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PENN STATE / Computer Engineering / CMPEN 331 / What are the steps used in multiplication hardware?

What are the steps used in multiplication hardware?

What are the steps used in multiplication hardware?

Description

School: Pennsylvania State University
Department: Computer Engineering
Course: Computer Organization and Design
Professor: Mohamed almekkawy
Term: Spring 2016
Tags: computer engineering, Penn State, and CMPEN 331
Cost: 25
Name: CMPEN 311 Notes Lectures 11-14
Description: Floating Points Operations
Uploaded: 03/06/2016
8 Pages 184 Views 0 Unlocks
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"}ul.lst-kix_vy9p1in95yw7-6{list-style-type:none}ul.lst-kix_vy9p1in95yw7-7{list-style-type:none}ul.lst-kix_vy9p1in95yw7-4{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-4.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-4 0}ul.lst-kix_vy9p1in95yw7-5{list-style-type:none}ul.lst-kix_vk8kglbrv5v-2{list-style-type:none}.lst-kix_e20ockdrdgvo-3 > li:before{content:"● "}.lst-kix_e20ockdrdgvo-4 > li:before{content:"○ "}ul.lst-kix_vk8kglbrv5v-1{list-style-type:none}.lst-kix_wdieg29y6egk-8 > li:before{content:"- "}ul.lst-kix_vk8kglbrv5v-0{list-style-type:none}.lst-kix_ptttsnkm2g53-2 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-2,lower-roman) ". "}.lst-kix_ptttsnkm2g53-4 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-4,lower-latin) ". 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"}ul.lst-kix_vk8kglbrv5v-3{list-style-type:none}.lst-kix_wdieg29y6egk-5 > li:before{content:"- "}.lst-kix_e20ockdrdgvo-0 > li:before{content:"● "}.lst-kix_e20ockdrdgvo-7 > li:before{content:"○ "}.lst-kix_e20ockdrdgvo-8 > li:before{content:"■ "}.lst-kix_wdieg29y6egk-4 > li:before{content:"- "}.lst-kix_wdieg29y6egk-6 > li:before{content:"- "}ul.lst-kix_vk8kglbrv5v-8{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-4.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-4 0}ul.lst-kix_vk8kglbrv5v-7{list-style-type:none}.lst-kix_e20ockdrdgvo-6 > li:before{content:"● "}.lst-kix_ptttsnkm2g53-3 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-3,decimal) ". "}.lst-kix_wdieg29y6egk-1 > li:before{content:"- "}.lst-kix_wdieg29y6egk-0 > li:before{content:"- "}.lst-kix_wdieg29y6egk-2 > li:before{content:"- "}.lst-kix_wdieg29y6egk-3 > li:before{content:"- "}ol.lst-kix_9gd9xs5k3gl0-5.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-5 0}.lst-kix_ptttsnkm2g53-6 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-6,decimal) ". "}.lst-kix_ptttsnkm2g53-8 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-8,lower-roman) ". "}ol.lst-kix_ptttsnkm2g53-5.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-5 0}.lst-kix_e20ockdrdgvo-2 > li:before{content:"■ "}.lst-kix_ptttsnkm2g53-7 > li:before{content:"" counter(lst-ctn-kix_ptttsnkm2g53-7,lower-latin) ". 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"}.lst-kix_cdkdltrbe8sb-1 > li:before{content:"" counter(lst-ctn-kix_cdkdltrbe8sb-1,lower-latin) ". "}.lst-kix_cdkdltrbe8sb-5 > li:before{content:"" counter(lst-ctn-kix_cdkdltrbe8sb-5,lower-roman) ". "}ol.lst-kix_9gd9xs5k3gl0-0{list-style-type:none}.lst-kix_cdkdltrbe8sb-0 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-0}.lst-kix_ptttsnkm2g53-2 > li{counter-increment:lst-ctn-kix_ptttsnkm2g53-2}ol.lst-kix_9gd9xs5k3gl0-1{list-style-type:none}.lst-kix_m6g8ecu3vgqd-7 > li:before{content:"- "}ol.lst-kix_9gd9xs5k3gl0-2{list-style-type:none}.lst-kix_vk8kglbrv5v-7 > li:before{content:"- "}ol.lst-kix_9gd9xs5k3gl0-3{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-4{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-5{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-6{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-7{list-style-type:none}ol.lst-kix_9gd9xs5k3gl0-8{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-6{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-5{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-4{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-3{list-style-type:none}ol.lst-kix_ptttsnkm2g53-8.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-8 0}.lst-kix_9gd9xs5k3gl0-4 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-4}ol.lst-kix_cdkdltrbe8sb-8{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-7{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-7.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-7 0}ol.lst-kix_9gd9xs5k3gl0-2.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-2 0}.lst-kix_9gd9xs5k3gl0-5 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-5,lower-roman) ". "}ol.lst-kix_cdkdltrbe8sb-2{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-1{list-style-type:none}ol.lst-kix_cdkdltrbe8sb-0{list-style-type:none}.lst-kix_pqf6up3d0e8u-1 > li:before{content:"○ "}.lst-kix_pqf6up3d0e8u-0 > li:before{content:"● "}.lst-kix_pqf6up3d0e8u-2 > li:before{content:"■ "}.lst-kix_9gd9xs5k3gl0-7 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-7,lower-latin) ". "}ul.lst-kix_yz4aqjyx22ao-7{list-style-type:none}.lst-kix_pqf6up3d0e8u-3 > li:before{content:"● "}ul.lst-kix_yz4aqjyx22ao-8{list-style-type:none}ul.lst-kix_yz4aqjyx22ao-5{list-style-type:none}ul.lst-kix_yz4aqjyx22ao-6{list-style-type:none}.lst-kix_9gd9xs5k3gl0-8 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-8,lower-roman) ". "}ul.lst-kix_yz4aqjyx22ao-3{list-style-type:none}.lst-kix_pqf6up3d0e8u-5 > li:before{content:"■ "}.lst-kix_9gd9xs5k3gl0-5 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-5}ul.lst-kix_yz4aqjyx22ao-4{list-style-type:none}ul.lst-kix_yz4aqjyx22ao-1{list-style-type:none}.lst-kix_pqf6up3d0e8u-4 > li:before{content:"○ "}ul.lst-kix_yz4aqjyx22ao-2{list-style-type:none}.lst-kix_9gd9xs5k3gl0-3 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-3,decimal) ". "}ul.lst-kix_yz4aqjyx22ao-0{list-style-type:none}.lst-kix_9gd9xs5k3gl0-2 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-2,lower-roman) ". "}.lst-kix_9gd9xs5k3gl0-1 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-1,lower-latin) ". "}.lst-kix_pqf6up3d0e8u-8 > li:before{content:"■ "}.lst-kix_pqf6up3d0e8u-6 > li:before{content:"● "}.lst-kix_pqf6up3d0e8u-7 > li:before{content:"○ "}.lst-kix_9gd9xs5k3gl0-0 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-0,decimal) ". "}ol.lst-kix_9gd9xs5k3gl0-7.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-7 0}.lst-kix_ptttsnkm2g53-6 > li{counter-increment:lst-ctn-kix_ptttsnkm2g53-6}.lst-kix_cdkdltrbe8sb-8 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-8}ol.lst-kix_ptttsnkm2g53-1{list-style-type:none}ol.lst-kix_ptttsnkm2g53-2{list-style-type:none}ol.lst-kix_ptttsnkm2g53-3.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-3 0}ol.lst-kix_ptttsnkm2g53-0{list-style-type:none}ol.lst-kix_ptttsnkm2g53-7{list-style-type:none}.lst-kix_m6g8ecu3vgqd-4 > li:before{content:"- "}ol.lst-kix_ptttsnkm2g53-8{list-style-type:none}ol.lst-kix_ptttsnkm2g53-5{list-style-type:none}ol.lst-kix_ptttsnkm2g53-6{list-style-type:none}ol.lst-kix_ptttsnkm2g53-3{list-style-type:none}.lst-kix_m6g8ecu3vgqd-3 > li:before{content:"- "}ol.lst-kix_ptttsnkm2g53-4{list-style-type:none}.lst-kix_m6g8ecu3vgqd-0 > li:before{content:"- "}.lst-kix_m6g8ecu3vgqd-2 > li:before{content:"- "}ol.lst-kix_cdkdltrbe8sb-6.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-6 0}.lst-kix_9gd9xs5k3gl0-3 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-3}.lst-kix_m6g8ecu3vgqd-1 > li:before{content:"- "}.lst-kix_cdkdltrbe8sb-1 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-1}.lst-kix_vy9p1in95yw7-8 > li:before{content:"■ "}.lst-kix_vy9p1in95yw7-4 > li:before{content:"○ "}.lst-kix_vy9p1in95yw7-3 > li:before{content:"● "}.lst-kix_vy9p1in95yw7-7 > li:before{content:"○ "}.lst-kix_vy9p1in95yw7-6 > li:before{content:"● "}.lst-kix_vy9p1in95yw7-5 > li:before{content:"■ "}.lst-kix_vy9p1in95yw7-0 > li:before{content:"● "}.lst-kix_vy9p1in95yw7-2 > li:before{content:"■ "}.lst-kix_vy9p1in95yw7-1 > li:before{content:"○ "}.lst-kix_ptttsnkm2g53-4 > li{counter-increment:lst-ctn-kix_ptttsnkm2g53-4}.lst-kix_yz4aqjyx22ao-1 > li:before{content:"○ "}.lst-kix_cdkdltrbe8sb-4 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-4}.lst-kix_yz4aqjyx22ao-3 > li:before{content:"● "}.lst-kix_ptttsnkm2g53-0 > li{counter-increment:lst-ctn-kix_ptttsnkm2g53-0}.lst-kix_cdkdltrbe8sb-3 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-3}ol.lst-kix_9gd9xs5k3gl0-6.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-6 0}ol.lst-kix_ptttsnkm2g53-4.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-4 0}.lst-kix_yz4aqjyx22ao-7 > li:before{content:"○ "}.lst-kix_yz4aqjyx22ao-5 > li:before{content:"■ "}ol.lst-kix_cdkdltrbe8sb-2.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-2 0}.lst-kix_9gd9xs5k3gl0-0 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-0}.lst-kix_cdkdltrbe8sb-2 > li{counter-increment:lst-ctn-kix_cdkdltrbe8sb-2}.lst-kix_vk8kglbrv5v-2 > li:before{content:"- "}.lst-kix_vk8kglbrv5v-4 > li:before{content:"- "}ol.lst-kix_ptttsnkm2g53-2.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-2 0}.lst-kix_cdkdltrbe8sb-0 > li:before{content:"" counter(lst-ctn-kix_cdkdltrbe8sb-0,decimal) ". "}.lst-kix_cdkdltrbe8sb-2 > li:before{content:"" counter(lst-ctn-kix_cdkdltrbe8sb-2,lower-roman) ". "}.lst-kix_cdkdltrbe8sb-4 > li:before{content:"" counter(lst-ctn-kix_cdkdltrbe8sb-4,lower-latin) ". "}.lst-kix_m6g8ecu3vgqd-6 > li:before{content:"- "}ol.lst-kix_cdkdltrbe8sb-0.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-0 0}.lst-kix_vk8kglbrv5v-6 > li:before{content:"- "}.lst-kix_vk8kglbrv5v-8 > li:before{content:"- "}.lst-kix_m6g8ecu3vgqd-8 > li:before{content:"- "}.lst-kix_9gd9xs5k3gl0-1 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-1}ol.lst-kix_ptttsnkm2g53-1.start{counter-reset:lst-ctn-kix_ptttsnkm2g53-1 0}.lst-kix_9gd9xs5k3gl0-7 > li{counter-increment:lst-ctn-kix_9gd9xs5k3gl0-7}.lst-kix_9gd9xs5k3gl0-4 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-4,lower-latin) ". "}ol.lst-kix_9gd9xs5k3gl0-8.start{counter-reset:lst-ctn-kix_9gd9xs5k3gl0-8 0}ol.lst-kix_cdkdltrbe8sb-1.start{counter-reset:lst-ctn-kix_cdkdltrbe8sb-1 0}.lst-kix_9gd9xs5k3gl0-6 > li:before{content:"" counter(lst-ctn-kix_9gd9xs5k3gl0-6,decimal) ". "}

OverflowWe also discuss several other topics like Intraplate earthquakes may be linked to what?
Don't forget about the age old question of What are the 4 types of AS/AD Model?

Signed 8-bit integers

→add + and -, no overflow

→add + and +, overflow if signed bit is 1

→add - and -, overflow if signed bit is 0

Multiplication Ex)                                Division Ex)If you want to learn more check out Define risk premium.

                1000                                                ____1001                        

        x        1001                                        1000⎡1001010We also discuss several other topics like What is the use of FIR filters?

                1000                                            -    1000We also discuss several other topics like What is an unincorporated business owned by one person?

                  0000                                                    10If you want to learn more check out What is the ionization energy trend in the periodic table?

       0000                                                    101

     1000___                                                    1010        

     1001000                                                -   1000

                                                                10

Multiplication

Multiplication  Hardware:

Steps used for each bit can be optimized to perform the 3 steps in parallel

                                        

  • Faster multiplication
  • Adders used in parallel

MIPS Multiplication

64-bit product:

HIMSB

LO32LSB

Mult rs,rt/ multv rs, rt

  • 64-bit products

Mfhi rd/ mflo rd

  • Move from HI/LO to rd

mul rd, rs, rt

  • 32 LSB in rd

MIPS Division

HI 32 bit remainder        mfhi and mflo are used same as multiply

LO 32 bit quotient                No overflow or div-by-zero cheeks

Div rs, rt / divu rs, rt                

Floating Point

Normalized 1 number to left of decimal point

Format:                                   ↳known as binary point for binary numbers         

        

        

            Single: 8 bits              Single: 23 bits        

        Double: 11 bits     Double: 52 bits

        

                        ↳ leading 1 bit to left of binary point is implied

  • Exponent has excess representation: actual exponents and bias
  • 00000000 and 11111111 exponents reserved
  • Significantleading implied bit and fraction bits 1.xxx …

Range:

Smallest

Largest

Single precision

  • Binary
  • Decimal

Double precision

  • Binary         
  • Decimal

        Single ≈ 6 decimal digit precision

        Double ≈ 16 decimal digit precision

        Ex) -0.75 decimal binary floating point

                 (single)

                Fraction:

                Exponent:  Bias

                        Single

Floating Point Addition

  1. Align Binary points

 shift number with smaller exponent to match exponent of a larger num

  1. Add significands
  2. Normalize and check for overflow / underflow

Overflow exponent too large

Underflow exponent too small

  1. Round and renormalize

Floating Point Multiplication

  1. Add exponents

for biased exponents, subtract bias from sum

  1. Multiply significands
  2. Normalize and check for overflow / underflow
  3. Round and renormalize
  4. Use operand signs to determine result sign

Exponent (single)        Fraction        object

        255                      0                    

        255                non zero        NaN

Ex)

  1. No over / underflow
  2. No rounding needed
  3. - sign, hide hidden bit before storing

Floating Point MIPS

  • 32  single-precision registers: $f0-$f31
  • Double precision pairs registers: $f0/$f1...

        FP instructions only work on FP registers

Load / Store

        lwc1, swc1 - load / store single-precision FP

        ldc1, sdc1 - load / store double-precision FP

        

        add.s, sub.s, mult.s, div.s - single-precision arithmetic

                ex) mul.s $f0, $f1, $f6 $f0 = $f1 x $f6

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