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<TITLE>Berkeley TestFloat General Documentation</TITLE>
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<H1>Berkeley TestFloat Release 3: General Documentation</H1>

<P>
John R. Hauser<BR>
2014 ______<BR>
</P>

<P>
*** CONTENT DONE.
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*** REPLACE QUOTATION MARKS.
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*** REPLACE APOSTROPHES.
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*** REPLACE EM DASH.
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<H2>Contents</H2>

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*** CHECK.<BR>
*** FIX FORMATTING.
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<PRE>
    Introduction
    Limitations
    Acknowledgments and License
    What TestFloat Does
    Executing TestFloat
    Operations Tested by TestFloat
        Conversion Operations
        Basic Arithmetic Operations
        Fused Multiply-Add Operations
        Remainder Operations
        Round-to-Integer Operations
        Comparison Operations
    Interpreting TestFloat Output
    Variations Allowed by the IEEE Floating-Point Standard
        Underflow
        NaNs
        Conversions to Integer
    Contact Information
</PRE>


<H2>1. Introduction</H2>

<P>
Berkeley TestFloat is a small collection of programs for testing that an
implementation of binary floating-point conforms to the IEEE Standard for 
Floating-Point Arithmetic.
All operations required by the original 1985 version of the IEEE Floating-Point
Standard can be tested, except for conversions to and from decimal.
The following binary formats can be tested:  <NOBR>32-bit</NOBR>
single-precision, <NOBR>64-bit</NOBR> double-precision, <NOBR>80-bit</NOBR>
double-extended-precision, and/or <NOBR>128-bit</NOBR> quadruple-precision.
TestFloat cannot test decimal floating-point.
</P>

<P>
Included in the TestFloat package are the <CODE>testsoftfloat</CODE> and
<CODE>timesoftfloat</CODE> programs for testing the Berkeley SoftFloat software
implementation of floating-point and for measuring its speed.
Information about SoftFloat can be found at the SoftFloat Web page,
<A HREF="http://www.jhauser.us/arithmetic/SoftFloat.html"><CODE>http://www.jhauser.us/arithmetic/SoftFloat.html</CODE></A>.
The <CODE>testsoftfloat</CODE> and <CODE>timesoftfloat</CODE> programs are
expected to be of interest only to people compiling the SoftFloat sources.
</P>

<P>
This document explains how to use the TestFloat programs.
It does not attempt to define or explain much of the IEEE Floating-Point
Standard.
Details about the standard are available elsewhere.
</P>

<P>
The current version of TestFloat is <NOBR>Release 3</NOBR>.
The set of TestFloat programs as well as the programs' arguments and behavior
have changed some compared to earlier TestFloat releases.
</P>


<H2>2. Limitations</H2>

<P>
TestFloat output is not always easily interpreted.
Detailed knowledge of the IEEE Floating-Point Standard and its vagaries is
needed to use TestFloat responsibly.
</P>

<P>
TestFloat performs relatively simple tests designed to check the fundamental
soundness of the floating-point under test.
TestFloat may also at times manage to find rarer and more subtle bugs, but it
will probably only find such bugs by chance.
Software that purposefully seeks out various kinds of subtle floating-point
bugs can be found through links posted on the TestFloat Web page
(<A HREF="http://www.jhauser.us/arithmetic/TestFloat.html"><CODE>http://www.jhauser.us/arithmetic/TestFloat.html</CODE></A>).
</P>


<H2>3. Acknowledgments and License</H2>

<P>
The TestFloat package was written by me, <NOBR>John R.</NOBR> Hauser.
<NOBR>Release 3</NOBR> of TestFloat is a completely new implementation
supplanting earlier releases.
This project was done in the employ of the University of California, Berkeley,
within the Department of Electrical Engineering and Computer Sciences, first
for the Parallel Computing Laboratory (Par Lab) and then for the ASPIRE Lab.
The work was officially overseen by Prof. Krste Asanovic, with funding provided
by these sources:
<BLOCKQUOTE>
<TABLE>
<TR>
<TD><NOBR>Par Lab:</NOBR></TD>
<TD>
Microsoft (Award #024263), Intel (Award #024894), and U.C. Discovery
(Award #DIG07-10227), with additional support from Par Lab affiliates Nokia,
NVIDIA, Oracle, and Samsung.
</TD>
</TR>
<TR>
<TD><NOBR>ASPIRE Lab:</NOBR></TD>
<TD>
DARPA PERFECT program (Award #HR0011-12-2-0016), with additional support from
ASPIRE industrial sponsor Intel and ASPIRE affiliates Google, Nokia, NVIDIA,
Oracle, and Samsung.
</TD>
</TR>
</TABLE>
</BLOCKQUOTE>
</P>

<P>
The following applies to the whole of TestFloat <NOBR>Release 3</NOBR> as well
as to each source file individually.
</P>

<P>
Copyright 2011, 2012, 2013, 2014 The Regents of the University of California
(Regents).
All Rights Reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
</P>

<P>
Redistributions of source code must retain the above copyright notice, this
list of conditions, and the following two paragraphs of disclaimer.
Redistributions in binary form must reproduce the above copyright notice, this
list of conditions, and the following two paragraphs of disclaimer in the
documentation and/or other materials provided with the distribution.
Neither the name of the Regents nor the names of its contributors may be used
to endorse or promote products derived from this software without specific
prior written permission.
</P>

<P>
IN NO EVENT SHALL REGENTS BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, SPECIAL,
INCIDENTAL, OR CONSEQUENTIAL DAMAGES, INCLUDING LOST PROFITS, ARISING OUT OF
THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF REGENTS HAS BEEN
ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
</P>

<P>
REGENTS SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
THE SOFTWARE AND ACCOMPANYING DOCUMENTATION, IF ANY, PROVIDED HEREUNDER IS
PROVIDED "<NOBR>AS IS</NOBR>".
REGENTS HAS NO OBLIGATION TO PROVIDE MAINTENANCE, SUPPORT, UPDATES,
ENHANCEMENTS, OR MODIFICATIONS.
</P>


<H2>4. What TestFloat Does</H2>

<P>
TestFloat is designed to test a floating-point implementation by comparing its
behavior with that of TestFloat's own internal floating-point implemented in
software.
For each operation to be tested, the TestFloat programs can generate a large
number of test cases, made up of simple pattern tests intermixed with weighted
random inputs.
The cases generated should be adequate for testing carry chain propagations,
plus the rounding of addition, subtraction, multiplication, and simple
operations like conversions.
TestFloat makes a point of checking all boundary cases of the arithmetic,
including underflows, overflows, invalid operations, subnormal inputs, zeros
(positive and negative), infinities, and NaNs.
For the interesting operations like addition and multiplication, millions of
test cases may be checked.
</P>

<P>
TestFloat is not remarkably good at testing difficult rounding cases for
division and square root.
It also makes no attempt to find bugs specific to SRT division and the like
(such as the infamous Pentium division bug).
Software that tests for such failures can be found through links on the
TestFloat Web page,
<A HREF="http://www.jhauser.us/arithmetic/TestFloat.html"><CODE>http://www.jhauser.us/arithmetic/TestFloat.html</CODE></A>.
</P>

<P>
NOTE!<BR>
It is the responsibility of the user to verify that the discrepancies TestFloat
finds actually represent faults in the implementation being tested.
Advice to help with this task is provided later in this document.
Furthermore, even if TestFloat finds no fault with a floating-point
implementation, that in no way guarantees that the implementation is bug-free.
</P>

<P>
For each operation, TestFloat can test all five rounding modes defined by the
IEEE Floating-Point Standard.
TestFloat verifies not only that the numeric results of an operation are
correct, but also that the proper floating-point exception flags are raised.
All five exception flags are tested, including the <I>inexact</I> flag.
TestFloat does not attempt to verify that the floating-point exception flags
are actually implemented as sticky flags.
</P>

<P>
For the <NOBR>80-bit</NOBR> double-extended-precision format, TestFloat can
test the addition, subtraction, multiplication, division, and square root
operations at all three of the standard rounding precisions.
The rounding precision can be set to <NOBR>32 bits</NOBR>, equivalent to
single-precision, to <NOBR>64 bits</NOBR>, equivalent to double-precision, or
to the full <NOBR>80 bits</NOBR> of the double-extended-precision.
Rounding precision control can be applied only to the double-extended-precision
format and only for the five basic arithmetic operations:  addition,
subtraction, multiplication, division, and square root.
Other operations can be tested only at full precision.
</P>

<P>
As a rule, TestFloat is not particular about the bit patterns of NaNs that
appear as operation results.
Any NaN is considered as good a result as another.
This laxness can be overridden so that TestFloat checks for particular bit
patterns within NaN results.
See <NOBR>section 8</NOBR> below, <I>Variations Allowed by the IEEE
Floating-Point Standard</I>, plus the <CODE>-checkNaNs</CODE> option documented
for programs <CODE>testfloat_ver</CODE> and <CODE>testfloat</CODE>.
</P>

<P>
TestFloat normally compares an implementation of floating-point against the
Berkeley SoftFloat software implementation of floating-point, also created by
me.
The SoftFloat functions are linked into each TestFloat program's executable.
Information about SoftFloat can be found at the Web page
<A HREF="http://www.jhauser.us/arithmetic/SoftFloat.html"><CODE>http://www.jhauser.us/arithmetic/SoftFloat.html</CODE></A>.
</P>

<P>
For testing SoftFloat itself, the TestFloat package includes a
<CODE>testsoftfloat</CODE> program that compares SoftFloat's floating-point
against <EM>another</EM> software floating-point implementation.
The second software floating-point is simpler and slower than SoftFloat, and is
completely independent of SoftFloat.
Although the second software floating-point cannot be guaranteed to be
bug-free, the chance that it would mimic any of SoftFloat's bugs is low.
Consequently, an error in one or the other floating-point version should appear
as an unexpected difference between the two implementations.
Note that testing SoftFloat should be necessary only when compiling a new
TestFloat executable or when compiling SoftFloat for some other reason.
</P>


<H2>5. Executing TestFloat</H2>

<P>
The TestFloat package consists of five programs, all intended to be executed
from a command-line interpreter:
<BLOCKQUOTE>
<TABLE>
<TR>
<TD>
<A HREF="testfloat_gen.html"><CODE>testfloat_gen</CODE></A><CODE>&nbsp;&nbsp;&nbsp;</CODE>
</TD>
<TD>
Generates test cases for a specific floating-point operation.
</TD>
</TR>
<TR>
<TD><A HREF="testfloat_ver.html"><CODE>testfloat_ver</CODE></A></TD>
<TD>
Verifies whether the results from executing a floating-point operation are as
expected.
</TD>
</TR>
<TR>
<TD><A HREF="testfloat.html"><CODE>testfloat</CODE></A></TD>
<TD>
An all-in-one program that generates test cases, executes floating-point
operations, and verifies whether the results match expectations.
</TD>
</TR>
<TR>
<TD>
<A HREF="testsoftfloat.html"><CODE>testsoftfloat</CODE></A><CODE>&nbsp;&nbsp;&nbsp;</CODE>
</TD>
<TD>
Like <CODE>testfloat</CODE>, but for testing SoftFloat.
</TD>
</TR>
<TR>
<TD>
<A HREF="timesoftfloat.html"><CODE>timesoftfloat</CODE></A><CODE>&nbsp;&nbsp;&nbsp;</CODE>
</TD>
<TD>
A program for measuring the speed of SoftFloat (included in the TestFloat
package for convenience).
</TD>
</TR>
</TABLE>
</BLOCKQUOTE>
Each program has its own page of documentation that can be opened through the
links in the table above.
</P>

<P>
To test a floating-point implementation other than SoftFloat, one of three
different methods can be used.
The first method pipes output from <CODE>testfloat_gen</CODE> to a program
that:
<NOBR>(a) reads</NOBR> the incoming test cases, <NOBR>(b) invokes</NOBR> the
floating-point operation being tested, and <NOBR>(c) writes</NOBR> the
operation results to output.
These results can then be piped to <CODE>testfloat_ver</CODE> to be checked for
correctness.
Assuming a vertical bar (<CODE>|</CODE>) indicates a pipe between programs, the
complete process could be written as a single command like so:
<PRE>
     testfloat_gen ... &lt;type&gt; | &lt;program-that-invokes-op&gt; | testfloat_ver ... &lt;function&gt;
</PRE>
The program in the middle is not supplied by TestFloat but must be created
independently.
If for some reason this program cannot take command-line arguments, the
<CODE>-prefix</CODE> option of <CODE>testfloat_gen</CODE> can communicate
parameters through the pipe.
</P>

<P>
A second method for running TestFloat is similar but has
<CODE>testfloat_gen</CODE> supply not only the test inputs but also the
expected results for each case.
With this additional information, the job done by <CODE>testfloat_ver</CODE>
can be folded into the invoking program to give the following command:
<PRE>
     testfloat_gen ... &lt;function&gt; | &lt;program-that-invokes-op-and-compares-results&gt;
</PRE>
Again, the program that actually invokes the floating-point operation is not
supplied by TestFloat but must be created independently.
Depending on circumstance, it may be preferable either to let
<CODE>testfloat_ver</CODE> check and report suspected errors (first method) or
to include this step in the invoking program (second method).
</P>

<P>
The third way to use TestFloat is the all-in-one <CODE>testfloat</CODE>
program.
This program can perform all the steps of creating test cases, invoking the
floating-point operation, checking the results, and reporting suspected errors.
However, for this to be possible, <CODE>testfloat</CODE> must be compiled to
contain the method for invoking the floating-point operations to test.
Each build of <CODE>testfloat</CODE> is therefore capable of testing
<EM>only</EM> the floating-point implementation it was built to invoke.
To test a new implementation of floating-point, a new <CODE>testfloat</CODE>
must be created, linked to that specific implementation.
By comparison, the <CODE>testfloat_gen</CODE> and <CODE>testfloat_ver</CODE>
programs are entirely generic;
one instance is usable for testing any floating-point implementation, because
implementation-specific details are segregated in the custom program that
follows <CODE>testfloat_gen</CODE>.
</P>

<P>
Program <CODE>testsoftfloat</CODE> is another all-in-one program specifically
for testing SoftFloat.
</P>

<P>
Programs <CODE>testfloat_ver</CODE>, <CODE>testfloat</CODE>, and
<CODE>testsoftfloat</CODE> all report status and error information in a common
way.
As it executes, each of these programs writes status information to the
standard error output, which should be the screen by default.
In order for this status to be displayed properly, the standard error stream
should not be redirected to a file.
Any discrepancies that are found are written to the standard output stream,
which is easily redirected to a file if desired.
Unless redirected, reported errors will appear intermixed with the ongoing
status information in the output.
</P>


<H2>6. Operations Tested by TestFloat</H2>

<P>
TestFloat can test all operations required by the original 1985 IEEE
Floating-Point Standard except for conversions to and from decimal.
These operations are:
<UL>
<LI>
conversions among the supported floating-point formats, and also between
integers (<NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR>, signed and unsigned) and
any of the floating-point formats;
<LI>
for each floating-point format, the usual addition, subtraction,
multiplication, division, and square root operations;
<LI>
for each format, the floating-point remainder operation defined by the IEEE
Standard;
<LI>
for each format, a ``round to integer'' operation that rounds to the nearest
integer value in the same format; and
<LI>
comparisons between two values in the same floating-point format.
</UL>
In addition, TestFloat can also test
<UL>
<LI>
for each floating-point format except <NOBR>80-bit</NOBR>
double-extended-precision, the fused multiply-add operation defined by the 2008
IEEE Standard.
</UL>
</P>

<P>
More information about all these operations is given below.
In the operation names used by TestFloat, <NOBR>32-bit</NOBR> single-precision
is called <CODE>f32</CODE>, <NOBR>64-bit</NOBR> double-precision is
<CODE>f64</CODE>, <NOBR>80-bit</NOBR> double-extended-precision is
<CODE>extF80</CODE>, and <NOBR>128-bit</NOBR> quadruple-precision is
<CODE>f128</CODE>.
TestFloat generally uses the same names for operations as Berkeley SoftFloat,
except that TestFloat's names never include the <CODE>M</CODE> that SoftFloat
uses to indicate that values are passed through pointers.
</P>

<H3>6.1. Conversion Operations</H3>

<P>
All conversions among the floating-point formats and all conversions between a
floating-point format and <NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR> integers
can be tested.
The conversion operations are:
<PRE>
     ui32_to_f32      ui64_to_f32      i32_to_f32       i64_to_f32
     ui32_to_f64      ui64_to_f64      i32_to_f64       i64_to_f64
     ui32_to_extF80   ui64_to_extF80   i32_to_extF80    i64_to_extF80
     ui32_to_f128     ui64_to_f128     i32_to_f128      i64_to_f128

     f32_to_ui32      f64_to_ui32      extF80_to_ui32   f128_to_ui32
     f32_to_ui64      f64_to_ui64      extF80_to_ui64   f128_to_ui64
     f32_to_i32       f64_to_i32       extF80_to_i32    f128_to_i32
     f32_to_i64       f64_to_i64       extF80_to_i64    f128_to_i64

     f32_to_f64       f64_to_f32       extF80_to_f32    f128_to_f32
     f32_to_extF80    f64_to_extF80    extF80_to_f64    f128_to_f64
     f32_to_f128      f64_to_f128      extF80_to_f128   f128_to_extF80
</PRE>
Abbreviations <CODE>ui32</CODE> and <CODE>ui64</CODE> indicate
<NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR> unsigned integer types, while
<CODE>i32</CODE> and <CODE>i64</CODE> indicate their signed counterparts.
These conversions all round according to the current rounding mode as relevant.
Conversions from a smaller to a larger floating-point format are always exact
and so require no rounding.
Likewise, conversions from <NOBR>32-bit</NOBR> integers to <NOBR>64-bit</NOBR>
double-precision or to any larger floating-point format are also exact, as are
conversions from <NOBR>64-bit</NOBR> integers to <NOBR>80-bit</NOBR>
double-extended-precision and <NOBR>128-bit</NOBR> quadruple-precision.
</P>

<P>
For the all-in-one <CODE>testfloat</CODE> program, this list of conversion
operations requires amendment.
For <CODE>testfloat</CODE> only, conversions to an integer type have names that
explicitly specify the rounding mode and treatment of inexactness.
Thus, instead of
<PRE>
     &lt;float&gt;_to_&lt;int&gt;
</PRE>
as listed above, operations converting to integer type have names of these
forms:
<PRE>
     &lt;float&gt;_to_&lt;int&gt;_r_&lt;round&gt;
     &lt;float&gt;_to_&lt;int&gt;_rx_&lt;round&gt;
</PRE>
The <CODE>&lt;round&gt;</CODE> component is one of `<CODE>near_even</CODE>',
`<CODE>near_maxMag</CODE>', `<CODE>minMag</CODE>', `<CODE>min</CODE>', or
`<CODE>max</CODE>', choosing the rounding mode.
Any other indication of rounding mode is ignored.
The operations with `<CODE>_r_</CODE>' in their names never raise the
<I>inexact</I> exception, while those with `<CODE>_rx_</CODE>' raise the
<I>inexact</I> exception whenever the result is not exact.
</P>

<P>
TestFloat assumes that conversions from floating-point to an integer type
should raise the <I>invalid</I> exception if the input cannot be rounded to an
integer representable by the result format.
In such a circumstance, if the result type is an unsigned integer, TestFloat
expects the result of the operation to be the type's largest integer value.
If the result type is a signed integer and conversion overflows, TestFloat
expects the result to be the largest-magnitude integer with the same sign as
the input.
Lastly, when a NaN is converted to a signed integer type, TestFloat allows
either the largest postive or largest-magnitude negative integer to be
returned.
Conversions to integer types are expected never to raise the <I>overflow</I>
exception.
</P>

<H3>6.2. Basic Arithmetic Operations</H3>

<P>
The following standard arithmetic operations can be tested:
<PRE>
     f32_add      f32_sub      f32_mul      f32_div      f32_sqrt
     f64_add      f64_sub      f64_mul      f64_div      f64_sqrt
     extF80_add   extF80_sub   extF80_mul   extF80_div   extF80_sqrt
     f128_add     f128_sub     f128_mul     f128_div     f128_sqrt
</PRE>
The double-extended-precision (<CODE>extF80</CODE>) operations can be rounded
to reduced precision under rounding precision control.
</P>

<H3>6.3. Fused Multiply-Add Operations</H3>

<P>
For all floating-point formats except <NOBR>80-bit</NOBR>
double-extended-precision, TestFloat can test the fused multiply-add operation
defined by the 2008 IEEE Floating-Point Standard.
The fused multiply-add operations are:
<PRE>
     f32_mulAdd
     f64_mulAdd
     f128_mulAdd
</PRE>
</P>

<P>
If one of the multiplication operands is infinite and the other is zero,
TestFloat expects the fused multiply-add operation to raise the <I>invalid</I>
exception even if the third operand is a NaN.
</P>

<H3>6.4. Remainder Operations</H3>

<P>
For each format, TestFloat can test the IEEE Standard's remainder operation.
These operations are:
<PRE>
     f32_rem
     f64_rem
     extF80_rem
     f128_rem
</PRE>
The remainder operations are always exact and so require no rounding.
</P>

<H3>6.5. Round-to-Integer Operations</H3>

<P>
For each format, TestFloat can test the IEEE Standard's round-to-integer
operation.
For most TestFloat programs, these operations are:
<PRE>
     f32_roundToInt
     f64_roundToInt
     extF80_roundToInt
     f128_roundToInt
</PRE>
</P>

<P>
Just as for conversions to integer types (<NOBR>section 6.1</NOBR> above), the
all-in-one <CODE>testfloat</CODE> program is again an exception.
For <CODE>testfloat</CODE> only, the round-to-integer operations have names of
these forms:
<PRE>
     &lt;float&gt;_roundToInt_r_&lt;round&gt;
     &lt;float&gt;_roundToInt_x
</PRE>
For the `<CODE>_r_</CODE>' versions, the <I>inexact</I> exception is never
raised, and the <CODE>&lt;round&gt;</CODE> component specifies the rounding
mode as one of `<CODE>near_even</CODE>', `<CODE>near_maxMag</CODE>',
`<CODE>minMag</CODE>', `<CODE>min</CODE>', or `<CODE>max</CODE>'.
The usual indication of rounding mode is ignored.
In contrast, the `<CODE>_x</CODE>' versions accept the usual indication of
rounding mode and raise the <I>inexact</I> exception whenever the result is not
exact.
This irregular system follows the IEEE Standard's precise specification for the
round-to-integer operations.
</P>

<H3>6.6. Comparison Operations</H3>

<P>
The following floating-point comparison operations can be tested:
<PRE>
     f32_eq      f32_le      f32_lt
     f64_eq      f64_le      f64_lt
     extF80_eq   extF80_le   extF80_lt
     f128_eq     f128_le     f128_lt
</PRE>
The abbreviation <CODE>eq</CODE> stands for ``equal'' (=), <CODE>le</CODE>
stands for ``less than or equal'' (&le;), and <CODE>lt</CODE> stands for
``less than'' (&lt;).
</P>

<P>
The IEEE Standard specifies that, by default, the less-than-or-equal and
less-than comparisons raise the <I>invalid</I> exception if either input is any
kind of NaN.
The equality comparisons, on the other hand, are defined by default to raise
the <I>invalid</I> exception only for signaling NaNs, not for quiet NaNs.
For completeness, the following additional operations can be tested if
supported:
<PRE>
     f32_eq_signaling      f32_le_quiet      f32_lt_quiet
     f64_eq_signaling      f64_le_quiet      f64_lt_quiet
     extF80_eq_signaling   extF80_le_quiet   extF80_lt_quiet
     f128_eq_signaling     f128_le_quiet     f128_lt_quiet
</PRE>
The <CODE>signaling</CODE> equality comparisons are identical to the standard
operations except that the <I>invalid</I> exception should be raised for any
NaN input.
Similarly, the <CODE>quiet</CODE> comparison operations should be identical to
their counterparts except that the <I>invalid</I> exception is not raised for
quiet NaNs.
</P>

<P>
Obviously, no comparison operations ever require rounding.
Any rounding mode is ignored.
</P>


<H2>7. Interpreting TestFloat Output</H2>

<P>
The ``errors'' reported by TestFloat programs may or may not really represent
errors in the system being tested.
For each test case tried, the results from the floating-point implementation
being tested could differ from the expected results for several reasons:
<UL>
<LI>
The IEEE Floating-Point Standard allows for some variation in how conforming
floating-point behaves.
Two implementations can sometimes give different results without either being
incorrect.
<LI>
The trusted floating-point emulation could be faulty.
This could be because there is a bug in the way the enulation is coded, or
because a mistake was made when the code was compiled for the current system.
<LI>
The TestFloat program may not work properly, reporting differences that do not
exist.
<LI>
Lastly, the floating-point being tested could actually be faulty.
</UL>
It is the responsibility of the user to determine the causes for the
discrepancies that are reported.
Making this determination can require detailed knowledge about the IEEE
Standard.
Assuming TestFloat is working properly, any differences found will be due to
either the first or last of the reasons above.
Variations in the IEEE Standard that could lead to false error reports are
discussed in <NOBR>section 8</NOBR>, <I>Variations Allowed by the IEEE
Floating-Point Standard</I>.
</P>

<P>
For each reported error (or apparent error), a line of text is written to the
default output.
If a line would be longer than 79 characters, it is divided.
The first part of each error line begins in the leftmost column, and any
subsequent ``continuation'' lines are indented with a tab.
</P>

<P>
Each error reported is of the form:
<PRE>
     &lt;inputs&gt;  => &lt;observed-output&gt;  expected: &lt;expected-output&gt;
</PRE>
The <CODE>&lt;inputs&gt;</CODE> are the inputs to the operation.
Each output (observed and expected) is shown as a pair:  the result value
first, followed by the exception flags.
</P>

<P>
For example, two typical error lines could be
<PRE>
     800.7FFF00  87F.000100  => 001.000000 ...ux  expected: 001.000000 ....x
     081.000004  000.1FFFFF  => 001.000000 ...ux  expected: 001.000000 ....x
</PRE>
In the first line, the inputs are <CODE>800.7FFF00</CODE> and
<CODE>87F.000100</CODE>, and the observed result is <CODE>001.000000</CODE>
with flags <CODE>...ux</CODE>.
The trusted emulation result is the same but with different flags,
<CODE>....x</CODE>.
Items such as <CODE>800.7FFF00</CODE> composed of hexadecimal digits and a
single period represent floating-point values (here <NOBR>32-bit</NOBR>
single-precision).
The two instances above were reported as errors because the exception flag
results differ.
</P>

<P>
Aside from the exception flags, there are nine data types that may be
represented.
Four are floating-point types:  <NOBR>32-bit</NOBR> single-precision,
<NOBR>64-bit</NOBR> double-precision, <NOBR>80-bit</NOBR>
double-extended-precision, and <NOBR>128-bit</NOBR> quadruple-precision.
The remaining five types are <NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR>
unsigned integers, <NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR> two's-complement
signed integers, and Boolean values (the results of comparison operations).
Boolean values are represented as a single character, either a <CODE>0</CODE>
or a <CODE>1</CODE>.
<NOBR>32-bit</NOBR> integers are represented as 8 hexadecimal digits.
Thus, for a signed <NOBR>32-bit</NOBR> integer, <CODE>FFFFFFFF</CODE> is -1,
and <CODE>7FFFFFFF</CODE> is the largest positive value.
<NOBR>64-bit</NOBR> integers are the same except with 16 hexadecimal digits.
</P>

<P>
Floating-point values are written in a correspondingly primitive form.
Values of the <NOBR>64-bit</NOBR> double-precision format are represented by 16
hexadecimal digits that give the raw bits of the floating-point encoding.
A period separates the 3rd and 4th hexadecimal digits to mark the division
between the exponent bits and fraction bits.
Some notable <NOBR>64-bit</NOBR> double-precision values include:
<PRE>
     000.0000000000000    +0
     3FF.0000000000000     1
     400.0000000000000     2
     7FF.0000000000000    +infinity

     800.0000000000000    -0
     BFF.0000000000000    -1
     C00.0000000000000    -2
     FFF.0000000000000    -infinity

     3FE.FFFFFFFFFFFFF    largest representable number less than +1
</PRE>
The following categories are easily distinguished (assuming the
<CODE>x</CODE>s are not all 0):
<PRE>
     000.xxxxxxxxxxxxx    positive subnormal (denormalized) numbers
     7FF.xxxxxxxxxxxxx    positive NaNs
     800.xxxxxxxxxxxxx    negative subnormal numbers
     FFF.xxxxxxxxxxxxx    negative NaNs
</PRE>
</P>

<P>
<NOBR>128-bit</NOBR> quadruple-precision values are written the same except
with 4 hexadecimal digits for the sign and exponent and 28 for the fraction.
Notable values include:
<PRE>
     0000.0000000000000000000000000000    +0
     3FFF.0000000000000000000000000000     1
     4000.0000000000000000000000000000     2
     7FFF.0000000000000000000000000000    +infinity

     8000.0000000000000000000000000000    -0
     BFFF.0000000000000000000000000000    -1
     C000.0000000000000000000000000000    -2
     FFFF.0000000000000000000000000000    -infinity

     3FFE.FFFFFFFFFFFFFFFFFFFFFFFFFFFF    largest representable number
                                              less than +1
</PRE>
</P>

<P>
<NOBR>80-bit</NOBR> double-extended-precision values are a little unusual in
that the leading bit of precision is not hidden as with other formats.
When correctly encoded, the leading significand bit of an <NOBR>80-bit</NOBR>
double-extended-precision value will be 0 if the value is zero or subnormal,
and will be 1 otherwise.
Hence, the same values listed above appear in <NOBR>80-bit</NOBR>
double-extended-precision as follows (note the leading <CODE>8</CODE> digit in
the significands):
<PRE>
     0000.0000000000000000    +0
     3FFF.8000000000000000     1
     4000.8000000000000000     2
     7FFF.8000000000000000    +infinity

     8000.0000000000000000    -0
     BFFF.8000000000000000    -1
     C000.8000000000000000    -2
     FFFF.8000000000000000    -infinity

     3FFE.FFFFFFFFFFFFFFFF    largest representable number less than +1
</PRE>
</P>

<P>
The representation of <NOBR>32-bit</NOBR> single-precision values is unusual
for a different reason.
Because the subfields of standard <NOBR>32-bit</NOBR> single-precision do not
fall on neat <NOBR>4-bit</NOBR> boundaries, single-precision outputs are
slightly perturbed.
These are written as 9 hexadecimal digits, with a period separating the 3rd and
4th hexadecimal digits.
Broken out into bits, the 9 hexademical digits cover the <NOBR>32-bit</NOBR>
single-precision subfields as follows:
<PRE>
     x000 .... ....  .  .... .... .... .... .... ....    sign       (1 bit)
     .... xxxx xxxx  .  .... .... .... .... .... ....    exponent   (8 bits)
     .... .... ....  .  0xxx xxxx xxxx xxxx xxxx xxxx    fraction  (23 bits)
</PRE>
As shown in this schematic, the first hexadecimal digit contains only the sign,
and will be either <CODE>0</CODE> <NOBR>or <CODE>8</CODE></NOBR>.
The next two digits give the biased exponent as an <NOBR>8-bit</NOBR> integer.
This is followed by a period and 6 hexadecimal digits of fraction.
The most significant hexadecimal digit of the fraction can be at most
<NOBR>a <CODE>7</CODE></NOBR>.
</P>

<P>
Notable single-precision values include:
<PRE>
     000.000000    +0
     07F.000000     1
     080.000000     2
     0FF.000000    +infinity

     800.000000    -0
     87F.000000    -1
     880.000000    -2
     8FF.000000    -infinity

     07E.7FFFFF    largest representable number less than +1
</PRE>
Again, certain categories are easily distinguished (assuming the
<CODE>x</CODE>s are not all 0):
<PRE>
     000.xxxxxx    positive subnormal (denormalized) numbers
     0FF.xxxxxx    positive NaNs
     800.xxxxxx    negative subnormal numbers
     8FF.xxxxxx    negative NaNs
</PRE>
</P>

<P>
Lastly, exception flag values are represented by five characters, one character
per flag.
Each flag is written as either a letter or a period (<CODE>.</CODE>) according
to whether the flag was set or not by the operation.
A period indicates the flag was not set.
The letter used to indicate a set flag depends on the flag:
<PRE>
     v    invalid exception
     i    infinite exception ("divide by zero")
     o    overflow exception
     u    underflow exception
     x    inexact exception
</PRE>
For example, the notation <CODE>...ux</CODE> indicates that the
<I>underflow</I> and <I>inexact</I> exception flags were set and that the other
three flags (<I>invalid</I>, <I>infinite</I>, and <I>overflow</I>) were not
set.
The exception flags are always written following the value returned as the
result of the operation.
</P>


<H2>8. Variations Allowed by the IEEE Floating-Point Standard</H2>

<P>
The IEEE Floating-Point Standard admits some variation among conforming
implementations.
Because TestFloat expects the two implementations being compared to deliver
bit-for-bit identical results under most circumstances, this leeway in the
standard can result in false errors being reported if the two implementations
do not make the same choices everywhere the standard provides an option.
</P>

<H3>8.1. Underflow</H3>

<P>
The standard specifies that the <I>underflow</I> exception flag is to be raised
when two conditions are met simultaneously:
<NOBR>(1) <I>tininess</I></NOBR> and <NOBR>(2) <I>loss of accuracy</I></NOBR>.
</P>

<P>
A result is tiny when its magnitude is nonzero yet smaller than any normalized
floating-point number.
The standard allows tininess to be determined either before or after a result
is rounded to the destination precision.
If tininess is detected before rounding, some borderline cases will be flagged
as underflows even though the result after rounding actually lies within the
normal floating-point range.
By detecting tininess after rounding, a system can avoid some unnecessary
signaling of underflow.
All the TestFloat programs support options <CODE>-tininessbefore</CODE> and
<CODE>-tininessafter</CODE> to control whether TestFloat expects tininess on
underflow to be detected before or after rounding.
One or the other is selected as the default when TestFloat is compiled, but
these command options allow the default to be overridden.
</P>

<P>
Loss of accuracy occurs when the subnormal format is not sufficient to
represent an underflowed result accurately.
The original 1985 version of the IEEE Standard allowed loss of accuracy to be
detected either as an <I>inexact result</I> or as a
<I>denormalization loss</I>;
however, few if any systems ever chose the latter.
The latest standard requires that loss of accuracy be detected as an inexact
result, and TestFloat can test only for this case.
</P>

<H3>8.2. NaNs</H3>

<P>
The IEEE Standard gives the floating-point formats a large number of NaN
encodings and specifies that NaNs are to be returned as results under certain
conditions.
However, the standard allows an implementation almost complete freedom over
<EM>which</EM> NaN to return in each situation.
</P>

<P>
By default, TestFloat does not check the bit patterns of NaN results.
When the result of an operation should be a NaN, any NaN is considered as good
as another.
This laxness can be overridden with the <CODE>-checkNaNs</CODE> option of
programs <CODE>testfloat_ver</CODE> and <CODE>testfloat</CODE>.
In order for this option to be sensible, TestFloat must have been compiled so
that its internal floating-point implementation (SoftFloat) generates the
proper NaN results for the system being tested.
</P>

<H3>8.3. Conversions to Integer</H3>

<P>
Conversion of a floating-point value to an integer format will fail if the
source value is a NaN or if it is too large.
The IEEE Standard does not specify what value should be returned as the integer
result in these cases.
Moreover, according to the standard, the <I>invalid</I> exception can be raised
or an unspecified alternative mechanism may be used to signal such cases.
</P>

<P>
TestFloat assumes that conversions to integer will raise the <I>invalid</I>
exception if the source value cannot be rounded to a representable integer.
In such cases, TestFloat expects the result value to be the largest-magnitude
positive or negative integer as detailed earlier in <NOBR>section 6.1</NOBR>,
<I>Conversion Operations</I>.
The current version of TestFloat provides no means to alter these expectations.
</P>


<H2>9. Contact Information</H2>

<P>
At the time of this writing, the most up-to-date information about TestFloat
and the latest release can be found at the Web page
<A HREF="http://www.jhauser.us/arithmetic/TestFloat.html"><CODE>http://www.jhauser.us/arithmetic/TestFloat.html</CODE></A>.
</P>


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