Operation For Mac

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:

1. Is There An Operation For Macular Degeneration
2. Operation Machine
3. Mac Operating System List
4. Apple - Support - Downloads

${\displaystyle a\leftarrow a+(b\times c)}$

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).

Run Nearly Any Operating System on a Mac. Running Windows on Mac is only the beginning. VMware Fusion lets you choose from hundreds of supported operating systems, from lesser-known Linux distributions to the latest Windows 10 release, or even macOS, to run side by side with the latest macOS release on your desktop. Global Nav Open Menu Global Nav Close Menu; Apple; Shopping Bag +.

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. Percy Ludgate was the first to conceive a MAC in his Analytical Machine of 1909,^[1] and the first to exploit a MAC for division (using multiplication seeded by reciprocal, via the convergent series (1+x)⁻¹). The first modern 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[edit]

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.^[2]

Fused multiply–add[edit]

A fused multiply–add (FMA or fmadd)^[3]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 expression 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:

• Polynomial evaluation (e.g., with Horner's rule)
• Newton's method for evaluating functions (from the inverse function)
• Convolutions and artificial neural networks

Fused multiply–add can usually be relied on to give more accurate results. However, William Kahan has pointed out that it can give problems if used unthinkingly.^[4] If x² − y² is evaluated as ((x×x) − y×y) using fused multiply–add, then the result may be negative even when 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 be faster than a multiply operation followed by an add. However, standard industrial implementations based on the original IBM RS/6000 design require a 2N-bit adder to compute the sum properly.^[5]

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.^[6]

Dot product instruction[edit]

Some machines combine multiple fused multiply add operations into a single step, e.g. performing a four-element dot-product on two 128-bit SIMD registers a0×b0+a1×b1+a2×b2+a3×b3 with single cycle throughput.

Support[edit]

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 multiply and add steps. Instruction descriptions do not specify whether the multiply and add are performed using a single FMA step.^[7] 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,^[8] but has been added to numerous other processors since then:

• HPPA-8000 (1996) and above
• HitachiSuperH SH-4 (1998)
• SCE-ToshibaEmotion Engine (1999)
• Intel Itanium (2001)
• STI Cell (2006)
• FujitsuSPARC64 VI (2007) and above
• (MIPS-compatible) Loongson-2F (2008)^[9]
• Elbrus-8SV (2018)
• x86 processors with FMA3 and/or FMA4 instruction set
  • AMD Bulldozer (2011, FMA4 only)
  • AMD Piledriver (2012, FMA3 and FMA4)^[10]
  • AMD Steamroller (2014)
  • AMD Excavator (2015)
  • AMD Zen (2017, FMA3 only)
  • Intel Haswell (2013, FMA3 only)^[11]
  • Intel Skylake (2015, FMA3 only)
• ARM processors with VFPv4 and/or NEONv2:
  • ARM Cortex-M4F (2010)
  • ARM Cortex-A5 (2012)
  • ARM Cortex-A7 (2013)
  • ARM Cortex-A15 (2012)
  • Qualcomm Krait (2012)
  • Apple A6 (2012)
  • All ARMv8 processors
• GPUs and GPGPU boards:
  • Advanced Micro Devices GPUs (2009) and newer
    • TeraScale 2 'Evergreen'-series based
    • Graphics Core Next-based
  • NVidia GPUs (2010) and newer
    • Fermi-based (2010)
    • Kepler-based (2012)
    • Maxwell-based (2014)
    • Pascal-based (2016)
    • Volta-based (2017)
  • Intel GPUs since Sandy Bridge
  • Intel MIC (2012)
  • ARM Mali T600 Series (2012) and above
• Vector Processors:

References[edit]

1. ^'The Feasibility of Ludgate's Analytical Machine'. Archived from the original on 2019-08-07. Retrieved 2020-08-30.
2. ^Whitehead, Nathan; Fit-Florea, Alex (2011). 'Precision & Performance: Floating Point and IEEE 754 Compliance for NVIDIA GPUs'(PDF). nvidia. Retrieved 2013-08-31.
3. ^'fmadd instrs'.
4. ^Kahan, William (1996-05-31). 'IEEE Standard 754 for Binary Floating-Point Arithmetic'.
5. ^Quinnell, Eric (May 2007). Floating-Point Fused Multiply–Add Architectures(PDF) (PhD thesis). Retrieved 2011-03-28.
6. ^Markstein, Peter (November 2004). Software Division and Square Root Using Goldschmidt's Algorithms(PDF). 6th Conference on Real Numbers and Computers. CiteSeerX10.1.1.85.9648.
7. ^'VAX instruction of the week: POLY'. Archived from the original on 2020-02-13.
8. ^Montoye, R. K.; Hokenek, E.; Runyon, S. L. (January 1990). 'Design of the IBM RISC System/6000 floating-point execution unit'. IBM Journal of Research and Development. 34 (1): 59–70. doi:10.1147/rd.341.0059.
9. ^'Godson-3 Emulates x86: New MIPS-Compatible Chinese Processor Has Extensions for x86 Translation'.
10. ^Hollingsworth, Brent (October 2012). 'New 'Bulldozer' and 'Piledriver' Instructions'. AMD Developer Central.
11. ^'Intel adds 22nm octo-core 'Haswell' to CPU design roadmap'. The Register. Archived from the original on 2012-03-27. Retrieved 2008-08-19.

macOS is the operating system that powers every Mac. It lets you do things you simply can’t with other computers. That’s because it’s designed specifically for the hardware it runs on — and vice versa. macOS comes with an entire suite of beautifully designed apps. It works hand in hand with iCloud to keep photos, documents, and other stuff up to date on all your devices. It makes your Mac work like magic with your iPhone and other Apple devices. And it’s been built from the ground up with privacy and security in mind.

Featuring all-new, dedicated apps for music, TV, and podcasts. Smart new features in the apps you use every day. And Sidecar, which lets you use iPad as a second Mac display.

Easy to Use When it’s simple to
do
everything, you can do anything.

On a Mac, everything is designed to work just the way you expect it to — from operating it with simple, intuitive gestures to asking Siri to find a file to having your apps automatically update themselves. Whatever you set out to do on a Mac, there’s much less to learn and that much more to accomplish.

The Finder lets you easily browse and organize the files on your Mac — and those stored in iCloud Drive. View files by icons, in a list, or with Gallery View, which helps you quickly locate a file by how it looks. Drag and drop files into folders in the sidebar. And use the Preview pane to see all of your files’ metadata, quickly create .zip files, and share using the handy Share menu.

Dark Mode adds a dramatic look to your desktop and apps that puts the focus on your content. Dynamic Desktop makes your Mac even more beautiful with time-shifting desktop pictures that match the time of day wherever you are. And Stacks keeps your desktop free of clutter by automatically organizing your files, images, documents, PDFs, and more into tidy groups.

Spotlight helps you quickly and effortlessly find what you want, like documents on your Mac, movie showtimes, and flight departure and arrival times. Just type a few keystrokes in the Spotlight search field and autocomplete immediately shows you relevant results.¹

A simple two-finger swipe from the right edge of your trackpad brings up Notification Center — the same one you use on iPhone — where you can see all the alerts that have come in recently. It’s also where you’ll find your Today view, which you can customize with helpful widgets or handy information pinned from Siri searches.

Siri helps you get things done just by using your voice. It also helps you get more things done at the same time. So you can finish your document while asking Siri to tell your coworker that it’s on the way. Looking for the presentation you worked on last week? Just ask Siri.² And with Apple Music, Siri can be your personal DJ and recommend music based on your tastes, and tell you more about songs, albums, and artists.³

Continuity All your devices.
One seamless experience.

Your Mac works with your other Apple devices in ways no other computer can. If you get a call on your iPhone, you can take it on your Mac. And when friends text you — regardless of the phone they have — you can respond from your Mac, iPad, or iPhone, whichever is closest. When you copy text or an image from one device, you can paste it into another with standard copy and paste commands. And with Sidecar, you can extend your workspace by using your iPad as a second Mac display. You can even unlock your Mac with your Apple Watch. No password typing required.

Privacy and Security We believe your data belongs to you. Period.

Everything you do with your Mac is protected by strong privacy and security features. That’s because we build data security into everything we make, right from the start.

Privacy. You trust our products with your most personal information, and we believe that you should be in complete control of it. We respect your privacy by enacting strict policies that govern how all data is handled. And when you browse the web, Safari helps prevent data companies from tracking the sites you visit.

Is There An Operation For Macular Degeneration

Security. We design Mac hardware and software with advanced technologies that work together to help you run apps safely, protect your data, and keep you safe on the web. The new Find My app helps you locate a missing Mac that’s lost or stolen — even if it’s offline or sleeping. And Gatekeeper makes it safer to download and install apps from the Mac App Store and the internet.

Built in Apps Apps as powerful and elegant as your Mac. Because they’re designed for it.

With every Mac, you get a collection of powerful apps. They’re the same apps you use on your iPhone or iPad, so they’ll feel instantly familiar. They all work with iCloud, so your schedule, contacts, and notes are always up to date everywhere. And because they’re native apps — not web apps in a browser — they take full advantage of the power of your Mac to deliver the fastest possible performance and more robust features.

Media. Experience music, TV, and podcasts in three all-new Mac apps. Get all the news that matters from sources you trust, all in one place.⁴ Track the market and customize your watchlist. Peruse the best book titles from your desktop. And find the latest apps and extensions for your Mac, right on your Mac.

• Podcasts

• Stocks

• App Store

Creativity. Enjoy your best photos and videos, showcased in an immersive, dynamic new look. Transform home videos into unforgettable movies or quickly share clips with your favorite people. Create music like the pros with a huge collection of sounds, instruments, amps, and a lineup of virtual session drummers and percussionists perfectly suited to play along with your song.

Productivity. Create beautiful documents with stunning simplicity. Visualize your data precisely and persuasively in spreadsheets with images, text, and shapes. Make stage-worthy presentations using powerful graphics tools and dazzling cinematic effects to bring your ideas to life. And collaborate with friends and coworkers in real time — whether they’re across town or across the globe.

Operation Machine

Communication. Easily manage all of your email accounts in one simple, powerful app that works with email services like iCloud, Gmail, Yahoo, AOL, and Microsoft Exchange. Send unlimited messages to anyone on any Apple device, using the same Messages app on your Mac as you do on your iPhone. Make unlimited high-quality video and audio calls right from your Mac with FaceTime. And securely locate a missing Mac using the new Find My app on Mac, iPhone, and iPad.

• Mail

• Messages

• FaceTime

Organization. A new gallery view and more powerful search help you find your notes more quickly than ever. Easily create, organize, and add attachments to reminders. Manage your iCloud, Google, and Exchange calendars in one place, create separate calendars for home, work, and school, and view them all or just the ones you want. Instantly capture personal reminders, class lectures, even interviews or song ideas with Voice Memos. Keep all your contact info up to date on all your devices, and easily import numbers, addresses, and photos from Google and Microsoft Exchange. And control all your HomeKit-enabled accessories from the comfort of your desktop.

• Notes

• Reminders

• Calendar

• Voice Memos

• Contacts

Internet. Surf the web seamlessly with an updated start page that helps you quickly access your favorite and frequently visited sites. Use Apple Pay to make purchases on the web in Safari with Touch ID on MacBook Pro and MacBook Air. Map out new destinations from your desktop, with gorgeous 3D city views like Flyover, point-to-point directions, and comprehensive transit directions. And instantly search your entire Mac or get results from the internet in just a few keystrokes.

• Spotlight

iCloud All your stuff on your Mac.
And everywhere else you want it.

Every Apple app uses iCloud — it’s the magic behind the seamless experience you have with all your Apple devices. So you always have your photos, videos, documents, email, notes, contacts, calendars, and more on whatever device you’re using. And any file you store in iCloud Drive can be shared with friends, family, and colleagues just by sending them a link. iCloud Drive also lets you access files from your Mac on your iPhone or iPad. It’s all done automatically. Just like that.⁵

Accessibility We make sure that everyone is able to use a Mac.

macOS comes standard with a wide range of assistive technologies that help people with disabilities experience what the Mac has to offer, providing many features you won’t find in other operating systems. Voice Control lets users control their Mac, iPad, and iPhone entirely with their voice. And features such as VoiceOver, Accessibility Keyboard, FaceTime,⁶ and Text to Speech help everyone get more out of Mac.

Technology Advanced to the core.

Mac Operating System List

macOS features powerful core technologies engineered for the most important functions of your Mac. Thanks to Mac Catalyst, you can now enjoy your favorite iPad apps on your Mac. With SwiftUI, developers have a simple way to build better apps across all Apple platforms with less code. Create ML gives developers a faster and easier way to build machine learning into apps. And support for virtual and augmented reality lets developers create immersive worlds for you to step into.

Compatibility Need to work with Windows?
Mac does that, too.

With macOS, it’s easy to transfer your files, photos, and other data from a Windows PC to your Mac. Work with popular file types such as JPEG, MP3, and PDF, as well as Microsoft Word, Excel, and PowerPoint documents. And, of course, you can run Microsoft Office natively on a Mac. If you want, you can even run Windows on your Mac.⁷

Find the right iMac for you.

21.5‑inch iMac

Starting at $1099

• Up to 21.5' Retina 4K display
• Up to 6-core Intel Core i7 processor
• Up to 32GB memory
• Up to 1TB SSD storage
• Up to Radeon Pro Vega 20 with 4GB of HBM2 memory

Apple - Support - Downloads

27‑inch iMac

Starting at $1799

• 27' Retina 5K display
• Up to 10-core Intel Core i9 processor
• Up to 128GB memory
• Up to 8TB SSD storage
• Up to Radeon Pro 5700 XT with 16GB of GDDR6 memory