...
Java
...
uses
...
the
...
IEEE
...
754
...
standard
...
for
...
floating
...
point
...
representation.
...
In
...
this
...
representation,
...
floats
...
are
...
encoded
...
using
...
1
...
sign
...
bit,
...
8
...
exponent
...
bits,
...
and
...
23
...
mantissa
...
bits.
...
Doubles
...
are
...
encoded
...
and
...
used
...
exactly
...
the
...
same
...
way,
...
except
...
they
...
use
...
1
...
sign
...
bit,
...
11
...
exponent
...
bits,
...
and
...
52
...
mantissa
...
bits.
...
These
...
bits
...
encode
...
the
...
values
...
of
...
s,
...
the
...
sign;
...
M,
...
the
...
significand;
...
and
...
E,
...
the
...
exponent.
...
Floating
...
point
...
numbers
...
are
...
then
...
calculated
...
as
...
(-1)
...
s *
...
M
...
*
...
2
...
E.
...
Ordinarily
...
all
...
of
...
the
...
mantissa
...
bits
...
are
...
used
...
to
...
express
...
significant
...
figures,
...
in
...
addition
...
to
...
a
...
leading
...
1,
...
which
...
is
...
implied
...
and,
...
therefore,
...
left
...
out.
...
Floats
...
consequently
...
have
...
24
...
significant
...
bits
...
of
...
precision;
...
doubles
...
have
...
53
...
significant
...
bits
...
of
...
precision.
...
Such
...
numbers
...
are
...
called
...
normalized
...
numbers.
...
All
...
floating
...
point
...
numbers
...
are
...
limited
...
in
...
this
...
sense
...
that
...
they
...
have
...
fixed
...
precision.
...
When
...
the
...
value
...
to
...
be
...
represented
...
is
...
too
...
small
...
to
...
encode
...
normally,
...
it
...
is
...
encoded
...
in
...
denormalized
...
form,
...
indicated
...
by
...
an
...
exponent
...
value
...
of
...
Float.MIN_EXPONENT
...
-
...
1
...
or
...
Double.MIN_EXPONENT
...
-
...
1
...
.
...
Denormalized
...
floating
...
point
...
numbers
...
have
...
an
...
assumed
...
0
...
in
...
the
...
ones
...
place,
...
and
...
have
...
zero
...
or
...
more
...
leading
...
zeros
...
in
...
represented
...
portion
...
of
...
their
...
mantissa.
...
These
...
leading
...
zero
...
bits
...
no
...
longer
...
function
...
as
...
significant
...
bits
...
of
...
precision;
...
consequently,
...
the
...
total
...
precision
...
of
...
denormalized
...
floating
...
point
...
numbers
...
is
...
less
...
than
...
that
...
of
...
normalized
...
floating
...
point
...
numbers.
...
Note
...
that
...
even
...
use
...
of
...
normalized
...
numbers
...
where
...
precision
...
is
...
required
...
can
...
pose
...
a
...
risk.
...
See
...
recommendation
...
...
...
...
...
...
...
...
...
...
...
...
.
...
for
...
more
...
information.
...
Use
...
of
...
denormalized
...
numbers
...
can
...
severely
...
impair
...
the
...
precision
...
of
...
floating
...
point
...
calculations;
...
consequently
...
denormalized
...
numbers
...
must
...
not
...
be
...
used.
...
Detecting
...
Denormalized
...
Numbers
...
The
...
following
...
code
...
tests
...
whether
...
a
...
float
...
value
...
is
...
denormalized
...
in
...
strictfp
...
mode,
...
or
...
for
...
platforms
...
that
...
lack
...
extended
...
range
...
support.
...
Testing
...
for
...
denormalized
...
numbers
...
in
...
the
...
presence
...
of
...
extended
...
range
...
support
...
is
...
platform
...
dependent;
...
see
...
...
...
...
...
...
...
...
...
...
...
...
...
for
...
additional
...
information.
| Code Block |
|---|
{{code}} strictfp public static boolean isDenormalized(float val) { if (val == 0) { return false; } if ((val > -Float.MIN_NORMAL) && (val < Float.MIN_NORMAL)) { return true; } return false; } |
Testing whether values of type double are denormalized is exactly analogous.
Print Representation of Denormalized Numbers
Denormalized numbers can also be troublesome because their printed representation is unusual. Floats and normalized doubles, when formatted with the %a specifier begin with a leading nonzero digit. Denormalized doubles can begin with a leading zero to the left of the decimal point in the mantissa.
The following program produces the following output:
| Code Block |
|---|
{{code}} Testing whether values of type {{double}} are denormalized is exactly analogous. h3. Print Representation of Denormalized Numbers Denormalized numbers can also be troublesome because their printed representation is unusual. Floats and normalized doubles, when formatted with the {{%a}} specifier begin with a leading nonzero digit. Denormalized doubles can begin with a leading zero to the left of the decimal point in the mantissa. The following program produces the following output: {code} class FloatingPointFormats { public static void main(String[] args) { float x = 0x1p-125f; double y = 0x1p-1020; System.out.format("normalized float with %%e : %e\n", x); System.out.format("normalized float with %%a : %a\n", x); x = 0x1p-140f; System.out.format("denormalized float with %%e : %e\n", x); System.out.format("denormalized float with %%a : %a\n", x); System.out.format("normalized double with %%e : %e\n", y); System.out.format("normalized double with %%a : %a\n", y); y = 0x1p-1050; System.out.format("denormalized double with %%e : %e\n", y); System.out.format("denormalized double with %%a : %a\n", y); } } {code} {code} |
| Code Block |
|---|
normalized float with %e : 2.350989e-38
normalized float with %a : 0x1.0p-125
denormalized float with %e : 7.174648e-43
denormalized float with %a : 0x1.0p-140
normalized double with %e : 8.900295e-308
normalized double with %a : 0x1.0p-1020
denormalized double with %e : 8.289046e-317
denormalized double with %a : 0x0.0000001p-1022
{code}
h2. Noncompliant Code Example
This code attempts to reduce a floating point number to a denormalized value and then restore the value.
{code:bgColor=#FFCCCC} |
Noncompliant Code Example
This code attempts to reduce a floating point number to a denormalized value and then restore the value.
| Code Block | ||
|---|---|---|
| ||
#include <stdio.h>
float x = 1/3.0f;
System.out.println("Original : " + x);
x = x * 7e-45f;
System.out.println("Denormalized? : " + x);
x = x / 7e-45f;
System.out.println("Restored : " + x);
{code}
|
This
...
operation
...
is
...
imprecise.
...
The
...
code
...
produces
...
the
...
following
...
output:
| Code Block |
|---|
} Original : 0.33333334 Denormalized? : 2.8E-45 Restored : 0.4 {code} h2. Compliant Solution Do not use code that could use denormalized numbers. When calculations using {{float}} produce denormalized numbers, use of {{double}} may provide sufficient precision. {code:bgColor=#ccccff} |
Compliant Solution
Do not use code that could use denormalized numbers. When calculations using float produce denormalized numbers, use of double may provide sufficient precision.
| Code Block | ||
|---|---|---|
| ||
#include <stdio.h>
double x = 1/3.0;
System.out.println("Original : " + x);
x = x * 7e-45;
System.out.println("Denormalized? : " + x);
x = x / 7e-45;
System.out.println("Restored : " + x);
{code}
|
This
...
code
...
produces
...
the
...
following
...
output:
| Code Block |
|---|
} Original : 0.3333333333333333 Denormalized? : 2.333333333333333E-45 Restored : 0.3333333333333333 {code} h2. Exceptions * |
Exceptions
NUM08-EX1:
...
Denormalized
...
numbers
...
are
...
acceptable
...
when
...
competent
...
numerical
...
analysis
...
demonstrates
...
that
...
the
...
computed
...
values
...
will
...
meet
...
all
...
accuracy
...
and
...
behavioral
...
requirements
...
that
...
are
...
appropriate
...
to
...
the
...
application.
...
Note
...
that
...
"competent
...
numerical
...
analysis"
...
generally
...
requires
...
a
...
specialized
...
professional
...
numerical
...
analyst;
...
lesser
...
levels
...
of
...
rigor
...
fail
...
to
...
qualify
...
for
...
this
...
exception.
...
Risk Assessment
Floating point numbers are an approximation; denormalized floating point numbers are a less precise approximation. Use of denormalized numbers can cause unexpected loss of precision, and consequently can lead to incorrect or unexpected results. Although the severity stated below for violations of this guideline is "low", applications that require accurate results should consider the severity of violations to be high.
Rule | Severity | Likelihood | Remediation Cost | Priority | Level |
|---|---|---|---|---|---|
NUM08-J | low | probable | high | P2 | L3 |
Related Guidelines
CERT C Secure Coding Standard FLP05-C. Don't use denormalized numbers
Bibliography
| Wiki Markup |
|---|
\[[IEEE 754|AA. Bibliography#IEEE 754 2006]\]
\[[Bryant 2003|AA. Bibliography#Bryant 03]\] Computer Systems: A Programmer's Perspective. Section 2.4 Floating Point |
...
...
...
...
...
...
...
...
...
...
...
...
required 03. Numeric Types and Operations (NUM) NUM09-J.
...
...
...
...
...
...
...
...
...
...
...
...