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 In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

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 The University of Lynchburg, formerly Lynchburg College, is a private university associated with the Christian Church (Disciples of Christ) and located in Lynchburg, Virginia. It has approximately 2,800 undergraduate and graduate students. T ...

. 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. The severity of the FDIV bug is debated. Though rarely encountered by most users (''

Byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...

'' 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. In December 1994, Intel recalled 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."

Description

In order to improve the speed of floating-point division calculations on the Pentium chip over the

486DX The Intel 486, officially named i486 and also known as 80486, is a microprocessor. It is a higher-performance follow-up to the Intel 386. The i486 was introduced in 1989. It represents the fourth generation of binary compatible CPUs following ...

, 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: . 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. 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. 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 Windows Calculator is a software calculator developed by Microsoft and included in Windows. In its Windows 10 incarnation it has four modes: standard, scientific, programmer, and a graphing mode. The standard mode includes a number pad and but ...

, would allow users to discover whether their Pentium chip was affected. The correct value of the calculation is: 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:

Discovery and response

Thomas Nicely, a professor of mathematics at Lynchburg College, had written code to enumerate primes,

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...

s, 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. 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. 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. The bug was quickly verified by others, and news of it spread quickly on the

Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, p ...

. The bug acquired the name "Pentium FDIV bug" from the x86 assembly language mnemonic for floating-point division, the most frequently used instruction affected. 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, 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. 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. The motive behind IBM's decision was questioned by some in the industry; IBM produced the

PowerPC PowerPC (with the backronym Performance Optimization With Enhanced RISC – Performance Computing, sometimes abbreviated as PPC) is a reduced instruction set computer (RISC) instruction set architecture (ISA) created by the 1991 Apple– IBM– ...

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. 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 flawed processors." This is equivalent to $ in . 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". A 1995 article in ''

Science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...

'' 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. Intel's response to the FDIV bug has been cited as a case of the

public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization (such as a business, government agency, or a nonprofit organization) to the public in order to influence their perception. ...

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 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". In the aftermath of the bug and subsequent recall, there was a marked increase in the use of

formal verification In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal met ...

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 architectures. In the development of the

Pentium 4 Pentium 4 is a series of single-core CPUs for desktops, laptops and entry-level servers manufactured by Intel. The processors were shipped from November 20, 2000 until August 8, 2008. The production of Netburst processors was active from 2000 ...

symbolic trajectory evaluation Symbolic trajectory evaluation (STE) is a lattice-based model checking technology that uses a form of symbolic simulation. STE is essentially used for computer hardware, that is circuit verification. The technique uses abstraction Abstracti ...

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 Nehalem, developed in 2008.

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.

Software patches

Various

software patch A patch is a set of changes to a computer program or its supporting data designed to update, fix, or improve it. This includes fixing security vulnerabilities and other bugs, with such patches usually being called bugfixes or bug fixes. Patche ...

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. 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. Microsoft offered operating system level workarounds in versions of

Windows Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for se ...

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.

References

External links