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In [[computing]], especially [[digital signal processing]], the '''multiply–accumulate operation''' is a common step that computes the product of two numbers and adds that product to an [[accumulator (computing)|accumulator]].  The hardware unit that performs the operation is known as a '''multiplier–accumulator''' (MAC, or MAC unit); the operation itself is also often called a MAC or a MAC operation. The MAC operation modifies an accumulator ''a'':
 
:<math>\ a \leftarrow a + ( b \times c )</math>
 
When done with [[floating point]] numbers, it might be performed with two [[rounding]]s (typical in many [[digital signal processors|DSP]]s), or with a single rounding.  When performed with a single rounding, it is called a '''fused multiply–add''' (FMA) or '''fused multiply–accumulate''' (FMAC).
 
Modern computers may contain a dedicated MAC, consisting of a multiplier implemented in [[combinational logic]] followed by an [[adder (electronics)|adder]] and an accumulator register that stores the result. The output of the register is fed back to one input of the adder, so that on each clock cycle, the output of the multiplier is added to the register. Combinational multipliers require a large amount of logic, but can compute a product much more quickly than the [[peasant multiplication|method of shifting and adding]] typical of earlier computers. The first processors to be equipped with MAC units were [[digital signal processor]]s, but the technique is now also common in general-purpose processors.
 
==In floating-point arithmetic==
 
When done with [[integer]]s, the operation is typically exact (computed [[modular arithmetic|modulo]] some [[power of two]]). However, [[floating-point]] numbers have only a certain amount of mathematical [[arithmetic precision|precision]]. That is, digital floating-point arithmetic is generally not [[associativity|associative]] or [[distributivity|distributive]]. (See [[Floating point#Accuracy problems]].)
Therefore, it makes a difference to the result whether the multiply–add is performed with two roundings, or in one operation with a single rounding (a fused multiply–add). [[IEEE 754-2008]] specifies that it must be performed with one rounding, yielding a more accurate result.<ref>{{cite web |url=https://developer.nvidia.com/sites/default/files/akamai/cuda/files/NVIDIA-CUDA-Floating-Point.pdf |title=Precision & Performance: Floating Point and IEEE 754 Compliance for NVIDIA GPUs |publisher=nvidia |first1=Nathan |last1=Whitehead |first2=Alex |last2=Fit-Florea |year=2011 |accessdate=2013-08-31}}</ref>
 
==Fused multiply–add==
 
A ''fused'' multiply–add is a floating-point multiply–add operation performed in one step, with a single rounding. That is, where an unfused multiply–add would compute the product ''b''×''c'', round it to ''N'' significant bits, add the result to ''a'', and round back to ''N'' significant bits, a fused multiply–add would compute the entire sum ''a''+''b''×''c'' to its full precision before rounding the final result down to ''N'' significant bits.
 
A fast FMA can speed up and improve the accuracy of many computations that involve the accumulation of products:
* [[Dot product]]
* [[Matrix multiplication]]
* [[Polynomial]] evaluation (e.g., with [[Horner's rule]])
* [[Newton's method]] for evaluating functions.
 
Fused multiply–add can usually be relied on to give more accurate results. However, Kahan has pointed out that it can give problems if used unthinkingly.<ref>{{cite web |title=IEEE Standard 754  for  Binary Floating-Point Arithmetic |author=W.Kahan | url=http://www.cs.berkeley.edu/~wkahan/ieee754status/ieee754.ps  | date=May 31, 1996}}</ref> If {{math|''x''<sup>2</sup> − ''y''<sup>2</sup>}} is evaluated as {{math| ((''x''×''x'') − ''y''×''y'')}} using fused multiply–add, then the result may be negative even when {{math|''x'' {{=}} ''y''}} due to the first multiplication discarding low significance bits. This could then lead to an error if, for instance, the square root of the result is then evaluated.
 
When implemented inside a [[microprocessor]], an FMA can actually be faster than a multiply operation followed by an add, even though standard industrial implementations based on the original IBM RS/6000 design require a 2''N''-bit adder to compute the sum properly.<ref name="bridged">{{cite web|url=http://users.ece.utexas.edu/~quinnell/Research/Bridged%20Floating-Point%20Fused%20Multiply-Add%20Design.pdf |title=Bridged Floating-Point Fused Multiply–Add Design|author=Eric Quinnell et al| date=undated, circa 2006}}{{Dead link|date=March 2011}}</ref><ref>{{cite thesis |url=http://repositories.lib.utexas.edu/bitstream/handle/2152/3082/quinnelle60861.pdf |date=May 2007 |title=Floating-Point Fused Multiply–Add Architectures |author=Eric Quinnell |degree=PhD |accessdate=2011-03-28}}</ref>
 
A useful benefit of including this instruction is that it allows an efficient software implementation of [[division (mathematics)|division]] (see [[division algorithm]]) and [[square root]] (see [[methods of computing square roots]]) operations, thus eliminating the need for dedicated hardware for those operations.<ref name="goldschmidt_algo">{{cite web|id = {{citeseerx|10.1.1.85.9648}}|title= Software Division and Square Root Using Goldschmidt's Algorithms|author= Peter Markstein|date= Nov. 2004}}</ref>
 
The FMA operation is included in [[IEEE 754-2008]].
 
The [[Digital Equipment Corporation|DEC]] [[VAX]]'s POLY instruction is used for evaluating polynomials with Horner's rule using a succession of fused multiply–add steps.<ref>{{Cite web|url=http://uranium.vaxpower.org/~isildur/vax/week.html|title= VAX instruction of the week: POLY}}</ref> This instruction has been a part of the VAX instruction set since its original 11/780 implementation in 1977.
 
The [[C99|1999 standard]] of the [[C (programming language)|C programming language]] supports the FMA operation through the <code>fma</code> standard math library function, and standard pragmas controlling optimizations based on FMA.
 
The fused multiply–add operation was introduced as ''multiply–add fused'' in the IBM [[POWER1]] (1990) processor,<ref>{{cite journal |first1=R. K. |last1=Montoye |first2=E. |last2=Hokenek |first3=S. L. |last3=Runyon |url=http://domino.research.ibm.com/tchjr/journalindex.nsf/4ac37cf0bdc4dd6a85256547004d47e1/e3d1d5353695231c85256bfa0067fa31?OpenDocument |title=Design of the IBM RISC System/6000 floating-point execution unit |journal=IBM Journal of Research and Development |volume=34 |issue=1 |date=January 1990 |pages=59–70 |issn=0018-8646 |doi=10.1147/rd.341.0059 }}</ref> but has been added to numerous other processors since then:
* [[Hewlett-Packard|HP]] [[PA-8000]] (1996) and above
* [[Hitachi, Ltd.|Hitachi]] [[SuperH#SH-4|SuperH SH-4]] (1998)
* [[Sony Computer Entertainment|SCE]]-[[Toshiba]] [[Emotion Engine]] (1999)
* Intel [[Itanium]] (2001)
* STI [[Cell (microprocessor)|Cell]] (2006)
* [[Fujitsu]] [[SPARC64 VI]] (2007) and above
* ([[MIPS architecture|MIPS]]-compatible) [[Loongson]]-2F (2008).<ref>{{cite web|url=http://www.mdronline.com/mpr/h/2008/1103/224401.html |title= Godson-3 Emulates x86: New MIPS-Compatible Chinese Processor Has Extensions for x86 Translation}}</ref>
* x86 processors with [[FMA instruction set|FMA3 and/or FMA4 instruction set]]
** AMD [[Bulldozer (processor)|Bulldozer]] (2011, FMA4 only)
** AMD [[Piledriver (microarchitecture)|Piledriver]] (2012, FMA3 and FMA4)
** [[Intel Haswell]] (2013, FMA3 only)<ref>{{cite web|url=http://www.reghardware.co.uk/2008/08/19/idf_intel_architecture_roadmap/ |title= Intel adds 22nm octo-core 'Haswell' to CPU design roadmap|work= The Register}}</ref>
* ARM processors with VFPv4 and/or NEONv2:
** [[ARM Cortex-A5]] (2012)
** [[ARM Cortex-A7 MPCore|ARM Cortex-A7]] (2013)
** [[ARM Cortex-A15 MPCore|ARM Cortex-A15]] (2012)
** [[Krait (CPU)|Qualcomm Krait]] (2012)
** [[Apple A6|Apple Swift]] (2012)
** All [[ARM architecture#ARMv8-A|ARMv8]] processors
* GPUs and GPGPU boards:
** AMD Evergreen GPUs (2009) and above
** [[NVidia]] Fermi GPUs (2010) and above
** [[Intel MIC]] (2012)
** [[Mali (GPU)|ARM Mali T600 Series]] (2012) and above
 
==References==
 
{{Reflist}}
 
{{DEFAULTSORT:Multiply-accumulate operation}}
[[Category:Computer arithmetic]]
[[Category:Digital signal processing]]

Revision as of 18:46, 15 January 2014

In computing, especially digital signal processing, the multiply–accumulate operation is a common step that computes the product of two numbers and adds that product to an accumulator. The hardware unit that performs the operation is known as a multiplier–accumulator (MAC, or MAC unit); the operation itself is also often called a MAC or a MAC operation. The MAC operation modifies an accumulator a:

When done with floating point numbers, it might be performed with two roundings (typical in many DSPs), or with a single rounding. When performed with a single rounding, it is called a fused multiply–add (FMA) or fused multiply–accumulate (FMAC).

Modern computers may contain a dedicated MAC, consisting of a multiplier implemented in combinational logic followed by an adder and an accumulator register that stores the result. The output of the register is fed back to one input of the adder, so that on each clock cycle, the output of the multiplier is added to the register. Combinational multipliers require a large amount of logic, but can compute a product much more quickly than the method of shifting and adding typical of earlier computers. The first processors to be equipped with MAC units were digital signal processors, but the technique is now also common in general-purpose processors.

In floating-point arithmetic

When done with integers, the operation is typically exact (computed modulo some power of two). However, floating-point numbers have only a certain amount of mathematical precision. That is, digital floating-point arithmetic is generally not associative or distributive. (See Floating point#Accuracy problems.) Therefore, it makes a difference to the result whether the multiply–add is performed with two roundings, or in one operation with a single rounding (a fused multiply–add). IEEE 754-2008 specifies that it must be performed with one rounding, yielding a more accurate result.[1]

Fused multiply–add

A fused multiply–add is a floating-point multiply–add operation performed in one step, with a single rounding. That is, where an unfused multiply–add would compute the product b×c, round it to N significant bits, add the result to a, and round back to N significant bits, a fused multiply–add would compute the entire sum a+b×c to its full precision before rounding the final result down to N significant bits.

A fast FMA can speed up and improve the accuracy of many computations that involve the accumulation of products:

Fused multiply–add can usually be relied on to give more accurate results. However, Kahan has pointed out that it can give problems if used unthinkingly.[2] If Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. due to the first multiplication discarding low significance bits. This could then lead to an error if, for instance, the square root of the result is then evaluated.

When implemented inside a microprocessor, an FMA can actually be faster than a multiply operation followed by an add, even though standard industrial implementations based on the original IBM RS/6000 design require a 2N-bit adder to compute the sum properly.[3][4]

A useful benefit of including this instruction is that it allows an efficient software implementation of division (see division algorithm) and square root (see methods of computing square roots) operations, thus eliminating the need for dedicated hardware for those operations.[5]

The FMA operation is included in IEEE 754-2008.

The DEC VAX's POLY instruction is used for evaluating polynomials with Horner's rule using a succession of fused multiply–add steps.[6] This instruction has been a part of the VAX instruction set since its original 11/780 implementation in 1977.

The 1999 standard of the C programming language supports the FMA operation through the fma standard math library function, and standard pragmas controlling optimizations based on FMA.

The fused multiply–add operation was introduced as multiply–add fused in the IBM POWER1 (1990) processor,[7] but has been added to numerous other processors since then:

References

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  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ Template:Cite webTemplate:Dead link
  4. ↑ Template:Cite thesis
  5. ↑ Template:Cite web
  6. ↑ Template:Cite web
  7. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. ↑ Template:Cite web
  9. ↑ Template:Cite web
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