Suppose 5% of ALU operations are multiply ops, and other ALU operations take 1 cycle. CP I alu = = 2.55! Half of the bits of the multiplicand are zero 64-bit adder is wasted 's inserted when multiplicand shifted left product LSBs don't change

6 Using A 32-Bit ALU 32 bit multiplicand bit multiplier shift_right 32b ALU alu_control lsb carry out of adder 32 shift_right control FSM top half write bit product

7 New Control start 1 check lsb add multiplicand to left product shift product reg right by 1 Bottom half of product register is zero initially. Each iteration: adds 1 product bit loses one multiplier bit don t forget to shift in the carry out when an add is performed! done shift multiplier right by 1 Y 32nd iteration? N Share storage for product register and multiplier!

8 Integer Multiplication Hardware 32 bit multiplicand 32 32b ALU alu_control carry out of adder 32 shift_right control FSM top half bit product write lsb initially, multiplier bits stored here.

9 Integer Multiplication Hardware b ALU alu_control 11 shift_right 1 1 control FSM write lsb

10 Integer Multiplication Hardware b ALU alu_control shift_right write control FSM lsb

11 Integer Multiplication Hardware b ALU alu_control shift_right 1 1 control FSM write lsb

12 Integer Multiplication Hardware b ALU alu_control 11 shift_right control FSM write lsb

13 Integer Multiplication Hardware b ALU alu_control shift_right 1 1 control FSM write lsb

14 Integer Multiplication Each step requires an add and shift MIPS: hi and lo registers correspond to the two parts of the product register Hardware implements multu Signed multiplication: Determine sign of the inputs, make inputs positive Use multu hardware, x up sign Better: Booth's algorithm

15 Booth Multiplication Example: multiplicand 1 1 multiplier Instead we could subtract early and add later... 6x = 2x + 4x = 2x + 8x 1111 = 1XXXX 1XXXX

16 Booth Multiplication run of 1s first "" after run of 1s add back! first "1" after subtract! Current Right Explanation 1 beginning of run of 1s 1 end of run of 1s 1 1 middle of run of 1s middle of run of s Originally for speed: shifts faster than adds

17 Booth Multiplication Depending on current and previous bits, do one of the following: : middle of a run of s no operation 1: end of a run of 1s add multiplicand to left half of product 1: start of a run of 1s subtract multiplicand from left half of product 11: middle of a run of 1s no operation As before, shift product register right by 1 bit per step.

18 Integer Division 11 quotient divisor dividend remainder Red: steps where subtracting would result in a negative number, i.e. quotient bit is zero.

19 Integer Division 11 quotient divisor dividend remainder Pad out the dividend and divisor to 8 bits.

20 Integer Division 64 bit divisor shift_right bit quotient shift_left 64b ALU alu_control result bit remainder write control FSM 64 msb

21 Integer Division start subtract divisor from remainder 1 check msb restore remainder quotient: shift in quotient: shift in 1 shift divisor right by 1 done Y 33rd iteration? N

22 Integer Division Observations: Half the bits in the divisor are zero 64-bit ALU wasted Instead of shifting divisor right, we can shift remainder left When does the rst iteration shift in a 1 into the quotient? save 1 iteration What is the initial value of the divisor?

23 Integer Division 32 bit divisor bit quotient shift_left 32b ALU alu_control result 32 shift_left control FSM top half 32 write 64 bit remainder msb

24 New Control start remainder := dividend << 1 subtract divisor from remainder Remainder loses one bit per iteration; 1 check msb Quotient gains one bit per iteration. restore remainder quotient: shift in quotient: shift in 1 => share registers! shift remainder left by 1 except last iteration! done Y 32nd iteration? N

25 Final Divider Hardware 32 bit divisor 32 32b ALU 32 alu_control shift_left control FSM top half bit remainder write result msb quotient bits go here

26 Mult/Div divisor/multiplicand 32 32b ALU alu_control carry out of adder 32 shift_left, shift_right top half bit remainder control FSM write msb result/lsb It's the same hardware...

27 Real Numbers How do we represent real numbers? Several issues: How many digits can we represent? What is the range? How accurate are mathematical operations? Consistency... Is a + b = b + a? Is (a + b) + c = a + (b + c)? Is (a + b) b = a?

28 Fixed Point Basic idea: radix point is here Choose a xed place in the binary number where the radix point is located. For the example above, the number is (1.11) 2 = = (2.3125) 1 How would you do mathematical operations?

29 Floating-Point Some problematic numbers Scienti c computations require a number of digits of precision... But they also need range permit the radix point to move Šoating-point numbers

30 Floating-Point: Scienti c Notation sign x 1 23 exponent mantissa radix (base) Number represented as: mantissa, exponent Arithmetic multiplication, division: perform operation on mantissa, add/subtract exponent addition, subtraction: convert operands to have the same exponent value, add/subtract mantissas