Die analysis of the 8087 math coprocessor's fast bit shifter

Floating-point numbers are very useful for scientific programming, but early microprocessors only supported integers directly.1 Although floating-point was common in mainframes back in the 1950s and 1960s, it wasn't until 1980 that Intel introduced the 8087 floating-point coprocessor for microcomputers.2 Adding this chip to a microcomputer such as the IBM PC made floating-point operations up to 100 times faster. This was a huge benefit for applications such as AutoCAD, spreadsheets, or flight simulators.3 The downside was the 8087 chip cost hundreds of dollars.4

It's hard to implement floating-point operations so they are computed quickly and accurately. Problems can arise from overflow, rounding, transcendental operations, and numerous edge cases. Prior to the 8087, each manufacturer had their own incompatible ad hoc implementation of floating point. Intel, however, enlisted numerical analysis expert William Kahan to design accurate floating point based on rigorous principles.5 The result was the floating-point architecture of the 8087. This became the IEEE 754 standard used in almost all modern computers, so I consider the 8087 one of the most influential chips ever designed.

Die of the Intel 8087 floating point unit chip, with main functional blocks labeled. The die is 5mm×6mm. The shifter is outlined in red. Click for a larger image.

To explore how the 8087 works, I opened up an 8087 chip and took photos of the silicon die with a microscope. Containing 40,000 transistors, the 8087 pushed chip manufacturing to the limit; in comparison, the companion 8086 microprocessor only had 29,000 transistors. To make the chip possible, Intel developed new techniques. In this article, I focus on the high-speed binary shifter (outlined in red above). The shifter takes up a large fraction of the chip's area, so minimizing its area was vital to making the 8087 possible.

A floating-point number consists of a fraction (also called significand or mantissa), an exponent, and a sign bit. (These are expressed in binary, but for a base-10 analogy, the number 6.02×10²³ has 6.02 as the fraction and 23 as the exponent.) The circuitry to process the fraction is at the bottom of the die photo. From left to right, the fraction circuitry consists of a constant ROM, a shifter (highlighted), adder/subtracters, and the register stack. The exponent processing circuitry is in the middle of the chip. Above it, the microcode engine and ROM control the chip.

The shifter

The role of the shifter is to shift binary numbers left or right, a task with several critical roles in floating-point operations. When two floating-point numbers are added or subtracted, the numbers must be shifted so the binary points line up. (The binary point is like the decimal point, but for a binary number.) The 8087's transcendental instructions are built around shift and add operations, using an algorithm called CORDIC. The shifter is also used to assemble a floating-point number from 16-bit chunks read from memory.8

Since shifts are so essential to performance, the 8087 uses a "barrel shifter", which can shift a number by any number of bits in a single step.6 Intel used a two-stage shifter design that kept its size manageable while still providing high performance. The first stage shifts the value by 0 to 7 bits, while the second stage shifts by 0 to 7 bytes. In combination, the two stages shift a value by any amount from 0 to 63 bits.

The bit shifter

I'll start by describing the bit shifter, which performs a shift of 0 to 7 bit positions. The diagram below outlines the structure of the bit shifter, showing five of the inputs and outputs; the full shifter supports 68 bits.7 The concept is that by activating a particular column, the input is shifted by the desired amount. Each circle indicates a transistor that can act as a switch between an input line and an output line. The vertical select lines are used to activate the desired transistors. Each input line is connected diagonally to eight transistors, allowing it to be directed to one of eight outputs. For example, the diagram shows shift select line 3 activated, turning on the associated transistors (green). The highlighted input 20 (orange) is directed to output 23 (blue). Similarly, the other inputs are connected to the corresponding outputs, yielding a shift by 3. By activating a different shift select line, the input will be shifted by a different amount between 0 and 7 bits.

Structure of the bit shifter. By energizing a shift select line, the inputs are connected to outputs with the desired bit shift.

To explain the internal construction of the shifter, I'll start by describing the NMOS transistors used in the 8087 chip. Transistors are built by doping areas of the silicon substrate with impurities to create "diffusion" regions with different electrical properties. The transistor can be considered a switch, controlling the flow of current between two regions called the source and drain. The transistor is activated by the gate, made of a special type of silicon called polysilicon, layered above the substrate silicon. Applying voltage to the gate lets current flow between the source and drain, which is otherwise blocked. Transistors are wired together by a metal layer on top, building a complex integrated circuit.

Structure of a MOSFET as implemented in an integrated circuit.

The photo below shows a transistor in the 8087 as it appears under the microscope. Its structure matches the diagram above, although its shape is more complex. The source, gate, and drain all continue out of the photo, connected to other transistors. In addition, wiring in the metal layer is connected to the silicon at the circular vias. (The metal layer was removed with acid for this photo.)

An NMOS transistor in the 8087 chip, as seen under the microscope.

Zooming out, the diagram below shows part of the bit shifter as implemented on the chip. About 48 transistors, similar to the one above, are in this photo. The orange and yellow diagonal corresponds to one of the inputs: the orange regions show transistors connected through the silicon, while the yellow lines show connections in the metal layer. (The metal layer is used to jump over the polysilicon select lines.) The green highlight shows the polysilicon line for shift-by-three. In the center, this polysilicon gate line turns on a transistor, connecting the input to the long yellow output line, shifting the highlighted input by three positions. (The other non-highlighted inputs are shifted similarly.) Thus, this circuit implements the shifter as described at the beginning of the section. The photo shows six of the 68 inputs, so the complete shifter is much taller.

Closeup of the silicon circuitry for the bit shifter. The path of one signal is shown, as controlled by the shift-by-three control (green).

The byte shifter

The byte shifter shifts its inputs by multiples of eight bits, rather than one bit. Its design is similar to the bit shifter, except each input connects to every eighth output. For instance, input 20 connects to outputs 20, 28, 36, and so forth, shifting by bytes. As a result, the diagonal connections are steep and packed tightly, with eight lines between each switch. In the diagram below, the line for shift-by-four is selected, with the connection from input 0 to output 32 highlighted. Note the lack of wires in the right half of the diagram because any bit shifted from beyond input 0 becomes zeroed. For instance, when shifting left by 4 bytes, low-order bits 31 and below become zero.

The structure of the byte shifter.

The die photo below shows part of the bit shifter and the byte shifter. This photo is zoomed-out to show the overall structure; individual transistors are barely visible. The bit shifter's area is densely packed with transistors, but the byte shifter consists mostly of wiring, with columns of transistors in between.9 Also note that the byte shifter is partially empty at the top, filling in with more wiring towards the bottom. The wiring layout isn't as orderly as in the diagram above, but is arranged for maximum efficiency.

The bit shifter and byte shifter in the 8087 chip.

The bidirectional drivers

So far, the bit and byte shifters only shift bits in one direction.11 However, bits need to be shifted in both directions. One of the key innovations of the 8087's shifter is its bidirectional design: data can be passed through the shifter in reverse to shift bits the opposite direction. This is possible because the shifter is constructed with pass transistors, not logic gates. Pass transistor logic uses transistors as switches that pass or block signals, so signals can travel in either direction. (In contrast, regular logic gates such as NOR gates have specific inputs and outputs.)

Special driver circuitry on the left and right sides of the shifter allows the shifter to operate in either direction. To send data from left to right, the left-hand driver reads data from the fraction bus and sends it into the shifter. The right-hand driver circuit receives this shifted data, latches it temporarily, and then writes it back to the fraction bus. To send data in the opposite direction, the driver circuits reverse roles: the right-hand driver sends data from the fraction bus into the shifter while the left-hand circuit receives the shifted data.10

The multiplexer / decoders

The final feature I'll describe is the circuitry that controlled the shifter. Three different sources control how many positions to shift. First, the microcode engine can specify the number directly. Second, the number can come from a loop counter; this is used as part of the CORDIC transcendental algorithms. Finally, the number can come from a leading zero counter; this allows numbers to be normalized by eliminating leading zeroes through shifting. Each of these sources provides a 6-bit shift number; the six multiplexers each select one bit from the desired source.12

The multiplexer/decoder circuitry.

Next, decoders activate one of eight bit-shift lines and one of eight byte-shift lines to control the appropriate pass transistors in the shifter. (Each decoder takes a 3-bit input and activates one of 8 output lines.) Because each decoder line controls a large column of pass transistors in the shifter, the decoder uses relatively large power transistors.13 At the bottom, the 16 control lines exit the circuitry.

Conclusion

The 8087 is a complex chip with many functional units. However, by examining the die closely, the circuits of the 8087 can be understood. This blog post described the 8087's fast barrel shifter, capable of shifting by up to 63 bits at a time.14 Intel received a patent on this innovative programmable bidirectional shifter.

The shifter was just one of the features that let the 8087 compute floating-point operations much faster than the 8086 processor could. The 8087 operates on 80 bits at a time instead of 16. The 8087 has 80-bit wide registers, reducing memory accesses during computations. The 8087 stores constants for transcendental operations in a ROM, also avoiding memory accesses. Hardware in the 8087 checked for NaN, underflow, overflow, etc., avoiding slow checks in code. The 8087's hardware made multiplication and division faster. I don't know the relative contributions of these factors, but in combination, they improved floating-point performance dramatically, by up to a factor of 100.

The benefits of floating point hardware are so great that Intel started integrating the floating-point unit into the processor with the 80486 (1989). Now, most processors include a floating-point unit and the expense of purchasing a separate floating-point coprocessor is a thing of the past.

Die photo of the 8087 with the metal layer removed. The colors are due to some of the oxide layer remaining. Click for a larger image.

For more information on the 8087, see my other articles: Extracting ROM constants from the 8087, The two-bit-per-transistor ROM and The substrate bias generator. I announce my latest blog posts on Twitter, so follow me @kenshirriff for future articles. I also have an RSS feed.

Notes and references

1. Even without floating-point hardware, early microcomputers could perform floating-point operations. The operations would be broken down into many integer operations, manipulating the exponent and fraction as necessary. In other words, floating-point support didn't make floating-point operations possible, it just made them much faster. (Another way to represent non-integers is fixed-point numbers, which have a fixed number of digits after the decimal. Fixed-point numbers are simpler than floating-point, but can't represent as large a range.) ↩

2. The 8087 wasn't the first floating-point chip. National Semiconductor introduced the MM57109 Number Cruncher Unit (that is the real name) in 1977. It was essentially a repackaged 12-digit scientific calculator chip, operating on binary-coded decimal values with values entered in Reverse Polish Notation. This chip was absurdly slow; a tangent, for instance, could take over a second. AMD introduced their floating-point chip, the Am9511, in 1978 (details). This chip supported 32-bit floating-point numbers and took up to 1.4 milliseconds for a tangent. (Intel ended up licensing the Am9511 from AMD and selling it as the 8231.) A 10-MHz 8087 in comparison, could do a tangent in 54 microseconds, operating on an 80-bit floating-point number. Thus, the 8087's performance and accuracy were far superior to previous chips. ↩

3. The original IBM PC (1981) had an empty socket on its motherboard for adding an 8087 coprocessor. a huge benefit for applications such as AutoCAD. The large empty socket is visible in the upper left below, above the 8088 microprocessor. A list of applications with support for the 8087 is here.

   

   Motherboard of the original IBM PC (1981). Photo from Wikimedia, CC BY-SA 3.0.

    ↩

4. I couldn't find the original price for the 8087, but it was expensive. At first, Intel only sold the 8087 as a matched and tested pair with an 8088, due to timing flakiness with the 8087. By 1982, Intel dropped the price of the 8087 to $230, equivalent to about $500 in current dollars. Compared to today's open-source world, it seems strange that customers also had to pay for software support: using the 8087 with the BASIC language cost another $150, while Intel's 8087 development library was $1250. ↩

5. The designers of the 8087 commented on the guidance offered by Professor Kahan: "We did not do as well as he wanted, but we did better than he expected." Kahan later received a Turing Award for his work on floating point.  ↩

6. Processors often include a variety of shift instructions, including rotate operations that shift bits from one end of the word to the other. The 8087 only performs straight shifts, not rotates. ↩

7. The shifter handles the 8087's 64-bit fraction, along with three extra bits for rounding accuracy, so it supports 67 bits. Unless I miscounted, the shifter also has an extra bit in the most significant position, making it 68 bits wide. ↩

8. Multiplication and division make heavy use of shifting; multiplication is performed by shifts and adds, while division uses shifts and subtracts. However, the 8087 does not use the general-purpose shifter for these operations, but has specialized shifters optimized for these operations. ↩

9. In order to pack the wiring as close together as possible, the shifter alternated wires of diffused silicon and wires of polysilicon. In the photo below, the diffused silicon wires are pinkish, while the polysilicon is yellowish. The 8087 was built with Intel's HMOS III process, which required a 4µm spacing for polysilicon and 5µm for diffusion, probably due to the resolution of the photolithography practice. However, the spacing between a diffusion line and a polysilicon line could be much smaller, probably because they were created with separate masks and were on separate layers. Thus, alternating diffusion and polysilicon lines could be packed together tightly, saving space.

   ↩

   Wiring in the byte shifter consists of alternating, tightly-packed silicon and polysilicon lines. The large rectangles on either side are pairs of transistors, controlled by vertical polysilicon lines.

10. The driver circuitry has a few subtleties. Instead of sending data directly into the shifter, bits are transferred in two steps. First, the shifter lines are pre-charged to a high level. Then, any 1-bit inputs cause the corresponding shifter lines to be pulled low. In other words, the shifter lines are active-low, with a low voltage representing a 1. Since any unused outputs keep their high voltage (a 0 bit), 0 bits are shifted into low bit positions automatically. I think the pre-charge technique also was a better match for NMOS circuitry, which was better at pulling a signal low than pulling it high, so pre-charging the lines helped performance, especially given their relatively high capacitance. The latch between the shifter and the fraction bus prevents an unwanted cycle with the shifted data immediately flowing back into the shifter and getting re-shifted. ↩

11. This footnote will clarify the physical shift versus the logical shift. On the die, the fraction circuitry is arranged with the most-significant bit at the bottom. Passing data through the shifter from left to right shifts bits physically downward. This corresponds to a left-shift of a binary number, moving bits to a higher position. In the opposite direction, passing data through the shifter from right to left performs a right-shift of the data. ↩

12. The left/right direction also needs to be selected from one of the three shift sources, but I haven't located the circuitry for that yet. ↩

13. Each decoder essentially consists of eight NOR gates: seven will be pulled low and only the one with all inputs low will be high. However, it's not implemented as a straightforward logic gate. Instead, all outputs are precharged high, and then the seven undesired outputs are pulled low. This sort of dynamic precharge logic is still used in modern circuits; see the book Synchronous Precharge Logic. The multiplexers are also implemented with precharge logic. ↩

14. Intel's x86 processors didn't include a barrel shifter until the 80386 (1985), which provided a 64-bit barrel shifter. Before that, the 8086 and descendants shifted one bit at a time, so shifts by many bit positions were much slower. ↩

Extracting ROM constants from the 8087 math coprocessor's die

Intel introduced the 8087 chip in 1980 to improve floating-point performance on the 8086 and 8088 processors, and it was used with the original IBM PC. Since early microprocessors operated only on integers, arithmetic with floating-point numbers was slow and transcendental operations such as arctangent or logarithms were even worse. Adding the 8087 co-processor chip to a system made floating-point operations up to 100 times faster.

I opened up an 8087 chip and took photos with a microscope. The photo below shows the chip's tiny silicon die. Around the edges of the chip, tiny bond wires connect the chip to the 40 external pins. The labels show the main functional blocks, based on my reverse engineering. By examining the chip closely, various constants can be read out of the chip's ROM, numbers such as pi that the chip uses in its calculations.

Die of the Intel 8087 floating point unit chip, with main functional blocks labeled. The constant ROM is outlined in green. Click for a larger image.

The top half of the chip contains the control circuitry. Performing a floating-point instruction might require 1000 steps; the 8087 used microcode to specify these steps. The die photo above shows the "engine" that ran the microcode program; it is basically a simple CPU. Next to it is the large ROM that holds the microcode.

The bottom half of the die holds the circuitry that processes floating-point numbers. A floating-point number consists of a fraction (also called significand or mantissa), an exponent, and a sign bit. (For a base-10 analogy, in the number 6.02×10²³, 6.02 is the fraction and 23 is the exponent.) The chip has separate circuitry to process the fraction and the exponent in parallel. The fraction processing circuitry supports 67-bit values, a 64-bit fraction with three extra bits for accuracy. From left to right, the fraction circuitry consists of a constant ROM, a shifter, adder/subtracters, and the register stack. The constant ROM (highlighted in green) is the subject of this post.

The 8087 operated as a co-processor with the 8086 processor. When the 8086 encountered a special floating-point instruction, the processor ignored it and let the 8087 execute the instruction in parallel.1 I won't explain in detail how the 8087 works internally, but as an overview, floating-point operations are implemented using integer adds/subtracts and shifts. To add or subtract two floating-point numbers, the 8087 shifts the numbers until the binary points (i.e. the decimal points but in binary) line up, and then adds or subtracts the fraction. Multiplication, division, and square root are performed through repeated shifts and adds or subtracts. Transcendental operations (tan, arctan, log, power) use CORDIC algorithms, which use shifts and adds of special constants for efficient computation.

Implementation of the ROM

This post describes the ROM that holds constants (not to be confused with the larger, four-level microcode ROM.2) The constant ROM holds the constants (such as pi, ln(2), and sqrt(2)) that the 8087 needs for its computations. The photo below shows part of the constant ROM. The metal layer has been removed to show the silicon underneath. The pinkish regions are silicon doped to have different properties, while the reddish and greenish lines are polysilicon, a special type of silicon wiring layered on top. Note the regular grid structure of the ROM. The ROM consists of two columns of transistors, holding the bits. To explain how the ROM works, I'll start by explaining how a transistor works.

Part of the constant ROM, with the metal layer removed. The three columns of larger transistors are used to select between rows.

High-density integrated circuits in the 1970s were usually built from a type of transistor known as NMOS. (Modern computers are built from CMOS, which consists of NMOS transistors along with opposite-polarity PMOS transistors.) The diagram below shows the structure of an NMOS transistor. An integrated circuit is constructed from a silicon substrate, with transistors built on it. Regions of the silicon are doped with impurities to create "diffusion" regions with desired electrical properties. The transistor can be viewed as a switch, allowing current to flow between two diffusion regions called the source and drain. The transistor is controlled by the gate, made of a special type of silicon called polysilicon. Applying voltage to the gate lets current flow between the source and drain, which is otherwise blocked. The die of the 8087 is fairly complex, with about 40,000 of these transistors.3

Structure of a MOSFET as implemented in an integrated circuit.

Zooming in on the ROM shows the individual transistors. The pinkish regions are the doped silicon, forming transistor sources and drains. The vertical polysilicon select lines form the gates of the transistors. The indicated silicon regions are connected to ground, pulling one side of each transistor low. The circles are connections called vias between the silicon and the metal lines above. (The metal lines have been removed; the orange line shows the position of one.)

A portion of the constant ROM. Each select line selects a particular constant. Transistors are indicated by the yellow symbols. An X indicates a missing transistor, corresponding to a 0 bit. The orange line indicates the position of a metal wire. (The metal layer was dissolved for this picture.)

The important feature of the ROM is that some of the transistors are missing, the first one in the upper row, and two marked with X in the lower row. Bits are programmed into the ROM by changing the silicon doping pattern, creating transistors or leaving insulating regions. Each transistor or missing transistor represents one bit. When a select line is activated, all the transistors in that column will turn on, pulling the corresponding output lines low. But if the transistor is missing from a selected position, the corresponding output line will remain high. Thus, a value is read from the ROM by activating a select line, reading that ROM value onto the output lines.

Contents of the ROM

The constant ROM has 134 rows of 21 columns.5 Under a microscope, the bit pattern of the ROM is visible and can be extracted.4 How to interpret the raw bits is not obvious, though. The first question is if a transistor (versus a gap) indicates a 0 or a 1. (It turns out that a transistor indicates a 1 bit.) The next issue is how to map the 134×21 grid of bits into values.6

The chip's data path consists of 67 horizontal rows, so it seemed pretty clear that the 134 rows in the ROM corresponded to two sets of 67-bit constants. I extracted one set of constants for the odd rows and one for the even rows, but the values didn't make any sense. After more thought, I determined that the rows do not alternate but are arranged in a repeating "ABBA" pattern.7 Using this pattern yielded a bunch of recognizable constants, including pi and 1. Bits from those constants are shown in the diagram below. (In this photo, a 1 bit appears as a green stripe, while a 0 bit appears as a red stripe.) In binary, pi is 11.001001... and this value is visible in the upper labeled bits. The bottom value is the constant 1.8

Bit values labeled in the constant ROM. The top bits are the first part of pi, while the lower bits are the constant 1, This diagram has been rotated 90 degrees compared to the other diagrams. The unlabeled bits form other constants.

The next difficulty in interpretation is that this ROM holds just the fractional parts of the numbers, not the exponents. (I haven't found the separate exponent ROM yet.) I experimented with various exponents until I got values that were sensible numbers. Some were straightforward: for instance, the constant 1.204120 yielded log₁₀(2) when the exponent 2^-2 was used. Others were harder,9 such as 1.734723. Eventually, I figured out that 1.734723×2⁵⁹ is 10¹⁸.10

The complete table of constants is in the footnotes.11 Physically, the constants are arranged in three groups. The first group is values that the user can load (1, pi, log₂10, log₂e, log₁₀2, and ln 2)12 along with values used internally (10¹⁸, ln(2)/3, 3*log₂(e), log₂(e), and sqrt(2)). The second group is sixteen arctan constants, and the third is fourteen log₂ constants. The last two groups of constants are used to compute transcendental functions using the CORDIC algorithm, which I will discuss next.

The CORDIC algorithms

The constants in the ROM reveal some details about the algorithms used by the 8087. The ROM contains 16 arctangent values, the arctans of 2^-n. It also contains 14 log values, the base-2 logs of (1+2^-n). These may seem like unusual values, but they are used in an efficient algorithm called CORDIC, which was invented in 1958.

The basic idea of CORDIC is to compute tangent and arctangent by breaking down an angle into smaller angles, and rotating a vector by these angles. The trick is that by carefully choosing the smaller angles, each rotation can be computed with efficient shifts and adds instead of trig functions. Specifically, suppose we want to find tan(z). We can break z into a sum of smaller angles: z ≈ {atan(2^-1) or 0} + {atan(2^-2) or 0} + {atan(2^-3) or 0} + ... + {atan(2^-16) or 0}. Now, rotating a vector by, say atan(2^-2), can be done by multiplying by 2^-2 and adding. The key thing is that multiplying by 2^-2 is just a fast bit shift. Putting this all together, computing tan(z) can be done by comparing z with the atan constants, and then doing 16 cycles of additions and shifts, which are fast to perform in hardware.13 To make the algorithm work, the atan constants are precomputed and stored in the constant ROM.14

Computing the base-2 log and base-2 exponential also use CORDIC algorithms, with the associated logarithmic constants. The key observation is that multiplying by (1 + 2^-n) can be done quickly with a shift and addition. By multiplying one side of the equation by the sequence of values, and adding the corresponding log constants to the other side, the log or exponential can be computed.15

The 8087's support for transcendental functions is more limited than you might expect. It only supports tangent and arctangent, not sine or cosine; the user must apply trig identities to compute sine or cosine. Logs and exponentials only support base 2; for base 10 or base e, the user must apply the appropriate scale factor. At the time, the 8087 pushed the limits of what could fit on a chip, so the instruction set was limited to the essentials.

Conclusion

The 8087 is a complex chip and at first it looks like a hopeless maze of circuitry. But much of it can be understood with careful study. It contains 42 constants in a ROM, and the values of these constants can be extracted under a microscope. Some of the constants (such as pi) are expected, while others (such as ln(2)/3) are more puzzling. Many of the constants are used for computing the tangent, arctangent, log, and power functions, using fast CORDIC algorithms.

Die photo of the 8087 with the metal layer removed. Click for a larger image.

Even though Intel's 8087 floating point unit chip was introduced 40 years ago, it still has a large influence today. It spawned the IEEE 754 floating-point standard used for most modern floating-point arithmetic, and the 8087's instructions remain a part of the x86 processors used in most computers.

For more information on the 8087, see my other articles: the two-bit-per-transistor ROM and the substrate bias generator. I announce my latest blog posts on Twitter, so follow me @kenshirriff for future articles. I also have an RSS feed.

Notes and references

1. The interaction between the 8086 processor and the 8087 floating point unit is somewhat tricky; I'll discuss some highlights. The simplified view is that the 8087 watches the 8086's instruction stream, and executes any instructions that are 8087 instructions. The complication is that the 8086 has an instruction prefetch buffer, so the instruction being fetched isn't the one being executed. Thus, the 8087 duplicates the 8086's prefetch buffer (or the 8088's smaller prefetch buffer), so it knows that the 8086 is doing. (A Twitter thread discusses this in detail.) Another complication is the complex addressing modes used by the 8086, which use registers inside the 8086. The 8087 can't perform these addressing modes since it doesn't have access to the 8086 registers. Instead, when the 8086 sees an 8087 instruction, it does a memory fetch from the addressed location and ignores the result. Meanwhile, the 8087 grabs the address off the bus so it can use the address if it needs it. If there is no 8087 present, you might expect a trap, but that's not what happens. Instead, for a system without an 8087, the linker rewrites the 8087 instructions, replacing them with subroutine calls to the emulation library. ↩

2. The 8087's microcode ROM is built with an unusual technique that stores two bits per transistor. It does this by using three different transistor sizes or no transistor in each position. The four possibilities at each position represent two bits. This complex technique was necessary in order to fit the large ROM onto the 8087 die. I wrote a blog post with more details. The constant ROM, in comparison, is built using standard techniques. ↩

3. Sources provide inconsistent values for the number of transistors in the 8087: Intel claims 40,000 transistors while Wikipedia claims 45,000. The discrepancy could be due to different ways of counting transistors. In particular, since the number of transistors in a ROM, PLA or similar structure depends on the data stored in it, sources often count "potential" transistors rather than the number of physical transistors. Other discrepancies can be due to whether or not pull-up transistors are counted and if high-current drivers are counted as multiple transistors in parallel or one large transistor. ↩

4. Instead of copying bits from the ROM by hand, I made a simple JavaScript program to help me read out the ROM. I clicked on the ROM image to indicate each transistor, and the program produced the corresponding pattern of 0's and 1's. ↩

5. The ROM has 134 rows of 21 bits, except there is a 6×6 chunk missing from the upper left. Thus, the physical size is of the constant ROM is 2946 bits.

   

   The upper-left corner of the constant ROM, showing the missing 6×6 section.

   Because of the ROM layout, this missing section means that the first 12 constants are 64 bits long, rather than 67 bits. These are the non-CORDIC constants, which apparently don't require the extra bits for accuracy. ↩

6. There are two ways to determine the encoding of the bits. The first is to trace out the circuitry that reads from the ROM and examine how the data is used. The second is to look for patterns in the raw data, and determine what makes sense for an encoding. Since the 8087 is very complex, I wanted to avoid a full reverse-engineering to understand the constants and I used the second approach. ↩

7. The organization of the rows follows the pattern ABBAABBAABBA..., where "A" rows hold bits for one set of constants and "B" rows hold bits for the second set of constants. This layout was probably used instead of alternating rows ("ABAB") because one connection can drive two neighboring selection transistors. That is, each "AA" or "BB" group can be selected with one wire. ↩

8. A bit more trial-and-error was necessary to pull the values out of the ROM. I determined three key factors. First, the bits started at the bottom of the ROM, going up. Second, a transistor indicated a 1, rather than a 0. Third, the constants did not have an implicit 1 bit at the beginning. (In other words, the constant format does not match the external data format used by the 8087.) ↩

9. Some of the exponents were tricky to determine. I used brute force for some of them, seeing if any exponent would yield the log or power of some number. One of the hardest numbers to figure out was ln(2)/3; I'm not sure why this value is important. ↩

10. Why does the 8087 contain the constant 10¹⁸? Probably because the 8087 supports a packed BCD datatype holding 18 digits, so it can hold up to 10¹⁸. ↩

11. The following table summarizes the contents of the constant ROM. The "meaning" column is my interpretation of the number.

    Constant	Decimal value	Meaning
    1.204120×2-2	0.3010300	log10(2)
    1.386294×2-1	0.6931472	ln(2)
    1.442695×20	1.4426950	log2(e)
    1.570796×21	3.1415927	Pi
    1.000000×20	1.0000000	1
    1.660964×21	3.3219281	log2(10)
    1.734723×259	1.000e+18	1018
    1.734723×259	1.000e+18	1018
    1.848392×2-3	0.2310491	ln(2)/3
    1.082021×22	4.3280851	3*log2(e)
    1.442695×20	1.4426950	log2(e)
    1.414214×20	1.4142136	sqrt(2)
    1.570796×2-1	0.7853982	atan(20)
    1.854590×2-2	0.4636476	atan(2-1)
    2.000000×2-15	0.0000610	atan(2-14)
    2.000000×2-16	0.0000305	atan(2-15)
    1.959829×2-3	0.2449787	atan(2-2)
    1.989680×2-4	0.1243550	atan(2-3)
    2.000000×2-13	0.0002441	atan(2-12)
    2.000000×2-14	0.0001221	atan(2-13)
    1.997402×2-5	0.0624188	atan(2-4)
    1.999349×2-6	0.0312398	atan(2-5)
    1.999999×2-11	0.0009766	atan(2-10)
    2.000000×2-12	0.0004883	atan(2-11)
    1.999837×2-7	0.0156237	atan(2-6)
    1.999959×2-8	0.0078123	atan(2-7)
    1.999990×2-9	0.0039062	atan(2-8)
    1.999997×2-10	0.0019531	atan(2-9)
    1.441288×2-9	0.0028150	log2(1+2-9)
    1.439885×2-8	0.0056245	log2(1+2-8)
    1.437089×2-7	0.0112273	log2(1+2-7)
    1.431540×2-6	0.0223678	log2(1+2-6)
    1.442343×2-11	0.0007043	log2(1+2-11)
    1.441991×2-10	0.0014082	log2(1+2-10)
    1.420612×2-5	0.0443941	log2(1+2-5)
    1.399405×2-4	0.0874628	log2(1+2-4)
    1.442607×2-13	0.0001761	log2(1+2-13)
    1.442519×2-12	0.0003522	log2(1+2-12)
    1.359400×2-3	0.1699250	log2(1+2-3)
    1.287712×2-2	0.3219281	log2(1+2-2)
    1.442673×2-15	0.0000440	log2(1+2-15)
    1.442651×2-14	0.0000881	log2(1+2-14)

    It's clear from the CORDIC constants that the values in the ROM are not physically stored in order, i.e. sequential rows are not addressed in order. I'm not sure why 10¹⁸ appears twice; probably one exponent is different. The binary exponents are not in the ROM that I examined, so I had to estimate them. ↩

12. The 8087 provides seven instructions to load constants directly. The instructions FDLZ, FLD1, FLDPI, FLD2T, FLD2E, FLDLG2, and FLDLN2 load onto the stack the constants 0, 1, pi, log₂10, log₂e, log₁₀2, and ln 2, respectively. Apart from 0, these constants can be found in the ROM. ↩

13. The 8087's CORDIC algorithm is described in Implementation of transcendental functions on a numerics processor. I wrote sample tangent code based on that description here. There are also a couple of multiplications and divisions in the 8087's full tan algorithm. It uses a simple rational approximation of tangent on the "leftover" angle, giving it a bit more accuracy than straight CORDIC. ↩

14. Computing the arctangent of an angle uses an algorithm that is similar to the tangent algorithm, but in reverse: as rotations are performed, the angles (from the constant ROM) are summed up to yield the resulting angle. ↩

15. I couldn't find documentation on the 8087's log and exponent algorithms. I think the algorithms are very similar to the ones on this page, except the 8087 uses base 2 instead of base e. I'm a bit puzzled why the 8087 doesn't need the constant log₂(1 + 2^-1), which is used by that algorithm. ↩