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Wednesday, September 3. 2008

Numbers and Magic Tricks

I was in a conference room waiting for a meeting to start the other day. Several people in the room were talking about an interesting numbers trick that Clarke Thacher had posted to his internal blog. All agreed that others might be intrigued by his "Magic Trick." I talked to Clarke and got his permission to make it available for you here. The following is Clarke's Magic Trick.

Here's a magic trick to amuse and befuddle your (geeky) friends.

Take any positive value.

Add a special value to it.

Subtract that value out again.

You have rounded your original value to the nearest whole integer.

Here's a log that demonstrates this magic:

25         data null;
26           format m n best20.;
27           magic = &magic;
28           input m @@;
29           put m= +1 @15 @;
30           n=m+magic;
31           put ' => ' @;
32           n=n-magic;
33           put n=;
34           cards;

m=0.25         => n=0
m=1.25         => n=1
m=1.5          => n=2
m=1.75         => n=2
m=2.25         => n=2
m=2.5          => n=2
m=2.75         => n=3
m=1000.0001    => n=1000
m=1000.5       => n=1000
m=2047.25      => n=2047

Now you can challenge your audience to guess the value of MAGIC. You might want to offer a prize to the first person who can give the correct answer. Note: This trick will not work on the mainframe.

The answer

Magic=4,503,599,627,370,496

4,503,599,627,370,496 is also 2⁵².

What happened?

Does this mean that SAS has a bug? No.

Is there are hardware problem with my computer? No.

Let's take one number and follow the calculations as they would be done by the floating point processor.

All double precision numbers on Windows, UNIX and OpenVMS are represented in IEEE 754 binary floating point format. IEEE 754 says that numbers are represented as

Sign*2^exp *(1+(b₁/2¹)+(b₂/2²)+(b₃/2³)+(b₄/2⁴)+(b₅/2⁵)...+(b₅₂/2⁵²))

If we use this formula to represent 1.25, we get 2⁰*(1+ 1/2²).

Magic (2⁵²) is represented as 2⁵²*1.

Transform 2⁰*(1 + 1/2²) to 2⁵²*(1/2⁵²+1/2⁵⁴),

Magic + 1.25 =2⁵²*(1 + 1/2⁵²+1/2⁵⁴)

Transform back into standard form 2⁵²*(1 + 1/2⁵²)

Subtract Magic2⁵²*(1/2⁵²)

Transform back into standard form 2⁰*1which is 1.0

A message to you

Clarke is a Senior Manager in R&D. You may know Clarke from his many presentations at users' group events. He hopes to incorporate this trick into a paper for SAS Global Forum 2009 on numeric precision issues. Leave Clarke a note, comment, correction, or suggestion. I know that he would love to hear from you.

Posted by Renee Harper in Tips & Samples at 09:49 | Comments (8) | Trackbacks (0)

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An interesting artifact, but your description is itself a bit imprecise. How?

I can follow your instructions *on paper*, handling the math symbolically, and get the "correct" (expected) answer: the exact number I started with. You neglected to state that this quirk occurs only in a SAS (or other?) computer program.

#1 Ann Onymous on 2008-10-14 18:55 (Reply)

This is not a SAS specific issue. It is a trait shared by all floating point processors that implement the IEEE 754 standard.

#1.1 Clarke Thacher on 2008-10-15 08:49 (Reply)

This is an interesting finding. Adding and subtracting this special number can lead to a loss of precision. My question is: Is this the only number that leads to such a 'magic trick'? Does this happen for adding and subtracting some other numbers?

#2 Jackson Li on 2008-10-15 02:03 (Reply)

This is not quiet true. Look at what happens with m=0.5, 2.5, 4.5 etc.
They are not rounded yo the nearest whole integer

m=0.25 => n=0
m=0.4999 => n=0
m=0.5 => n=0
m=0.5001 => n=1
m=0.75 => n=1
m=1 => n=1
m=1.5 => n=2
m=2.5 => n=2
m=3.5 => n=4
m=4.5 => n=4

#3 Jos Verhees on 2008-10-15 03:26 (Reply)

Values that end in .5 are exactly between the two bounding integers. So you have to have a rule to handle ties.

Here's what William Kahan (father of IEEE 754) says:
"Half-way cases are rounded to Nearest Even, which means that the neighbor with last digit 0 is chosen."

Here's what is happening for m=2.5.

2.5 = 2**1 * (1 + 1/2**2)
which transforms into
= 2**52*(1/2**51 + 1/2**53)
add magic
= 2**52*(1 + 1/2**51 + 1/2**53)
(can't store 1/2**53)
subtract magic
= 2**52*(1/2**51)
= 2**1(1)

#3.1 Clarke Thacher on 2008-10-15 09:52 (Reply)

You should always be careful when adding very large and very small numbers - by definition the computer is using a representation of the number you want, using a limited number of bits. Normally it's not an issue, but it can lead to loss of precision.

#4 Anonymous on 2008-10-15 10:34 (Reply)

Good to know that. However if I move the code line
put ' => ' @;
down one line, so it becomes

n=m+magic;
n=n-magic;
put ' => ' @;

Then the magic will not happen. Does this mean there is no trigger for the floating number processor to store the n, so nothing is lost?

#5 Sam on 2008-10-22 02:02 (Reply)

I'm guessing that you are running on Windows or Linux. This is because the x86 architecture does some calculations in extended precision (80 bit) floating point registers. The datastep compiler will attempt to keep intermediate results in registers.

The PUT statement forced the intermediate value to be stored in 64 bit format.

#5.1 Clarke Thacher on 2008-10-22 11:21 (Reply)

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Welcome to the blog about SAS online support. Renee Harper (that's me) will keep you up-to-date about new and updated content on support.sas.com, as well as support services and software releases. I'll try to include relevant examples you can use -- sample programs and information about how others use SAS. I’ll be able to do this better if you join me – this is a place to share your ideas, successes, and frustrations.