John R. Hauser
2018 January 20
1. Introduction 2. Limitations 3. Acknowledgments and License 4. Types and Functions 4.1. Boolean and Integer Types 4.2. Floating-Point Types 4.3. Supported Floating-Point Functions 4.4. Non-canonical Representations in extFloat80_t
4.5. Conventions for Passing Arguments and Results 5. Reserved Names 6. Mode Variables 6.1. Rounding Mode 6.2. Underflow Detection 6.3. Rounding Precision for the 80-Bit Extended Format7. Exceptions and Exception Flags 8. Function Details 8.1. Conversions from Integer to Floating-Point 8.2. Conversions from Floating-Point to Integer 8.3. Conversions Among Floating-Point Types 8.4. Basic Arithmetic Functions 8.5. Fused Multiply-Add Functions 8.6. Remainder Functions 8.7. Round-to-Integer Functions 8.8. Comparison Functions 8.9. Signaling NaN Test Functions 8.10. Raise-Exception Function 9. Changes from SoftFloat Release 2 9.1. Name Changes 9.2. Changes to Function Arguments 9.3. Added Capabilities 9.4. Better Compatibility with the C Language 9.5. New Organization as a Library 9.6. Optimization Gains (and Losses) 10. Future Directions 11. Contact Information
Berkeley SoftFloat is a software implementation of binary floating-point that
conforms to the IEEE Standard for Floating-Point Arithmetic.
The current release supports five binary formats:
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.
The current version of SoftFloat is
The previous
Among earlier releases, 3b was notable for adding support for the
SoftFloat-history.html
The functional interface of SoftFloat
SoftFloat assumes the computer has an addressable byte size of 8 or
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
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
The SoftFloat package was written by me,
Par Lab: 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. ASPIRE Lab: 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.
The following applies to the whole of SoftFloat
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 The Regents of the University of California. All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.
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.
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.
THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS “AS IS”, 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.
The types and functions of SoftFloat are declared in header file
softfloat.h
.
Header file softfloat.h
depends on standard headers
<stdbool.h>
and <stdint.h>
to define type
bool
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 latest ISO C Standard.
For older or nonstandard compilers, a port of SoftFloat may have substitutes
for these headers.
Header softfloat.h
depends only on the name bool
from
<stdbool.h>
and on these type names from
<stdint.h>
:
uint16_t uint32_t uint64_t int32_t int64_t uint_fast8_t uint_fast32_t uint_fast64_t int_fast32_t int_fast64_t
The softfloat.h
header defines five floating-point types:
The non-extended types are each exactly the size specified:
float16_t
16-bit half-precision binary formatfloat32_t
32-bit single-precision binary formatfloat64_t
64-bit double-precision binary formatextFloat80_t
80-bit double-extended-precision binary format (old Intel or Motorola format)float128_t
128-bit quadruple-precision binary format
float16_t
, float32_t
, float64_t
, and
float128_t
.
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 float
,
double
, and long
double
.
Header file softfloat.h
also defines a structure,
struct
extFloat80M
, for the representation of
extFloat80_t
and contains
at least these two fields (not necessarily in this order):
Fielduint16_t signExp; uint64_t signif;
signExp
contains the sign and exponent of the floating-point
value, with the sign in the most significant bit (signif
is the complete SoftFloat implements these arithmetic operations for its floating-point types:
extFloat80_t
, the fused multiply-add
operation defined by the IEEE Standard;
The following operations required by the 2008 IEEE Floating-Point Standard are
not supported in SoftFloat
extFloat80_t
Because the extFloat80_t
, stores an explicit leading significand bit, many
finite floating-point numbers are encodable in this type in multiple equivalent
forms.
Of these multiple encodings, there is always a unique one with the least
encoded exponent value, and this encoding is considered the canonical
representation of the floating-point number.
Any other equivalent representations (having a higher encoded exponent value)
are non-canonical.
For a value in the subnormal range (including zero), the canonical
representation always has an encoded exponent of zero and a leading significand
bit
For an infinity or NaN, the leading significand bit is similarly expected to
extFloat80_t
must have a leading significand bit
SoftFloat’s functions are not guaranteed to operate as expected when
inputs of type extFloat80_t
are non-canonical.
Assuming all of a function’s extFloat80_t
inputs (if any)
are canonical, function outputs of type extFloat80_t
will always
be canonical.
Values that are at most
float64_t f64_add( float64_t, float64_t );
The story is more complex when function inputs and outputs are
void f128M_add( const float128_t *, const float128_t *, float128_t * );
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 M
in the name f128M_add
is mnemonic for the fact
that the
All ports of SoftFloat implement these pass-by-pointer functions for
types extFloat80_t
and float128_t
.
At the same time, SoftFloat ports may also implement alternate versions of
these same functions that pass extFloat80_t
and
float128_t
by value, like the smaller formats.
Thus, besides the function with name f128M_add
shown above, a
SoftFloat port may also supply an equivalent function with this signature:
float128_t f128_add( float128_t, float128_t );
As a general rule, on computers where the machine word size is
f128M_add
) are provided for types extFloat80_t
and float128_t
, because passing such large types directly can have
significant extra cost.
On computers where the word size is f128M_add
and f128_add
) 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.
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
‘softfloat_
’.
When a program includes header softfloat.h
or links with the
SoftFloat library, all names with prefix ‘softfloat_
’
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.
The following global variables control rounding mode, underflow detection, and
the
These mode variables are covered in the next several subsections. For some SoftFloat ports, these variables may be per-thread (declaredsoftfloat_roundingMode
softfloat_detectTininess
extF80_roundingPrecision
thread_local
), meaning that different execution threads have their
own separate copies of the variables.
All five rounding modes defined by the 2008 IEEE Floating-Point Standard are implemented for all operations that require rounding. Some ports of SoftFloat may also implement the round-to-odd mode.
The rounding mode is selected by the global variable
uint_fast8_t softfloat_roundingMode;
This variable may be set to one of the values
Variable
softfloat_round_near_even
round to nearest, with ties to even softfloat_round_near_maxMag
round to nearest, with ties to maximum magnitude (away from zero) softfloat_round_minMag
round to minimum magnitude (toward zero) softfloat_round_min
round to minimum (down) softfloat_round_max
round to maximum (up) softfloat_round_odd
round to odd (jamming), if supported by the SoftFloat port
softfloat_roundingMode
is initialized to
softfloat_round_near_even
.
When softfloat_round_odd
is the rounding mode for a function that
rounds to an integer value (either conversion to an integer format or a
‘roundToInt
’ function), if the input is not already an
integer, the rounded result is the closest odd integer.
For other operations, this rounding mode acts as though the floating-point
result is first rounded to minimum magnitude, the same as
softfloat_round_minMag
, and then, if the result is inexact, the
least-significant bit of the result is set
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
uint_fast8_t softfloat_detectTininess;
which can be set to either
Detecting tininess after rounding is usually 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.softfloat_tininess_beforeRounding
softfloat_tininess_afterRounding
For extFloat80_t
only, the rounding precision of the basic
arithmetic operations is controlled by the global variable
uint_fast8_t extF80_roundingPrecision;
The operations affected are:
WhenextF80_add
extF80_sub
extF80_mul
extF80_div
extF80_sqrt
extF80_roundingPrecision
is set to its default value of 80,
these operations are rounded to the full precision of the extF80_roundingPrecision
to 32 or to 64 causes the
operations listed to be rounded to float32_t
) or to float64_t
), respectively.
When rounding to reduced precision, additional bits in the result significand
beyond the rounding point are set to zero.
The consequences of setting extF80_roundingPrecision
to a value
other than 32, 64, or 80 is not specified.
Operations other than the ones listed above are not affected by
extF80_roundingPrecision
.
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
uint_fast8_t softfloat_exceptionFlags;
The positions of the exception flag bits within this variable are determined by
the bit masks
Variablesoftfloat_flag_inexact
softfloat_flag_underflow
softfloat_flag_overflow
softfloat_flag_infinite
softfloat_flag_invalid
softfloat_exceptionFlags
is initialized to all zeros,
meaning no exceptions.
For some SoftFloat ports, softfloat_exceptionFlags
may be
per-thread (declared thread_local
), meaning that different
execution threads have their own separate instances of it.
An individual exception flag can be cleared with the statement
softfloat_exceptionFlags &= ~softfloat_flag_<exception>;
where <exception>
is the appropriate name.
To raise a floating-point exception, function softfloat_raiseFlags
should normally be used.
When SoftFloat detects an exception other than inexact, it calls
softfloat_raiseFlags
.
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 softfloat_raiseFlags
.
A program may also supply its own softfloat_raiseFlags
function to
override the one from the SoftFloat library.
Because inexact results occur frequently under most circumstances (and thus are
hardly exceptional), SoftFloat does not ordinarily call
softfloat_raiseFlags
for inexact exceptions.
It does always raise the inexact exception flag as required.
In this section, <float>
appears in function names as
a substitute for one of these abbreviations:
The circumstances under which values of floating-point types
f16
indicates float16_t
, passed by valuef32
indicates float32_t
, passed by valuef64
indicates float64_t
, passed by valueextF80M
indicates extFloat80_t
, passed indirectly via pointersextF80
indicates extFloat80_t
, passed by valuef128M
indicates float128_t
, passed indirectly via pointersf128
indicates float128_t
, passed by value
extFloat80_t
and float128_t
may be passed either by
value or indirectly via pointers was discussed earlier in
All conversions from a
Conversions fromui32_to_<float>
ui64_to_<float>
i32_to_<float>
i64_to_<float>
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:
float64_t i32_to_f64( int32_t a );void i32_to_f128M( int32_t a, float128_t *destPtr );
Conversions from a floating-point format to a
The functions have signatures as follows, depending on whether the floating-point input is passed by value or via pointers:<float>_to_ui32
<float>_to_ui64
<float>_to_i32
<float>_to_i64
int_fast32_t f64_to_i32( float64_t a, uint_fast8_t roundingMode, bool exact );int_fast32_t f128M_to_i32( const float128_t *aPtr, uint_fast8_t roundingMode, bool exact );
The roundingMode
argument specifies the rounding mode for
the conversion.
The variable that usually indicates rounding mode,
softfloat_roundingMode
, is ignored.
Argument exact
determines whether the inexact
exception flag is raised if the conversion is not exact.
If exact
is true
, the inexact flag may
be raised;
otherwise, it will not be, even if the conversion is inexact.
A conversion from floating-point to integer format raises the invalid exception if the source value cannot be rounded to a representable integer of the desired size (32 or 64 bits). In such circumstances, the integer result returned is determined by the particular port of SoftFloat, although typically this value will be either the maximum or minimum value of the integer format. The functions that convert to integer types never raise the floating-point overflow exception.
Because languages such
These functions round only toward zero (to minimum magnitude). The signatures for these functions are the same as above without the redundant<float>_to_ui32_r_minMag
<float>_to_ui64_r_minMag
<float>_to_i32_r_minMag
<float>_to_i64_r_minMag
roundingMode
argument:
int_fast32_t f64_to_i32_r_minMag( float64_t a, bool exact );int_fast32_t f128M_to_i32_r_minMag( const float128_t *aPtr, bool exact );
Conversions between floating-point formats are done by functions with these names:
<float>_to_<float>
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:
float32_t f64_to_f32( float64_t a );float32_t f128M_to_f32( const float128_t *aPtr );void f32_to_f128M( float32_t a, float128_t *destPtr );void extF80M_to_f128M( const extFloat80_t *aPtr, float128_t *destPtr );
Conversions from a smaller to a larger floating-point format are always exact and so require no rounding.
The following basic arithmetic functions are provided:
Each floating-point operation takes two operands, except for<float>_add
<float>_sub
<float>_mul
<float>_div
<float>_sqrt
sqrt
(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:
When floating-point values are passed indirectly through pointers, argumentsfloat64_t f64_add( float64_t a, float64_t b );void f128M_add( const float128_t *aPtr, const float128_t *bPtr, float128_t *destPtr );float64_t f64_sqrt( float64_t a );void f128M_sqrt( const float128_t *aPtr, float128_t *destPtr );
aPtr
and bPtr
point to the input
operands, and the last argument, destPtr
, points to the
location where the result is stored.
Rounding of the extFloat80_t
) functions is affected by variable
extF80_roundingPrecision
, as explained earlier in
The 2008 version of the IEEE Floating-Point Standard defines a fused multiply-add operation that does a combined multiplication and addition with only a single rounding. SoftFloat implements fused multiply-add with functions
<float>_mulAdd
Unlike other operations, fused multiple-add is not supported for the
extFloat80_t
.
Depending on whether floating-point values are passed by value or via pointers, the fused multiply-add functions have signatures of these forms:
The functions computefloat64_t f64_mulAdd( float64_t a, float64_t b, float64_t c );void f128M_mulAdd( const float128_t *aPtr, const float128_t *bPtr, const float128_t *cPtr, float128_t *destPtr );
a
× b
)
+ c
aPtr
, bPtr
, and
cPtr
point to operands a
,
b
, and c
respectively, and
destPtr
points to the location where the result is stored.
If one of the multiplication operands a
and
b
is infinite and the other is zero, these functions raise
the invalid exception even if operand c
is a quiet NaN.
For each format, SoftFloat implements the remainder operation defined by the IEEE Floating-Point Standard. The remainder functions have names
<float>_rem
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:
When floating-point values are passed indirectly through pointers, argumentsfloat64_t f64_rem( float64_t a, float64_t b );void f128M_rem( const float128_t *aPtr, const float128_t *bPtr, float128_t *destPtr );
aPtr
and bPtr
point to operands
a
and b
respectively, and
destPtr
points to the location where the result is stored.
The IEEE Standard remainder operation computes the value
a
− n × b
a
÷ b
a
÷ b
a
÷ b
Depending on the relative magnitudes of the operands, the remainder functions can take considerably longer to execute than the other SoftFloat functions. This is an inherent characteristic of the remainder operation itself and is not a flaw in the SoftFloat implementation.
For each format, SoftFloat implements the round-to-integer operation specified by the IEEE Floating-Point Standard. These functions are named
<float>_roundToInt
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.)
The signatures of the round-to-integer functions are similar to those for conversions to an integer type:
When floating-point values are passed indirectly through pointers,float64_t f64_roundToInt( float64_t a, uint_fast8_t roundingMode, bool exact );void f128M_roundToInt( const float128_t *aPtr, uint_fast8_t roundingMode, bool exact, float128_t *destPtr );
aPtr
points to the input operand and
destPtr
points to the location where the result is stored.
The roundingMode
argument specifies the rounding mode to
apply.
The variable that usually indicates rounding mode,
softfloat_roundingMode
, is ignored.
Argument exact
determines whether the inexact
exception flag is raised if the conversion is not exact.
If exact
is true
, the inexact flag may
be raised;
otherwise, it will not be, even if the conversion is inexact.
For each format, the following floating-point comparison functions are provided:
Each comparison takes two operands of the same type and returns a Boolean. The abbreviation<float>_eq
<float>_le
<float>_lt
eq
stands for “equal” (=);
le
stands for “less than or equal” (≤);
and lt
stands for “less than” (<).
Depending on whether the floating-point operands are passed by value or via
pointers, the comparison functions have signatures of these forms:
bool f64_eq( float64_t a, float64_t b );bool f128M_eq( const float128_t *aPtr, const float128_t *bPtr );
The usual greater-than (>), greater-than-or-equal (≥), and not-equal (≠) 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.
The IEEE Floating-Point Standard specifies that the less-than-or-equal and less-than comparisons by default raise the invalid exception if either operand is any kind of NaN. Equality comparisons, on the other hand, are defined by default to raise the invalid exception only for signaling NaNs, not quiet NaNs. For completeness, SoftFloat provides these complementary functions:
The<float>_eq_signaling
<float>_le_quiet
<float>_lt_quiet
signaling
equality comparisons are identical to the default
equality comparisons except that the invalid exception is raised for any
NaN input, not just for signaling NaNs.
Similarly, the quiet
comparison functions are identical to their
default counterparts except that the invalid exception is not raised for
quiet NaNs.
Functions for testing whether a floating-point value is a signaling NaN are provided with these names:
<float>_isSignalingNaN
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
bool f64_isSignalingNaN( float64_t a );bool f128M_isSignalingNaN( const float128_t *aPtr );
SoftFloat provides a single function for raising floating-point exceptions:
Thevoid softfloat_raiseFlags( uint_fast8_t exceptions );
exceptions
argument is a mask indicating the set of
exceptions to raise.
(See earlier section 7, Exceptions and Exception Flags.)
In addition to setting the specified exception flags in variable
softfloat_exceptionFlags
, the softfloat_raiseFlags
function may cause a trap or abort appropriate for the current system.
Apart from a change in the legal use license,
The most obvious and pervasive difference compared to
old name, Release 2: new name, Release 3: float32
float32_t
float64
float64_t
floatx80
extFloat80_t
float128
float128_t
float_rounding_mode
softfloat_roundingMode
float_round_nearest_even
softfloat_round_near_even
float_round_to_zero
softfloat_round_minMag
float_round_down
softfloat_round_min
float_round_up
softfloat_round_max
float_detect_tininess
softfloat_detectTininess
float_tininess_before_rounding
softfloat_tininess_beforeRounding
float_tininess_after_rounding
softfloat_tininess_afterRounding
floatx80_rounding_precision
extF80_roundingPrecision
float_exception_flags
softfloat_exceptionFlags
float_flag_inexact
softfloat_flag_inexact
float_flag_underflow
softfloat_flag_underflow
float_flag_overflow
softfloat_flag_overflow
float_flag_divbyzero
softfloat_flag_infinite
float_flag_invalid
softfloat_flag_invalid
float_raise
softfloat_raiseFlags
Furthermore,
Thus, for example, the function to add two
used in names in Release 2:
used in names in Release 3: int32
i32
int64
i64
float32
f32
float64
f64
floatx80
extF80
float128
f128
float32_add
in f32_add
.
Lastly, there have been a few other changes to function names:
used in names in Release 2:
used in names in Release 3:
relevant functions: _round_to_zero
_r_minMag
conversions from floating-point to integer ( section 8.2 )round_to_int
roundToInt
round-to-integer functions ( section 8.7 )is_signaling_nan
isSignalingNaN
signaling NaN test functions ( section 8.9 )
Besides simple name changes, some operations were given a different interface
in
Since <stdint.h>
, such as
uint32_t
, whereas previously their types could be defined
differently for each port of SoftFloat, usually using traditional C types such
as unsigned
int
.
Likewise, functions in bool
from <stdbool.h>
, whereas
previously these were again passed as a port-specific type (usually
int
).
As explained earlier in
Functions that round to an integer have additional
roundingMode
and exact
arguments that
they did not have in softfloat_roundingMode
but previously known as
float_rounding_mode
).
Also, for softfloat_roundingMode
for argument
roundingMode
and true
for argument
exact
.
With
A port of SoftFloat can now define any of the floating-point types
float32_t
, float64_t
, extFloat80_t
, and
float128_t
as aliases for C’s standard floating-point types
float
, double
, and long
double
, using either #define
or typedef
.
This potential convenience was not supported under
(Note, however, that there may be a performance cost to defining SoftFloat’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.)
float16_t
, is supported.
extFloat80_t
.
softfloat_round_near_maxMag
(round to nearest, with ties to
maximum magnitude, away from zero), and, as of softfloat_round_odd
(round to odd, also known as
jamming).
Starting with
Individual SoftFloat functions have been variously improved in
extFloat80_t
and
float128_t
, code size has also generally been reduced.
However, because
The following improvements are anticipated for future releases of SoftFloat:
extFloat80_t
(discussed in extFloat80_t
).
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
http://www.jhauser.us/arithmetic/SoftFloat.html