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class='fauxcolumn-outer fauxcolumn-right-outer'> <div class='cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left'> <div class='fauxborder-right'></div> <div class='fauxcolumn-inner'> </div> </div> <div class='cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div>  <div class='columns-inner'> <div class='column-center-outer'> <div class='column-center-inner'> <div class='main section' id='main'><div class='widget Blog' data-version='1' id='Blog1'> <div class='blog-posts hfeed'> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='https://static.righto.com/images/pentium-microcode1/pentium-labeled-w500.jpg' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='3129002752159663932' itemprop='postId'/> <a name='3129002752159663932'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2025/03/pentium-microcde-rom-circuitry.html'>Notes on the Pentium's microcode circuitry</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-3129002752159663932' itemprop='description articleBody'> <p>Most people think of machine instructions as the fundamental steps that a computer performs. However, many processors have another layer of software underneath: microcode. With microcode, instead of building the processor's control circuitry from complex logic gates, the control logic is implemented with code known as microcode, stored in the microcode ROM. To execute a machine instruction, the computer internally executes several simpler micro-instructions, specified by the microcode. In this post, I examine the microcode ROM in the original Pentium, looking at the low-level circuitry.</p> <p>The photo below shows the Pentium's thumbnail-sized silicon die under a microscope. I've labeled the main functional blocks. The microcode ROM is highlighted at the right. If you look closely, you can see that the microcode ROM consists of two rectangular banks, one above the other.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/pentium-labeled.jpg"><img alt="This die photo of the Pentium shows the location of the microcode ROM. Click this image (or any other) for a larger version." class="hilite" height="524" src="https://static.righto.com/images/pentium-microcode1/pentium-labeled-w500.jpg" title="This die photo of the Pentium shows the location of the microcode ROM. Click this image (or any other) for a larger version." width="500" /></a><div class="cite">This die photo of the Pentium shows the location of the microcode ROM. Click this image (or any other) for a larger version.</div></p> <p>The image below shows a closeup of the two microcode ROM banks. Each bank provides 45 bits of output; together they implement a micro-instruction that is 90 bits long. Each bank consists of a grid of transistors arranged into 288 rows and 720 columns. The microcode ROM holds 4608 micro-instructions, 414,720 bits in total. At this magnification, the ROM appears featureless, but it is covered with horizontal wires, each just 1.5 碌m thick.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/rom-output-lines.jpg"><img alt="The 90 output lines from the ROM, with a closeup of six lines exiting the ROM." class="hilite" height="470" src="https://static.righto.com/images/pentium-microcode1/rom-output-lines-w500.jpg" title="The 90 output lines from the ROM, with a closeup of six lines exiting the ROM." width="500" /></a><div class="cite">The 90 output lines from the ROM, with a closeup of six lines exiting the ROM.</div></p> <p>The ROM's 90 output lines are collected into a bundle of wires between the banks, as shown above. The detail shows how six of the bits exit from the banks and join the bundle. This bundle exits the ROM to the left, travels to various parts of the chip, and controls the chip's circuitry. The output lines are in the chip's top metal layer (M3): the Pentium has three layers of metal wiring with M1 on the bottom, M2 in the middle, and M3 on top.</p> <p>The Pentium has a large number of bits in its micro-instruction, 90 bits compared to 21 bits in the <a href="https://www.righto.com/2022/11/how-8086-processors-microcode-engine.html">8086</a>. Presumably, the Pentium has a "horizontal" microcode architecture, where the microcode bits correspond to low-level control signals, as opposed to "vertical" microcode, where the bits are encoded into denser micro-instructions. I don't have any information on the Pentium's encoding of microcode; unlike the 8086, the Pentium's patents don't provide any clues. The 8086's microcode ROM holds 512 micro-instructions, much less than the Pentium's 4608 micro-instructions. This makes sense, given the much greater complexity of the Pentium's instruction set, including the floating-point unit on the chip.</p>  <p>The image below shows a closeup of the Pentium's microcode ROM. For this image, I removed the three layers of metal and the polysilicon layer to expose the chip's underlying silicon. The pattern of silicon doping is visible, showing the transistors and thus the data stored in the ROM. If you have enough time, you can extract the bits from the ROM by examining the silicon and seeing where transistors are present.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/rom-closeup.jpg"><img alt="A closeup of the ROM showing how bits are encoded in the layout of transistors." class="hilite" height="469" src="https://static.righto.com/images/pentium-microcode1/rom-closeup-w500.jpg" title="A closeup of the ROM showing how bits are encoded in the layout of transistors." width="500" /></a><div class="cite">A closeup of the ROM showing how bits are encoded in the layout of transistors.</div></p> <p>Before explaining the ROM's circuitry, I'll review how an NMOS transistor is constructed. A transistor can be considered a switch between the source and drain, controlled by the gate. The source and drain regions (green) consist of silicon doped with impurities to change its semiconductor properties, forming N+ silicon. (These regions are visible in the photo above.) The gate consists of a layer of polysilicon (red), separated from the silicon by a very thin insulating oxide layer. Whenever polysilicon crosses active silicon, a transistor is formed. </p> <p><a href="https://static.righto.com/images/pentium-microcode1/mosfet-n.jpg"><img alt="Diagram showing the structure of an NMOS transistor." class="hilite" height="231" src="https://static.righto.com/images/pentium-microcode1/mosfet-n-w400.jpg" title="Diagram showing the structure of an NMOS transistor." width="400" /></a><div class="cite">Diagram showing the structure of an NMOS transistor.</div></p> <p>Bits are stored in the ROM through the pattern of transistors in the grid. The presence or absence of a transistor stores a 0 or 1 bit.<span id="fnref:ambiguity"><a class="ref" href="#fn:ambiguity">1</a></span> The closeup below shows eight bits of the microcode ROM. There are four transistors present and four gaps where transistors are missing. Thus, this part of the ROM holds four 0 bits and four 1 bits. For the diagram below, I removed the three metal layers and the polysilicon to show the underlying silicon. I colored doped (active) silicon regions green, and drew in the horizontal polysilicon lines in red. As explained above, a transistor is created if polysilicon crosses doped silicon. Thus, the contents of the ROM are defined by the pattern of silicon regions, which creates the transistors.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/rom-transistors.jpg"><img alt="Eight bits of the microcode ROM, with four transistors present." class="hilite" height="211" src="https://static.righto.com/images/pentium-microcode1/rom-transistors-w500.jpg" title="Eight bits of the microcode ROM, with four transistors present." width="500" /></a><div class="cite">Eight bits of the microcode ROM, with four transistors present.</div></p> <p>The horizontal silicon lines are used as wiring to provide ground to the transistors, while the horizontal polysilicon lines select one of the rows in the ROM. The transistors in that row will turn on, pulling the associated output lines low. That is, the presence of a transistor in a row causes the output to be pulled low, while the absence of a transistor causes the output line to remain high.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/rom-schematic.jpg"><img alt="A schematic corresponding to the eight bits above." class="hilite" height="225" src="https://static.righto.com/images/pentium-microcode1/rom-schematic-w300.jpg" title="A schematic corresponding to the eight bits above." width="300" /></a><div class="cite">A schematic corresponding to the eight bits above.</div></p> <p>The diagram below shows the silicon, polysilicon, and bottom metal (M1) layers. I removed the metal from the left to reveal the silicon and polysilicon underneath, but the pattern of vertical metal lines continues there. As shown earlier, the silicon pattern forms transistors. Each horizontal metal line has a connection to ground through a metal line (not shown). The horizontal polysilicon lines select a row. When polysilicon lines cross doped silicon, the gate of a transistor is formed. Two transistors may share the drain, as in the transistor pair on the left.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/m1-diagram.jpg"><img alt="Diagram showing the silicon, polysilicon, and M1 layers." class="hilite" height="330" src="https://static.righto.com/images/pentium-microcode1/m1-diagram-w500.jpg" title="Diagram showing the silicon, polysilicon, and M1 layers." width="500" /></a><div class="cite">Diagram showing the silicon, polysilicon, and M1 layers.</div></p> <p>The vertical metal wires form the outputs. The circles are contacts between the metal wire and the silicon of a transistor.<span id="fnref:contacts"><a class="ref" href="#fn:contacts">2</a></span> Short metal jumpers connect the polysilicon lines to the metal layer above, which will be described next.</p> <p>The image below shows the upper left corner of the ROM. The yellowish metal lines are the top metal layer (M3), while the reddish metal lines are the middle metal layer (M2). The thick yellowish M3 lines distribute ground to the ROM. Underneath the horizontal M3 line, a horizontal M2 line also distributes ground. The grids of black dots are numerous contacts between the M3 line and the M2 line, providing a low-resistance connection. The M2 line, in turn, connects to vertical M1 ground lines underneath—these wide vertical lines are faintly visible. These M1 lines connect to the silicon, as shown earlier, providing ground to each transistor. This illustrates the complexity of power distribution in the Pentium: the thick top metal (M3) is the primary distribution of +5 volts and ground through the chip, but power must be passed down through M2 and M1 to reach the transistors.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/rom-m3.jpg"><img alt="The upper left corner of the ROM." class="hilite" height="419" src="https://static.righto.com/images/pentium-microcode1/rom-m3-w600.jpg" title="The upper left corner of the ROM." width="600" /></a><div class="cite">The upper left corner of the ROM.</div></p> <p>The other important feature above is the horizontal metal lines, which help distribute the row-select signals. As shown earlier, horizontal polysilicon lines provide the row-select signals to the transistors. However, polysilicon is not as good a conductor as metal, so long polysilicon lines have too much resistance. The solution is to run metal lines in parallel, periodically connected to the underlying polysilicon lines and reducing the overall resistance. Since the vertical metal output lines are in the M1 layer, the horizontal row-select lines run in the M2 layer so they don't collide. Short "jumpers" in the M1 layer connect the M2 lines to the polysilicon lines.</p> <p>To summarize, each ROM bank contains a grid of transistors and transistor vacancies to define the bits of the ROM. The ROM is carefully designed so the different layers—silicon, polysilicon, M1, and M2—work together to maximize the ROM's performance and density.</p> <h2>Microcode Address Register</h2> <p>As the Pentium executes an instruction, it provides the address of each micro-instruction to the microcode ROM. The Pentium holds this address—the micro-address—in the Microcode Address Register (MAR). The MAR is a 13-bit register located above the microcode ROM. </p> <p>The diagram below shows the Microcode Address Register above the upper ROM bank. It consists of 13 bits; each bit has multiple latches to hold the value as well as any pushed subroutine micro-addresses. Between bits 7 and 8, some buffer circuitry amplifies the control signals that go to each bit's circuitry. At the right, drivers amplify the outputs from the MAR, sending the signals to the row drivers and column-select circuitry that I will discuss below. To the left of the MAR is a 32-bit register that is apparently unrelated to the microcode ROM, although I haven't determined its function.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/MAR.jpg"><img alt="The Microcode Address Register is located above the upper ROM bank." class="hilite" height="226" src="https://static.righto.com/images/pentium-microcode1/MAR-w600.jpg" title="The Microcode Address Register is located above the upper ROM bank." width="600" /></a><div class="cite">The Microcode Address Register is located above the upper ROM bank.</div></p> <p>The outputs from the Microcode Address Register select rows and columns in the microcode ROM, as I'll explain below. Bits 12 through 7 of the MAR select a block of 8 rows, while bits 6 through 4 select a row in this block. Bits 3 through 0 select one column out of each group of 16 columns to select an output bit. Thus, the microcode address controls what word is provided by the ROM.</p> <p>Several different operations can be performed on the Microcode Address Register. When executing a machine instruction, the MAR must be loaded with the address of the corresponding microcode routine. (I haven't determined how this address is generated.) As microcode is executed, the MAR is usually incremented to move to the next micro-instruction. However, the MAR can branch to a new micro-address as required. The MAR also supports microcode subroutine calls; it will push the current micro-address and jump to the new micro-address. At the end of the micro-subroutine, the micro-address is popped so execution returns to the previous location. The MAR supports three levels of subroutine calls, as it contains three registers to hold the stack of pushed micro-addresses.</p> <p>The MAR receives control signals and addresses from <a href="https://www.righto.com/2024/07/pentium-standard-cells.html">standard-cell logic</a> located above the MAR. Strangely, in Intel's published <a href="https://doi.org/10.1109/40.216745">floorplans</a> for the Pentium, this standard-cell logic is labeled as part of the branch prediction logic, which is above it. However, carefully tracing the signals from the standard-cell logic shows that is connected to the Microcode Address Register, not the branch predictor.</p> <h2>Row-select drivers</h2> <p>As explained above, each ROM bank has 288 rows of transistors, with polysilicon lines to select one of the rows. To the right of the ROM is circuitry that activates one of these row-select lines, based on the micro-address. Each row matches a different 9-bit address. A straightforward implementation would use a 9-input AND gate for each row, matching a particular pattern of 9 address bits or their complements.</p> <p>However, this implementation would require 576 very large AND gates, so it is impractical. Instead, the Pentium uses an optimized implementation with one 6-input AND gate for each group of 8 rows. The remaining three address bits are decoded once at the top of the ROM. As a result, each row only needs one gate, detecting if its group of eight rows is selected and if the particular one of eight is selected.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/row-driver-schematic.jpg"><img alt="Simplified schematic of the row driver circuitry." class="hilite" height="453" src="https://static.righto.com/images/pentium-microcode1/row-driver-schematic-w500.jpg" title="Simplified schematic of the row driver circuitry." width="500" /></a><div class="cite">Simplified schematic of the row driver circuitry.</div></p> <p>The schematic above shows the circuitry for a group of eight rows, slightly simplified.<span id="fnref:simplified-rows"><a class="ref" href="#fn:simplified-rows">3</a></span> At the top, three address bits are decoded, generating eight output lines with one active at a time. The remaining six address bits are inverted, providing the bit and its complement to the decoding circuitry. Thus, the 9 bits are converted into 20 signals that flow through the decoders, a large number of wires, but not unmanageable. Each group of eight rows has a 6-input AND gate that matches a particular 6-bit address, determined by which inputs are complemented and which are not.<span id="fnref:binary"><a class="ref" href="#fn:binary">4</a></span> The NAND gate and inverter at the left combine the 3-bit decoding and the 6-bit decoding, activating the appropriate row.</p> <p>Since there are up to 720 transistors in each row, the row-select lines need to be driven with high current. Thus, the row-select drivers use large transistors, roughly 25 times the size of a regular transistor. To fit these transistors into the same vertical spacing as the rest of the decoding circuitry, a tricky packing is used. The drivers for each group of 8 rows are packed into a 3×3 grid, except the first column has two drivers (since there are 8 drivers in the group, not 9). To avoid a gap, the drivers in the first column are larger vertically and squashed horizontally.</p> <h2>Output circuitry</h2> <p>The schematic below shows the multiplexer circuit that selects one of 16 columns for a microcode output bit. The first stage has four 4-to-1 multiplexers. Next, another 4-to-1 multiplexer selects one of the outputs. Finally, a BiCMOS driver amplifies the output for transmission to the rest of the processor.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/output-mux.jpg"><img alt="The 16-to-1 multiplexer/output driver." class="hilite" height="272" src="https://static.righto.com/images/pentium-microcode1/output-mux-w700.jpg" title="The 16-to-1 multiplexer/output driver." width="700" /></a><div class="cite">The 16-to-1 multiplexer/output driver.</div></p> <p>In more detail, the ROM and the first multiplexer are essentially NMOS circuits, rather than CMOS. Specifically, the ROM's grid of transistors is constructed from NMOS transistors that can pull a column line low, but there are no PMOS transistors in the grid to pull the line high (since that would double the size of the ROM). Instead, the multiplexer includes precharge transistors to pull the lines high, presumably in the clock phase before the ROM is read. The capacitance of the lines will keep the line high unless it is pulled low by a transistor in the grid. One of the four transistors in the multiplexer is activated (by control signal <code>a</code>, <code>b</code>, <code>c</code>, or <code>d</code>) to select the desired line. The output goes to a "keeper" circuit, which keeps the output high unless it is pulled low. The keeper uses an inverter with a weak PMOS transistor that can only provide a small pull-up current. A stronger low input will overpower this transistor, switching the state of the keeper. </p> <p>The output of this multiplexer, along with the outputs of three other multiplexers, goes to the second-stage multiplexer,<span id="fnref:mux"><a class="ref" href="#fn:mux">5</a></span> which selects one of its four inputs, based on control signals <code>e</code>, <code>f</code>, <code>g</code>, and <code>h</code>. The output of this multiplexer is held in a latch built from two inverters. The second latch has weak transistors so the latch can be easily forced into the desired state. The output from the first latch goes through a CMOS switch into a second latch, creating a flip-flop.</p> <p>The output from the second latch goes to a BiCMOS driver, which drives one of the 90 microcode output lines. Most processors are built from CMOS circuitry (i.e. NMOS and PMOS transistors), but the Pentium is built from BiCMOS circuitry: bipolar transistors as well as CMOS. At the time, bipolar transistors improved performance for high-current drivers; see my article on <a href="https://www.righto.com/2025/01/pentium-reverse-engineering-bicmos.html">the Pentium's BiCMOS circuitry</a>.</p> <p>The diagram below shows three bits of the microcode output. This circuitry is for the upper ROM bank; the circuitry is mirrored for the lower bank. The circuitry matches the schematic above. Each of the three blocks has 16 input lines from the ROM grid. Four 4-to-1 multiplexers reduce this to 4 lines, and the second multiplexer selects a single line. The result is latched and amplified by the output driver. (Note the large square shape of the bipolar transistors.) Next is the shift register that processes the microcode ROM outputs for testing. The shift register uses XOR logic for its feedback; unlike the rest of the circuitry, the XOR logic is irregular since only some bits are fed into XOR gates.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/output-die.jpg"><img alt="Three bits of output from the microcode, I removed the three metal layers to show the polysilicon and silicon." class="hilite" height="523" src="https://static.righto.com/images/pentium-microcode1/output-die-w500.jpg" title="Three bits of output from the microcode, I removed the three metal layers to show the polysilicon and silicon." width="500" /></a><div class="cite">Three bits of output from the microcode, I removed the three metal layers to show the polysilicon and silicon.</div></p> <h3>Circuitry for testing</h3> <p>Why does the microcode ROM have shift registers and XOR gates? The reason is that a chip such as the Pentium is very difficult to test: if one out of 3.1 million transistors goes bad, how do you detect it? For a simple processor like the 8086, you can run through the instruction set and be fairly confident that any problem would turn up. But with a complex chip, it is almost impossible to design an instruction sequence that would test every bit of the microcode ROM, every bit of the cache, and so forth. Starting with the 386, Intel added circuitry to the processor solely to make testing easier; about 2.7% of the transistors in the 386 were for testing.</p> <p>The Pentium has this testing circuitry for many ROMs and PLAs, including the division PLA that caused the infamous <a href="https://www.righto.com/2024/12/this-die-photo-of-pentium-shows.html">FDIV bug</a>. To test a ROM inside the processor, Intel added circuitry to scan the entire ROM and checksum its contents. Specifically, a pseudo-random number generator runs through each address, while another circuit computes a checksum of the ROM output, forming a "signature" word. At the end, if the signature word has the right value, the ROM is almost certainly correct. But if there is even a single bit error, the checksum will be wrong and the chip will be rejected.</p> <p>The pseudo-random numbers and the checksum are both implemented with linear feedback shift registers (LFSR), a shift register along with a few XOR gates to feed the output back to the input. For more information on testing circuitry in the 386, see <a href="https://doi.org/10.1109/MDT.1987.295165">Design and Test of the 80386</a>, written by Pat Gelsinger, who became Intel's CEO years later.</p> <h2>Conclusions</h2> <p>You'd think that implementing a ROM would be straightforward, but the Pentium's microcode ROM is surprisingly complex due to its optimized structure and its circuitry for testing. I haven't been able to determine much about how the microcode works, except that the micro-instruction is 90 bits wide and the ROM holds 4608 micro-instructions in total. But hopefully you've found this look at the circuitry interesting.</p> <p>Disclaimer: this should all be viewed as slightly speculative and there are probably some errors. I didn't want to prefix every statement with "I think that..." but you should pretend it is there. I plan to write more about the implementation of the Pentium, so follow me on Bluesky (<a href="https://bsky.app/profile/righto.com">@righto.com</a>) or <a href="https://www.righto.com/feeds/posts/default">RSS</a> for updates. Peter Bosch has done some reverse engineering of the Pentium II microcode; his information is <a href="https://pbx.sh/pentiumii-part1/">here</a>.</p> <h2>Footnotes and references</h2> <div class="footnote"> <ol> <li id="fn:ambiguity"> <p>It is arbitrary if a transistor corresponds to a 0 bit or a 1 bit. A transistor will pull the output line low (i.e. a 0 bit), but the signal could be inverted before it is used. More analysis of the circuitry or ROM contents would clear this up. <a class="footnote-backref" href="#fnref:ambiguity" title="Jump back to footnote 1 in the text">↩</a></p> </li> <li id="fn:contacts"> <p>When looking at a ROM like this, the contact pattern seems like it should tell you the contents of the ROM. Unfortunately, this doesn't work. Since a contact can be attached to one or two transistors, the contact pattern doesn't give you enough information. You need to see the silicon to determine the transistor pattern and thus the bits. <a class="footnote-backref" href="#fnref:contacts" title="Jump back to footnote 2 in the text">↩</a></p> </li> <li id="fn:simplified-rows"> <p>I simplified the row driver schematic. The most interesting difference is that the NAND gates are optimized to use three transistors each, instead of four transistors. The trick is that one of the NMOS transistors is essentially shared across the group of 8 drivers; an inverter drives the low side of all eight gates. The second simplification is that the 6-input AND gate is implemented with two 3-input NAND gates and a NOR gate for electrical reasons.</p> <p>Also, the decoder that converts 3 bits into 8 select lines is located between the banks, at the right, not at the top of the ROM as I showed in the schematic. Likewise, the inverters for the 6 row-select bits are not at the top. Instead, there are 6 inverters and 6 buffers arranged in a column to the right of the ROM, which works better for the layout. These are BiCMOS drivers so they can provide the high-current outputs necessary for the long wires and numerous transistor gates that they must drive. <a class="footnote-backref" href="#fnref:simplified-rows" title="Jump back to footnote 3 in the text">↩</a></p> </li> <li id="fn:binary"> <p>The inputs to the 6-input AND gate are arranged in a binary counting pattern, selecting each row in sequence. This binary arrangment is standard for a ROM's decoder circuitry and is a good way to recognize a ROM on a die. The Pentium has 36 row decoders, rather than the 64 that you'd expect from a 6-bit input. The ROM was made to the size necessary, rather than a full power of two. In most ROMs, it's difficult to determine if the ROM is addressed bottom-to-top or top-to-bottom. However, because the microcode ROM's counting pattern is truncated, one can see that the top bank starts with 0 at the top and counts downward, while the bottom bank is reversed, starting with 0 at the bottom and counting upward. <a class="footnote-backref" href="#fnref:binary" title="Jump back to footnote 4 in the text">↩</a></p> </li> <li id="fn:mux"> <p>A note to anyone trying to read the ROM contents: it appears that the order of entries in a group of 16 is inconsistent, so a straightforward attempt to visually read the ROM will end up with scrambled data. That is, some of the groups are reversed. I don't see any obvious pattern in which groups are reversed.</p> <p><a href="https://static.righto.com/images/pentium-microcode1/output-mux-detail.jpg"><img alt="A closeup of the first stage output mux. This image shows the M1 metal layer." class="hilite" height="319" src="https://static.righto.com/images/pentium-microcode1/output-mux-detail-w600.jpg" title="A closeup of the first stage output mux. This image shows the M1 metal layer." width="600" /></a><div class="cite">A closeup of the first stage output mux. This image shows the M1 metal layer.</div></p> <p>In the diagram above, look at the contacts from the select lines, connecting the select lines to the mux transistors. The contacts on the left are the mirror image of the contacts on the right, so the columns will be accessed in the opposite order. This mirroring pattern isn't consistent, though; sometimes neighboring groups are mirrored and sometimes they aren't.</p> <p>I don't know why the circuitry has this layout. Sometimes mirroring adjacent groups makes the layout more efficient, but the inconsistent mirroring argues against this. Maybe an automated layout system decided this was the best way. Or maybe Intel did this to provide a bit of obfuscation against reverse engineering. <a class="footnote-backref" href="#fnref:mux" title="Jump back to footnote 5 in the text">↩</a></p> </li> </ol> </div> <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'><span class='post-comment-link'> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/6264947694886887540/3129002752159663932' onclick=''> 1 comment: </a> </span> <span class='post-icons'> <span class='item-action'> <a href='https://www.blogger.com/email-post/6264947694886887540/3129002752159663932' title='Email Post'> <img alt='' class='icon-action' height='13' src='http://img1.blogblog.com/img/icon18_email.gif' width='18'/> </a> </span> <span class='item-control blog-admin pid-1138732533'> <a href='https://www.blogger.com/post-edit.g?blogID=6264947694886887540&postID=3129002752159663932&from=pencil' title='Edit Post'> <img alt='' class='icon-action' height='18' src='https://resources.blogblog.com/img/icon18_edit_allbkg.gif' width='18'/> </a> </span> </span> <span class='post-backlinks post-comment-link'> </span> <div class='post-share-buttons goog-inline-block'> <a class='goog-inline-block share-button sb-email' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=3129002752159663932&target=email' target='_blank' title='Email This'><span class='share-button-link-text'>Email This</span></a><a class='goog-inline-block share-button sb-blog' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=3129002752159663932&target=blog' onclick='window.open(this.href, "_blank", "height=270,width=475"); return false;' target='_blank' title='BlogThis!'><span class='share-button-link-text'>BlogThis!</span></a><a class='goog-inline-block share-button sb-twitter' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=3129002752159663932&target=twitter' target='_blank' title='Share to X'><span class='share-button-link-text'>Share to X</span></a><a class='goog-inline-block share-button sb-facebook' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=3129002752159663932&target=facebook' onclick='window.open(this.href, "_blank", "height=430,width=640"); return false;' target='_blank' title='Share to Facebook'><span class='share-button-link-text'>Share to Facebook</span></a><a class='goog-inline-block share-button sb-pinterest' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=3129002752159663932&target=pinterest' target='_blank' title='Share to Pinterest'><span class='share-button-link-text'>Share to Pinterest</span></a> </div> </div> <div class='post-footer-line post-footer-line-2'><span class='post-labels'> Labels: <a href='http://www.righto.com/search/label/intel' rel='tag'>intel</a>, <a href='http://www.righto.com/search/label/microcode' rel='tag'>microcode</a>, <a href='http://www.righto.com/search/label/Pentium' rel='tag'>Pentium</a>, <a href='http://www.righto.com/search/label/reverse-engineering' rel='tag'>reverse-engineering</a> </span> </div> <div class='post-footer-line post-footer-line-3'></div> </div> </div> </div> </div></div> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='https://static.righto.com/images/engelbart/interface-w500.jpg' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='4116959493954575947' itemprop='postId'/> <a name='4116959493954575947'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2025/03/mother-of-all-demos-usb-keyset-interface.html'>A USB interface to the "Mother of All Demos" keyset</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-4116959493954575947' itemprop='description articleBody'> <p>In the early 1960s, Douglas Engelbart started investigating how computers could augment human intelligence:  "If, in your office, you as an intellectual worker were supplied with a computer display backed up by a computer that was alive for you all day and was instantly responsive to every action you had, how much value could you derive from that?" Engelbart developed many features of modern computing that we now take for granted: the mouse,<span id="fnref:mouse"><a class="ref" href="#fn:mouse">1</a></span> hypertext, shared documents, windows, and a graphical user interface. At the 1968 Joint Computer Conference, Engelbart demonstrated these innovations in a groundbreaking presentation, now known as "The Mother of All Demos."</p>  <p><a href="https://static.righto.com/images/engelbart/interface.jpg"><img alt="The keyset with my prototype USB interface." class="hilite" height="364" src="https://static.righto.com/images/engelbart/interface-w500.jpg" title="The keyset with my prototype USB interface." width="500" /></a><div class="cite">The keyset with my prototype USB interface.</div></p> <p>Engelbart's demo also featured an input device known as the keyset, but unlike his other innovations, the keyset failed to catch on. The 5-finger keyset lets you type without moving your hand, entering characters by pressing multiple keys simultaneously as a chord. Christina Englebart, his daughter, loaned one of Engelbart's keysets to me. I constructed an interface to connect the keyset to USB, so that it can be used with a modern computer. The video below shows me typing with the keyset, using the mouse buttons to select upper case and special characters.<span id="fnref:keys"><a class="ref" href="#fn:keys">2</a></span></p> <iframe width="560" height="315" src="https://www.youtube.com/embed/DpshKBKt_os?si=gzyYjd-2_ltR9oeI" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <p>I wrote this blog post to describe my USB keyset interface. Along the way, however, I got sidetracked by the history of The Mother of All Demos and how it obtained that name. It turns out that Engelbart's demo isn't the first demo to be called "The Mother of All Demos".</p> <h2>Engelbart and The Mother of All Demos</h2>  <p>Engelbart's work has its roots in Vannevar Bush's 1945 visionary essay, "<a href="https://worrydream.com/refs/Bush%20-%20As%20We%20May%20Think%20(Life%20Magazine%209-10-1945).pdf">As We May Think</a>." Bush envisioned thinking machines, along with the "memex", a compact machine holding a library of collective knowledge with hypertext-style links: "The Encyclopedia Britannica could be reduced to the volume of a matchbox." The memex could search out information based on associative search, building up a hypertext-like trail of connections.</p> <p>In the early 1960s, Engelbart was inspired by Bush's essay and set out to develop means to augment human intellect: "increasing the capability of a man to approach a complex problem situation, to gain comprehension to suit his particular needs, and to derive solutions to problems."<span id="fnref:1962"><a class="ref" href="#fn:1962">3</a></span> Engelbart founded the Augmentation Research Center at the Stanford Research Institute (now SRI), where he and his team created a system called NLS (oN-Line System).</p> <p><a href="https://static.righto.com/images/engelbart/shopping-list.jpg"><img alt="Engelbart editing a hierarchical shopping list." class="hilite" height="351" src="https://static.righto.com/images/engelbart/shopping-list-w500.jpg" title="Engelbart editing a hierarchical shopping list." width="500" /></a><div class="cite">Engelbart editing a hierarchical shopping list.</div></p> <p>In 1968, Engelbart demonstrated NLS to a crowd of two thousand people at the Fall Joint Computer Conference. Engelbart gave the demo from the stage, wearing a crisp shirt and tie and a headset microphone. Engelbart created hierarchical documents, such as the shopping list above, and moved around them with hyperlinks. He demonstrated how text could be created, moved, and edited with the keyset and mouse. Other documents included graphics, crude line drawing by today's standards but cutting-edge for the time. The computer's output was projected onto a giant screen, along with video of Engelbart.</p> <p><a href="https://static.righto.com/images/engelbart/keyset-video.jpg"><img alt="Engelbart using the keyset to edit text. Note that the display doesn't support lowercase text; instead, uppercase is indicated by a line above the character. Adapted from The Mother of All Demos." class="hilite" height="354" src="https://static.righto.com/images/engelbart/keyset-video-w500.jpg" title="Engelbart using the keyset to edit text. Note that the display doesn't support lowercase text; instead, uppercase is indicated by a line above the character. Adapted from The Mother of All Demos." width="500" /></a><div class="cite">Engelbart using the keyset to edit text. Note that the display doesn't support lowercase text; instead, uppercase is indicated by a line above the character. Adapted from <a href="https://youtu.be/UhpTiWyVa6k?si=cqfTbRsOxTy8eE01">The Mother of All Demos</a>.</div></p> <p>Engelbart sat at a specially-designed Herman Miller desk<span id="fnref:herman-miller"><a class="ref" href="#fn:herman-miller">6</a></span> that held the keyset, keyboard, and mouse, shown above. While Engelbart was on stage in San Francisco, the SDS 940<span id="fnref:sds940"><a class="ref" href="#fn:sds940">4</a></span> computer that ran the NLS software was 30 miles to the south in Menlo Park.<span id="fnref:moad-video"><a class="ref" href="#fn:moad-video">5</a></span></p> <p>To the modern eye, the demo resembles a PowerPoint presentation over Zoom, as Engelbart collaborated with Jeff Rulifson and Bill Paxton, miles away in Menlo Park. (Just like a modern Zoom call, the remote connection started with "We're not hearing you. How about now?") Jeff Rulifson browsed the NLS code, jumping between code files with hyperlinks and expanding subroutines by clicking on them. NLS was written in custom <a href="https://bitsavers.org/pdf/sri/arc/NLS_Programmers_Guide_Jan76.pdf">high-level languages</a>, which they developed with a "compiler compiler" called <a href="https://en.wikipedia.org/wiki/TREE-META">TREE-META</a>. The NLS system held interactive documentation as well as tracking bugs and changes. Bill Paxton interactively drew a diagram and then demonstrated how NLS could be used as a database, retrieving information by searching on keywords. (Although Engelbart was stressed by the live demo, Paxton told me that he was "too young and inexperienced to be concerned.")</p> <p><a href="https://static.righto.com/images/engelbart/demo-english.jpg"><img alt="Bill Paxton, in Menlo Park, communicating with the conference in San Francisco." class="hilite" height="326" src="https://static.righto.com/images/engelbart/demo-english-w500.jpg" title="Bill Paxton, in Menlo Park, communicating with the conference in San Francisco." width="500" /></a><div class="cite">Bill Paxton, in Menlo Park, communicating with the conference in San Francisco.</div></p> <p>Bill English, an electrical engineer, not only built the first mouse for Engelbart but was also the hardware mastermind behind the demo. In San Francisco, the screen images were projected on a 20-foot screen by a Volkswagen-sized Eidophor projector, bouncing light off a modulated oil film. Numerous cameras, video switchers and mixers created the video image. Two leased microwave links and half a dozen antennas connected SRI in Menlo Park to the demo in San Francisco. High-speed modems send the mouse, keyset, and keyboard signals from the demo back to SRI. Bill English spent months assembling the hardware and network for the demo and then managed the demo behind the scenes, assisted by a team of about 17 people.</p> <p>Another participant was the famed counterculturist Stewart Brand, known for the <a href="https://en.wikipedia.org/wiki/Whole_Earth_Catalog">Whole Earth Catalog</a> and the WELL, one of the oldest online virtual communities. Brand advised Engelbart on the presentation, as well as running a camera. He'd often point the camera at a monitor to generate swirling psychedelic feedback patterns, reminiscent of the LSD that he and Engelbart had experimented with.</p> <p>The demo received press attention such as a San Francisco Chronicle article titled "Fantastic World of Tomorrow's Computer". It stated, "The most fantastic glimpse into the computer future was taking place in a windowless room on the third floor of the Civic Auditorium" where Engelbart "made a computer in Menlo Park do secretarial work for him that ten efficient secretaries couldn't do in twice the time." His goal: "We hope to help man do better what he does—perhaps by as much as 50 per cent." However, the demo received little attention in the following decades.<span id="fnref:attention"><a class="ref" href="#fn:attention">7</a></span></p> <p>Engelbart continued his work at SRI for almost a decade, but as Engelbart commented with frustration, “There was a slightly less than universal perception of our value at SRI”.<span id="fnref:levy"><a class="ref" href="#fn:levy">8</a></span> In 1977, SRI sold the Augmentation Research Center to Tymshare, a time-sharing computing company. (Timesharing was the cloud computing of the 1970s and 1980s, where companies would use time on a centralized computer.) At Tymshare, Engelbart's system was renamed AUGMENT and marketed as an office automation service, but Engelbart himself was sidelined from development, a situation that he <a href="https://stanford.edu/dept/SUL/sites/engelbart/engfmst3-ntb.html">described</a> as sitting in a corner and becoming invisible.</p> <p>Meanwhile, Bill English and some other SRI researchers<span id="fnref:researchers"><a class="ref" href="#fn:researchers">9</a></span> migrated four miles south to Xerox PARC and worked on the Xerox Alto computer. The Xerox Alto incorporated many ideas from the Augmentation Research Center including the graphical user interface, the mouse, and the keyset. The Alto's keyset was almost identical to the Engelbart keyset, as can be seen in the photo below. The Alto's keyset was most popular for the networked 3D shooter game "<a href="https://www.digibarn.com/collections/games/xerox-maze-war/index.html">Maze War</a>", with the clicking of keysets echoing through the hallways of Xerox PARC.</p> <p><a href="https://static.righto.com/images/engelbart/alto.jpg"><img alt="A Xerox Alto with a keyset on the left." class="hilite" height="359" src="https://static.righto.com/images/engelbart/alto-w500.jpg" title="A Xerox Alto with a keyset on the left." width="500" /></a><div class="cite">A Xerox Alto with a keyset on the left.</div></p> <p>Xerox famously failed to commercialize the ideas from the Xerox Alto, but Steve Jobs recognized the importance of interactivity, the graphical user interface, and the mouse when he visited Xerox PARC in 1979. Steve Jobs provided the Apple Lisa and Macintosh ended up with a graphical user interface and the mouse (streamlined to one button instead of three), but he left the keyset behind.<span id="fnref:parc"><a class="ref" href="#fn:parc">10</a></span></p> <p>When McDonnell Douglas acquired Tymshare in 1984, Engelbart and his software—now called Augment—had a new home.<span id="fnref:augment"><a class="ref" href="#fn:augment">11</a></span> In 1987, McDonnell Douglas released a text editor and outline processor for the IBM PC called <a href="https://archive.org/details/1987-augment-mini-base-users-guide_202503">MiniBASE</a>, one of the few PC applications that supported a keyset. The functionality of MiniBASE was almost identical to Engelbart's 1968 demo, but in 1987, MiniBASE was competing against GUI-based word processors such as MacWrite and Microsoft Word, so MiniBASE had little impact. Engelbart left McDonnell Douglas in 1988, forming a research foundation called the <a href="https://www.nytimes.com/1988/09/05/business/business-people-computer-scientist-forming-a-foundation.html">Bootstrap Institute</a> to continue his research independently.</p> <h2>The name: "The Mother of All Demos"</h2> <p>The name "The Mother of All Demos" has its roots in the Gulf War. In August 1990, Iraq invaded Kuwait, leading to war between Iraq and a coalition of the United States and 41 other countries. During the months of buildup prior to active conflict, Iraq's leader, Saddam Hussein, exhorted the Iraqi people to prepare for "<a href="https://www.nytimes.com/1990/09/22/world/confrontation-in-the-gulf-leaders-bluntly-prime-iraq-for-mother-of-all-battles.html">the mother of all battles</a>",<span id="fnref:mother"><a class="ref" href="#fn:mother">12</a></span> a phrase that caught the attention of the media. The battle didn't proceed as Hussein hoped: during <a href="https://www.nytimes.com/1991/02/28/world/war-gulf-president-bush-halts-offensive-combat-kuwait-freed-iraqis-crushed.html">exactly 100 hours</a> of ground combat, the US-led coalition liberated Kuwait, pushed into Iraq, crushed the Iraqi forces, and declared a ceasefire.<span id="fnref:gulf-war"><a class="ref" href="#fn:gulf-war">13</a></span> Hussein's mother of all battles became the <a href="https://www.nytimes.com/1991/02/27/arts/critic-s-notebook-human-images-help-add-drama-to-war-coverage.html">mother of all surrenders</a>.</p> <p>The phrase "mother of all ..." became the 1990s equivalent of a meme, used as a slightly-ironic superlative. It was applied to everything from <a href="https://www.nytimes.com/1993/06/18/sports/us-open-golf-notebook-fore-the-mother-of-all-traffic-jams.html">The Mother of All Traffic Jams</a> to <a href="https://amzn.to/4bzQ7Tc">The Mother of All Windows Books</a>, from <a href="https://cooking.nytimes.com/recipes/1132-the-mother-of-all-butter-cookies">The Mother of All Butter Cookies</a> to Apple calling mobile devices <a href="https://www.nytimes.com/1992/07/19/business/the-executive-computer-mother-of-all-markets-or-a-pipe-dream-driven-by-greed.html">The Mother of All Markets</a>.<span id="fnref:mobile"><a class="ref" href="#fn:mobile">14</a></span></p> <p>In 1991, this superlative was applied to a computer demo, but it wasn't Engelbart's demo. Andy Grove, Intel's president, gave a keynote speech at Comdex 1991 entitled <a href="https://www.youtube.com/watch?v=CwvOeKqXv18">The Second Decade: Computer-Supported Collaboration</a>, a live demonstration of his vision for PC-based video conferencing and wireless communication in the PC's second decade. This complex hour-long demo required almost six months to prepare, with 15 companies collaborating. Intel called this demo "The Mother of All Demos", a name repeated in the New York Times, San Francisco Chronicle, Fortune, and PC Week.<span id="fnref:intel"><a class="ref" href="#fn:intel">15</a></span> Andy Grove's demo was a hit, with over 20,000 people requesting a video tape, but the demo was soon forgotten.</p> <p><a href="https://static.righto.com/images/engelbart/nytimes-moad.jpg"><img alt="On the eve of Comdex, the New York Times wrote about Intel's "Mother of All Demos". Oct 21, 1991, D1-D2." class="hilite" height="357" src="https://static.righto.com/images/engelbart/nytimes-moad-w350.jpg" title="On the eve of Comdex, the New York Times wrote about Intel's "Mother of All Demos". Oct 21, 1991, D1-D2." width="350" /></a><div class="cite">On the eve of Comdex, the New York Times <a href="https://www.nytimes.com/1991/10/21/business/computer-industry-gathers-amid-chaos.html">wrote</a> about Intel's "Mother of All Demos". Oct 21, 1991, D1-D2.</div></p> <p>In 1994, <em>Wired</em> writer Steven Levy wrote <a href="https://amzn.to/4kCE63A">Insanely Great: The Life and Times of Macintosh, the Computer that Changed Everything</a>.<span id="fnref2:levy"><a class="ref" href="#fn:levy">8</a></span> In the second chapter of this comprehensive book, Levy explained how Vannevar Bush and Doug Engelbart "sparked a chain reaction" that led to the Macintosh. The chapter described Engelbart's 1968 demo in detail including a throwaway line saying, "<a href="https://archive.org/details/insanely_great_levy_hard_cover_1994_pdf__mlib/page/42/mode/1up">It was the mother of all demos.</a>"<span id="fnref:vandam"><a class="ref" href="#fn:vandam">16</a></span> Based on my research, I think this is the source of the name "The Mother of All Demos" for Engelbart's demo.</p> <p>By the end of the century, multiple publications echoed Levy's catchy phrase. In February 1999, the San Jose Mercury News had a <a href="https://web.archive.org/web/19991003082606/http://www.mercurycenter.com/svtech/news/special/engelbart/part4.htm">special article</a> on Engelbart, saying that the demonstration was "still called 'the mother of all demos'", a description echoed by the industry publication <a href="https://archive.org/details/sim_computerworld_1999-05-10_33_19/page/n83/mode/1up">Computerworld</a>.<span id="fnref:still"><a class="ref" href="#fn:still">17</a></span> The book <a href="https://archive.org/details/nerds20100step/page/124/mode/2up">Nerds: A Brief History of the Internet</a> stated that the demo "has entered legend as 'the mother of all demos'". By this point, Engelbart's fame for the "mother of all demos" was cemented and the phrase became near-obligatory when writing about him. The classic Silicon Valley history <a href="https://archive.org/details/fireinvalleymaki0000frei">Fire in the Valley</a> (1984), for example, didn't even mention Engelbart but in the <a href="https://archive.org/details/fireinvalleymaki00frei_0/page/303">second edition</a> (2000), "The Mother of All Demos" had its own chapter.</p> <h2>Interfacing the keyset to USB</h2> <p>Getting back to the keyset interface, the keyset consists of five microswitches, triggered by the five levers. The switches are wired to a standard DB-25 connector. I used a <a href="https://www.pjrc.com/store/teensy36.html">Teensy 3.6</a> microcontroller board for the interface, since this board can act both as a USB device and as a USB host. As a USB device, the Teensy can emulate a standard USB keyboard. As a USB host, the Teensy can receive input from a standard USB mouse.</p> <p>Connecting the keyset to the Teensy is (almost) straightforward, wiring the switches to five data inputs on the Teensy and the common line connected to ground. The Teensy's input lines can be configured with pullup resistors inside the microcontroller. The result is that a data line shows <code>1</code> by default and <code>0</code> when the corresponding key is pressed. One complication is that the keyset apparently has a 1.5 k惟 between the leftmost button and ground, maybe to indicate that the device is plugged in. This resistor caused that line to always appear low to the Teensy. To counteract this and allow the Teensy to read the pin, I connected a 1 k惟 pullup resistor to that one line.</p> <h3>The interface code</h3> <p>Reading the keyset and sending characters over USB is mostly straightforward, but there are a few complications. First, it's unlikely that the user will press multiple keyset buttons at exactly the same time. Moreover, the button contacts may bounce. To deal with this, I wait until the buttons have a stable value for 100 ms (a semi-arbitrary delay) before sending a key over USB.</p> <p>The second complication is that with five keys, the keyset only supports 32 characters. To obtain upper case, numbers, special characters, and control characters, the keyset is designed to be used in conjunction with mouse buttons. Thus, the interface needs to act as a USB host, so I can plug in a USB mouse to the interface. If I want the mouse to be usable as a mouse, not just buttons in conjunction with the keyset, the interface mus forward mouse events over USB. But it's not that easy, since mouse clicks in conjunction with the keyset shouldn't be forwarded. Otherwise, unwanted clicks will happen while using the keyset.</p> <p>To emulate a keyboard, the code uses the <a href="https://docs.arduino.cc/language-reference/en/functions/usb/Keyboard/">Keyboard</a> library. This library provides an API to send characters to the destination computer. Inconveniently, the simplest method, <code>print()</code>, supports only regular characters, not special characters like <code>ENTER</code> or <code>BACKSPACE</code>. For those, I needed to use the lower-level <code>press()</code> and <code>release()</code> methods. To read the mouse buttons, the code uses the <a href="https://github.com/PaulStoffregen/USBHost_t36">USBHost_t36</a> library, the Teensy version of the <a href="https://docs.arduino.cc/libraries/usb-host-shield-library-2.0/">USB Host</a> library. Finally, to pass mouse motion through to the destination computer, I use the <a href="https://docs.arduino.cc/language-reference/en/functions/usb/Mouse/">Mouse</a> library.</p> <p>If you want to make your own keyset, Eric Schlaepfer has a model <a href="https://github.com/schlae/engelbart-keyset">here</a>.</p> <h2>Conclusions</h2> <p>Engelbart claimed  that learning a keyset wasn't difficult—a six-year-old kid could learn it in less than a week—but I'm not willing to invest much time into learning it. In my brief use of the keyset, I found it very difficult to use physically. Pressing four keys at once is difficult, with the worst being all fingers except the ring finger. Combining this with a mouse button or two at the same time gave me the feeling that I was sight-reading a difficult piano piece. Maybe it becomes easier with use, but I noticed that Alto programs tended to treat the keyset as function keys, rather than a mechanism for typing with chords.<span id="fnref:alto"><a class="ref" href="#fn:alto">18</a></span> David Liddle of Xerox PARC <a href="https://archive.computerhistory.org/resources/access/text/2020/06/102792010-05-01-acc.pdf#page=9">said</a>, "We found that [the keyset] was tending to slow people down, once you got away from really hot [stuff] system programmers. It wasn't quite so good if you were giving it to other engineers, let alone clerical people and so on."</p> <p>If anyone else has a keyset that they want to connect via USB (unlikely as it may be), my code is on <a href="https://github.com/shirriff/keyset-to-usb-interface">github</a>.<span id="fnref:hackaday"><a class="ref" href="#fn:hackaday">19</a></span> Thanks to Christina Engelbart for loaning me the keyset. Thanks to Bill Paxton for answering my questions. Follow me on Bluesky (<a href="https://bsky.app/profile/righto.com">@righto.com</a>) or <a href="https://www.righto.com/feeds/posts/default">RSS</a> for updates.</p> <h2>Footnotes and references</h2> <div class="footnote"> <ol> <li id="fn:mouse"> <p>Engelbart's use of the mouse wasn't arbitrary, but based on research. In 1966, shortly after inventing the mouse, Engelbart carried out a <a href="https://archive.org/details/nasa_techdoc_19660020914">NASA-sponsored study</a> that evaluated six input devices: two types of joysticks, a Graphacon positioner, the mouse, a light pen, and a control operated by the knees (leaving the hands free). The mouse, knee control, and light pen performed best, with users finding the mouse satisfying to use. Although inexperienced subjects had some trouble with the mouse, experienced subjects considered it the best device.</p> <p><a href="https://static.righto.com/images/engelbart/devices.jpg"><img alt="A joystick, Graphacon, mouse, knee control, and light pen were examined as input devices. Photos from the study." class="hilite" height="546" src="https://static.righto.com/images/engelbart/devices-w600.jpg" title="A joystick, Graphacon, mouse, knee control, and light pen were examined as input devices. Photos from the study." width="600" /></a><div class="cite">A joystick, Graphacon, mouse, knee control, and light pen were examined as input devices. Photos from <a href="https://archive.org/details/nasa_techdoc_19660020914">the study</a>.</div></p> <p> <a class="footnote-backref" href="#fnref:mouse" title="Jump back to footnote 1 in the text">↩</a></p> </li> <li id="fn:keys"> <p>The information sheet below from the Augmentation Research Center shows what keyset chords correspond to each character. I used this encoding for my interface software. Each column corresponds to a different combination of mouse buttons.</p> <p><a href="https://static.righto.com/images/engelbart/keyset-sheet-front.jpg"><img alt="The information sheet for the keyset specifies how to obtain each character." class="hilite" height="626" src="https://static.righto.com/images/engelbart/keyset-sheet-front-w400.jpg" title="The information sheet for the keyset specifies how to obtain each character." width="400" /></a><div class="cite">The information sheet for the keyset specifies how to obtain each character.</div></p> <p>The special characters above are <code><CD></code> (Command Delete, i.e. cancel a partially-entered command), <code><BC></code> (Backspace Character), <code><OK></code> (confirm command), <code><BW></code>(Backspace Word), <code><RC></code> (Replace Character), <code><ESC></code> (which does filename completion).</p> <p>NLS and the Augment software have the concept of a <a href="https://dougengelbart.org/content/view/218/">viewspec</a>, a view specification that controls the view of a file. For instance, viewspecs can expand or collapse an outline to show more or less detail, filter the content, or show authorship of sections. The keyset can select viewspecs, as shown below.</p> <p><a href="https://static.righto.com/images/engelbart/keyset-sheet-back.jpg"><img alt="Back of the keyset information sheet." class="hilite" height="621" src="https://static.righto.com/images/engelbart/keyset-sheet-back-w400.jpg" title="Back of the keyset information sheet." width="400" /></a><div class="cite">Back of the keyset information sheet.</div></p> <p>Viewsets are explained in more detail in <a href="https://youtu.be/UhpTiWyVa6k?si=FsrEOWVd4QCszEGI&t=316">The Mother of All Demos</a>. For my keyset interface, I ignored viewspecs since I don't have software to use these inputs, but it would be easy to modify the code to output the desired viewspec characters.</p> <p> <a class="footnote-backref" href="#fnref:keys" title="Jump back to footnote 2 in the text">↩</a></p> </li> <li id="fn:1962"> <p>See <a href="https://www.dougengelbart.org/pubs/augment-3906.html">Augmenting Human Intellect: A Conceptual Framework</a>, Engelbart's 1962 report. <a class="footnote-backref" href="#fnref:1962" title="Jump back to footnote 3 in the text">↩</a></p> </li> <li id="fn:sds940"> <p>Engelbart <a href="https://dougengelbart.org/pubs/papers/scanned-original/1968-augment-3954-A-Research-Center-for-Augmenting-Human-Intellect.pdf">used</a> an SDS 940 computer running the Berkeley Timesharing System. The computer had 64K words of core memory, with 4.5 MB of drum storage for swapping and 96 MB of disk storage for files. For displays, the computer drove twelve 5" high-resolution CRTs, but these weren't viewed directly. Instead, each CRT had a video camera pointed at it and the video was redisplayed on a larger display in a work station in each office.</p> <p>The SDS 940 was a large 24-bit scientific computer, built by Scientific Data Systems. Although SDS built the first integrated-circuit-based commercial computer in 1965 (the <a href="https://en.wikipedia.org/wiki/Scientific_Data_Systems#SDS_92">SDS 92</a>), the SDS 940 was a transistorized system. It consisted of multiple refrigerator-sized cabinets, as shown below. Since each memory cabinet held 16K words and the computer at SRI had 64K, SRI's computer had two additional cabinets of memory.</p> <p><a href="https://static.righto.com/images/engelbart/sds940.jpg"><img alt="Front view of an SDS 940 computer. From the Theory of Operation manual." class="hilite" height="370" src="https://static.righto.com/images/engelbart/sds940-w800.jpg" title="Front view of an SDS 940 computer. From the Theory of Operation manual." width="800" /></a><div class="cite">Front view of an SDS 940 computer. From the <a href="http://www.bitsavers.org/pdf/sds/9xx/940/980126A_940_TheoryOfOperation_Mar67.pdf">Theory of Operation</a> manual.</div></p> <p>In the late 1960s, Xerox wanted to get into the computer industry, so Xerox <a href="https://www.nytimes.com/1969/05/16/archives/xerox-joins-computer-industry-xerox-entering-computer-field.html">bought</a> Scientific Data Systems in 1969 for $900 million (about $8 billion in current dollars). The acquisition was a disaster. After steadily losing money, Xerox decided to <a href="https://www.nytimes.com/1975/07/22/archives/computer-making-will-end-at-xerox-844million-writeoff-is-taken-in.html">exit</a> the mainframe computer business in 1975. Xerox's CEO summed up the purchase: "With hindsight, we would not have done the same thing." <a class="footnote-backref" href="#fnref:sds940" title="Jump back to footnote 4 in the text">↩</a></p> </li> <li id="fn:moad-video"> <p>The Mother of All Demos is on <a href="https://www.youtube.com/watch?v=UhpTiWyVa6k">YouTube</a>, as well as a five-minute <a href="https://www.youtube.com/watch?v=B6rKUf9DWRI">summary</a> for the impatient. <a class="footnote-backref" href="#fnref:moad-video" title="Jump back to footnote 5 in the text">↩</a></p> </li> <li id="fn:herman-miller"> <p>The desk for the keyset and mouse was designed by Herman Miller, the office furniture company. Herman Miller worked with SRI to design the desks, chairs, and office walls as part of their plans for the office of the future. Herman Miller invented the cubicle office in 1964, creating a modern replacement for the commonly used open office arrangement. <a class="footnote-backref" href="#fnref:herman-miller" title="Jump back to footnote 6 in the text">↩</a></p> </li> <li id="fn:attention"> <p>Engelbart's demo is famous now, but for many years it was ignored. For instance, Electronic Design had a long <a href="https://archive.org/details/bitsavers_ElectronicignV17N0319690201_71033514/page/25/mode/1up">article</a> on Engelbart's work in 1969 (putting the system on the cover), but there was no mention of the demo.</p> <p><a href="https://static.righto.com/images/engelbart/electronic-design.jpg"><img alt="Engelbart's system was featured on the cover of Electronic Design. Feb 1, 1969. (slightly retouched)" class="hilite" height="398" src="https://static.righto.com/images/engelbart/electronic-design-w500.jpg" title="Engelbart's system was featured on the cover of Electronic Design. Feb 1, 1969. (slightly retouched)" width="500" /></a><div class="cite">Engelbart's system was featured on the <a href="https://archive.org/details/bitsavers_ElectronicignV17N0319690201_71033514/mode/1up">cover</a> of Electronic Design. Feb 1, 1969. (slightly retouched)</div></p> <p>But by the 1980s, the Engelbart demo started getting attention. The 1986 documentary <a href="https://archive.org/details/XD303_86KTEH54_SiliconVllyBoomtown?start=1884.5">Silicon Valley Boomtown</a> had a long section on Engelbart's work and the demo. By 1988, the New York Times was referring to the demo as <a href="https://www.nytimes.com/1988/09/05/business/business-people-computer-scientist-forming-a-foundation.html">legendary</a>. <a class="footnote-backref" href="#fnref:attention" title="Jump back to footnote 7 in the text">↩</a></p> </li> <li id="fn:levy"> <p>Levy had written about Engelbart a decade earlier, in the May 1984 issue of the magazine <a href="https://guidebookgallery.org/articles/ofmiceandmen">Popular Computing</a>. The article focused on the mouse, recently available to the public through the Apple Lisa and the IBM PC (as an option). The big issue at the time was how many buttons a mouse should have: three like Engelbart's mouse, the one button that Apple used, or two buttons as Bill Gates preferred. But Engelbart's larger vision also came through in Levy's interview along with his frustration that most of his research had been ignored, overshadowed by the mouse. Notably, there was no mention of Engelbart's 1968 demo in the article. <a class="footnote-backref" href="#fnref:levy" title="Jump back to footnote 8 in the text">↩</a><a class="footnote-backref" href="#fnref2:levy" title="Jump back to footnote 8 in the text">↩</a></p> </li> <li id="fn:researchers"> <p>The SRI researchers who moved to Xerox include Bill English, Charles Irby, Jeff Rulifson, Bill Duval, and Bill Paxton (<a href="https://web.stanford.edu/class/history34q/readings/Engelbart/Engelbart_AugmentWorkshop.html">details</a>). <a class="footnote-backref" href="#fnref:researchers" title="Jump back to footnote 9 in the text">↩</a></p> </li> <li id="fn:parc"> <p>In 2023, Xerox donated the entire Xerox PARC research center to SRI. The research center remained in Palo Alto but became part of SRI. In a sense, this closed the circle, since many of the people and ideas from SRI had gone to PARC in the 1970s. However, both PARC and SRI had changed radically since the 1970s, with the cutting edge of computer research moving elsewhere. <a class="footnote-backref" href="#fnref:parc" title="Jump back to footnote 10 in the text">↩</a></p> </li> <li id="fn:augment"> <p>For a detailed discussion of the Augment system, see <a href="https://archive.org/details/seyboldreportonw00medi">Tymshare's Augment: Heralding a New Era</a>, Oct 1978. Augment provided a "broad range of information handling capability" that was not available elsewhere. Unlike other word processing systems, Augment was targeted at the professional, not clerical workers, people who were "eager to explore the open-ended possibilities" of the interactive process.</p> <p>The main complaints about Augment were its price and that it was not easy to use. Accessing Engelbart's NLS system over ARPANET cost an eye-watering $48,000 a year (over $300,000 a year in current dollars). Tymshare's Augment service was cheaper (about $80 an hour in current dollars), but still much more expensive than a standard word processing service.</p> <p>Overall, the article found that Augment users were delighted with the system: "It is stimulating to belong to the electronic intelligentsia." Users found it to be "a way of life—an absorbing, enriching experience". <a class="footnote-backref" href="#fnref:augment" title="Jump back to footnote 11 in the text">↩</a></p> </li> <li id="fn:mother"> <p>William Safire provided background in the New York Times, <a href="https://www.nytimes.com/1991/02/24/magazine/on-language-degrading-attrition.html">explaining</a> that "the mother of all battles" originally referred to the battle of Qadisiya in A.D. 636, and Saddam Hussein was referencing that ancient battle. A translator <a href="https://www.nytimes.com/1991/03/07/opinion/l-mother-of-battles-mistranslates-arabic-834791.html">responded</a>, however, that the Arabic expression would be better translated as "the great battle" than "the mother of all battles." <a class="footnote-backref" href="#fnref:mother" title="Jump back to footnote 12 in the text">↩</a></p> </li> <li id="fn:gulf-war"> <p>The end of the Gulf War left Saddam Hussein in control of Iraq and left thousands of US troops in Saudi Arabia. These factors would turn out to be catastrophic in the following years. <a class="footnote-backref" href="#fnref:gulf-war" title="Jump back to footnote 13 in the text">↩</a></p> </li> <li id="fn:mobile"> <p>At the Mobile '92 conference, Apple's CEO, John Sculley, said personal communicators could be "the mother of all markets," while Andy Grove of Intel said that the idea of a wireless personal communicator in every pocket is "a pipe dream driven by greed" (<a href="https://www.nytimes.com/1992/07/19/business/the-executive-computer-mother-of-all-markets-or-a-pipe-dream-driven-by-greed.html">link</a>). In hindsight, Sculley was completely right and Grove was completely wrong. <a class="footnote-backref" href="#fnref:mobile" title="Jump back to footnote 14 in the text">↩</a></p> </li> <li id="fn:intel"> <p>Some references to Intel's "Mother of all demos" are <a href="https://www.nytimes.com/1991/10/21/business/computer-industry-gathers-amid-chaos.html">Computer Industry Gathers Amid Chaos</a>, New York Times, Oct 21, 1991 and "Intel's High-Tech Vision of the Future: Chipmaker proposes using computers to dramatically improve productivity", San Francisco Chronicle, Oct 21, 1991, p24. The title of an article in Microprocessor Report, "Intel Declares Victory in the Mother of All Demos" (Nov. 20, 1991), alluded to the recently-ended war. <a href="https://archive.org/details/fortune135janluce/page/n401/mode/1up">Fortune</a> wrote about Intel's demo in the Feb 17, 1997 issue. A longer description of Intel's demo is in the book <a href="https://books.google.com/books?id=VazSDwAAQBAJ&pg=PA264">Strategy is Destiny</a>. <a class="footnote-backref" href="#fnref:intel" title="Jump back to footnote 15 in the text">↩</a></p> </li> <li id="fn:vandam"> <p>Several sources claim that Andy van Dam was the first to call Engelbart's demo "The Mother of All Demos." Although van Dam attended the 1968 demo, I couldn't find any evidence that he coined the phrase. John Markoff, a technology journalist for The New York Times, wrote a book <a href="https://books.google.com/books?id=cTyfxP-g2IIC&pg=PT228&dq=%22van+dam%22+%22mother+of+all+demos%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiC4ajp7JKMAxWKLkQIHTMiGLoQ6AF6BAgGEAM#v=onepage&q=%22van%20dam%22%20%22mother%20of%20all%20demos%22&f=false">What the Dormouse Said: How the Sixties Counterculture Shaped the Personal Computer Industry</a>. In this book, Markoff wrote about Engelbart's demo, saying "Years later, his talk remained 'the mother of all demos' in the words of Andries van Dam, a Brown University computer scientist." As far as I can tell, van Dam used the phrase but only after it had already been popularized by Levy. <a class="footnote-backref" href="#fnref:vandam" title="Jump back to footnote 16 in the text">↩</a></p> </li> <li id="fn:still"> <p>It's curious to write that the demonstration was <em>still</em> called the "mother of all demos" when the phrase was just a few years old. <a class="footnote-backref" href="#fnref:still" title="Jump back to footnote 17 in the text">↩</a></p> </li> <li id="fn:alto"> <p>The photo below shows a keyset from the Xerox Alto. The five keys are labeled with separate functions—Copy, Undelete, Move, Draw, and Fine— for use with <a href="https://xeroxparcarchive.computerhistory.org/indigo/da/AlePaper.dm!1_/.Ale.paper.html">ALE</a>, a program for IC design. ALE supported <a href="https://xeroxparcarchive.computerhistory.org/ivy/sweet/alto/ale/.ALE.press!1.pdf">keyset chording</a> in combination with the mouse.</p> <p><a class="footnote-backref" href="#fnref:alto" title="Jump back to footnote 18 in the text">↩</a><a href="https://static.righto.com/images/engelbart/alto-keyset.jpg"><img alt="Keyset from a Xerox Alto, courtesy of Digibarn." class="hilite" height="415" src="https://static.righto.com/images/engelbart/alto-keyset-w500.jpg" title="Keyset from a Xerox Alto, courtesy of Digibarn." width="500" /></a><div class="cite">Keyset from a Xerox Alto, courtesy of Digibarn.</div></p> </li> <li id="fn:hackaday"> <p>After I implemented this interface, I came across a project that constructed a 3D-printed chording keyset, also using a Teensy for the USB interface. You can find that project <a href="https://www.pjrc.com/engelbart-chording-keyset/">here</a>. <a class="footnote-backref" href="#fnref:hackaday" title="Jump back to footnote 19 in the text">↩</a></p> </li> </ol> </div> <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'><span class='post-comment-link'> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/6264947694886887540/4116959493954575947' onclick=''> 17 comments: </a> </span> <span class='post-icons'> <span class='item-action'> <a href='https://www.blogger.com/email-post/6264947694886887540/4116959493954575947' title='Email Post'> <img alt='' class='icon-action' height='13' src='http://img1.blogblog.com/img/icon18_email.gif' width='18'/> </a> </span> <span class='item-control blog-admin pid-1138732533'> <a href='https://www.blogger.com/post-edit.g?blogID=6264947694886887540&postID=4116959493954575947&from=pencil' title='Edit Post'> <img alt='' class='icon-action' height='18' src='https://resources.blogblog.com/img/icon18_edit_allbkg.gif' width='18'/> </a> </span> </span> <span class='post-backlinks post-comment-link'> </span> <div class='post-share-buttons goog-inline-block'> <a class='goog-inline-block share-button sb-email' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=4116959493954575947&target=email' target='_blank' title='Email This'><span class='share-button-link-text'>Email This</span></a><a class='goog-inline-block share-button sb-blog' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=4116959493954575947&target=blog' onclick='window.open(this.href, "_blank", "height=270,width=475"); return false;' target='_blank' title='BlogThis!'><span class='share-button-link-text'>BlogThis!</span></a><a class='goog-inline-block share-button sb-twitter' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=4116959493954575947&target=twitter' target='_blank' title='Share to X'><span class='share-button-link-text'>Share to X</span></a><a class='goog-inline-block share-button sb-facebook' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=4116959493954575947&target=facebook' onclick='window.open(this.href, "_blank", "height=430,width=640"); return false;' target='_blank' title='Share to Facebook'><span class='share-button-link-text'>Share to Facebook</span></a><a class='goog-inline-block share-button sb-pinterest' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=4116959493954575947&target=pinterest' target='_blank' title='Share to Pinterest'><span class='share-button-link-text'>Share to Pinterest</span></a> </div> </div> <div class='post-footer-line post-footer-line-2'><span class='post-labels'> Labels: <a href='http://www.righto.com/search/label/alto' rel='tag'>alto</a>, <a href='http://www.righto.com/search/label/electronics' rel='tag'>electronics</a>, <a href='http://www.righto.com/search/label/reverse-engineering' rel='tag'>reverse-engineering</a> </span> </div> <div class='post-footer-line post-footer-line-3'></div> </div> </div> </div> </div></div> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='https://static.righto.com/images/pentium-mult3/pentium-labeled-w500.jpg' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='3533328182698832872' itemprop='postId'/> <a name='3533328182698832872'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2025/03/pentium-multiplier-adder-reverse-engineered.html'>The Pentium contains a complicated circuit to multiply by three</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-3533328182698832872' itemprop='description articleBody'> <p>In 1993, Intel released the high-performance Pentium processor, the start of the long-running Pentium line. I've been examining the Pentium's circuitry in detail and I came across a circuit to multiply by three, a complex circuit with thousands of transistors. Why does the Pentium have a circuit to multiply specifically by three? Why is it so complicated? In this article, I examine this multiplier—which I'll call the ×3 circuit—and explain its purpose and how it is implemented.</p> <p>It turns out that this multiplier is a small part of the Pentium's floating-point multiplier circuit. In particular, the Pentium multiplies two 64-bit numbers using base-8 multiplication, which is faster than binary multiplication.<span id="fnref:speed"><a class="ref" href="#fn:speed">1</a></span> However, multiplying by 3 needs to be handled as a special case. Moreover, since the rest of the multiplication process can't start until the multiplication by 3 finishes, this circuit must be very fast. If you've studied digital design, you may have heard of techniques such as carry lookahead, Kogge-Stone addition, and carry-select addition. I'll explain how the ×3 circuit combines all these techniques to maximize performance.</p> <p>The photo below shows the Pentium's thumbnail-sized silicon die under a microscope. I've labeled the main functional blocks. In the center is the integer execution unit that performs most instructions. On the left, the code and data caches improve memory performance. The floating point unit, in the lower right, performs floating point operations. Almost half of the floating point unit is occupied by the multiplier, which uses an array of adders to rapidly multiply two 64-bit numbers. The focus of this article is the ×3 circuit, highlighted in yellow near the top of the multiplier. As you can see, the ×3 circuit takes up a nontrivial amount of the Pentium die, especially considering that its task seems simple.</p> <p><a href="https://static.righto.com/images/pentium-mult3/pentium-labeled.jpg"><img alt="This die photo of the Pentium shows the location of the multiplier." class="hilite" height="524" src="https://static.righto.com/images/pentium-mult3/pentium-labeled-w500.jpg" title="This die photo of the Pentium shows the location of the multiplier." width="500" /></a><div class="cite">This die photo of the Pentium shows the location of the multiplier.</div></p> <h2>Why does the Pentium use base-8 to multiply numbers?</h2> <p>Multiplying two numbers in binary is conceptually straightforward. You can think of binary multiplication as similar to grade-school long multiplication, but with binary numbers instead of decimal numbers. The example below shows how 5×6 is computed in binary: the three terms are added to produce the result. Conveniently, each term is either the multiplicand (101 in this case) or 0, shifted appropriately, so computing the terms is easy.</p> <pre style="border:none"> 101 ×110 ――― 000 <span style="font-family:serif;font-style:italic">i.e. 0×101</span> 101 <span style="font-family:serif;font-style:italic">i.e. 1×101</span> +101 <span style="font-family:serif;font-style:italic">i.e. 1×101</span> ――――― 11110 </pre> <p>Unfortunately, this straightforward multiplication approach is slow. With the three-bit numbers above, there are three terms to add. But if you multiply two 64-bit numbers, you have 64 terms to add, requiring a lot of time and/or circuitry.</p> <p>The Pentium uses a more complicated approach, computing multiplication in base 8. The idea is to consider the multiplier in groups of three bits, so instead of multiplying by 0 or 1 in each step, you multiply by a number from 0 to 7. Each term that gets added is still in binary, but the number of terms is reduced by a factor of three. Thus, instead of adding 64 terms, you add 22 terms, providing a substantial reduction in the circuitry required. (I'll describe the full details of the Pentium multiplier in a future article.<span id="fnref:details"><a class="ref" href="#fn:details">2</a></span>)</p> <p>The downside to radix-8 multiplication is that multiplying by a number from 0 to 7 is much more complicated than multiplying by 0 or 1, which is almost trivial. Fortunately, there are some shortcuts. Note that multiplying by 2 is the same as shifting the number to the left by 1 bit position, which is very easy in hardware—you wire each bit one position to the left. Similarly, to multiply by 4, shift the multiplicand two bit positions to the left.</p> <p>Multiplying by 7 seems inconvenient, but there is a trick, known as Booth's multiplication algorithm. Instead of multiplying by 7, you add 8 times the number and subtract the number, ending up with 7 times the number. You might think this requires two steps, but the trick is to multiply by one more in the (base-8) digit to the left, so you get the factor of 8 without an additional step. (A base-10 analogy is that if you want to multiply by 19, you can multiply by 20 and subtract the multiplicand.) Thus, you can get the ×7 by subtracting. Similarly, for a ×6 term, you can subtract a ×2 multiple and add ×8 in the next digit. Thus, the only difficult multiple is ×3. (What about ×5? If you can compute ×3, you can subtract that from ×8 to get ×5.)</p> <p>To summarize, the Pentium's radix-8 Booth's algorithm is a fast way to multiply, but it requires a special circuit to produce the ×3 multiple of the multiplicand.</p> <h2>Implementing a fast ×3 circuit with carry lookahead</h2> <p>Multiplying a number by three is straightforward in binary: add the number to itself, shifted to the left one position. (As mentioned above, shifting to the left is the same as multiplying by two and is easy in hardware.) Unfortunately, using a simple adder is too slow.</p> <p>The problem with addition is that carries make addition slow. Consider calculating 99999+1 by hand. You'll start with 9+1=10, then carry the one, generating another carry, which generates another carry, and so forth, until you go through all the digits. Computer addition has the same problem: If you're adding two numbers, the low-order bits can generate a carry that then propagates through all the bits. An adder that works this way—known as a ripple carry adder—will be slow because the carry has to ripple through all the bits. As a result, CPUs use special circuits to make addition faster.</p> <p>One solution is the carry-lookahead adder. In this adder, all the carry bits are computed in parallel, before computing the sums. Then, the sum bits can be computed in parallel, using the carry bits. As a result, the addition can be completed quickly, without waiting for the carries to ripple through the entire sum.</p> <p>It may seem impossible to compute the carries without computing the sum first, but there's a way to do it. For each bit position, you determine signals called "carry generate" and "carry propagate". These signals can then be used to determine all the carries in parallel. The <em>generate</em> signal indicates that the position generates a carry. For instance, if you add binary <code>1xx</code> and <code>1xx</code> (where <code>x</code> is an arbitrary bit), a carry will be generated from the top bit, regardless of the unspecified bits. On the other hand, adding <code>0xx</code> and <code>0xx</code> will never generate a carry. Thus, the <em>generate</em> signal is produced for the first case but not the second.</p> <p>But what about <code>1xx</code> plus <code>0xx</code>? We might get a carry, for instance, <code>111+001</code>, but we might not, for instance, <code>101+001</code>. In this "maybe" case, we set the <em>carry propagate</em> signal, indicating that a carry into the position will get propagated out of the position. For example, if there is a carry out of the middle position, <code>1xx+0xx</code> will have a carry from the top bit. But if there is no carry out of the middle position, then there will not be a carry from the top bit. In other words, the <em>propagate</em> signal indicates that a carry into the top bit will be propagated out of the top bit.</p> <p>To summarize, adding <code>1+1</code> will generate a carry. Adding <code>0+1</code> or <code>1+0</code> will propagate a carry. Thus, the <em>generate</em> signal is formed at each position by <em>G<sub>n</sub> = A<sub>n</sub>·B<sub>n</sub></em>, where <em>A</em> and <em>B</em> are the inputs. The <em>propagate</em> signal is <em>P<sub>n</sub> = A<sub>n</sub>+B<sub>n</sub></em>, the logical-OR of the inputs.<span id="fnref:propagate"><a class="ref" href="#fn:propagate">3</a></span></p> <p>Now that the <em>propagate</em> and <em>generate</em> signals are defined, some moderately complex logic<span id="fnref:carry"><a class="ref" href="#fn:carry">4</a></span> can compute the carry <em>C<sub>n</sub></em> into each bit position. The important thing is that all the carry bits can be computed in parallel, without waiting for the carry to ripple through each bit position. Once each carry is computed, the sum bits can be computed in parallel: <em>S<sub>n</sub> = A<sub>n</sub> ⊕ B<sub>n</sub> ⊕ C<sub>n</sub></em>. In other words, the two input bits and the computed carry are combined with exclusive-or. Thus, the entire sum can be computed in parallel by using carry lookahead. However, there are complications.</p> <h2>Implementing carry lookahead with a parallel prefix adder</h2> <p>The carry bits can be generated directly from the <em>G</em> and <em>P</em> signals. However, the straightforward approach requires too much hardware as the number of bits increases. Moreover, this approach needs gates with many inputs, which are slow for electrical reasons. For these reasons, the Pentium uses two techniques to keep the hardware requirements for carry lookahead tractable. First, it uses a "parallel prefix adder" algorithm for carry lookahead across 8-bit chunks.<span id="fnref:parallel-prefix"><a class="ref" href="#fn:parallel-prefix">7</a></span> Second, it uses a two-level hierarchical approach for carry lookahead: the upper carry-lookahead circuit handles eight 8-bit chunks, using the same 8-bit algorithm.<span id="fnref:bytes"><a class="ref" href="#fn:bytes">5</a></span></p> <p>The photo below shows the complete ×3 circuit; you can see that the circuitry is divided into blocks of 8 bits. (Although I'm calling this a 64-bit circuit, it really produces a 69-bit output: there are 5 "extra" bits on the left to avoid overflow and to provide additional bits for rounding.)</p> <p><a href="https://static.righto.com/images/pentium-mult3/wide-view.jpg"><img alt="The full ×3 adder circuit under a microscope." class="hilite" height="65" src="https://static.righto.com/images/pentium-mult3/wide-view-w800.jpg" title="The full ×3 adder circuit under a microscope." width="800" /></a><div class="cite">The full ×3 adder circuit under a microscope.</div></p> <p>The idea of the parallel-prefix adder is to produce the <em>propagate</em> and <em>generate</em> signals across ranges of bits, not just single bits as before. For instance, the <em>propagate</em> signal <em>P<sub>32</sub></em> indicates that a carry in to bit 2 would be propagated out of bit 3, (This would happen with <code>10xx+01xx</code>, for example.) And <em>G<sub>30</sub></em> indicates that bits 3 to 0 generate a carry out of bit 3. (This would happen with <code>1011+0111</code>, for example.)</p> <p>Using some mathematical tricks,<span id="fnref:pg"><a class="ref" href="#fn:pg">6</a></span> you can take the <em>P</em> and <em>G</em> values for two smaller ranges and merge them into the <em>P</em> and <em>G</em> values for the combined range. For instance, you can start with the <em>P</em> and <em>G</em> values for bits 0 and 1, and produce <em>P<sub>10</sub></em> and <em>G<sub>10</sub></em>, the <em>propagate</em> and <em>generate</em> signals describing two bits. These could be merged with <em>P<sub>32</sub></em> and <em>G<sub>32</sub></em> to produce <em>P<sub>30</sub></em> and <em>G<sub>30</sub></em>, indicating if a carry is propagated across bits 3-0 or generated by bits 3-0. Note that <em>G<sub>n0</sub></em> tells us if a carry is generated into bit <em>n+1</em> from all the lower bits, which is the <em>C<sub>n+1</sub></em> carry value that we need to compute the final sum. This merging process is more efficient than the "brute force" implementation of the carry-lookahead logic since logic subexpressions can be reused.</p> <p>There are many different ways that you can combine the <em>P</em> and <em>G</em> terms to generate the necessary terms.<span id="fnref:brent-kung"><a class="ref" href="#fn:brent-kung">8</a></span> The Pentium uses an approach called <a href="https://en.wikipedia.org/wiki/Kogge%E2%80%93Stone_adder">Kogge-Stone</a> that attempts to minimize the total delay while keeping the amount of circuitry reasonable. The diagram below is the standard diagram that illustrates how a Kogge-Stone adder works. It's rather abstract, but I'll try to explain it. The diagram shows how the <em>P</em> and <em>G</em> signals are merged to produce each output at the bottom. Each square box at the top generates the <em>P</em> and <em>G</em> signals for that bit. Each line corresponds to both the <em>P</em> and the <em>G</em> signal. Each diamond combines two ranges of <em>P</em> and <em>G</em> signals to generate new <em>P</em> and <em>G</em> signals for the combined range. Thus, the signals cover wider ranges of bits as they progress downward, ending with the <em>G<sub>n0</sub></em> outputs that indicate carries.</p> <p><a href="https://static.righto.com/images/pentium-mult3/kogge-stone.jpg"><img alt="A diagram of an 8-bit Kogge-Stone adder highlighting the carry out of bit 6 (green) and out of bit 2 (purple). Modification of the diagram by Robey Pointer, Wikimedia Commons." class="hilite" height="437" src="https://static.righto.com/images/pentium-mult3/kogge-stone-w500.jpg" title="A diagram of an 8-bit Kogge-Stone adder highlighting the carry out of bit 6 (green) and out of bit 2 (purple). Modification of the diagram by Robey Pointer, Wikimedia Commons." width="500" /></a><div class="cite">A diagram of an 8-bit Kogge-Stone adder highlighting the carry out of bit 6 (green) and out of bit 2 (purple). Modification of the diagram by Robey Pointer, <a href="https://commons.wikimedia.org/wiki/File:Kogge-stone-8-bit.png">Wikimedia Commons</a>.</div></p> <p>I've labeled a few of the intermediate signals so you can get an idea of how it works. Circuit "A" combines <em>P<sub>7</sub></em> and <em>G<sub>7</sub></em> with <em>P<sub>6</sub></em> and <em>G<sub>6</sub></em> to produce the signals describing two bits: <em>P<sub>76</sub></em> and <em>G<sub>76</sub></em>. Similarly, circuit "B" combines <em>P<sub>76</sub></em> and <em>G<sub>76</sub></em> with <em>P<sub>54</sub></em> and <em>G<sub>54</sub></em> to produce the signals describing four bits: <em>P<sub>74</sub></em> and <em>G<sub>74</sub></em>. Finally, circuit "C" produces the final outputs for bit 7: <em>P<sub>70</sub></em> and <em>G<sub>70</sub></em>. Note that most of the intermediate results are used twice, reducing the amount of circuitry. Moreover, there are at most three levels of combination circuitry, reducing the delay compared to a deeper network.</p> <p>The key point is the <em>P</em> and <em>G</em> values are computed in parallel so the carry bits can all be computed in parallel, without waiting for the carry to ripple through all the bits. (If this explanation doesn't make sense, see my discussion of the Kogge-Stone adder in the <a href="https://www.righto.com/2025/01/pentium-carry-lookahead-reverse-engineered.html">Pentium's division circuit</a> for a different—but maybe still confusing—explanation.)</p> <h2>Recursive Kogge-Stone lookahead</h2> <p>The Kogge-Stone approach can be extended to 64 bits, but the amount of circuitry and wiring becomes overwhelming. Instead, the Pentium uses a recursive, hierarchical approach with two levels of Kogge-Stone lookahead. The lower layer uses eight Kogge-Stone adders as described above, supporting 64 bits in total.</p> <p>The upper layer uses a single eight-bit Kogge-Stone lookahead circuit, treating each of the lower chunks as a single bit. That is, a lower chunk has a propagate signal <em>P</em> indicating that a carry into the chunk will be propagated out, as well as a generate signal <em>G</em> indicating that the chunk generates a carry. The upper Kogge-Stone circuit combines these chunked signals to determine if carries will be generated or propagated by groups of chunks.<span id="fnref:recursive"><a class="ref" href="#fn:recursive">9</a></span></p> <p>To summarize, each of the eight lower lookahead circuits computes the carries within an 8-bit chunk. The upper lookahead circuit computes the carries into and out of each 8-bit chunk. In combination, the circuits rapidly provide all the carries needed to compute the 64-bit sum.</p> <h2>The carry-select adder</h2> <p>Suppose you're on a game show: "What is 553 + 246 + <em>c</em>? In 10 seconds, I'll tell you if <em>c</em> is 0 or 1 and whoever gives the answer first wins $1000." Obviously, you shouldn't just sit around until you get <em>c</em>. You should do the two sums now, so you can hit the buzzer as soon as <em>c</em> is announced. This is the concept behind the carry-select adder: perform two additions—with a carry-in and without--and then supply the correct answer as soon as the carry is available. The carry-select adder requires additional hardware—two adders along with a multiplexer to select the result—but it overlaps the time to compute the sum with the time to compute the carry. In effect, the addition and the carry lookahead operations are performed in parallel, with the multiplexer combining the results from each.</p> <p>The Pentium uses a carry-select adder for each 8-bit chunk in the ×3 circuit. The carry from the second-level carry-lookahead selects which sum should be produced for the chunk. Thus, the time to compute the carry is overlapped with the time to compute the sum.</p> <h2>Putting the adder pieces together</h2> <p>The image below zooms in on an 8-bit chunk of the ×3 multiplier, implementing an 8-bit adder. Eight input lines are at the top (along with some unrelated wires). Note that each input line splits with a signal going to the adder on the left and a signal going to the right. This is what causes the adder to multiply by 3: it adds the input and the input shifted one bit to the left, i.e. multiplied by two. The top part of the adder has eight circuits to produce the <em>propagate</em> and <em>generate</em> signals. These signals go into the 8-bit Kogge-Stone lookahead circuit. Although most of the adder consists of a circuit block repeated eight times, the Kogge-Stone circuitry appears chaotic. This is because each bit of the Kogge-Stone circuit is different—higher bits are more complicated to compute than lower bits.</p> <p><a href="https://static.righto.com/images/pentium-mult3/block-poly-labeled.jpg"><img alt="One 8-bit block of the ×3 circuit." class="hilite" height="323" src="https://static.righto.com/images/pentium-mult3/block-poly-labeled-w500.jpg" title="One 8-bit block of the ×3 circuit." width="500" /></a><div class="cite">One 8-bit block of the ×3 circuit.</div></p> <p>The lower half of the circuit block contains an 8-bit carry-select adder. This circuit produces two sums, with multiplexers selecting the correct sum based on the carry into the block. Note that the carry-select adder blocks are narrower than the other circuitry.<span id="fnref:cell"><a class="ref" href="#fn:cell">10</a></span> This makes room for a Kogge-Stone block on the left. The second level Kogge-Stone circuitry is split up; the 8-bit carry-lookahead circuitry has one bit implemented in each block of the adder, and produces the carry-in signal for that adder block. In other words, the image above includes 1/8 of the second-level Kogge-Stone circuit. Finally, eight driver circuits amplify the output bits before they are sent to the rest of the floating-point multiplier.</p> <p>The block diagram below shows the pieces are combined to form the ×3 multiplier. The multiplier has eight 8-bit adder blocks (green boxes, corresponding to the image above). Each block computes eight bits of the total sum. Each block provides <em>P<sub>70</sub></em> and <em>G<sub>70</sub></em> signals to the second-level lookahead, which determines if each block receives a carry in. The key point to this architecture is that everything is computed in parallel, making the addition fast.</p> <p><a href="https://static.righto.com/images/pentium-mult3/overall-diagram.jpg"><img alt="A block diagram of the multiplier." class="hilite" height="312" src="https://static.righto.com/images/pentium-mult3/overall-diagram-w600.jpg" title="A block diagram of the multiplier." width="600" /></a><div class="cite">A block diagram of the multiplier.</div></p> <p>In the diagram above, the first 8-bit block is expanded to show its contents. The 8-bit lookahead circuit generates the <em>P</em> and <em>G</em> signals that determine the internal carry signals. The carry-select adder contains two 8-bit adders that use the carry lookahead values. As described earlier, one adder assumes that the block's carry-in is 1 and the second assumes the carry-in is 0. When the real carry in value is provided by the second-level lookahead circuit, the multiplexer selects the correct sum.</p> <p>The photo below shows how the complete multiplier is constructed from 8-bit blocks. The multiplier produces a 69-bit output; there are 5 "extra" bits on the left. Note that the second-level Kogge-Stone blocks are larger on the right than the left since the lookahead circuitry is more complex for higher-order bits.</p> <p><a href="https://static.righto.com/images/pentium-mult3/wide-view.jpg"><img alt="The full adder circuit. This is the same image as before, but hopefully it makes more sense at this point." class="hilite" height="65" src="https://static.righto.com/images/pentium-mult3/wide-view-w800.jpg" title="The full adder circuit. This is the same image as before, but hopefully it makes more sense at this point." width="800" /></a><div class="cite">The full adder circuit. This is the same image as before, but hopefully it makes more sense at this point.</div></p> <p>Going back to the full ×3 circuit above, you can see that the 8 bits on the right have significantly simpler circuitry. Because there is no carry-in to this block, the carry-select circuitry can be omitted. The block's internal carries, generated by the Kogge-Stone lookahead circuitry, are added using exclusive-NOR gates. The diagram below shows the implementation of an XNOR gate, using inverters and a multiplexer.</p> <h2>The XNOR circuit</h2> <p>I'll now describe one of the multiplier's circuits at the transistor level, in particular an XNOR gate. It's interesting to look at XNOR because XNOR (like XOR) is a tricky gate to implement and different processors use very different approaches. For instance, the Intel 386 implements XOR from AND-NOR gates (<a href="https://www.righto.com/2023/12/386-xor-circuits.html">details</a>) while the Z-80 uses pass transistors (<a href="https://www.righto.com/2013/09/understanding-z-80-processor-one-gate.html">details</a>). The Pentium, on the other hand, uses a multiplexer.</p> <p><a href="https://static.righto.com/images/pentium-mult3/xnor-diagram.jpg"><img alt="An exclusive-NOR gate with the components labeled. This is a focus-stacked image." class="hilite" height="271" src="https://static.righto.com/images/pentium-mult3/xnor-diagram-w500.jpg" title="An exclusive-NOR gate with the components labeled. This is a focus-stacked image." width="500" /></a><div class="cite">An exclusive-NOR gate with the components labeled. This is a focus-stacked image.</div></p> <p>The diagram above shows one of the XNOR gates in the adder's low bits.<span id="fnref:low-bits"><a class="ref" href="#fn:low-bits">11</a></span> The gate is constructed from four inverters and a pass-transistor multiplexer. Input B selects one of the multiplexer's two inputs: input A or input A inverted. The result is the XNOR function. (Inverter 1 buffers the input, inverter 5 buffers the output, and inverter 4 provides the complemented B signal to drive the multiplexer.)</p> <p>For the photo, I removed the top two metal layers from the chip, leaving the bottom metal layer, called M1. The doped silicon regions are barely visible beneath the metal. When a polysilicon line crosses doped silicon, it forms the gate of a transistor. This CMOS circuit has NMOS transistors at the top and PMOS transistors at the bottom. Each inverter consists of two transistors, while the multiplexer consists of four transistors.</p> <h2>The BiCMOS output drivers</h2> <p>The outputs from the ×3 circuit require high current. In particular, each signal from the ×3 circuit can drive up to 22 terms in the floating-point multiplier. Moreover, the destination circuits can be a significant distance from the ×3 circuit due to the size of the multiplier. Since the ×3 signals are connected to many transistor gates through long wires, the capacitance is high, requiring high current to change the signals quickly.</p> <p>The Pentium is constructed with a somewhat unusual process called BiCMOS, which combines bipolar transistors and CMOS on the same chip. The Pentium extensively uses BiCMOS circuits since they reduced signal delays by up to 35%. Intel also used BiCMOS for the Pentium Pro, Pentium II, Pentium III, and Xeon processors. However, as chip voltages dropped, the benefit from bipolar transistors dropped too and BiCMOS was eventually abandoned.</p> <p>The schematic below shows a simplified BiCMOS driver that inverts its input. A 0 input turns on the upper inverter, providing current into the bipolar (NPN) transistor's base. This turns on the transistor, causing it to pull the output high strongly and rapidly. A 1 input, on the other hand, will stop the current flow through the NPN transistor's base, turning it off. At the same time, the lower inverter will pull the output low. (The NPN transistor can only pull the output high.)</p> <p>Note the asymmetrical construction of the inverters. Since the upper inverter must provide a large current into the NPN transistor's base, it is designed to produce a strong (high-current) positive output and a weak low output. The lower inverter, on the other hand, is responsible for pulling the output low. Thus, it is constructed to produce a strong low output, while the high output can be weak.</p> <p><a href="https://static.righto.com/images/pentium-mult3/bicmos-driver.jpg"><img alt="The basic circuit for a BiCMOS driver." class="hilite" height="150" src="https://static.righto.com/images/pentium-mult3/bicmos-driver-w200.jpg" title="The basic circuit for a BiCMOS driver." width="200" /></a><div class="cite">The basic circuit for a BiCMOS driver.</div></p> <p>The driver of the ×3 circuit goes one step further: it uses a BiCMOS driver to drive a second BiCMOS driver. The motivation is that the high-current inverters have fairly large transistor gates, so they need to be driven with high current (but not as much as they produce, so there isn't an infinite regress).<span id="fnref:logical-effort"><a class="ref" href="#fn:logical-effort">12</a></span></p> <p>The schematic below shows the BiCMOS driver circuit that the ×3 multiplier uses. Note the large, box-like appearance of the NPN transistors, very different from the regular MOS transistors. Each box contains two NPN transistors sharing collectors: a larger transistor on the left and a smaller one on the right. You might expect these transistors to work together, but the contiguous transistors are part of two separate circuits. Instead, the small NPN transistor to the left and the large NPN transistor to the right are part of the same circuit.</p> <p><a href="https://static.righto.com/images/pentium-mult3/driver-diagram.jpg"><img alt="One of the output driver circuits, showing the polysilicon and silicon." class="hilite" height="292" src="https://static.righto.com/images/pentium-mult3/driver-diagram-w800.jpg" title="One of the output driver circuits, showing the polysilicon and silicon." width="800" /></a><div class="cite">One of the output driver circuits, showing the polysilicon and silicon.</div></p> <p>The inverters are constructed as standard CMOS circuits with PMOS transistors to pull the output high and NMOS transistors to pull the output low. The inverters are carefully structured to provide asymmetrical current, making them more interesting than typical inverters. Two pullup transistors have a long gate, making these transistors unusually weak. Other parts of the inverters have multiple transistors in parallel, providing more current. Moreover, the inverters have unusual layouts, with the NMOS and PMOS transistors widely separated to make the layout more efficient. For more on BiCMOS in the Pentium, see my article on <a href="https://www.righto.com/2025/01/pentium-reverse-engineering-bicmos.html">interesting BiCMOS circuits in the Pentium</a>.</p> <h2>Conclusions</h2> <p>Hardware support for computer multiplication has a long history going back to the 1950s.<span id="fnref:history"><a class="ref" href="#fn:history">13</a></span> Early microprocessors, though, had very limited capabilities, so microprocessors such as the 6502 didn't have hardware support for multiplication; users had to implement multiplication in software through shifts and adds. As hardware advanced, processors provided multiplication instructions but they were still slow. For example, the Intel 8086 processor (1978) implemented multiplication in microcode, performing a slow shift-and-add loop internally. Processors became exponentially more powerful over time, as described by Moore's Law, allowing later processors to include dedicated multiplication hardware. The 386 processor (1985) included a <a href="https://bitsavers.trailing-edge.com/components/intel/80386/231746-001_Introduction_to_the_80386_Apr86.pdf#page=9">multiply unit</a>, but it was still slow, taking up to 41 clock cycles for a multiplication instruction.</p> <p>By the time of the Pentium (1993), microprocessors contained millions of transistors, opening up new possibilities for design. With a seemingly unlimited number of transistors, chip architects could look at complicated new approaches to squeeze more performance out of a system. This ×3 multiplier contains roughly 9000 transistors, a bit more than an entire Z80 microprocessor (1976). Keep in mind that the ×3 multiplier is a small part of the floating-point multiplier, which is part of the floating-point unit in the Pentium. Thus, this small piece of a feature is more complicated than an entire microprocessor from 17 years earlier, illustrating the incredible growth in processor complexity.</p> <p>I plan to write more about the implementation of the Pentium, so follow me on Bluesky (<a href="https://bsky.app/profile/righto.com">@righto.com</a>) or <a href="https://www.righto.com/feeds/posts/default">RSS</a> for updates. (I'm no longer on Twitter.) The <a href="https://www.righto.com/2024/08/pentium-navajo-fairchild-shiprock.html">Pentium Navajo rug</a> inspired me to examine the Pentium in more detail.</p> <h2>Footnotes and references</h2> <div class="footnote"> <ol> <li id="fn:speed"> <p>A floating-point multiplication on the Pentium takes three clock cycles, of which the multiplication circuitry is busy for two cycles. (See Agner Fog's <a href="https://www.agner.org/optimize/instruction_tables.pdf#page=164">optimization manual</a>.) In comparison, integer multiplication (<code>MUL</code>) is much slower, taking 11 cycles. The Nehalem microarchitecture (2008) reduced floating-point multiplication time to 1 cycle. <a class="footnote-backref" href="#fnref:speed" title="Jump back to footnote 1 in the text">↩</a></p> </li> <li id="fn:details"> <p>I'll give a quick outline of the Pentium's floating-point multiplier as a preview. The multiplier is built from a tree of ten carry-save adders to sum the terms. Each carry-save adder is a 4:2 compression adder, taking four input bits and producing two output bits. The output from the carry-save adder is converted to the final result by an adder using Kogge-Stone lookahead and carry select. Multiplying two 64-bit numbers yields 128 bits, but the Pentium produces a 64-bit result. (There are actually a few more bits for rounding.) The low 64 bits can't simply be discarded because they could produce a carry into the preserved bits. Thus, the low 64 bits go into another Kogge-Stone lookahead circuit that doesn't produce a sum, but indicates if there is a carry. Since the datapath is 64 bits wide, but the product is 128 bits, there are many shift stages to move the bits to the right column. Moreover, the adders are somewhat wider than 64 bits as needed to hold the intermediate sums. <a class="footnote-backref" href="#fnref:details" title="Jump back to footnote 2 in the text">↩</a></p> </li> <li id="fn:propagate"> <p>The bits <code>1+1</code> will set <em>generate</em>, but should <em>propagate</em> be set too? It doesn't make a difference as far as the equations. This adder sets <em>propagate</em> for <code>1+1</code> but some other adders do not. The answer depends on if you use an inclusive-or or exclusive-or gate to produce the <em>propagate</em> signal. <a class="footnote-backref" href="#fnref:propagate" title="Jump back to footnote 3 in the text">↩</a></p> </li> <li id="fn:carry"> <p>The carry <em>C<sub>n</sub></em> at each bit position <em>n</em> can be computed from the <em>G</em> and <em>P</em> signals by considering the various cases:</p> <p><em>C<sub>1</sub> = G<sub>0</sub></em>: a carry into bit 1 occurs if a carry is generated from bit 0. <br><em>C<sub>2</sub> = G<sub>1</sub> + G<sub>0</sub>P<sub>1</sub></em>: A carry into bit 2 occurs if bit 1 generates a carry or bit 1 propagates a carry from bit 0. <br><em>C<sub>3</sub> = G<sub>2</sub> + G<sub>1</sub>P<sub>2</sub> + G<sub>0</sub>P<sub>1</sub>P<sub>2</sub></em>: A carry into bit 3 occurs if bit 2 generates a carry, or bit 2 propagates a carry generated from bit 1, or bits 2 and 1 propagate a carry generated from bit 0. <br><em>C<sub>4</sub> = G<sub>3</sub> + G<sub>2</sub>P<sub>3</sub> + G<sub>1</sub>P<sub>2</sub>P<sub>3</sub> + G<sub>0</sub>P<sub>1</sub>P<sub>2</sub>P<sub>3</sub></em>: A carry into bit 4 occurs if a carry is generated from bit 3, 2, 1, or 0 along with the necessary propagate signals. <br>And so on...</p> <p>Note that the formula gets more complicated for each bit position. The circuit complexity is approximately <em>O(N<sup>3</sup>)</em>, depending on how you measure it. Thus, implementing the carry lookahead formula directly becomes impractical as the number of bits gets large. The Kogge-Stone approach uses approximately <em>O(N log N)</em> transistors, but the wiring becomes excessive for large <em>N</em> since there are <em>N/2</em> wires of length <em>N/2</em>. Using a tree of Kogge-Stone circuits reduces the amount of wiring. <a class="footnote-backref" href="#fnref:carry" title="Jump back to footnote 4 in the text">↩</a></p> </li> <li id="fn:bytes"> <p>The 8-bit chunks in the circuitry have nothing to do with bytes. The motivation is that 8 bits is a reasonable size for a chunk, as well as providing a nice breakdown into 8 chunks of 8 bits. Other systems have used 4-bit chunks for carry lookahead (such as minicomputers based on the 74181 ALU chip). <a class="footnote-backref" href="#fnref:bytes" title="Jump back to footnote 5 in the text">↩</a></p> </li> <li id="fn:pg"> <p>I won't go into the mathematics of merging <em>P</em> and <em>G</em> signals; see, for example, <a href="https://bpb-us-w2.wpmucdn.com/sites.coecis.cornell.edu/dist/4/81/files/2019/06/4740_lecture21-adder-circuits.pdf#page=14">Adder Circuits</a> or <a href="https://personal.utdallas.edu/~ivor/ce6305/m4.pdf">Carry Lookahead Adders</a> for additional details. The important factor is that the carry merge operator is associative (actually a monoid), so the sub-ranges can be merged in any order. This flexibility is what allows different algorithms with different tradeoffs. <a class="footnote-backref" href="#fnref:pg" title="Jump back to footnote 6 in the text">↩</a></p> </li> <li id="fn:parallel-prefix"> <p>The idea behind a prefix adder is that we want to see if there is a carry out of bit 0, bits 0-1, bits 0-2, bits 0-3, 0-4, and so forth. These are all the prefixes of the word. Since the prefixes are computed in parallel, it's called a parallel prefix adder. <a class="footnote-backref" href="#fnref:parallel-prefix" title="Jump back to footnote 7 in the text">↩</a></p> </li> <li id="fn:brent-kung"> <p>The lookahead merging process can be implemented in many ways, including <a href="https://en.wikipedia.org/wiki/Kogge%E2%80%93Stone_adder">Kogge-Stone</a>, <a href="https://en.wikipedia.org/wiki/Brent%E2%80%93Kung_adder">Brent-Kung</a>, and Ladner-Fischer, with different tradeoffs. For one example, the diagram below shows that Brent-Kung uses fewer "diamonds" but more layers. Thus, a Brent-Kung adder uses less circuitry but is slower. (You can follow each output upward to verify that the tree reaches the correct inputs.)</p> <p><a href="https://static.righto.com/images/pentium-mult3/brent-kung.png"><img alt="A diagram of an 8-bit Brent-Kung adder. Diagram by Robey Pointer, Wikimedia Commons." class="hilite" height="300" src="https://static.righto.com/images/pentium-mult3/brent-kung-w300.png" title="A diagram of an 8-bit Brent-Kung adder. Diagram by Robey Pointer, Wikimedia Commons." width="300" /></a><div class="cite">A diagram of an 8-bit Brent-Kung adder. Diagram by Robey Pointer, <a href="https://commons.wikimedia.org/wiki/File:Brent-kung-8-bit.png">Wikimedia Commons</a>.</div></p> <p> <a class="footnote-backref" href="#fnref:brent-kung" title="Jump back to footnote 8 in the text">↩</a></p> </li> <li id="fn:recursive"> <p>The higher-level Kogge-Stone lookahead circuit uses the eight <em>P<sub>70</sub></em> and <em>G<sub>70</sub></em> signals from the eight lower-level lookahead circuits. Note that <em>P<sub>70</sub></em> and <em>G<sub>70</sub></em> indicate that an 8-bit chunk will propagate or generate a carry. The higher-level lookahead circuit treats 8-bit chunks as a unit, while the lower-level lookahead circuit treats 1-bit chunks as a unit. Thus, the higher-level and lower-level lookahead circuits are essentially identical, acting on 8-bit values. <a class="footnote-backref" href="#fnref:recursive" title="Jump back to footnote 9 in the text">↩</a></p> </li> <li id="fn:cell"> <p>The floating-point unit is built from fixed-width columns, one for each bit. Each column is 38.5 碌m wide, so the circuitry in each column must be designed to fit that width. For the most part, the same circuitry is repeated for each of the 64 (or so) bits. The carry-select adder is unusual since it doesn't follow the column width of the rest of the floating-point unit. Instead, it crams 8 circuits into the width of 6.5 regular circuits. This leaves room for one Kogge-Stone circuitry block. <a class="footnote-backref" href="#fnref:cell" title="Jump back to footnote 10 in the text">↩</a></p> </li> <li id="fn:low-bits"> <p>Because there is no carry-in to the lowest 8-bit block of the ×3 circuit, the carry-select circuit is not needed. Instead, each output bit can be computed using an XNOR gate. <a class="footnote-backref" href="#fnref:low-bits" title="Jump back to footnote 11 in the text">↩</a></p> </li> <li id="fn:logical-effort"> <p>The principle of <a href="https://en.wikipedia.org/wiki/Logical_effort">Logical Effort</a> explains that for best performance, you don't want to jump from a small signal to a high-current signal in one step. Instead, a small signal produces a medium signal, which produces a larger signal. By using multiple stages of circuitry, the overall delay can be reduced. <a class="footnote-backref" href="#fnref:logical-effort" title="Jump back to footnote 12 in the text">↩</a></p> </li> <li id="fn:history"> <p>The <a href="https://www.ece.ucdavis.edu/~bbaas/281/papers/Booth.1951.pdf">Booth multiplication technique</a> was described in 1951, while parallel multipliers were proposed in the mid-1960s by <a href="https://doi.org/10.1109/PGEC.1964.263830">Wallace</a> and <a href="https://ieeemilestones.ethw.org/w/images/8/82/Some_schemes_for_parallel_multipliers_%28reprint%29.pdf">Dadda</a>. Jumping ahead to higher-radix multiplication, a 1992 paper <a href="https://doi.org/10.1109/MWSCAS.1992.271307">A Fast Hybrid Multiplier Combining Booth and Wallace/Dadda Algorithms</a> from Motorola discusses radix-4 and radix-8 algorithms for a 32-bit multiplier, but decides that computing the ×3 multiple makes radix-8 impractical. IBM discussed a 32-bit multiplier in 1997: <a href="https://doi.org/10.1109/ARITH.1997.614873">A Radix-8 CMOS S/390 Multiplier</a>. Bewick's 1994 PhD thesis <a href="http://i.stanford.edu/pub/cstr/reports/csl/tr/94/617/CSL-TR-94-617.pdf">Fast Multiplication: Algorithms and Implementation</a> describes numerous algorithms.</p> <p>For adders, <a href="https://pdfs.semanticscholar.org/9da8/de2627aa0d4669995c430210c6ea9844ddf1.pdf">Two-Operand Addition</a> is an interesting presentation on different approaches. <a href="https://pages.hmc.edu/harris/cmosvlsi/4e/cmosvlsidesign_4e_ch11.pdf">CMOS VLSI Design</a> has a good discussion of addition and various lookahead networks. It summarizes the tradeoffs: "Brent-Kung has too many logic levels. Sklansky has too much fanout. And Kogge-Stone has too many wires. Between these three extremes, the Han-Carlson, Ladner-Fischer, and Knowles trees fill out the design space with different compromises between number of stages, fanout, and wire count." The approach used in the Pentium's ×3 multiplier is sometimes called a sparse-tree adder. <a class="footnote-backref" href="#fnref:history" title="Jump back to footnote 13 in the text">↩</a></p> </li> </ol> </div> <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'><span class='post-comment-link'> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/6264947694886887540/3533328182698832872' onclick=''> 16 comments: </a> </span> <span class='post-icons'> <span class='item-action'> <a href='https://www.blogger.com/email-post/6264947694886887540/3533328182698832872' title='Email Post'> <img alt='' class='icon-action' height='13' src='http://img1.blogblog.com/img/icon18_email.gif' width='18'/> </a> </span> <span class='item-control blog-admin pid-1138732533'> <a href='https://www.blogger.com/post-edit.g?blogID=6264947694886887540&postID=3533328182698832872&from=pencil' title='Edit Post'> <img alt='' 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class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/02/'> February </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/'> 2023 </a> <span class='post-count' dir='ltr'>(35)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/12/'> December </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/09/'> September </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/08/'> August </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/07/'> July </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/03/'> March </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/02/'> February </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/01/'> January </a> <span class='post-count' dir='ltr'>(8)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/'> 2022 </a> <span class='post-count' dir='ltr'>(18)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/07/'> July </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/04/'> April </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/02/'> February </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/01/'> January </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/'> 2021 </a> <span class='post-count' dir='ltr'>(26)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/12/'> December </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/09/'> September </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/06/'> June </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/03/'> March </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/02/'> February </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/'> 2020 </a> <span class='post-count' dir='ltr'>(33)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/10/'> October </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/08/'> August </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/06/'> June </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/05/'> May </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/03/'> March </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/01/'> January </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/'> 2019 </a> <span class='post-count' dir='ltr'>(18)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/10/'> October </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/09/'> September </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/07/'> July </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/01/'> January </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/'> 2018 </a> <span class='post-count' dir='ltr'>(17)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/03/'> March </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/01/'> January </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/'> 2017 </a> <span class='post-count' dir='ltr'>(21)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/12/'> December </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/06/'> June </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/01/'> January </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/'> 2016 </a> <span class='post-count' dir='ltr'>(34)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/10/'> October </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/09/'> September </a> <span class='post-count' dir='ltr'>(8)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/08/'> August </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/07/'> July </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/06/'> June </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/02/'> February </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/'> 2015 </a> <span class='post-count' dir='ltr'>(12)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/11/'> November </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/'> 2014 </a> <span class='post-count' dir='ltr'>(13)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/10/'> October </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/09/'> September </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/02/'> February </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/'> 2013 </a> <span class='post-count' dir='ltr'>(24)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/08/'> August </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/07/'> July </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/06/'> June </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/02/'> February </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/'> 2012 </a> <span class='post-count' dir='ltr'>(10)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/11/'> November </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/10/'> October </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/'> 2011 </a> <span class='post-count' dir='ltr'>(11)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/02/'> February </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/'> 2010 </a> <span class='post-count' dir='ltr'>(22)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/11/'> November </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/04/'> April </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/03/'> March </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/01/'> January </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2009/'> 2009 </a> <span class='post-count' dir='ltr'>(22)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a 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