# Pentium FDIV bug

The Pentium FDIV bug is a hardware bug affecting the floating-point unit (FPU) of the early Intel Pentium processors. Because of the bug, the processor would return incorrect binary floating point results when dividing certain pairs of high-precision numbers. The bug was discovered in 1994 by Thomas R. Nicely, a professor of mathematics at Lynchburg College.[1] Missing values in a lookup table used by the FPU's floating-point division algorithm led to calculations acquiring small errors. While these errors would in most use-cases only occur rarely and result in small deviations from the correct output values, in certain circumstances the errors can occur frequently and lead to more significant deviations.[2]

66 MHz Intel Pentium (sSpec=SX837) with the FDIV bug

The severity of the FDIV bug is debated. Though rarely encountered by most users (Byte magazine estimated that 1 in 9 billion floating point divides with random parameters would produce inaccurate results),[3] both the flaw and Intel's initial handling of the matter were heavily criticized by the tech community.

In December 1994, Intel recalled the defective processors in what was the first full recall of a computer chip.[4] In January 1995, Intel announced "a pre-tax charge of \$475 million against earnings, ostensibly the total cost associated with replacement of the flawed processors."[5]

## Description

Graph showing one manifestation of the FDIV bug. Each data point should be ~3.1789 x 10−8 higher on the y-axis than its predecessor to the left, but in the region 4195834.4 < x < 4195835.9 the result differs from the expected value by ~8.14 x 10−5.

In order to improve the speed of floating-point division calculations on the Pentium chip over the 486DX, Intel opted to replace the shift-and-subtract division algorithm with the Sweeney, Robertson, and Tocher (SRT) algorithm. The SRT algorithm can generate two bits of the division result per clock cycle, whereas the 486's algorithm could only generate one. It is implemented using a programmable logic array with 2,048 cells, of which 1,066 cells should have been populated with one of five values: −2, −1, 0, +1, +2. When the original array for the Pentium was compiled, five values were not correctly downloaded into the equipment that etches the arrays into the chips – thus five of the array cells contained zero when they should have contained +2.[6]

As a result, calculations that rely on these five cells acquire errors; these errors can accumulate repeatedly owing to the recursive nature of the SRT algorithm. In pathological cases the error can reach the fourth significant digit of the result, although this is rare. The error is usually confined to the ninth or tenth significant digit.[3]

Only certain combinations of numerator and denominator trigger the bug. One commonly-reported example is dividing 4,195,835 by 3,145,727. Performing this calculation in any software that used the floating-point coprocessor, such as Windows Calculator, would allow users to discover whether their Pentium chip was affected.[7]

The correct value of the calculation is:

${\displaystyle \textstyle {\dfrac {4{,}195{,}835}{3{,}145{,}727}}=1.333820449136241002}$

When converted to the hexadecimal value used by the processor, 4,195,835 = 0x4005FB and 3,145,727 = 0x2FFFFF. The '5' in 0x4005FB triggers the access to the 'empty' array cells. As a result, the value returned by a flawed Pentium processor is incorrect at or beyond four digits:[8]

${\displaystyle \textstyle {\dfrac {4{,}195{,}835}{3{,}145{,}727}}=1.333{\color {red}{739068902037589}}}$

## Discovery and response

Thomas Nicely, a professor of mathematics at Lynchburg College, had written code to enumerate primes, twin primes, prime triplets, and prime quadruplets. Nicely noticed some inconsistencies in the calculations on June 13, 1994, shortly after adding a Pentium system to his group of computers, but was unable to eliminate other factors (such as programming errors, motherboard chipsets, etc.) until October 19, 1994.[1] On October 24, 1994, he reported the issue to Intel.[9] Intel had reportedly become aware of the issue independently by June 1994, and had begun fixing it at this point, but chose not to publicly disclose any details or recall affected CPUs.[10]

On October 30, 1994, Nicely sent an email describing the bug to various academic contacts, requesting reports of testing for the flaw on 486-DX4s, Pentiums and Pentium clones.[9] The bug was quickly verified by others, and news of it spread quickly on the Internet. The bug acquired the name "Pentium FDIV bug" from the x86 assembly language mnemonic for floating-point division, the most frequently used instruction affected.[9]

The story first appeared in the press on November 7, 1994, in an article in Electronic Engineering Times, "Intel fixes a Pentium FPU glitch" by Alexander Wolfe,[11] and was subsequently picked up by CNN in a segment aired on November 22. It was also reported on by the New York Times and the Boston Globe, making the front page in the latter.[10][12]

At this point, Intel acknowledged the floating-point flaw, but claimed that it was not serious and would not affect most users. Intel offered to replace processors to users who could prove that they were affected. However, although most independent estimates found that the bug would have a very limited impact on most users, it caused significant negative press for the company. IBM paused the sale of PCs containing Intel CPUs, and Intel's stock price decreased significantly.[13] The motive behind IBM's decision was questioned by some in the industry; IBM produced the PowerPC CPUs at the time, and potentially stood to benefit from any reputational damage to the Pentium or Intel as a company. However, the decision led to corporate buyers of PC equipment demanding replacements of existing Pentium CPUs, and soon afterwards other PC manufacturers began offering "no questions asked" replacements of flawed Pentium chips.[4]

The growing dissatisfaction with Intel's response led to the company offering to replace all flawed Pentium processors on request on December 20.[14] On January 17, 1995, Intel announced "a pre-tax charge of \$475 million against earnings, ostensibly the total cost associated with replacement of the flawed processors."[9] This is equivalent to \$743 million in 2019.[15] Intel was criticised for barring resellers and OEMs from participating in the recall programme, requiring end-users to replace chips themselves. Intel's justification for this, posted on its support web page, was that "it is the individual decision of the end user to determine if the flaw is affecting their application accuracy".[13]

A 1995 article in Science describes the value of number theory problems in discovering computer bugs and gives the mathematical background and history of Brun's constant, the problem Nicely was working on when he discovered the bug.[16]

Intel's response to the FDIV bug has been cited as a case of the public relations impact of a problem eclipsing the practical impact of said problem on customers.[17] While most users were unlikely to encounter the flaw in their day-to-day computing, the company's initial reaction to not replace chips unless customers could guarantee they were affected caused pushback from a vocal minority of industry experts. The subsequent publicity generated shook consumer confidence in the CPUs, and led to a demand for action even from people unlikely to be affected by the issue. Andrew Grove, Intel's CEO at the time was quoted in the Wall Street Journal as saying "I think the kernel of the issue we missed [...] was that we presumed to tell somebody what they should or shouldn't worry about, or should or shouldn't do".[4]

In the aftermath of the bug and subsequent recall, there was a marked increase in the use of formal verification of hardware floating point operations across the semiconductor industry. Prompted by the discovery of the bug, a technique applicable to the SRT algorithm called "word-level model checking" was developed in 1996.[18] Intel went on to use formal verification extensively in the development of later CPU architectures. In the development of the Pentium 4, symbolic trajectory evaluation and theorem proving were used to find a number of bugs that could have led to a similar recall incident had they gone undetected.[19] The first Intel microarchitecture to use formal verification as the primary method of validation was Nehalem, developed in 2008.[20]

## Affected models

The FDIV bug affects the 60 and 66 MHz Pentium P5 800 in stepping levels prior to D1, and the 75, 90, and 100 MHz Pentium P54C 600 in steppings prior to B5. The 120 MHz P54C and P54CQS CPUs are unaffected.[21][22]

## Software patches

Various software patches were produced by manufacturers to work around the bug. One specific algorithm, outlined in a paper in IEEE Computational Science & Engineering, is to check for numerators and denominators that will trigger the access to the programmable logic array cells that erroneously contain zero, and if found, multiply both numbers by 15/16. This takes them out of the 'buggy' range. However, doing this has a significant speed penalty – in some cases doubling the time taken to perform the calculation.[8]

The main challenge faced by software companies was implementing the fix in pre-existing software, much of which relied on libraries outside their control. Some companies, such as Wolfram Research, opted to directly patch the machine code of existing executables to replace the FDIV opcode with an illegal instruction. This would then trigger an exception that an exception handler (also patched in) would catch. From here, arbitrary code could be executed to work around the bug.[2]

Microsoft offered operating system level workarounds in versions of Windows up to Windows XP. Utilities were included with the operating system to check for the presence of the bug and disable the FPU if found.[23][24]

## References

1. ^ a b Edelman, Alan (January 1, 1997). "The Mathematics of the Pentium Division Bug" (PDF). SIAM Review. 39 (1): 54–67. Bibcode:1997SIAMR..39...54E. doi:10.1137/S0036144595293959. Retrieved April 11, 2021.
2. ^ a b "'A Discussion of and Fix for the Pentium FDIV Bug' from the Notebook Archive (2002)". notebookarchive.org. Wolfram Research, Inc. Retrieved April 11, 2021.
3. ^ a b Tom R. Halfhill (March 1995). "An error in a lookup table created the infamous bug in Intel's latest processor". BYTE (March 1995). Archived from the original on February 9, 2006. Retrieved December 19, 2006.
4. ^ a b c Carlton, Jim; Yoder, Stephen K. (December 21, 1994). "Computers: Humble Pie: Intel to Replace its Pentium Chips". Wall Street Journal (Eastern edition ed., December 21, 1994). p. B1.
5. ^ "1994 - Annual Report". Intel. June 20, 2020. Archived from the original on February 26, 2017. Retrieved June 20, 2020.
6. ^ Sharangpani, H. P.; Barton, M. L. (November 30, 1994). Statistical Analysis of Floating Point Flaw in the Pentium Processor (1994) (PDF) (Report). Intel Corporation. Retrieved April 11, 2021.
7. ^ "Pentium FDIV bug – a Picture". Kansas University Institute for Policy and Social Research. November 30, 1994. Retrieved November 3, 2010.
8. ^ a b Coe, T.; Mathisen, T.; Moler, C.; Pratt, V. (1995). "Computational aspects of the Pentium affair" (PDF). IEEE Computational Science and Engineering. 2 (1): 18–30. doi:10.1109/99.372929. Retrieved April 13, 2021.
9. ^ a b c d Nicely, Thomas (August 19, 2011). "Pentium FDIV flaw FAQ". trnicely.net. Archived from the original on June 18, 2019. Retrieved June 18, 2019.
10. ^ a b Markoff, John (November 24, 1994). "COMPANY NEWS; Flaw Undermines Accuracy of Pentium Chips". The New York Times. Retrieved April 11, 2021.
11. ^ Alexander Wolfe (November 9, 1994). "Intel fixes a Pentium FPU glitch". Electronic Engineering Times.
12. ^ Moler, Cleve (Winter 1995). "A Tale of Two Numbers" (PDF). MATLAB News and Notes. MathWorks. Retrieved April 21, 2021.
13. ^ a b Yeraswork, Zewde (March 30, 2011). "Lessons Learned: Pentium Flaws Aid Intel In Sandy Bridge Chipset Recall". CRN. Retrieved April 11, 2021.
14. ^ "Intel adopts upon-request replacement policy on Pentium processors with floating point flaw; Will take Q4 charge against earnings". Business Wire. December 20, 1994. Archived from the original on July 10, 2012. Retrieved December 24, 2006.
15. ^ Thomas, Ryland; Williamson, Samuel H. (2020). "What Was the U.S. GDP Then?". MeasuringWorth. Retrieved September 22, 2020. United States Gross Domestic Product deflator figures follow the Measuring Worth series.
16. ^ Cipra, Barry Arthur (January 13, 1995). "How number theory got the best of the Pentium chip". Science. 267 (5195): 175. Bibcode:1995Sci...267..175C. doi:10.1126/science.267.5195.175. PMID 17791336. S2CID 19898103.
17. ^ Price, D. (April 1995). "Pentium FDIV flaw-lessons learned". IEEE Micro. 15 (2): 86–88. doi:10.1109/40.372360.
18. ^ Clarke, E. M.; Khaira, M.; Zhao, X. (1996). "Word level model checking—avoiding the Pentium FDIV error". Proceedings of the 33rd Annual Conference on Design Automation Conference – DAC '96. Dac '96: 645–648. doi:10.1145/240518.240640. ISBN 0897917790. S2CID 2500033. Retrieved April 29, 2021.
19. ^ O'Leary, J. (2004). "Formal verification in intel cpu design". Proceedings. Second ACM and IEEE International Conference on Formal Methods and Models for Co-Design, 2004. MEMOCODE '04.: 152. doi:10.1109/MEMCOD.2004.1459841. ISBN 0-7803-8509-8. Retrieved April 29, 2021.
20. ^ Kaivola, Roope; Ghughal, Rajnish; Narasimhan, Naren; Telfer, Amber; Whittemore, Jesse; Pandav, Sudhindra; Slobodová, Anna; Taylor, Christopher; Frolov, Vladimir; Reeber, Erik; Naik, Armaghan (2009). "Replacing Testing with Formal Verification in Intel \$^{\scriptsize\circledR}\$ CoreTM i7 Processor Execution Engine Validation". Computer Aided Verification. 5643: 414–429. doi:10.1007/978-3-642-02658-4_32. Retrieved April 29, 2021.
21. ^ "P5 (586) Fifth-Generation Processors | Microprocessor Types and Specifications | InformIT". www.informit.com. June 8, 2001. Retrieved April 13, 2021.
22. ^ "FDIV Replacement Program: Frequently asked questions". Intel. March 20, 2009. Solution ID CS-012748. Archived from the original on May 11, 2009. Retrieved November 10, 2009.
23. ^ Slob, Arie. "Windows 95 Troubleshooting: How to Check for a Faulty Math Coprocessor". www.helpwithwindows.com. Retrieved April 23, 2019.
24. ^ "Pentnt". Microsoft TechNet. Microsoft. September 11, 2009. Retrieved April 23, 2019.