Pentium FPU Error

1
Pentium FPU Error

• Samuel Hishmeh
• EE 585 Fault Tolerant Computing
• October 12th 2006

2
History

• Intel 486 Upgraded
• Pentium I Released
• 486 vs. Pentium
• Pentium FDIV Bug Discovered

3
The Problem

• If we have
• x 4195835
• y 3145727
• z x - (x/y)y
• z 0??
• Pentium says
• 256!!

4
Examples

• 5505001 / 294911 18.66665197
• Pentium 18.66600093
• (824633702441.0)(1/824633702441.0) 1
• Pentium 0.999999996274709702

5
No Big Deal?

• 1/(1/x) x,
• for x 824,633,702,418 lt x lt 824,633,702,449
• Pentium 3,072!
• 5505001 / 294911 18.66665197
• Pentium 18.66600093
• Multiply by 3,422,378
• Theoretical 63,884,339.03578466
• Pentium 63,882,110.93081154
• Theoretical Pentium 2,228.10497312!
• (Assuming my P4 did the calculation correctly)

6
What Happened?

• Better Performance using SRT Algorithm
• Floating-Point Scalar Code 3x Improvement
• Vector Code 5x Improvement
• 2 Quotient Bits per Clock Cycle
• Algorithm used LUTs
• Programming Error
• 5 Table Entries Missing!

7
Instructions Affected

• FDIV
• FDIVP
• FDIVR
• FDIVRP
• FIDIV
• FIDIVR
• FPREM

8
Honorable Mentions

• Dan 0411 - Pentium II/Pro Bug
• Ariane 5

9
The Solution

• Better Test Benches
• Used Old Code

10
The Cost

• Catastrophic Potential
• Money
• Integrity
• 475 Million

11
References

• http//www.websters-online-dictionary.org/
• http//www.trnicely.net/index.html
• http//www.maa.org/
• http//support.intel.com/
• http//www.cs.earlham.edu/dusko/
• http//www.x86.org/secrets/
• http//www.ddj.com/