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 <HTML>

 <HEAD>
 <TITLE>Berkeley SoftFloat Library Interface</TITLE>
 </HEAD>

 <BODY>

 <H1>Berkeley SoftFloat Release 3a: Library Interface</H1>

 <P>
 John R. Hauser<BR>
 2015 October 23<BR>
 </P>


 <H2>Contents</H2>

 <BLOCKQUOTE>
 <TABLE BORDER=0 CELLSPACING=0 CELLPADDING=0>
 <COL WIDTH=25>
 <COL WIDTH=*>
 <TR><TD COLSPAN=2>1. Introduction</TD></TR>
 <TR><TD COLSPAN=2>2. Limitations</TD></TR>
 <TR><TD COLSPAN=2>3. Acknowledgments and License</TD></TR>
 <TR><TD COLSPAN=2>4. Types and Functions</TD></TR>
 <TR><TD></TD><TD>4.1. Boolean and Integer Types</TD></TR>
 <TR><TD></TD><TD>4.2. Floating-Point Types</TD></TR>
 <TR><TD></TD><TD>4.3. Supported Floating-Point Functions</TD></TR>
 <TR>
   <TD></TD>
   <TD>4.4. Non-canonical Representations in <CODE>extFloat80_t</CODE></TD>
 </TR>
 <TR><TD></TD><TD>4.5. Conventions for Passing Arguments and Results</TD></TR>
 <TR><TD COLSPAN=2>5. Reserved Names</TD></TR>
 <TR><TD COLSPAN=2>6. Mode Variables</TD></TR>
 <TR><TD></TD><TD>6.1. Rounding Mode</TD></TR>
 <TR><TD></TD><TD>6.2. Underflow Detection</TD></TR>
 <TR>
   <TD></TD>
   <TD>6.3. Rounding Precision for the <NOBR>80-Bit</NOBR> Extended Format</TD>
 </TR>
 <TR><TD COLSPAN=2>7. Exceptions and Exception Flags</TD></TR>
 <TR><TD COLSPAN=2>8. Function Details</TD></TR>
 <TR><TD></TD><TD>8.1. Conversions from Integer to Floating-Point</TD></TR>
 <TR><TD></TD><TD>8.2. Conversions from Floating-Point to Integer</TD></TR>
 <TR><TD></TD><TD>8.3. Conversions Among Floating-Point Types</TD></TR>
 <TR><TD></TD><TD>8.4. Basic Arithmetic Functions</TD></TR>
 <TR><TD></TD><TD>8.5. Fused Multiply-Add Functions</TD></TR>
 <TR><TD></TD><TD>8.6. Remainder Functions</TD></TR>
 <TR><TD></TD><TD>8.7. Round-to-Integer Functions</TD></TR>
 <TR><TD></TD><TD>8.8. Comparison Functions</TD></TR>
 <TR><TD></TD><TD>8.9. Signaling NaN Test Functions</TD></TR>
 <TR><TD></TD><TD>8.10. Raise-Exception Function</TD></TR>
 <TR><TD COLSPAN=2>9. Changes from SoftFloat <NOBR>Release 2</NOBR></TD></TR>
 <TR><TD></TD><TD>9.1. Name Changes</TD></TR>
 <TR><TD></TD><TD>9.2. Changes to Function Arguments</TD></TR>
 <TR><TD></TD><TD>9.3. Added Capabilities</TD></TR>
 <TR><TD></TD><TD>9.4. Better Compatibility with the C Language</TD></TR>
 <TR><TD></TD><TD>9.5. New Organization as a Library</TD></TR>
 <TR><TD></TD><TD>9.6. Optimization Gains (and Losses)</TD></TR>
 <TR><TD COLSPAN=2>10. Future Directions</TD></TR>
 <TR><TD COLSPAN=2>11. Contact Information</TD></TR>
 </TABLE>
 </BLOCKQUOTE>


 <H2>1. Introduction</H2>

 <P>
 Berkeley SoftFloat is a software implementation of binary floating-point that
 conforms to the IEEE Standard for Floating-Point Arithmetic.
 The current release supports four binary formats:  <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 following functions are supported for each format:
 <UL>
 <LI>
 addition, subtraction, multiplication, division, and square root;
 <LI>
 fused multiply-add as defined by the IEEE Standard, except for
 <NOBR>80-bit</NOBR> double-extended-precision;
 <LI>
 remainder as defined by the IEEE Standard;
 <LI>
 round to integral value;
 <LI>
 comparisons;
 <LI>
 conversions to/from other supported formats; and
 <LI>
 conversions to/from <NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR> integers,
 signed and unsigned.
 </UL>
 All operations required by the original 1985 version of the IEEE Floating-Point
 Standard are implemented, except for conversions to and from decimal.
 </P>

 <P>
 This document gives information about the types defined and the routines
 implemented by SoftFloat.
 It does not attempt to define or explain the IEEE Floating-Point Standard.
 Information about the standard is available elsewhere.
 </P>

 <P>
 The current version of SoftFloat is <NOBR>Release 3a</NOBR>.
 The only difference between this version and the previous
 <NOBR>Release 3</NOBR> is the replacement of the license text supplied by the
 University of California.
 </P>

 <P>
 The functional interface of SoftFloat <NOBR>Release 3</NOBR> and afterward
 differs in many details from that of earlier releases.
 For specifics of these differences, see <NOBR>section 9</NOBR> below,
 <I>Changes from SoftFloat <NOBR>Release 2</NOBR></I>.
 </P>


 <H2>2. Limitations</H2>

 <P>
 SoftFloat assumes the computer has an addressable byte size of 8 or
 <NOBR>16 bits</NOBR>.
 (Nearly all computers in use today have <NOBR>8-bit</NOBR> bytes.)
 </P>

 <P>
 SoftFloat is written in C and is designed to work with other C code.
 The C compiler used must conform at a minimum to the 1989 ANSI standard for the
 C language (same as the 1990 ISO standard) and must in addition support basic
 arithmetic on <NOBR>64-bit</NOBR> integers.
 Earlier releases of SoftFloat included implementations of <NOBR>32-bit</NOBR>
 single-precision and <NOBR>64-bit</NOBR> double-precision floating-point that
 did not require <NOBR>64-bit</NOBR> integers, but this option is not supported
 starting with <NOBR>Release 3</NOBR>.
 Since 1999, ISO standards for C have mandated compiler support for
 <NOBR>64-bit</NOBR> integers.
 A compiler conforming to the 1999 C Standard or later is recommended but not
 strictly required.
 </P>

 <P>
 Most operations not required by the original 1985 version of the IEEE
 Floating-Point Standard but added in the 2008 version are not yet supported in
 SoftFloat <NOBR>Release 3a</NOBR>.
 </P>


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

 <P>
 The SoftFloat package was written by me, <NOBR>John R.</NOBR> Hauser.
 <NOBR>Release 3</NOBR> of SoftFloat was a completely new implementation
 supplanting earlier releases.
 The project to create <NOBR>Release 3</NOBR> (and <NOBR>now 3a</NOBR>) 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>
 <COL>
 <COL WIDTH=10>
 <COL>
 <TR>
 <TD VALIGN=TOP><NOBR>Par Lab:</NOBR></TD>
 <TD></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 VALIGN=TOP><NOBR>ASPIRE Lab:</NOBR></TD>
 <TD></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 SoftFloat <NOBR>Release 3a</NOBR> as well
 as to each source file individually.
 </P>

 <P>
 Copyright 2011, 2012, 2013, 2014, 2015 The Regents of the University of
 California.
 All rights reserved.
 </P>

 <P>
 Redistribution and use in source and binary forms, with or without
 modification, are permitted provided that the following conditions are met:
 <OL>

 <LI>
 <P>
 Redistributions of source code must retain the above copyright notice, this
 list of conditions, and the following disclaimer.
 </P>

 <LI>
 <P>
 Redistributions in binary form must reproduce the above copyright notice, this
 list of conditions, and the following disclaimer in the documentation and/or
 other materials provided with the distribution.
 </P>

 <LI>
 <P>
 Neither the name of the University nor the names of its contributors may be
 used to endorse or promote products derived from this software without specific
 prior written permission.
 </P>

 </OL>
 </P>

 <P>
 THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS &ldquo;AS IS&rdquo;,
 AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, ARE
 DISCLAIMED.
 IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT,
 INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
 BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
 LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE
 OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
 ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 </P>


 <H2>4. Types and Functions</H2>

 <P>
 The types and functions of SoftFloat are declared in header file
 <CODE>softfloat.h</CODE>.
 </P>

 <H3>4.1. Boolean and Integer Types</H3>

 <P>
 Header file <CODE>softfloat.h</CODE> depends on standard headers
 <CODE>&lt;stdbool.h&gt;</CODE> and <CODE>&lt;stdint.h&gt;</CODE> to define type
 <CODE>bool</CODE> and several integer types.
 These standard headers have been part of the ISO C Standard Library since 1999.
 With any recent compiler, they are likely to be supported, even if the compiler
 does not claim complete conformance to the ISO C Standard.
 For older or nonstandard compilers, a port of SoftFloat may have substitutes
 for these headers.
 Header <CODE>softfloat.h</CODE> depends only on the name <CODE>bool</CODE> from
 <CODE>&lt;stdbool.h&gt;</CODE> and on these type names from
 <CODE>&lt;stdint.h&gt;</CODE>:
 <BLOCKQUOTE>
 <PRE>
 uint16_t
 uint32_t
 uint64_t
 int32_t
 int64_t
 uint_fast8_t
 uint_fast32_t
 uint_fast64_t
 </PRE>
 </BLOCKQUOTE>
 </P>


 <H3>4.2. Floating-Point Types</H3>

 <P>
 The <CODE>softfloat.h</CODE> header defines four floating-point types:
 <BLOCKQUOTE>
 <TABLE CELLSPACING=0 CELLPADDING=0>
 <TR>
 <TD><CODE>float32_t</CODE></TD>
 <TD><NOBR>32-bit</NOBR> single-precision binary format</TD>
 </TR>
 <TR>
 <TD><CODE>float64_t</CODE></TD>
 <TD><NOBR>64-bit</NOBR> double-precision binary format</TD>
 </TR>
 <TR>
 <TD><CODE>extFloat80_t&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD><NOBR>80-bit</NOBR> double-extended-precision binary format (old Intel or
 Motorola format)</TD>
 </TR>
 <TR>
 <TD><CODE>float128_t</CODE></TD>
 <TD><NOBR>128-bit</NOBR> quadruple-precision binary format</TD>
 </TR>
 </TABLE>
 </BLOCKQUOTE>
 The non-extended types are each exactly the size specified:
 <NOBR>32 bits</NOBR> for <CODE>float32_t</CODE>, <NOBR>64 bits</NOBR> for
 <CODE>float64_t</CODE>, and <NOBR>128 bits</NOBR> for <CODE>float128_t</CODE>.
 Aside from these size requirements, the definitions of all these types may
 differ for different ports of SoftFloat to specific systems.
 A given port of SoftFloat may or may not define some of the floating-point
 types as aliases for the C standard types <CODE>float</CODE>,
 <CODE>double</CODE>, and <CODE>long</CODE> <CODE>double</CODE>.
 </P>

 <P>
 Header file <CODE>softfloat.h</CODE> also defines a structure,
 <CODE>struct</CODE> <CODE>extFloat80M</CODE>, for the representation of
 <NOBR>80-bit</NOBR> double-extended-precision floating-point values in memory.
 This structure is the same size as type <CODE>extFloat80_t</CODE> and contains
 at least these two fields (not necessarily in this order):
 <BLOCKQUOTE>
 <PRE>
 uint16_t signExp;
 uint64_t signif;
 </PRE>
 </BLOCKQUOTE>
 Field <CODE>signExp</CODE> contains the sign and exponent of the floating-point
 value, with the sign in the most significant bit (<NOBR>bit 15</NOBR>) and the
 encoded exponent in the other <NOBR>15 bits</NOBR>.
 Field <CODE>signif</CODE> is the complete <NOBR>64-bit</NOBR> significand of
 the floating-point value.
 (In the usual encoding for <NOBR>80-bit</NOBR> extended floating-point, the
 leading <NOBR>1 bit</NOBR> of normalized numbers is not implicit but is stored
 in the most significant bit of the significand.)
 </P>

 <H3>4.3. Supported Floating-Point Functions</H3>

 <P>
 SoftFloat implements these arithmetic operations for its floating-point types:
 <UL>
 <LI>
 conversions between any two floating-point formats;
 <LI>
 for each floating-point format, conversions to and from signed and unsigned
 <NOBR>32-bit</NOBR> and <NOBR>64-bit</NOBR> integers;
 <LI>
 for each format, the usual addition, subtraction, multiplication, division, and
 square root operations;
 <LI>
 for each format except <CODE>extFloat80_t</CODE>, the fused multiply-add
 operation defined by the IEEE Standard;
 <LI>
 for each format, the floating-point remainder operation defined by the IEEE
 Standard;
 <LI>
 for each format, a &ldquo;round to integer&rdquo; 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>
 </P>

 <P>
 The following operations required by the 2008 IEEE Floating-Point Standard are
 not supported in SoftFloat <NOBR>Release 3a</NOBR>:
 <UL>
 <LI>
 <B>nextUp</B>, <B>nextDown</B>, <B>minNum</B>, <B>maxNum</B>, <B>minNumMag</B>,
 <B>maxNumMag</B>, <B>scaleB</B>, and <B>logB</B>;
 <LI>
 conversions between floating-point formats and decimal or hexadecimal character
 sequences;
 <LI>
 all &ldquo;quiet-computation&rdquo; operations (<B>copy</B>, <B>negate</B>,
 <B>abs</B>, and <B>copySign</B>, which all involve only simple copying and/or
 manipulation of the floating-point sign bit); and
 <LI>
 all &ldquo;non-computational&rdquo; operations other than <B>isSignaling</B>
 (which is supported).
 </UL>
 </P>

 <H3>4.4. Non-canonical Representations in <CODE>extFloat80_t</CODE></H3>

 <P>
 Because the <NOBR>80-bit</NOBR> double-extended-precision format,
 <CODE>extFloat80_t</CODE>, stores an explicit leading significand bit, many
 floating-point numbers are encodable in this type in equivalent normalized and
 denormalized forms.
 Zeros and values in the subnormal range have each only a single possible
 encoding, for which the leading significand bit must <NOBR>be 0</NOBR>.
 For other finite values (outside the subnormal range), a unique normalized
 representation, with leading significand bit set <NOBR>to 1</NOBR>, always
 exists, and is considered the <I>canonical</I> representation of the value.
 Any equivalent denormalized representations (having leading significand bit
 <NOBR>of 0</NOBR>) are <I>non-canonical</I>.
 Similarly, the leading significand bit is expected to <NOBR>be 1</NOBR> for
 infinities and NaNs as well;
 any infinity or NaN with a leading significand bit <NOBR>of 0</NOBR> is again
 considered non-canonical.
 In short, for an <CODE>extFloat80_t</CODE> representation to be canonical, the
 leading significand bit must <NOBR>be 1</NOBR> unless it is required to
 <NOBR>be 0</NOBR> because the encoded value is zero or a subnormal.
 </P>

 <P>
 Functions are not guaranteed to operate as expected when inputs of type
 <CODE>extFloat80_t</CODE> are non-canonical.
 Assuming all of a function&rsquo;s <CODE>extFloat80_t</CODE> inputs (if any)
 are canonical, function outputs of type <CODE>extFloat80_t</CODE> will always
 be canonical.
 </P>

 <H3>4.5. Conventions for Passing Arguments and Results</H3>

 <P>
 Values that are at most <NOBR>64 bits</NOBR> in size (i.e., not the
 <NOBR>80-bit</NOBR> or <NOBR>128-bit</NOBR> floating-point formats) are in all
 cases passed as function arguments by value.
 Likewise, when an output of a function is no more than <NOBR>64 bits</NOBR>, it
 is always returned directly as the function result.
 Thus, for example, the SoftFloat function for adding two <NOBR>64-bit</NOBR>
 floating-point values has this simple signature:
 <BLOCKQUOTE>
 <CODE>float64_t f64_add( float64_t, float64_t );</CODE>
 </BLOCKQUOTE>
 </P>

 <P>
 The story is more complex when function inputs and outputs are
 <NOBR>80-bit</NOBR> and <NOBR>128-bit</NOBR> floating-point.
 For these types, SoftFloat always provides a function that passes these larger
 values into or out of the function indirectly, via pointers.
 For example, for adding two <NOBR>128-bit</NOBR> floating-point values,
 SoftFloat supplies this function:
 <BLOCKQUOTE>
 <CODE>void f128M_add( const float128_t *, const float128_t *, float128_t * );</CODE>
 </BLOCKQUOTE>
 The first two arguments point to the values to be added, and the last argument
 points to the location where the sum will be stored.
 The <CODE>M</CODE> in the name <CODE>f128M_add</CODE> is mnemonic for the fact
 that the <NOBR>128-bit</NOBR> inputs and outputs are &ldquo;in memory&rdquo;,
 pointed to by pointer arguments.
 </P>

 <P>
 All ports of SoftFloat implement these <I>pass-by-pointer</I> functions for
 types <CODE>extFloat80_t</CODE> and <CODE>float128_t</CODE>.
 At the same time, SoftFloat ports may also implement alternate versions of
 these same functions that pass <CODE>extFloat80_t</CODE> and
 <CODE>float128_t</CODE> by value, like the smaller formats.
 Thus, besides the function with name <CODE>f128M_add</CODE> shown above, a
 SoftFloat port may also supply an equivalent function with this signature:
 <BLOCKQUOTE>
 <CODE>float128_t f128_add( float128_t, float128_t );</CODE>
 </BLOCKQUOTE>
 </P>

 <P>
 As a general rule, on computers where the machine word size is
 <NOBR>32 bits</NOBR> or smaller, only the pass-by-pointer versions of functions
 (e.g., <CODE>f128M_add</CODE>) are provided for types <CODE>extFloat80_t</CODE>
 and <CODE>float128_t</CODE>, because passing such large types directly can have
 significant extra cost.
 On computers where the word size is <NOBR>64 bits</NOBR> or larger, both
 function versions (<CODE>f128M_add</CODE> and <CODE>f128_add</CODE>) are
 provided, because the cost of passing by value is then more reasonable.
 Applications that must be portable accross both classes of computers must use
 the pointer-based functions, as these are always implemented.
 However, if it is known that SoftFloat includes the by-value functions for all
 platforms of interest, programmers can use whichever version they prefer.
 </P>


 <H2>5. Reserved Names</H2>

 <P>
 In addition to the variables and functions documented here, SoftFloat defines
 some symbol names for its own private use.
 These private names always begin with the prefix
 &lsquo;<CODE>softfloat_</CODE>&rsquo;.
 When a program includes header <CODE>softfloat.h</CODE> or links with the
 SoftFloat library, all names with prefix &lsquo;<CODE>softfloat_</CODE>&rsquo;
 are reserved for possible use by SoftFloat.
 Applications that use SoftFloat should not define their own names with this
 prefix, and should reference only such names as are documented.
 </P>


 <H2>6. Mode Variables</H2>

 <P>
 The following variables control rounding mode, underflow detection, and the
 <NOBR>80-bit</NOBR> extended format&rsquo;s rounding precision:
 <BLOCKQUOTE>
 <CODE>softfloat_roundingMode</CODE><BR>
 <CODE>softfloat_detectTininess</CODE><BR>
 <CODE>extF80_roundingPrecision</CODE>
 </BLOCKQUOTE>
 These mode variables are covered in the next several subsections.
 </P>

 <H3>6.1. Rounding Mode</H3>

 <P>
 All five rounding modes defined by the 2008 IEEE Floating-Point Standard are
 implemented for all operations that require rounding.
 The rounding mode is selected by the global variable
 <BLOCKQUOTE>
 <CODE>uint_fast8_t softfloat_roundingMode;</CODE>
 </BLOCKQUOTE>
 This variable may be set to one of the values
 <BLOCKQUOTE>
 <TABLE CELLSPACING=0 CELLPADDING=0>
 <TR>
 <TD><CODE>softfloat_round_near_even</CODE></TD>
 <TD>round to nearest, with ties to even</TD>
 </TR>
 <TR>
 <TD><CODE>softfloat_round_near_maxMag&nbsp;&nbsp;</CODE></TD>
 <TD>round to nearest, with ties to maximum magnitude (away from zero)</TD>
 </TR>
 <TR>
 <TD><CODE>softfloat_round_minMag</CODE></TD>
 <TD>round to minimum magnitude (toward zero)</TD>
 </TR>
 <TR>
 <TD><CODE>softfloat_round_min</CODE></TD>
 <TD>round to minimum (down)</TD>
 </TR>
 <TR>
 <TD><CODE>softfloat_round_max</CODE></TD>
 <TD>round to maximum (up)</TD>
 </TR>
 </TABLE>
 </BLOCKQUOTE>
 Variable <CODE>softfloat_roundingMode</CODE> is initialized to
 <CODE>softfloat_round_near_even</CODE>.
 </P>

 <H3>6.2. Underflow Detection</H3>

 <P>
 In the terminology of the IEEE Standard, SoftFloat can detect tininess for
 underflow either before or after rounding.
 The choice is made by the global variable
 <BLOCKQUOTE>
 <CODE>uint_fast8_t softfloat_detectTininess;</CODE>
 </BLOCKQUOTE>
 which can be set to either
 <BLOCKQUOTE>
 <CODE>softfloat_tininess_beforeRounding</CODE><BR>
 <CODE>softfloat_tininess_afterRounding</CODE>
 </BLOCKQUOTE>
 Detecting tininess after rounding is better because it results in fewer
 spurious underflow signals.
 The other option is provided for compatibility with some systems.
 Like most systems (and as required by the newer 2008 IEEE Standard), SoftFloat
 always detects loss of accuracy for underflow as an inexact result.
 </P>

 <H3>6.3. Rounding Precision for the <NOBR>80-Bit</NOBR> Extended Format</H3>

 <P>
 For <CODE>extFloat80_t</CODE> only, the rounding precision of the basic
 arithmetic operations is controlled by the global variable
 <BLOCKQUOTE>
 <CODE>uint_fast8_t extF80_roundingPrecision;</CODE>
 </BLOCKQUOTE>
 The operations affected are:
 <BLOCKQUOTE>
 <CODE>extF80_add</CODE><BR>
 <CODE>extF80_sub</CODE><BR>
 <CODE>extF80_mul</CODE><BR>
 <CODE>extF80_div</CODE><BR>
 <CODE>extF80_sqrt</CODE>
 </BLOCKQUOTE>
 When <CODE>extF80_roundingPrecision</CODE> is set to its default value of 80,
 these operations are rounded to the full precision of the <NOBR>80-bit</NOBR>
 double-extended-precision format, like occurs for other formats.
 Setting <CODE>extF80_roundingPrecision</CODE> to 32 or to 64 causes the
 operations listed to be rounded to <NOBR>32-bit</NOBR> precision (equivalent to
 <CODE>float32_t</CODE>) or to <NOBR>64-bit</NOBR> precision (equivalent to
 <CODE>float64_t</CODE>), respectively.
 When rounding to reduced precision, additional bits in the result significand
 beyond the rounding point are set to zero.
 The consequences of setting <CODE>extF80_roundingPrecision</CODE> to a value
 other than 32, 64, or 80 is not specified.
 Operations other than the ones listed above are not affected by
 <CODE>extF80_roundingPrecision</CODE>.
 </P>


 <H2>7. Exceptions and Exception Flags</H2>

 <P>
 All five exception flags required by the IEEE Floating-Point Standard are
 implemented.
 Each flag is stored as a separate bit in the global variable
 <BLOCKQUOTE>
 <CODE>uint_fast8_t softfloat_exceptionFlags;</CODE>
 </BLOCKQUOTE>
 The positions of the exception flag bits within this variable are determined by
 the bit masks
 <BLOCKQUOTE>
 <CODE>softfloat_flag_inexact</CODE><BR>
 <CODE>softfloat_flag_underflow</CODE><BR>
 <CODE>softfloat_flag_overflow</CODE><BR>
 <CODE>softfloat_flag_infinite</CODE><BR>
 <CODE>softfloat_flag_invalid</CODE>
 </BLOCKQUOTE>
 Variable <CODE>softfloat_exceptionFlags</CODE> is initialized to all zeros,
 meaning no exceptions.
 </P>

 <P>
 An individual exception flag can be cleared with the statement
 <BLOCKQUOTE>
 <CODE>softfloat_exceptionFlags &= ~softfloat_flag_&lt;<I>exception</I>&gt;;</CODE>
 </BLOCKQUOTE>
 where <CODE>&lt;<I>exception</I>&gt;</CODE> is the appropriate name.
 To raise a floating-point exception, function <CODE>softfloat_raise</CODE>
 should normally be used.
 </P>

 <P>
 When SoftFloat detects an exception other than <I>inexact</I>, it calls
 <CODE>softfloat_raise</CODE>.
 The default version of this function simply raises the corresponding exception
 flags.
 Particular ports of SoftFloat may support alternate behavior, such as exception
 traps, by modifying the default <CODE>softfloat_raise</CODE>.
 A program may also supply its own <CODE>softfloat_raise</CODE> function to
 override the one from the SoftFloat library.
 </P>

 <P>
 Because inexact results occur frequently under most circumstances (and thus are
 hardly exceptional), SoftFloat does not ordinarily call
 <CODE>softfloat_raise</CODE> for <I>inexact</I> exceptions.
 It does always raise the <I>inexact</I> exception flag as required.
 </P>


 <H2>8. Function Details</H2>

 <P>
 In this section, <CODE>&lt;<I>float</I>&gt;</CODE> appears in function names as
 a substitute for one of these abbreviations:
 <BLOCKQUOTE>
 <TABLE CELLSPACING=0 CELLPADDING=0>
 <TR>
 <TD><CODE>f32</CODE></TD>
 <TD>indicates <CODE>float32_t</CODE>, passed by value</TD>
 </TR>
 <TR>
 <TD><CODE>f64</CODE></TD>
 <TD>indicates <CODE>float64_t</CODE>, passed by value</TD>
 </TR>
 <TR>
 <TD><CODE>extF80M&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD>indicates <CODE>extFloat80_t</CODE>, passed indirectly via pointers</TD>
 </TR>
 <TR>
 <TD><CODE>extF80</CODE></TD>
 <TD>indicates <CODE>extFloat80_t</CODE>, passed by value</TD>
 </TR>
 <TR>
 <TD><CODE>f128M</CODE></TD>
 <TD>indicates <CODE>float128_t</CODE>, passed indirectly via pointers</TD>
 </TR>
 <TR>
 <TD><CODE>f128</CODE></TD>
 <TD>indicates <CODE>float128_t</CODE>, passed by value</TD>
 </TR>
 </TABLE>
 </BLOCKQUOTE>
 The circumstances under which values of floating-point types
 <CODE>extFloat80_t</CODE> and <CODE>float128_t</CODE> may be passed either by
 value or indirectly via pointers was discussed earlier in
 <NOBR>section 4.5</NOBR>, <I>Conventions for Passing Arguments and Results</I>.
 </P>

 <H3>8.1. Conversions from Integer to Floating-Point</H3>

 <P>
 All conversions from a <NOBR>32-bit</NOBR> or <NOBR>64-bit</NOBR> integer,
 signed or unsigned, to a floating-point format are supported.
 Functions performing these conversions have these names:
 <BLOCKQUOTE>
 <CODE>ui32_to_&lt;<I>float</I>&gt;</CODE><BR>
 <CODE>ui64_to_&lt;<I>float</I>&gt;</CODE><BR>
 <CODE>i32_to_&lt;<I>float</I>&gt;</CODE><BR>
 <CODE>i64_to_&lt;<I>float</I>&gt;</CODE>
 </BLOCKQUOTE>
 Conversions from <NOBR>32-bit</NOBR> integers to <NOBR>64-bit</NOBR>
 double-precision and larger formats are always exact, and likewise conversions
 from <NOBR>64-bit</NOBR> integers to <NOBR>80-bit</NOBR>
 double-extended-precision and <NOBR>128-bit</NOBR> quadruple-precision are also
 always exact.
 </P>

 <P>
 Each conversion function takes one input of the appropriate type and generates
 one output.
 The following illustrates the signatures of these functions in cases when the
 floating-point result is passed either by value or via pointers:
 <BLOCKQUOTE>
 <PRE>
 float64_t i32_to_f64( int32_t <I>a</I> );
 </PRE>
 <PRE>
 void i32_to_f128M( int32_t <I>a</I>, float128_t *<I>destPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 </P>

 <H3>8.2. Conversions from Floating-Point to Integer</H3>

 <P>
 Conversions from a floating-point format to a <NOBR>32-bit</NOBR> or
 <NOBR>64-bit</NOBR> integer, signed or unsigned, are supported with these
 functions:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_to_ui32</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_ui64</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_i32</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_i64</CODE>
 </BLOCKQUOTE>
 The functions have signatures as follows, depending on whether the
 floating-point input is passed by value or via pointers:
 <BLOCKQUOTE>
 <PRE>
 int_fast32_t f64_to_i32( float64_t <I>a</I>, uint_fast8_t <I>roundingMode</I>, bool <I>exact</I> );
 </PRE>
 <PRE>
 int_fast32_t
  f128M_to_i32( const float128_t *<I>aPtr</I>, uint_fast8_t <I>roundingMode</I>, bool <I>exact</I> );
 </PRE>
 </BLOCKQUOTE>
 The <CODE><I>roundingMode</I></CODE> argument specifies the rounding mode for
 the conversion.
 The variable that usually indicates rounding mode,
 <CODE>softfloat_roundingMode</CODE>, is ignored.
 Argument <CODE><I>exact</I></CODE> determines whether the <I>inexact</I>
 exception flag is raised if the conversion is not exact.
 If <CODE><I>exact</I></CODE> is <CODE>true</CODE>, the <I>inexact</I> flag may
 be raised;
 otherwise, it will not be, even if the conversion is inexact.
 </P>

 <P>
 Conversions from floating-point to integer raise the <I>invalid</I> exception
 if the source value cannot be rounded to a representable integer of the desired
 size (32 or 64 bits).
 In such a circumstance, if the floating-point input is a NaN or if the
 conversion is to an unsigned integer type, the largest positive integer is
 returned;
 otherwise, the largest integer with the same sign as the input is returned.
 The functions that convert to integer types never raise the <I>overflow</I>
 exception.
 </P>

 <P>
 Note that, when converting to an unsigned integer type, if the <I>invalid</I>
 exception is raised because the input floating-point value would round to a
 negative integer, the value returned is the <EM>maximum positive unsigned
 integer</EM>.
 Zero is not returned when the <I>invalid</I> exception is raised, even when
 zero is the closest integer to the original floating-point value.
 </P>

 <P>
 Because languages such <NOBR>as C</NOBR> require that conversions to integers
 be rounded toward zero, the following functions are provided for improved speed
 and convenience:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_to_ui32_r_minMag</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_ui64_r_minMag</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_i32_r_minMag</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_to_i64_r_minMag</CODE>
 </BLOCKQUOTE>
 These functions round only toward zero (to minimum magnitude).
 The signatures for these functions are the same as above without the redundant
 <CODE><I>roundingMode</I></CODE> argument:
 <BLOCKQUOTE>
 <PRE>
 int_fast32_t f64_to_i32_r_minMag( float64_t <I>a</I>, bool <I>exact</I> );
 </PRE>
 <PRE>
 int_fast32_t f128M_to_i32_r_minMag( const float128_t *<I>aPtr</I>, bool <I>exact</I> );
 </PRE>
 </BLOCKQUOTE>
 </P>

 <H3>8.3. Conversions Among Floating-Point Types</H3>

 <P>
 Conversions between floating-point formats are done by functions with these
 names:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_to_&lt;<I>float</I>&gt;</CODE>
 </BLOCKQUOTE>
 All combinations of source and result type are supported where the source and
 result are different formats.
 There are four different styles of signature for these functions, depending on
 whether the input and the output floating-point values are passed by value or
 via pointers:
 <BLOCKQUOTE>
 <PRE>
 float32_t f64_to_f32( float64_t <I>a</I> );
 </PRE>
 <PRE>
 float32_t f128M_to_f32( const float128_t *<I>aPtr</I> );
 </PRE>
 <PRE>
 void f32_to_f128M( float32_t <I>a</I>, float128_t *<I>destPtr</I> );
 </PRE>
 <PRE>
 void extF80M_to_f128M( const extFloat80_t *<I>aPtr</I>, float128_t *<I>destPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 </P>

 <P>
 Conversions from a smaller to a larger floating-point format are always exact
 and so require no rounding.
 </P>

 <H3>8.4. Basic Arithmetic Functions</H3>

 <P>
 The following basic arithmetic functions are provided:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_add</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_sub</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_mul</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_div</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_sqrt</CODE>
 </BLOCKQUOTE>
 Each floating-point operation takes two operands, except for <CODE>sqrt</CODE>
 (square root) which takes only one.
 The operands and result are all of the same floating-point format.
 Signatures for these functions take the following forms:
 <BLOCKQUOTE>
 <PRE>
 float64_t f64_add( float64_t <I>a</I>, float64_t <I>b</I> );
 </PRE>
 <PRE>
 void
  f128M_add(
      const float128_t *<I>aPtr</I>, const float128_t *<I>bPtr</I>, float128_t *<I>destPtr</I> );
 </PRE>
 <PRE>
 float64_t f64_sqrt( float64_t <I>a</I> );
 </PRE>
 <PRE>
 void f128M_sqrt( const float128_t *<I>aPtr</I>, float128_t *<I>destPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 When floating-point values are passed indirectly through pointers, arguments
 <CODE><I>aPtr</I></CODE> and <CODE><I>bPtr</I></CODE> point to the input
 operands, and the last argument, <CODE><I>destPtr</I></CODE>, points to the
 location where the result is stored.
 </P>

 <P>
 Rounding of the <NOBR>80-bit</NOBR> double-extended-precision
 (<CODE>extFloat80_t</CODE>) functions is affected by variable
 <CODE>extF80_roundingPrecision</CODE>, as explained earlier in
 <NOBR>section 6.3</NOBR>,
 <I>Rounding Precision for the <NOBR>80-Bit</NOBR> Extended Format</I>.
 </P>

 <H3>8.5. Fused Multiply-Add Functions</H3>

 <P>
 The 2008 version of the IEEE Floating-Point Standard defines a <I>fused
 multiply-add</I> operation that does a combined multiplication and addition
 with only a single rounding.
 SoftFloat implements fused multiply-add with functions
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_mulAdd</CODE>
 </BLOCKQUOTE>
 Unlike other operations, fused multiple-add is supported only for the
 non-extended formats, <CODE>float32_t</CODE>, <CODE>float64_t</CODE>, and
 <CODE>float128_t</CODE>.
 No fused multiple-add function is currently provided for the
 <NOBR>80-bit</NOBR> double-extended-precision type, <CODE>extFloat80_t</CODE>.
 </P>

 <P>
 Depending on whether floating-point values are passed by value or via pointers,
 the fused multiply-add functions have signatures of these forms:
 <BLOCKQUOTE>
 <PRE>
 float64_t f64_mulAdd( float64_t <I>a</I>, float64_t <I>b</I>, float64_t <I>c</I> );
 </PRE>
 <PRE>
 void
  f128M_mulAdd(
      const float128_t *<I>aPtr</I>,
      const float128_t *<I>bPtr</I>,
      const float128_t *<I>cPtr</I>,
      float128_t *<I>destPtr</I>
  );
 </PRE>
 </BLOCKQUOTE>
 The functions compute
 <NOBR>(<CODE><I>a</I></CODE> &times; <CODE><I>b</I></CODE>)
  + <CODE><I>c</I></CODE></NOBR>
 with a single rounding.
 When floating-point values are passed indirectly through pointers, arguments
 <CODE><I>aPtr</I></CODE>, <CODE><I>bPtr</I></CODE>, and
 <CODE><I>cPtr</I></CODE> point to operands <CODE><I>a</I></CODE>,
 <CODE><I>b</I></CODE>, and <CODE><I>c</I></CODE> respectively, and
 <CODE><I>destPtr</I></CODE> points to the location where the result is stored.
 </P>

 <P>
 If one of the multiplication operands <CODE><I>a</I></CODE> and
 <CODE><I>b</I></CODE> is infinite and the other is zero, these functions raise
 the invalid exception even if operand <CODE><I>c</I></CODE> is a quiet NaN.
 </P>

 <H3>8.6. Remainder Functions</H3>

 <P>
 For each format, SoftFloat implements the remainder operation defined by the
 IEEE Floating-Point Standard.
 The remainder functions have names
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_rem</CODE>
 </BLOCKQUOTE>
 Each remainder operation takes two floating-point operands of the same format
 and returns a result in the same format.
 Depending on whether floating-point values are passed by value or via pointers,
 the remainder functions have signatures of these forms:
 <BLOCKQUOTE>
 <PRE>
 float64_t f64_rem( float64_t <I>a</I>, float64_t <I>b</I> );
 </PRE>
 <PRE>
 void
  f128M_rem(
      const float128_t *<I>aPtr</I>, const float128_t *<I>bPtr</I>, float128_t *<I>destPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 When floating-point values are passed indirectly through pointers, arguments
 <CODE><I>aPtr</I></CODE> and <CODE><I>bPtr</I></CODE> point to operands
 <CODE><I>a</I></CODE> and <CODE><I>b</I></CODE> respectively, and
 <CODE><I>destPtr</I></CODE> points to the location where the result is stored.
 </P>

 <P>
 The IEEE Standard remainder operation computes the value
 <NOBR><CODE><I>a</I></CODE>
  &minus; <I>n</I> &times; <CODE><I>b</I></CODE></NOBR>,
 where <I>n</I> is the integer closest to
 <NOBR><CODE><I>a</I></CODE> &divide; <CODE><I>b</I></CODE></NOBR>.
 If <NOBR><CODE><I>a</I></CODE> &divide; <CODE><I>b</I></CODE></NOBR> is exactly
 halfway between two integers, <I>n</I> is the <EM>even</EM> integer closest to
 <NOBR><CODE><I>a</I></CODE> &divide; <CODE><I>b</I></CODE></NOBR>.
 The IEEE Standard&rsquo;s remainder operation is always exact and so requires
 no rounding.
 </P>

 <P>
 Depending on the relative magnitudes of the operands, the remainder
 functions can take considerably longer to execute than the other SoftFloat
 functions.
 This is inherent in the remainder operation itself and is not a flaw in the
 SoftFloat implementation.
 </P>

 <H3>8.7. Round-to-Integer Functions</H3>

 <P>
 For each format, SoftFloat implements the round-to-integer operation specified
 by the IEEE Floating-Point Standard.
 These functions are named
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_roundToInt</CODE>
 </BLOCKQUOTE>
 Each round-to-integer operation takes a single floating-point operand.
 This operand is rounded to an integer according to a specified rounding mode,
 and the resulting integer value is returned in the same floating-point format.
 (Note that the result is not an integer type.)
 </P>

 <P>
 The signatures of the round-to-integer functions are similar to those for
 conversions to an integer type:
 <BLOCKQUOTE>
 <PRE>
 float64_t f64_roundToInt( float64_t <I>a</I>, uint_fast8_t <I>roundingMode</I>, bool <I>exact</I> );
 </PRE>
 <PRE>
 void
  f128M_roundToInt(
      const float128_t *<I>aPtr</I>,
      uint_fast8_t <I>roundingMode</I>,
      bool <I>exact</I>,
      float128_t *<I>destPtr</I>
  );
 </PRE>
 </BLOCKQUOTE>
 The <CODE><I>roundingMode</I></CODE> argument specifies the rounding mode to
 apply.
 The variable that usually indicates rounding mode,
 <CODE>softfloat_roundingMode</CODE>, is ignored.
 Argument <CODE><I>exact</I></CODE> determines whether the <I>inexact</I>
 exception flag is raised if the conversion is not exact.
 If <CODE><I>exact</I></CODE> is <CODE>true</CODE>, the <I>inexact</I> flag may
 be raised;
 otherwise, it will not be, even if the conversion is inexact.
 When floating-point values are passed indirectly through pointers,
 <CODE><I>aPtr</I></CODE> points to the input operand and
 <CODE><I>destPtr</I></CODE> points to the location where the result is stored.
 </P>

 <H3>8.8. Comparison Functions</H3>

 <P>
 For each format, the following floating-point comparison functions are
 provided:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_eq</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_le</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_lt</CODE>
 </BLOCKQUOTE>
 Each comparison takes two operands of the same type and returns a Boolean.
 The abbreviation <CODE>eq</CODE> stands for &ldquo;equal&rdquo; (=);
 <CODE>le</CODE> stands for &ldquo;less than or equal&rdquo; (&le;);
 and <CODE>lt</CODE> stands for &ldquo;less than&rdquo; (&lt;).
 Depending on whether the floating-point operands are passed by value or via
 pointers, the comparison functions have signatures of these forms:
 <BLOCKQUOTE>
 <PRE>
 bool f64_eq( float64_t <I>a</I>, float64_t <I>b</I> );
 </PRE>
 <PRE>
 bool f128M_eq( const float128_t *<I>aPtr</I>, const float128_t *<I>bPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 </P>

 <P>
 The usual greater-than (&gt;), greater-than-or-equal (&ge;), and not-equal
 (&ne;) comparisons are easily obtained from the functions provided.
 The not-equal function is just the logical complement of the equal function.
 The greater-than-or-equal function is identical to the less-than-or-equal
 function with the arguments in reverse order, and likewise the greater-than
 function is identical to the less-than function with the arguments reversed.
 </P>

 <P>
 The IEEE Floating-Point Standard specifies that the less-than-or-equal and
 less-than comparisons by default raise the <I>invalid</I> exception if either
 operand is any kind of NaN.
 Equality comparisons, on the other hand, are defined by default to raise the
 <I>invalid</I> exception only for signaling NaNs, not quiet NaNs.
 For completeness, SoftFloat provides these complementary functions:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_eq_signaling</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_le_quiet</CODE><BR>
 <CODE>&lt;<I>float</I>&gt;_lt_quiet</CODE>
 </BLOCKQUOTE>
 The <CODE>signaling</CODE> equality comparisons are identical to the default
 equality comparisons except that the <I>invalid</I> exception is raised for any
 NaN input, not just for signaling NaNs.
 Similarly, the <CODE>quiet</CODE> comparison functions are identical to their
 default counterparts except that the <I>invalid</I> exception is not raised for
 quiet NaNs.
 </P>

 <H3>8.9. Signaling NaN Test Functions</H3>

 <P>
 Functions for testing whether a floating-point value is a signaling NaN are
 provided with these names:
 <BLOCKQUOTE>
 <CODE>&lt;<I>float</I>&gt;_isSignalingNaN</CODE>
 </BLOCKQUOTE>
 The functions take one floating-point operand and return a Boolean indicating
 whether the operand is a signaling NaN.
 Accordingly, the functions have the forms
 <BLOCKQUOTE>
 <PRE>
 bool f64_isSignalingNaN( float64_t <I>a</I> );
 </PRE>
 <PRE>
 bool f128M_isSignalingNaN( const float128_t *<I>aPtr</I> );
 </PRE>
 </BLOCKQUOTE>
 </P>

 <H3>8.10. Raise-Exception Function</H3>

 <P>
 SoftFloat provides a single function for raising floating-point exceptions:
 <BLOCKQUOTE>
 <PRE>
 void softfloat_raise( uint_fast8_t <I>exceptions</I> );
 </PRE>
 </BLOCKQUOTE>
 The <CODE><I>exceptions</I></CODE> argument is a mask indicating the set of
 exceptions to raise.
 (See earlier section 7, <I>Exceptions and Exception Flags</I>.)
 In addition to setting the specified exception flags in variable
 <CODE>softfloat_exceptionFlags</CODE>, the <CODE>softfloat_raise</CODE>
 function may cause a trap or abort appropriate for the current system.
 </P>


 <H2>9. Changes from SoftFloat <NOBR>Release 2</NOBR></H2>

 <P>
 Apart from a change in the legal use license, <NOBR>Release 3</NOBR> of
 SoftFloat introduced numerous technical differences compared to earlier
 releases.
 </P>

 <H3>9.1. Name Changes</H3>

 <P>
 The most obvious and pervasive difference compared to <NOBR>Release 2</NOBR>
 is that the names of most functions and variables have changed, even when the
 behavior has not.
 First, the floating-point types, the mode variables, the exception flags
 variable, the function to raise exceptions, and various associated constants
 have been renamed as follows:
 <BLOCKQUOTE>
 <TABLE>
 <TR>
 <TD>old name, Release 2:</TD>
 <TD>new name, Release 3:</TD>
 </TR>
 <TR>
 <TD><CODE>float32</CODE></TD>
 <TD><CODE>float32_t</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float64</CODE></TD>
 <TD><CODE>float64_t</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>floatx80</CODE></TD>
 <TD><CODE>extFloat80_t</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float128</CODE></TD>
 <TD><CODE>float128_t</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_rounding_mode</CODE></TD>
 <TD><CODE>softfloat_roundingMode</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_round_nearest_even</CODE></TD>
 <TD><CODE>softfloat_round_near_even</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_round_to_zero</CODE></TD>
 <TD><CODE>softfloat_round_minMag</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_round_down</CODE></TD>
 <TD><CODE>softfloat_round_min</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_round_up</CODE></TD>
 <TD><CODE>softfloat_round_max</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_detect_tininess</CODE></TD>
 <TD><CODE>softfloat_detectTininess</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_tininess_before_rounding&nbsp;&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD><CODE>softfloat_tininess_beforeRounding</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_tininess_after_rounding</CODE></TD>
 <TD><CODE>softfloat_tininess_afterRounding</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>floatx80_rounding_precision</CODE></TD>
 <TD><CODE>extF80_roundingPrecision</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_exception_flags</CODE></TD>
 <TD><CODE>softfloat_exceptionFlags</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_flag_inexact</CODE></TD>
 <TD><CODE>softfloat_flag_inexact</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_flag_underflow</CODE></TD>
 <TD><CODE>softfloat_flag_underflow</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_flag_overflow</CODE></TD>
 <TD><CODE>softfloat_flag_overflow</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_flag_divbyzero</CODE></TD>
 <TD><CODE>softfloat_flag_infinite</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_flag_invalid</CODE></TD>
 <TD><CODE>softfloat_flag_invalid</CODE></TD>
 </TR>
 <TR>
 <TD><CODE>float_raise</CODE></TD>
 <TD><CODE>softfloat_raise</CODE></TD>
 </TR>
 </TABLE>
 </BLOCKQUOTE>
 </P>

 <P>
 Furthermore, <NOBR>Release 3</NOBR> adopted the following new abbreviations for
 function names:
 <BLOCKQUOTE>
 <TABLE>
 <TR>
 <TD>used in names in Release 2:<CODE>&nbsp;&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD>used in names in Release 3:</TD>
 </TR>
 <TR> <TD><CODE>int32</CODE></TD>    <TD><CODE>i32</CODE></TD>    </TR>
 <TR> <TD><CODE>int64</CODE></TD>    <TD><CODE>i64</CODE></TD>    </TR>
 <TR> <TD><CODE>float32</CODE></TD>  <TD><CODE>f32</CODE></TD>    </TR>
 <TR> <TD><CODE>float64</CODE></TD>  <TD><CODE>f64</CODE></TD>    </TR>
 <TR> <TD><CODE>floatx80</CODE></TD> <TD><CODE>extF80</CODE></TD> </TR>
 <TR> <TD><CODE>float128</CODE></TD> <TD><CODE>f128</CODE></TD>   </TR>
 </TABLE>
 </BLOCKQUOTE>
 Thus, for example, the function to add two <NOBR>32-bit</NOBR> floating-point
 numbers, previously called <CODE>float32_add</CODE> in <NOBR>Release 2</NOBR>,
 is now <CODE>f32_add</CODE>.
 Lastly, there have been a few other changes to function names:
 <BLOCKQUOTE>
 <TABLE>
 <TR>
 <TD>used in names in Release 2:<CODE>&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD>used in names in Release 3:<CODE>&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD>relevant functions:</TD>
 </TR>
 <TR>
 <TD><CODE>_round_to_zero</CODE></TD>
 <TD><CODE>_r_minMag</CODE></TD>
 <TD>conversions from floating-point to integer (<NOBR>section 8.2</NOBR>)</TD>
 </TR>
 <TR>
 <TD><CODE>round_to_int</CODE></TD>
 <TD><CODE>roundToInt</CODE></TD>
 <TD>round-to-integer functions (<NOBR>section 8.7</NOBR>)</TD>
 </TR>
 <TR>
 <TD><CODE>is_signaling_nan&nbsp;&nbsp;&nbsp;&nbsp;</CODE></TD>
 <TD><CODE>isSignalingNaN</CODE></TD>
 <TD>signaling NaN test functions (<NOBR>section 8.9</NOBR>)</TD>
 </TR>
 </TABLE>
 </BLOCKQUOTE>
 </P>

 <H3>9.2. Changes to Function Arguments</H3>

 <P>
 Besides simple name changes, some operations were given a different interface
 in <NOBR>Release 3</NOBR> than they had in <NOBR>Release 2</NOBR>:
 <UL>

 <LI>
 <P>
 Since <NOBR>Release 3</NOBR>, integer arguments and results of functions have
 standard types from header <CODE>&lt;stdint.h&gt;</CODE>, such as
 <CODE>uint32_t</CODE>, whereas previously their types could be defined
 differently for each port of SoftFloat, usually using traditional C types such
 as <CODE>unsigned</CODE> <CODE>int</CODE>.
 Likewise, functions in <NOBR>Release 3</NOBR> and later pass Booleans as
 standard type <CODE>bool</CODE> from <CODE>&lt;stdbool.h&gt;</CODE>, whereas
 previously these were again passed as a port-specific type (usually
 <CODE>int</CODE>).
 </P>

 <LI>
 <P>
 As explained earlier in <NOBR>section 4.5</NOBR>, <I>Conventions for Passing
 Arguments and Results</I>, SoftFloat functions in <NOBR>Release 3</NOBR> and
 later may pass <NOBR>80-bit</NOBR> and <NOBR>128-bit</NOBR> floating-point
 values through pointers, meaning that functions take pointer arguments and then
 read or write floating-point values at the locations indicated by the pointers.
 In <NOBR>Release 2</NOBR>, floating-point arguments and results were always
 passed by value, regardless of their size.
 </P>

 <LI>
 <P>
 Functions that round to an integer have additional
 <CODE><I>roundingMode</I></CODE> and <CODE><I>exact</I></CODE> arguments that
 they did not have in <NOBR>Release 2</NOBR>.
 Refer to sections 8.2 <NOBR>and 8.7</NOBR> for descriptions of these functions
 since <NOBR>Release 3</NOBR>.
 For <NOBR>Release 2</NOBR>, the rounding mode, when needed, was taken from the
 same global variable that affects the basic arithmetic operations (now called
 <CODE>softfloat_roundingMode</CODE> but previously known as
 <CODE>float_rounding_mode</CODE>).
 Also, for <NOBR>Release 2</NOBR>, if the original floating-point input was not
 an exact integer value, and if the <I>invalid</I> exception was not raised by
 the function, the <I>inexact</I> exception was always raised.
 <NOBR>Release 2</NOBR> had no option to suppress raising <I>inexact</I> in this
 case.
 Applications using SoftFloat <NOBR>Release 3</NOBR> or later can get the same
 effect as <NOBR>Release 2</NOBR> by passing variable
 <CODE>softfloat_roundingMode</CODE> for argument
 <CODE><I>roundingMode</I></CODE> and <CODE>true</CODE> for argument
 <CODE><I>exact</I></CODE>.
 </P>

 </UL>
 </P>

 <H3>9.3. Added Capabilities</H3>

 <P>
 With <NOBR>Release 3</NOBR>, some new features have been added that were not
 present in <NOBR>Release 2</NOBR>:
 <UL>

 <LI>
 <P>
 A port of SoftFloat can now define any of the floating-point types
 <CODE>float32_t</CODE>, <CODE>float64_t</CODE>, <CODE>extFloat80_t</CODE>, and
 <CODE>float128_t</CODE> as aliases for C&rsquo;s standard floating-point types
 <CODE>float</CODE>, <CODE>double</CODE>, and <CODE>long</CODE>
 <CODE>double</CODE>, using either <CODE>#define</CODE> or <CODE>typedef</CODE>.
 This potential convenience was not supported under <NOBR>Release 2</NOBR>.
 </P>

 <P>
 (Note, however, that there may be a performance cost to defining
 SoftFloat&rsquo;s floating-point types this way, depending on the platform and
 the applications using SoftFloat.
 Ports of SoftFloat may choose to forgo the convenience in favor of better
 speed.)
 </P>

 <P>
 <LI>
 Functions have been added for converting between the floating-point types and
 unsigned integers.
 <NOBR>Release 2</NOBR> supported only signed integers, not unsigned.
 </P>

 <P>
 <LI>
 A new, fifth rounding mode, <CODE>softfloat_round_near_maxMag</CODE> (round to
 nearest, with ties to maximum magnitude, away from zero) is now supported for
 all cases involving rounding.
 </P>

 <P>
 <LI>
 Fused multiply-add functions have been added for the non-extended formats,
 <CODE>float32_t</CODE>, <CODE>float64_t</CODE>, and <CODE>float128_t</CODE>.
 </P>

 </UL>
 </P>

 <H3>9.4. Better Compatibility with the C Language</H3>

 <P>
 <NOBR>Release 3</NOBR> of SoftFloat was written to conform better to the ISO C
 Standard&rsquo;s rules for portability.
 For example, older releases of SoftFloat employed type conversions in ways
 that, while commonly practiced, are not fully defined by the C Standard.
 Such problematic type conversions have generally been replaced by the use of
 unions, the behavior around which is more strictly regulated these days.
 </P>

 <H3>9.5. New Organization as a Library</H3>

 <P>
 Starting with <NOBR>Release 3</NOBR>, SoftFloat now builds as a library.
 Previously, SoftFloat compiled into a single, monolithic object file containing
 all the SoftFloat functions, with the consequence that a program linking with
 SoftFloat would get every SoftFloat function in its binary file even if only a
 few functions were actually used.
 With SoftFloat in the form of a library, a program that is linked by a standard
 linker will include only those functions of SoftFloat that it needs and no
 others.
 </P>

 <H3>9.6. Optimization Gains (and Losses)</H3>

 <P>
 Individual SoftFloat functions have been variously improved in
 <NOBR>Release 3</NOBR> compared to earlier releases.
 In particular, better, faster algorithms have been deployed for the operations
 of division, square root, and remainder.
 For functions operating on the larger <NOBR>80-bit</NOBR> and
 <NOBR>128-bit</NOBR> formats, <CODE>extFloat80_t</CODE> and
 <CODE>float128_t</CODE>, code size has also generally been reduced.
 </P>

 <P>
 However, because <NOBR>Release 2</NOBR> compiled all of SoftFloat together as a
 single object file, compilers could make optimizations across function calls
 when one SoftFloat function calls another.
 Now that the functions of SoftFloat are compiled separately and only afterward
 linked together into a program, there is not usually the same opportunity to
 optimize across function calls.
 Some loss of speed has been observed due to this change.
 </P>


 <H2>10. Future Directions</H2>

 <P>
 The following improvements are anticipated for future releases of SoftFloat:
 <UL>
 <LI>
 support for the common <NOBR>16-bit</NOBR> &ldquo;half-precision&rdquo;
 floating-point format;
 <LI>
 more functions from the 2008 version of the IEEE Floating-Point Standard;
 <LI>
 consistent, defined behavior for non-canonical representations of extended
 format <CODE>extFloat80_t</CODE> (discussed in <NOBR>section 4.4</NOBR>,
 <I>Non-canonical Representations in <CODE>extFloat80_t</CODE></I>).

 </UL>
 </P>


 <H2>11. Contact Information</H2>

 <P>
 At the time of this writing, the most up-to-date information about SoftFloat
 and the latest release 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>


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