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aggressively with the makers of higher-end processors for engineering
workstations. The Pentium was an immediate success.
In the summer of 1994, Dr. Thomas Nicely of Lynchburg College in Virginia ran
an extensive set of computations to analyze the sums of reciprocals of certain
sequences of prime numbers. The results were not always what his theory
predicted, even after he took into account the inevitable roundoff errors. Then Dr.
Nicely noted that the same program did produce the correct results when run on
the slower 486 processor, which preceded the Pentium in Intel's lineup. This
should not have happened. The optimal roundoff behavior of floating-point
calculations had been standardized by the Institute of Electrical and Electronics
Engineers (IEEE), and Intel claimed to adhere to the IEEE standard in both the 486
and the Pentium processors. Upon further checking, Dr. Nicely discovered that
indeed there was a very small set of numbers for which the product of two
numbers was computed differently on the two processors. For example,
4,195,835 = ((4,195,835 / 3,145,727) ΒΆ 3,145,727)
is mathematically equal to 0, and it did compute as 0 on a 486 processor. On a
Pentium processor, however, the result was 256.
As it turned out, Intel had independently discovered the bug in its testing and had
started to produce chips that fixed it. (Subsequent versions of the Pentium, such as
the Pentium III and IV, are free of the problem.) The bug was caused by an error in
a table that was used to speed up the floating-point multiplication algorithm of the
processor. Intel determined that the problem was exceedingly rare. They claimed
that under normal use a typical consumer would only notice the problem once
every 27,000 years. Unfortunately for Intel, Dr. Nicely had not been a normal user.
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Now Intel had a real problem on its hands. It figured that replacing all the Pentium
processors that it had already sold would cost it a great deal of money. Intel
already had more orders for the chip than it could produce, and it would be
particularly galling to have to give out the scarce chips as free replacements
instead of selling them. Intel's management decided to punt on the issue and
initially offered to replace the processors only for those customers who could
prove that their work required absolute precision in mathematical calculations.
Naturally, that did not go over well with the hundreds of thousands of customers
who had paid retail prices of $700 and more for a Pentium chip and did not want to
