The Pentium division Flaw - Chapter 5
In July 1995 Tim Coe of Vitesse Semiconductor Corporation and Ping Tak
Tang of Argonne National Laboratory published a paper titled "It Takes
Six Ones To Reach a Flaw" in which they proved the binary divisor must
have a run of at least six ones starting at the 5th bit after the radix
point if it is to hit an error cell. i.e. the divisor must be one of
the following:
| 1.0001 1111 11__ ... |
| 1.0100 1111 11__ ... |
| 1.0111 1111 11__ ... |
| 1.1010 1111 11__ ... |
| 1.1101 1111 11__ ... |
They also prove that any of the 5 error cells can not be hit earlier
than the 9th iteration. This means the first 8 quotient digits will
always be correct.
Although a run of at least 6 ones is required to hit an error cell,
it appears there is a relationship between how long the run of ones is
(starting at the 5th bit after the radix), and what the earliest
iteration an error cell can be hit. Tim Coe believes the following
relationship to be true but has not rigorously proved it:
| # of 1's | | earliest iteration
error cell can be hit |
|---|
| ------- | | --------------------- |
| 6 ones | -- | 15 iterations |
| 7 ones | -- | cannot address flaw |
| 8 ones | -- | 10 iterations |
| 9 ones | -- | 10 iterations |
| 10 ones | -- | 10 iterations |
| 11 ones | -- | 9 iterations |
