Post

Intel launched the Pentium processor in 1993. Unfortunately, dividing sometimes gave a slightly wrong answer, the famous FDIV bug. Replacing the faulty chips cost Intel $475 million. I reverse-engineered the circuitry and can explain the bug. 1/9

The Pentium uses a division algorithm called SRT. It generates two bits at a time, making division twice as fast. SRT's secret is quotient digits can be negative: -2, -1, 0, 1, 2. A 2048-entry table gives the digit for a particular divisor and remainder. Unfortunately, 5 entries (red) were wrong. 2/

The table is stored in a PLA (Programmable Logic Array). A PLA stores logic equations in two grids of transistors: the "AND plane" and the "OR plane". Logic equations are defined by putting a transistor (or not) at each grid point. This is much more compact than a ROM: 112 rows instead of 2048. 3/9

I studied the transistor grids under a microscope and extracted the pattern. From this, I reverse-engineered the lookup table for division. The photos show a small part of the grids. A transistor is formed by a polysilicon line crossing doped silicon. No crossing, no transistor. 4/9

Smart mathematicians figured out Pentium's division algorithm and the missing entries in 1995 by examining the pattern of errors. But I can confirm it in silicon. Moreover, I see 16 missing entries in the table, not just 5, but 11 of them don't cause errors due to luck. 5/9

Intel claimed the bug was due to an error in a script to download the entries into the PLA. But due to the 16 missing entries, I think they made a mathematical error in constructing the table, misjudging the effect of a 7-bit adder. Here's the adder, just above the PLA. 6/9

You'd expect that Intel fixed the problem by adding the 5 missing entries. Instead, they filled *all* the unused entries with 2's. This made the table easier to store in a PLA, shrinking it by 1/3. The fixed PLA has lots of unused rows at the bottom. 7/9

Intel said the FDIV bug was unimportant, but the public disagreed. Newspapers and TV discussed the bug. Intel claimed the bug would happen every 27,000 years; IBM said every 24 days and stopped selling Pentiums. Intel gave in and replaced Pentiums at a cost of $475 million. 8/9

I hope to have a blog post with more details on the Pentium FDIV bug soon. Until then, you can read about the Pentium Navajo rug: bsky.app/profile/righ... 9/9

What sort of microscope?