Fearful Floats!
Problems for programmers
IEEE-754 Floating Point
representation
Lecture prepared by Tony Greening and Judy Kay,
and delivered by Bob Kummerfeld
1
Fractional representation!
The task:
Use 6 bits to represent fractional numbers
using binary representation.
Binary to decimal!
2-1
2-2
2-6
2-5
2-4
2-3
The floating point approach
1 sign
bit
n exponent
bits
6-n-1 mantissa
bits
This is the approach used in scientific notation . Thus, in
decimal, 315.82 could be expressed as 3.1582 x 102.
mantissa =
3.1582
exponent =
2
IEEE-754 Floating Point standard
We now look at the details of floating point
representation
Using the IEEE-754 standard
General structure
Floating point widths are commonly referred to as:
single-precision
32b
double-precision
64b
extended-precision
80b
Sign bit
Exponent
Mantissa
Normalisation!
Normalisation is a convention that leaves the mantissa
in the form:
1.xxxxx
where x represents a binary digit.
Thus,
1101.101
normalises to
1.101101000 x 23
Representation of the sign bit
A single, leading sign-bit is used:
0 represents positive
1 represents negative
Representation of the exponent
Excess notation
The exponent is represented using excess (or bias)
notation.
Specifically, for 32-bit IEEE-754, excess-127 notation
Excess-127 simply means you add 127 to the actual
(signed) integer before conversion to binary
Then write the number as an (unsigned) integer
9
Representation of the exponent
(Activity)
Represent the following numbers in
excess-127 notation, using an 8-bit data width
-127
0
128
10
De-normalised numbers!
An exponent field that is filled with zeros
denotes a denormalised number.
This means that the assumption of a leading
1 in the mantissa does not apply
Apart from other things, this enables the value
0.0 to be represented...
11
Infinity!
An exponent field that is filled with ones,
together with a mantissa field that is filled
with zeros denoted infinity.
Note the use of the sign bit allows both
positive and negative infinity to be defined.
12
Design considerations
Range
It is desirable for the representation method to
include as wide a range (maximum - minimum) as
possible
13
Design considerations
Density
It is not possible to represent all numbers. It is desirable to have
as high a density of computer-representable numbers as
possible.
To make the point, consider the numbers able to be represented
by the following fixed-point system:
0
The same situation applies in floating-point, with a
clustering effect instead of periodic.
Representable
numbers:
7.75, 7.5, 7.25, 7.0,
6.75, 6.5, 6.25, 6.0,
etc., etc.,
0.75, 0.5, 0.25, 0.0,
-0.25, -0.5, -0.75,
etc., etc.,
-7.0, -7.25, -7.5, -7.75
14
Calculations in the comfort zone !
Sqrt(3) squared = ?
3
0.1 * 10 = ?
1.0
20! = ?
a really big number
(something around 2.432902008176640e+018)
BUT in the real world of computing,
these pleasantries may not hold...
15
! !!What if we started with zero (in floating
point representation) and then added 0.1
to it 1000 times?
What would the result be?
!
!
16
17
What!does!this!print?!
!float x = 0.1;
printf("%10.1f\n", x);
printf("%10.5f\n", x);
printf("%20.15f\n", x);
printf("%50.45f\n", x);
!!!!!!!!
18
19
Catastrophic!cancella2on!
Two!operands!are!close!to!each!other!
They!cancel!out!because!of!order!of!the!
opera2on,!eg!
bignum!<!bignum!+!smallnum!
Quadra2c!equa2on!discriminant!
20
Catastrophic!cancella2on!
b = 3.34
a = 1.22
c = 2.28
exact (b * b) - 4 * a * c
.0292
(b * b) rounds to 11.2
4ac to 11,1
answer 0.1
21
Benign!cancella2on!
Avoid!the!problems!by!reorganising!the!
expression!
Example:!catastrophic!cancella2on!
(x!*!x)!<!(y!*!y)!has!catastrophic!
Becomes!benign!(more!accurate)!as!
(x!<!y)(x!+!y)!
22
About!errors!
Sources!
The!raw!data!eg!7.3!
Propaga2on!error!eg!7.3!+!8.15!
Representa2onal!errors!(niteness)!
Awareness!
Signicant!gures!
Prin2ng!
calcula2ons!
Measures!
Absolute!error!
Rela2ve!error!
23
Software errors are
everywhere
Let s look at some which occur due to the
effects of floating point data
representation
24
Software errors are
everywhere
In 1979, NORAD defense radar misinterpreted the moon as an
incoming missile
In 1983, a software bug resulted in an F-14 flying off the
edge of an aircraft carrier and into the North Sea.
In 1984, a 180-degree error caused a Soviet test missile to
head towards Hamburg instead of the Arctic.
The splashdown point of Gemini V was off by over 100 miles
because the guidance system neglected movement of the
Earth around the Sun.
25
June 4, 1996
Ariane 5 is launched
Ariane 4 had made 23 successful launches...
Ariane 5 was faster and able to carry heavier
payloads...
26
Things didn t go quite to plan...
27
Oops! An $(US)8-billion
fireworks display!!
Ariane 5 exploded in its 37th second of flight. Its payload of 4 satellites
was uninsured (as you do!).
It was later found to be due to a small data-representation problem...28
!Computed
horizontal velocity as floating point number
!Converted to 16-bit integer
!Worked OK for Ariane 4
!Overflowed for Ariane 5
"Used same software
29
February 25, 1991
The (first) Gulf War!
30
An Iraqi Scud missile penetrates US
On February 25th, 1991, an Iraqi Scud hit the barracks in
Patriot
Dahran, Saudi Arabia,
killing defenses
28 soldiers from the US
Army's 14th Quartermaster detatchment.
28 soldiers died and over 100 were injured as a result.
It was later found to be due to a small data-representation problem...
31
A government investigation revealed that the failed
intercept at Dhahran had been caused by a software error
in the system's clock. The Patriot missile battery at
Dhahran had been in operation for 100 hours, by which
time the system's internal clock had drifted by one third of
a second. Due to the closure speed of the interceptor and
the target, this resulted in a miss distance of 600 meters.
32
Summary!
Appreciation of floating point problems:
big disasters may result from small errors
small errors are hard to find
Common pitfalls to be aware of
Expect that floats may be inaccurate: the 0.1 problem due to 0.1 being
between two representable floats
the out of range problem - number too big or too small
Never, ever test floats for equality
!
33
end of segment
34
The radar system had successfully detected the Scud and
predicted where to look for it next, but because of the time
error, looked in the wrong part of the sky and found no
missile. With no missile, the initial detection was assumed
to be a spurious track and the missile was removed from
the system. No interception was attempted, and the missile
impacted on a barracks killing 28 soldiers.
At the time, the Israelis had already identified the problem
and informed the US Army and the PATRIOT Project
Office (the software manufacturer) on February 11th, 1991,
but no upgrade was present at the time. As a stopgap
measure, the Israelis recommended rebooting the system's
computers regularly, however, Army officials did not
understand how often they needed to do so. The
manufacturer supplied updated software to the Army on
February 26th, the day after the Scud struck the Army
barracks.
35
1982+
The Vancouver Stock Exchange!
36
The Vancouver STX goes
psycho
(for
2
years!)
Question:
How did a Vancouver Stock Exchange index gain around 574 points while the
stock prices were unchanged?
Partial Answer:
In 1982 the Vancouver STX introduced an index with an initial value of
1000.000
After 22 months the index was listed as 1524.881
But this was incorrect! Its true value was 1098.811
It was later found to be due to a small data-representation problem...
37
Sources!
David!Goldberg,!What!Every!Computer!Scien2st!Should!Know!about!Floa2ng<
Point!Arithme2c,!covers!important!aspects!of!oa2ng<point!arithme2c!from!the!
perspec2ve!of!IEEE!754!from!the!IEEE!web!site.!
Kernighan!and!Pike,!Prac2ce!of!Programming.!
Ariane 5 video
Ariane 5 explosion video (converted from qt format to MPEG by ye trusty ol SGI
O2):
http://www.abc.net.au/news/imageLibrary/00/04/18/g00041820000418stocks.jpg
Gulf war photo:
http://www.mpia-hd.mpg.de/SUW/SuW/1996/10-96/S713Abb1.html
STX photo:
http://graffiti.cribx1.u-bordeaux.fr/roussel/anim-e_07f.shtml
ariane 5 explosion photo
http://www.estec.esa.nl/spdwww/cluster/restored/pics/ar5lnch.mpg
http://www.washingtonpost.com/wp-srv/inatl/images/fogofwarpics/analysis/bombing1.jpg
Patriot missile photo:
http://www.ima.umn.edu/~arnold/disasters/patriot.html
38
end of segment
39
Representation of the exponent
(Activity)
Represent the following numbers in
excess-127 notation, using an 8-bit data width
-127
0
128
00000000
127
01111111
255
11111111
40
