In 1980, Intel introduced the 8087 floating-point coprocessor chip. Plugging this chip into your IBM PC made math 100 times faster, good for CAD and spreadsheets. I opened up the chip to reverse-engineer exactly how it does complicated calculations. Let's take a look inside...
Under a microscope, you can see the circuitry of the 8087. The 8087 performs its calculations using low-level software called microcode, stored in the large ROM in the middle. Along with the "Opcode Collective", I'm trying to understand the algorithms in the microcode, from addition to arctangents.
Zooming in on the microcode storage ROM, you can see the individual transistors, the pinkish blobs. The 8087 is very unusual, using four transistor sizes, giving it double density: 2 bits per transistor. The Opcode Collective examined all 13184 transistors, revealing the 0s and 1s of the microcode.
Knowing the contents of the 8087's microcode is just the first step; we've been working for months to assign meaning to the 0s and 1s. I've been reverse-engineering the circuitry in the 8087, units such as an adder, shifter, and registers, to understand what the microcode is doing.
I can now explain the microcode for an instruction called FSCALE that scales a number by a power of 2. It sounds simple, but the 8087 has an appalling number of special cases: positive and negative 0, two infinities, four rounding modes, three number sizes, six exceptions... so the code is complex.
The 8087's microcode is too complicated and low-level to explain here. For details, see my latest article:
https://www.righto.com/2026/09/8087-microcode-reverse-engineering-fscale.html
Credits: I've been working on this with the members of the "Opcode Collective", especially Smartest Blob and @gloriouscow, who converted the ROM images to microcode data and extensively analyzed the contents.
7/7
@kenshirriff Fascinating project.
It’s been too long since I worked on x87 microcode to remember anything useful, other than needing … 6? … extra bits of mantissa to be able to newton-raphson all the trig functions to a full 80 bit accuracy with reasonable number of cycles.
@kenshirriff @gloriouscow
Cool story, read like a who-dunnit. Revealing all those riddles.
@kenshirriff @gloriouscow it's so amazing what you and nand2mario are doing - bringing these old x86 cpu's back to life in fpga's!
@kenshirriff @gloriouscow
In the 8086/8087 era I ‘devoured’ those thick tomes produced by Intel that described all the low level stuff. Soon, it just became too complex to follow, at that level, for a mere mortal on the basis of ‘fascination’.
Bowers Avenue, Santa Clara - when I finally had a trip to ‘Silicon Valley, I made a ‘bee-line’ there!
@kenshirriff Thank God that you're not a Gynecologist! 😆
@kenshirriff Generic question:
If you have a program that calculates pi to infirnity. Does the 8086 know to pass certain opcodes to the 8087 and wait for it to complete before continuing to next instruction? do both share the same registers?
@jfmezei It's complicated. The 8087 more-or-less watches the instruction stream and grabs instructions to execute. The 8086 runs in parallel until it executes a WAIT instruction to sync up. The 8086 and 8087 have independent registers. So how does the 8087 access memory if the address is in an 8086 register? The trick is that the 8086 does a fake access to the memory location and the 8087 grabs the address off the bus.
@kenshirriff thanks.
At the time, this bad boy would cost nearly $600 in today's money. Which is sadly in line with a baller PC upgrade in 2026
@kenshirriff This work is so absolutely cool. I really had no idea how much went into the co-processor. Thanks so much for sharing!
Over complexity is denormal for Intel. 🥁
Wow! I had no idea the 8087 was such a complex beast.
My exposure to microcode was with the old Gould SEL 32-bit mini-computers I worked with in 1980. They had an add-on 'floating point array processor' from Floating Point Systems (aka a 'vector processor'), which one programmed using pipelined microcode. You loaded one vector (array of samples) as first input, then another one (same) as second input.
You could do an add, a multiply, a load, and a store in one clock (IIRC), so you had to 'prime the pipe (the instruction pipeline) then run the loop, then 'empty the pipe' to pop the last results out.
Immense fun for this bright young novice programmer (21yr old-ish), who had been dropped in the deep end with FORTRAN, assembler, and this microcode, and not much more support than a (very good) set of manuals!
Opcode Collective would be a great name for a band or a world class design group 🙂🤷♂️
@kenshirriff thank you for this gift of a thread you made! I had fun diving together with you on this #FediFriend ✌️💙
Is this the beginning of true open source hardware? That would be great.
But what we really need is DIY memory chips. One chip now costs as much as my computer.
Did you find F00F ?
@kenshirriff It looks like half a harmonica. Just sayin'... 🙂
Sep 13, 2026, 8:34 AM@kenshirriff 8088 It was my first PC with a 4 MHz processor and a 10 MB hard drive