View source for Pentium FDIV bug

{{redirect|Pentium bug|the 1997 bug affecting Pentium processors|Pentium F00F bug}}
{{Use mdy dates|date=May 2021}}
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{{Short description|Bug in the Intel P5 Pentium floating-point unit}}
[[Image:KL Intel Pentium A80501.jpg|right|thumb|66 MHz Intel Pentium (sSpec=SX837) with the FDIV bug]]
The '''Pentium FDIV bug''' is a [[hardware bug]] affecting the [[floating-point unit]] (FPU) of the [[P5 (microarchitecture)|early Intel Pentium processors]]. Because of the bug, the processor would return incorrect binary [[floating point]] results when dividing certain pairs of [[Significant_figures|high-precision]] numbers. The bug was discovered in 1994 by Thomas R. Nicely, a professor of mathematics at [[Lynchburg College]].<ref name="siam">{{cite journal |last1=Edelman |first1=Alan |title=The Mathematics of the Pentium Division Bug |journal=SIAM Review |date=January 1, 1997 |volume=39 |issue=1 |pages=54–67 |doi=10.1137/S0036144595293959 |bibcode=1997SIAMR..39...54E |access-date=April 11, 2021 |url=http://www-math.mit.edu/~edelman/homepage/papers/pentiumbug.pdf |archive-date=August 14, 2024 |archive-url=https://web.archive.org/web/20240814062858/http://www-math.mit.edu/~edelman/homepage/papers/pentiumbug.pdf |url-status=live }}</ref> 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.<ref name="wolfram">{{cite web |title='A Discussion of and Fix for the Pentium FDIV Bug' from the Notebook Archive (2002) |url=https://notebookarchive.org/a-discussion-of-and-fix-for-the-pentium-fdiv-bug--2018-10-10r833w/ |website=notebookarchive.org |publisher=Wolfram Research, Inc. |access-date=April 11, 2021 |language=en}}</ref>

The severity of the FDIV bug is debated. Though rarely encountered by most users (''[[Byte (magazine)|Byte]]'' magazine estimated that 1 in 9 billion floating point divides with random parameters would produce inaccurate results),<ref name="halfhill-199503">{{cite magazine |author=Tom R. Halfhill |date=March 1995 |title=An error in a lookup table created the infamous bug in Intel's latest processor |magazine=[[BYTE]] |issue=March 1995 |url=http://www.byte.com/art/9503/sec13/art1.htm |access-date=December 19, 2006 |url-status=dead |archive-url=https://web.archive.org/web/20060209005434/http://www.byte.com/art/9503/sec13/art1.htm |archive-date=February 9, 2006 }}</ref> both the flaw and Intel's initial handling of the matter were heavily criticized by the tech community.  

In December 1994, Intel [[product recall|recalled]] the defective processors in what was the first full recall of a computer chip.<ref name="wsjhumblepie">{{cite news |last1=Carlton |first1=Jim |last2=Yoder |first2=Stephen K. |title=Computers: Humble Pie: Intel to Replace its Pentium Chips. |work=The Wall Street Journal |edition=Eastern |date=December 21, 1994 |page=B1}}</ref> In its 1994 annual report, Intel said it incurred "a $475 million pre-tax charge&nbsp;... to recover replacement and write-off of these microprocessors."<ref>{{Cite web |date=June 20, 2020 |title=1994 - Annual Report |publisher=Intel |url=https://www.intel.com/content/www/us/en/history/history-1994-annual-report.html|url-status=live|archive-url=https://web.archive.org/web/20170226113353/https://www.intel.com/content/www/us/en/history/history-1994-annual-report.html|archive-date=February 26, 2017|access-date=June 20, 2020}}</ref>

==Description==

In order to improve the speed of floating-point division calculations on the Pentium chip over the [[Intel 80486|486DX]], Intel opted to replace the shift-and-subtract division algorithm with the [[Division_algorithm#SRT_division|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{{Citation needed|reason=That's not how programmable logic arrays work.|date=November 2024}}, of which 1,066 cells should have been populated with one of five values: {{nowrap|−2, −1, 0, +1, +2}}. When the original array for the Pentium was compiled, five values were not correctly sent to the equipment that etches the arrays into the chips{{Citation needed|reason=This is not how chips are created. There are many design steps in between.|date=November 2024}} – thus five of the array cells contained zero when they should have contained +2.<ref name="intelreport">{{Cite report |url=https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/fdiv_bug/intel_white11.pdf |title=Statistical Analysis of Floating Point Flaw in the Pentium Processor (1994) |last1=Sharangpani |first1=H. P. |last2=Barton |first2=M. L. |date=November 30, 1994 |publisher=Intel Corporation |publication-date=November 30, 1994 |access-date=April 11, 2021 |archive-date=March 19, 2022 |archive-url=https://web.archive.org/web/20220319184123/https://users.fmi.uni-jena.de/~nez/rechnerarithmetik_5/fdiv_bug/intel_white11.pdf |url-status=dead }}</ref>

As a result, calculations that rely on these five cells acquire errors; these errors can accumulate repeatedly owing to the [[Recursion_(computer_science)|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.<ref name="halfhill-199503" />

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.<ref name="kansas">{{cite web|publisher=Kansas University Institute for Policy and Social Research|title=Pentium FDIV bug – a Picture|date=November 30, 1994|url=http://www.ipsr.ku.edu/stafffil/hoyle/pentium_fdiv/|access-date=November 3, 2010|archive-date=November 3, 2021|archive-url=https://web.archive.org/web/20211103175303/http://www.ipsr.ku.edu/stafffil/hoyle/pentium_fdiv/|url-status=dead}}</ref>

The correct value of the calculation is:

{{block indent|<math>\textstyle \dfrac{4{,}195{,}835}{3{,}145{,}727} = 1.333820449136241002</math>}}

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:<ref name="compaspects">{{cite journal |last1=Coe |first1=T. |last2=Mathisen |first2=T. |last3=Moler |first3=C. |last4=Pratt |first4=V. |title=Computational aspects of the Pentium affair |journal=IEEE Computational Science and Engineering |date=1995 |volume=2 |issue=1 |pages=18–30 |doi=10.1109/99.372929 |url=https://people.cs.vt.edu/~naren/Courses/CS3414/assignments/pentium.pdf |access-date=April 13, 2021 |archive-date=June 23, 2021 |archive-url=https://web.archive.org/web/20210623203121/https://people.cs.vt.edu/~naren/Courses/CS3414/assignments/pentium.pdf |url-status=live }}</ref>

{{block indent|<math>\textstyle \dfrac{4{,}195{,}835}{3{,}145{,}727} = 1.333{\color{red}{739068902037589}}</math>}}

which is actually the value of <math>\textstyle \dfrac{4{,}195{,}579}{3{,}145{,}727} = \dfrac{4{,}195{,}835 - 256}{3{,}145{,}727}</math>.

==Discovery and response==
Thomas Nicely, a professor of mathematics at Lynchburg College, had written code to enumerate [[prime number|primes]], [[twin prime]]s, [[prime triplet]]s, and [[prime quadruplet]]s. 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.<ref name="siam"/> On October 24, 1994, he reported the issue to Intel.<ref name="NicelyFAQ">{{cite web|url=http://www.trnicely.net/pentbug/pentbug.html|title=Pentium FDIV flaw FAQ|author=Nicely|first=Thomas|date=August 19, 2011|website=trnicely.net|archive-url=https://web.archive.org/web/20190618044444/http://www.trnicely.net/pentbug/pentbug.html|archive-date=June 18, 2019|url-status=dead|access-date=June 18, 2019}}</ref> 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.<ref name="nytcompany">{{cite web |last1=Markoff |first1=John |title=COMPANY NEWS; Flaw Undermines Accuracy of Pentium Chips |url=https://www.nytimes.com/1994/11/24/business/company-news-flaw-undermines-accuracy-of-pentium-chips.html |website=The New York Times |access-date=April 11, 2021 |date=November 24, 1994 |archive-date=August 14, 2024 |archive-url=https://web.archive.org/web/20240814063016/https://www.nytimes.com/1994/11/24/business/company-news-flaw-undermines-accuracy-of-pentium-chips.html |url-status=live }}</ref>

On October 30, 1994, Nicely sent an email describing the bug to various academic contacts, requesting reports of testing for the flaw on [[Intel 80486DX4|486-DX4s]], Pentiums and [[Pentium compatible processor|Pentium clones]].<ref name="NicelyFAQ" /> 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 instruction listings|x86 assembly language mnemonic]] for floating-point division, the most frequently used instruction affected.<ref name="NicelyFAQ" />

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,<ref>{{cite web|title=Intel fixes a Pentium FPU glitch|url=http://davefaq.com/Opinions/Stupid/Pentium.html#glitch|author=Alexander Wolfe|date=November 9, 1994|website=Electronic Engineering Times|access-date=January 19, 2011|archive-date=December 18, 2010|archive-url=https://web.archive.org/web/20101218150110/http://davefaq.com/Opinions/Stupid/Pentium.html#glitch|url-status=live}}</ref> 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.<ref name="nytcompany" /><ref name="clevemoler">{{cite web |last1=Moler |first1=Cleve |title=A Tale of Two Numbers |url=https://www.mathworks.com/content/dam/mathworks/tag-team/Objects/a/72895_92024v00Cleve_Tale_Two_Numbers_Win_1995.pdf |website=MATLAB News and Notes |publisher=MathWorks |access-date=April 21, 2021 |date=Winter 1995 |archive-date=August 14, 2024 |archive-url=https://web.archive.org/web/20240814062858/https://www.mathworks.com/content/dam/mathworks/tag-team/Objects/a/72895_92024v00Cleve_Tale_Two_Numbers_Win_1995.pdf |url-status=live }}</ref>

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. During a 2019 talk, while reflecting on development of ''[[Quake_(video_game)|Quake]]'', [[John Romero]] described how frequently and persistently this bug could be reproduced by [[Michael Abrash]]. Abrash spent hours tracking down exact conditions needed to produce the bug, which would result in parts of a game level appearing unexpectedly when viewed from certain camera angles.<ref>{{cite web|url=https://www.youtube.com/watch?v=22FU31ZUgNA&t=1519|title=BTD12: The Programming Principles of Id Software|access-date=July 17, 2023|date=August 6, 2019|publisher=TNG Technology Consulting GmbH|archive-date=August 25, 2023|archive-url=https://web.archive.org/web/20230825100711/https://www.youtube.com/watch?v=22FU31ZUgNA&t=1519|url-status=live}}</ref> [[IBM]] paused the sale of PCs containing Intel CPUs, and Intel's stock price decreased significantly.<ref name="CRN">{{cite web |last1=Yeraswork |first1=Zewde |title=Lessons Learned: Pentium Flaws Aid Intel In Sandy Bridge Chipset Recall |url=https://www.crn.com/news/components-peripherals/229400535/lessons-learned-pentium-flaws-aid-intel-in-sandy-bridge-chipset-recall.htm |website=CRN |access-date=April 11, 2021 |date=March 30, 2011 |archive-date=August 14, 2024 |archive-url=https://web.archive.org/web/20240814063035/https://www.crn.com/news/components-peripherals/229400535/lessons-learned-pentium-flaws-aid-intel-in-sandy-bridge-chipset-recall |url-status=live }}</ref> 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.<ref name="wsjhumblepie" />

The growing dissatisfaction with Intel's response led to the company offering to replace all flawed Pentium processors on request on December 20.<ref>{{cite web|url=http://findarticles.com/p/articles/mi_m0EIN/is_1994_Dec_20/ai_15939945 |archive-url=https://archive.today/20120710222033/http://findarticles.com/p/articles/mi_m0EIN/is_1994_Dec_20/ai_15939945 |url-status=dead |archive-date=July 10, 2012 |title=Intel adopts upon-request replacement policy on Pentium processors with floating point flaw; Will take Q4 charge against earnings |access-date=December 24, 2006 |date=December 20, 1994 |publisher=Business Wire }}</ref> 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.<ref name="NicelyFAQ" /> This is equivalent to ${{Format price|{{Inflation|US-GDP|475000000|1995}}}} in {{Inflation/year|US-GDP}}.{{Inflation/fn|US-GDP}} Intel was criticised for barring resellers and OEMs from participating in the recall program, 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".<ref name="CRN" />

A 1995 article in ''[[Science (journal)|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.<ref name="Cipra Pentium Bug">{{cite journal
|last= Cipra  |first= Barry Arthur  |author-link = Barry Arthur Cipra
 | date = January 13, 1995
 | title = How number theory got the best of the Pentium chip
 | journal = Science
 | volume = 267
 | issue = 5195
 | pages = 175
 | doi = 10.1126/science.267.5195.175
 | pmid = 17791336
|bibcode= 1995Sci...267..175C |s2cid= 19898103 }}</ref>

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.<ref name="ieee">{{cite journal |last1=Price |first1=D. |title=Pentium FDIV flaw-lessons learned |journal=IEEE Micro |date=April 1995 |volume=15 |issue=2 |pages=86–88 |doi=10.1109/40.372360}}</ref> 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. [[Andy 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".<ref name="wsjhumblepie" />

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.<ref name="wordlevel">{{cite book |last1=Clarke |first1=E. M. |last2=Khaira |first2=M. |last3=Zhao |first3=X. |title=Proceedings of the 33rd annual conference on Design automation conference - DAC '96 |chapter=Word level model checking---avoiding the Pentium FDIV error |date=1996 |pages=645–648 |doi=10.1145/240518.240640 |isbn=0897917790 |s2cid=2500033 |chapter-url=https://dl.acm.org/doi/10.1145/240518.240640 |access-date=April 29, 2021 |archive-date=April 29, 2021 |archive-url=https://web.archive.org/web/20210429164257/https://dl.acm.org/doi/10.1145/240518.240640 |url-status=live }}</ref> 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.<ref name="memcod">{{cite book |last1=O'Leary |first1=J. |title=Proceedings. Second ACM and IEEE International Conference on Formal Methods and Models for Co-Design, 2004. MEMOCODE '04 |chapter=Formal verification in intel cpu design |date=2004 |pages=152 |doi=10.1109/MEMCOD.2004.1459841 |isbn=0-7803-8509-8 |chapter-url=https://www.computer.org/csdl/proceedings-article/memcod/2004/01459841/12OmNzBOimc |access-date=April 29, 2021 |archive-date=April 29, 2021 |archive-url=https://web.archive.org/web/20210429164258/https://www.computer.org/csdl/proceedings-article/memcod/2004/01459841/12OmNzBOimc |url-status=live }}</ref> The first Intel microarchitecture to use formal verification as the primary method of validation was [[Nehalem (microarchitecture)|Nehalem]], developed in 2008.<ref>{{cite journal |last1=Kaivola |first1=Roope |last2=Ghughal |first2=Rajnish |last3=Narasimhan |first3=Naren |last4=Telfer |first4=Amber |last5=Whittemore |first5=Jesse |last6=Pandav |first6=Sudhindra |last7=Slobodová |first7=Anna |last8=Taylor |first8=Christopher |last9=Frolov |first9=Vladimir |last10=Reeber |first10=Erik |last11=Naik |first11=Armaghan |title=Replacing Testing with Formal Verification in Intel® Core™ i7 Processor Execution Engine Validation |journal=Computer Aided Verification |date=2009 |volume=5643 |pages=414–429 |doi=10.1007/978-3-642-02658-4_32 |doi-access=free }}</ref>

==Affected models==
The FDIV bug affects the 60 and 66 MHz Pentium P5 800 in [[stepping level]]s 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.<ref name="informit">{{cite web |title=P5 (586) Fifth-Generation Processors {{!}} Microprocessor Types and Specifications {{!}} InformIT |url=https://www.informit.com/articles/article.aspx?p=130978&seqNum=29 |website=www.informit.com |access-date=April 13, 2021 |date=June 8, 2001 |archive-date=April 13, 2021 |archive-url=https://web.archive.org/web/20210413112752/https://www.informit.com/articles/article.aspx?p=130978&seqNum=29 |url-status=live }}</ref><ref name="IntelFaq">{{cite web |publisher=[[Intel]] |title=FDIV Replacement Program: Frequently asked questions |date=March 20, 2009 |url=http://www.intel.com/support/processors/pentium/sb/CS-012748.htm |id=Solution ID CS-012748 |access-date=November 10, 2009 |archive-url=https://web.archive.org/web/20090511094631/http://www.intel.com/support/processors/pentium/sb/CS-012748.htm |archive-date=May 11, 2009 |url-status=dead }}</ref>

== Software patches ==
Various [[software patch]]es 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 divisors that can trigger the access to the programmable logic array cells that erroneously contain zero, and if found, multiply both numerator and denominator by 15/16. This takes them out of the 'buggy' range. This fix does carry a measurable speed penalty - worst case for a program doing nothing but FDIV operations with bad divisors the running time would double since each FDIV would take about 80 instead of 40 clock cycles. With more random divisors the average time per FDIV was approximately 50 clock cycles, i.e. 10 cycles added to check the divisor: Only 5 out of 1024 random divisors would trigger the scaling fixup. Since FDIV is a rare operation in most programs, the normal slowdown with the fix installed was typically a percent or less.<ref name="compaspects" />

The main challenge faced by software companies was implementing the fix in pre-existing software, much of which relied on [[library (computing)|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.<ref name="wolfram" />

Microsoft offered operating system level workarounds in versions of [[Microsoft Windows|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.<ref>{{Cite web|url=http://www.helpwithwindows.com/windows95/troub-58.html|title=Windows 95 Troubleshooting: How to Check for a Faulty Math Coprocessor|last=Slob|first=Arie|website=www.helpwithwindows.com|access-date=April 23, 2019|archive-date=August 14, 2024|archive-url=https://web.archive.org/web/20240814063044/https://www.helpwithwindows.com/windows95/troub-58.html|url-status=live}}</ref><ref>{{cite web|url=https://technet.microsoft.com/en-us/library/bb490967.aspx|title=Pentnt|date=September 11, 2009|website=[[Microsoft TechNet]]|publisher=[[Microsoft]]|access-date=April 23, 2019|archive-date=February 3, 2018|archive-url=https://web.archive.org/web/20180203082401/https://technet.microsoft.com/en-us/library/bb490967.aspx|url-status=live}}</ref>

==See also==
*[[Pentium F00F bug]]
*[[MOS Technology 6502#Bugs and quirks|MOS Technology 6502 bugs and quirks]]
*[[Floating point#Accuracy problems|Accuracy problems in floating point operations]]
*[[MaverickCrunch]]

==References==
{{reflist|30em}}

==External links==
*[https://faculty.lynchburg.edu/~nicely/#PENT Personal website of Dr. Nicely, who discovered the bug]
*[http://www.cs.earlham.edu/~dusko/cs63/fdiv.html A page with precise information, also about the cause]
**[https://www.mathworks.com/matlabcentral/fileexchange/1666-pentium-division-bug-documents ZIP-file containing more details] (See [[ZIP file format]] for details on the file)
*[https://web.archive.org/web/20010429062848/http://support.intel.com/support/processors/pentium/fdiv/ Archive of Intel's official information page about the bug]
*[http://www.cpucollection.se/details.php?image_id=784 Unopened Intel CPU box from the FDIV replacement program]

{{DEFAULTSORT:Pentium Fdiv Bug}}
[[Category:X86 architecture]]
[[Category:Hardware bugs]]
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