442:, 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.
40:
364:
flawed processors. This is equivalent to $ 868 million in 2023. 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".
75:. 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.
359:
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
400:
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. Intel went on to use formal verification extensively in the development of later CPU
384:
impact of a problem eclipsing the practical impact of said problem on customers. 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
363:
The growing dissatisfaction with Intel's response led to the company offering to replace all flawed
Pentium processors on request on December 20. On January 17, 1995, Intel announced a pre-tax charge of $ 475 million against earnings, ostensibly the total cost associated with replacement of the
336:
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
292:
chipsets, etc.) until
October 19, 1994. On October 24, 1994, he reported the issue to Intel. 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.
181:
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:
228:
457:
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.
92:
the defective processors in what was the first full recall of a computer chip. In its 1994 annual report, Intel said it incurred "a $ 475 million pre-tax charge ... to recover replacement and write-off of these microprocessors."
132:
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
409:
and theorem proving were used to find a number of bugs that could have led to a similar recall incident had they gone undetected. The first Intel microarchitecture to use formal verification as the primary method of validation was
177:
121:. 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 – thus five of the array cells contained zero when they should have contained +2.
84:
magazine estimated that 1 in 9 billion floating point divides with random parameters would produce inaccurate results), both the flaw and Intel's initial handling of the matter were heavily criticized by the tech community.
288:. 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,
128:
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.
968:
1240:
Kaivola, Roope; Ghughal, Rajnish; Narasimhan, Naren; Telfer, Amber; Whittemore, Jesse; Pandav, Sudhindra; Slobodová, Anna; Taylor, Christopher; Frolov, Vladimir; Reeber, Erik; Naik, Armaghan (2009).
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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.
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paused the sale of PCs containing Intel CPUs, and Intel's stock price decreased significantly. The motive behind IBM's decision was questioned by some in the industry; IBM produced the
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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".
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limited impact on most users, it caused significant negative press for the company. During a 2019 talk, while reflecting on development of
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On
October 30, 1994, Nicely sent an email describing the bug to various academic contacts, requesting reports of testing for the flaw on
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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.
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426:
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.
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spent hours tracking down that would result in parts of a game level appearing unexpectedly when viewed from certain camera angles.
991:"Intel adopts upon-request replacement policy on Pentium processors with floating point flaw; Will take Q4 charge against earnings"
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describes the value of number theory problems in discovering computer bugs and gives the mathematical background and history of
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As a result, calculations that rely on these five cells acquire errors; these errors can accumulate repeatedly owing to the
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The main challenge faced by software companies was implementing the fix in pre-existing software, much of which relied on
601:
1197:
Proceedings. Second ACM and IEEE International
Conference on Formal Methods and Models for Co-Design, 2004. MEMOCODE '04
317:
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125:
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1454:
653:
638:
Carlton, Jim; Yoder, Stephen K. (December 21, 1994). "Computers: Humble Pie: Intel to
Replace its Pentium Chips".
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223:{\displaystyle \textstyle {\dfrac {4{,}195{,}835}{3{,}145{,}727}}=1.333{\color {red}{739068902037589}}}
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were produced by manufacturers to work around the bug. One specific algorithm, outlined in a paper in
1242:"Replacing Testing with Formal Verification in Intel® Core™ i7 Processor Execution Engine Validation"
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In order to improve the speed of floating-point division calculations on the
Pentium chip over the
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715:. Kansas University Institute for Policy and Social Research. November 30, 1994. Archived from
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described how frequently and persistently this bug could be reproduced by describing behavior
272:
Thomas Nicely, a professor of mathematics at Lynchburg College, had written code to enumerate
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In the aftermath of the bug and subsequent recall, there was a marked increase in the use of
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1274:"P5 (586) Fifth-Generation Processors | Microprocessor Types and Specifications | InformIT"
321:, "Intel fixes a Pentium FPU glitch" by Alexander Wolfe, and was subsequently picked up by
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with 2,048 cells, of which 1,066 cells should have been populated with one of five values:
71:
numbers. The bug was discovered in 1994 by Thomas R. Nicely, a professor of mathematics at
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172:{\displaystyle \textstyle {\dfrac {4{,}195{,}835}{3{,}145{,}727}}=1.333820449136241002}
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64:
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1438:
572:"'A Discussion of and Fix for the Pentium FDIV Bug' from the Notebook Archive (2002)"
281:
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109:(SRT) algorithm. The SRT algorithm can generate two bits of the division result per
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Proceedings of the 33rd annual conference on Design automation conference - DAC '96
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The severity of the FDIV bug is debated. Though rarely encountered by most users (
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1241:
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602:"An error in a lookup table created the infamous bug in Intel's latest processor"
113:, whereas the 486's algorithm could only generate one. It is implemented using a
27:"Pentium bug" redirects here. For the 1997 bug affecting Pentium processors, see
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304:. The bug was quickly verified by others, and news of it spread quickly on the
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The story first appeared in the press on November 7, 1994, in an article in
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for floating-point division, the most frequently used instruction affected.
105:, Intel opted to replace the shift-and-subtract division algorithm with the
1085:
681:
Statistical Analysis of Floating Point Flaw in the Pentium Processor (1994)
1147:
1056:(January 13, 1995). "How number theory got the best of the Pentium chip".
961:"Lessons Learned: Pentium Flaws Aid Intel In Sandy Bridge Chipset Recall"
305:
137:, would allow users to discover whether their Pentium chip was affected.
1335:"Windows 95 Troubleshooting: How to Check for a Faulty Math Coprocessor"
356:
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Microsoft offered operating system level workarounds in versions of
17:
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in a segment aired on November 22. It was also reported on by the
63:. Because of the bug, the processor would return incorrect binary
38:
380:
Intel's response to the FDIV bug has been cited as a case of the
606:
377:, the problem Nicely was working on when he discovered the bug.
352:
322:
1136:"Word level model checking---avoiding the Pentium FDIV error"
1107:
Price, D. (April 1995). "Pentium FDIV flaw-lessons learned".
1425:
Archive of Intel's official information page about the bug
833:"COMPANY NEWS; Flaw Undermines Accuracy of Pentium Chips"
422:
The FDIV bug affects the 60 and 66 MHz Pentium P5 800 in
1430:
Unopened Intel CPU box from the FDIV replacement program
308:. The bug acquired the name "Pentium FDIV bug" from the
1310:. March 20, 2009. Solution ID CS-012748. Archived from
679:
Sharangpani, H. P.; Barton, M. L. (November 30, 1994).
1403:
Personal website of Dr. Nicely, who discovered the bug
1304:"FDIV Replacement Program: Frequently asked questions"
191:
149:
1408:
A page with precise information, also about the cause
193:
190:
151:
148:
741:
Coe, T.; Mathisen, T.; Moler, C.; Pratt, V. (1995).
43:
66 MHz Intel Pentium (sSpec=SX837) with the FDIV bug
993:. Business Wire. December 20, 1994. Archived from
933:. TNG Technology Consulting GmbH. August 6, 2019.
931:"BTD12: The Programming Principles of Id Software"
222:
171:
595:
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1016:Johnston, Louis; Williamson, Samuel H. (2023).
449:outside their control. Some companies, such as
36:Bug in the Intel P5 Pentium floating-point unit
633:
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629:
485:Accuracy problems in floating point operations
826:
824:
743:"Computational aspects of the Pentium affair"
736:
734:
513:"The Mathematics of the Pentium Division Bug"
360:asked" replacements of flawed Pentium chips.
8:
1134:Clarke, E. M.; Khaira, M.; Zhao, X. (1996).
440:IEEE Computational Science & Engineering
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686:(Report). Intel Corporation. Archived from
750:IEEE Computational Science and Engineering
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1193:"Formal verification in intel cpu design"
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401:architectures. In the development of the
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140:The correct value of the calculation is:
389:, Intel's CEO at the time was quoted in
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333:, making the front page in the latter.
67:results when dividing certain pairs of
954:
952:
874:from the original on December 18, 2010
660:from the original on February 26, 2017
506:
504:
1381:from the original on February 3, 2018
610:. No. March 1995. Archived from
7:
1345:from the original on August 14, 2024
971:from the original on August 14, 2024
937:from the original on August 25, 2023
911:from the original on August 14, 2024
862:Alexander Wolfe (November 9, 1994).
843:from the original on August 14, 2024
549:from the original on August 14, 2024
1284:from the original on April 13, 2021
1221:from the original on April 29, 2021
1172:from the original on April 29, 2021
959:Yeraswork, Zewde (March 30, 2011).
831:Markoff, John (November 24, 1994).
480:MOS Technology 6502 bugs and quirks
864:"Intel fixes a Pentium FPU glitch"
797:Nicely, Thomas (August 19, 2011).
771:from the original on June 23, 2021
25:
511:Edelman, Alan (January 1, 1997).
211:
1413:ZIP-file containing more details
1037:Gross Domestic Product deflator
642:(Eastern ed.). p. B1.
232:which is actually the value of
713:"Pentium FDIV bug – a Picture"
600:Tom R. Halfhill (March 1995).
453:, opted to directly patch the
407:symbolic trajectory evaluation
310:x86 assembly language mnemonic
107:Sweeney, Robertson, and Tocher
61:early Intel Pentium processors
1:
1018:"What Was the U.S. GDP Then?"
1259:10.1007/978-3-642-02658-4_32
1078:10.1126/science.267.5195.175
893:Moler, Cleve (Winter 1995).
868:Electronic Engineering Times
318:Electronic Engineering Times
1246:Computer Aided Verification
1205:10.1109/MEMCOD.2004.1459841
1476:
26:
540:10.1137/S0036144595293959
1419:for details on the file)
656:. Intel. June 20, 2020.
115:programmable logic array
88:In December 1994, Intel
1339:www.helpwithwindows.com
895:"A Tale of Two Numbers"
799:"Pentium FDIV flaw FAQ"
640:The Wall Street Journal
578:. Wolfram Research, Inc
391:The Wall Street Journal
1377:. September 11, 2009.
654:"1994 - Annual Report"
268:Discovery and response
224:
173:
44:
1148:10.1145/240518.240640
902:MATLAB News and Notes
414:, developed in 2008.
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1191:O'Leary, J. (2004).
1142:. pp. 645–648.
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166:1.333820449136241002
146:
1070:1995Sci...267..175C
1054:Cipra, Barry Arthur
1039:figures follow the
719:on November 3, 2021
614:on February 9, 2006
576:notebookarchive.org
532:1997SIAMR..39...54E
398:formal verification
57:floating-point unit
837:The New York Times
367:A 1995 article in
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135:Windows Calculator
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1455:1994 in computing
1370:Microsoft TechNet
1121:10.1109/40.372360
762:10.1109/99.372929
693:on March 19, 2022
286:prime quadruplets
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119:−2, −1, 0, +1, +2
73:Lynchburg College
16:(Redirected from
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430:Software patches
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29:Pentium F00F bug
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723:November 3,
520:SIAM Review
345:John Romero
290:motherboard
278:twin primes
111:clock cycle
97:Description
1439:Categories
1157:0897917790
1109:IEEE Micro
496:References
387:Andy Grove
1385:April 23,
1375:Microsoft
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447:libraries
403:Pentium 4
259:3,145,727
243:3,145,727
237:4,195,579
199:3,145,727
196:4,195,835
157:3,145,727
154:4,195,835
126:recursive
1379:Archived
1365:"Pentnt"
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1094:19898103
1086:17791336
1043:series.
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469:See also
434:Various
329:and the
306:Internet
298:486-DX4s
90:recalled
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880:2011
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607:BYTE
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1254:doi
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