# Berkeley SoftFloat Release 3e: Library Interface

John R. Hauser
2018 January 20

## Contents

> 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 Format
> 7. 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

## 1. Introduction

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: 16-bit half-precision, 32-bit single-precision, 64-bit double-precision, 80-bit double-extended-precision, and 128-bit quadruple-precision. The following functions are supported for each format:

- addition, subtraction, multiplication, division, and square root;
- fused multiply-add as defined by the IEEE Standard, except for 80-bit double-extended-precision;
- remainder as defined by the IEEE Standard;
- round to integral value;
- comparisons;
- conversions to/from other supported formats; and
- conversions to/from 32-bit and 64-bit integers, signed and unsigned.

All operations required by the original 1985 version of the IEEE Floating-Point Standard are implemented, except for conversions to and from decimal.

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 Release 3e. This release modifies the behavior of the rarely used odd rounding mode (round to odd, also known as jamming), and also adds some new specialization and optimization examples for those compiling SoftFloat.

The previous Release 3d fixed bugs that were found in the square root functions for the 64-bit, 80-bit, and 128-bit floating-point formats. (Thanks to Alexei Sibidanov at the University of Victoria for reporting an incorrect result.) The bugs affected all prior Release-3 versions of SoftFloat through 3c. The flaw in the 64-bit floating-point square root function was of very minor impact, causing a 1-ulp error (1 unit in the last place) a few times out of a billion. The bugs in the 80-bit and 128-bit square root functions were more serious. Although incorrect results again occurred only a few times out of a billion, when they did occur a large portion of the less-significant bits could be wrong.

Among earlier releases, 3b was notable for adding support for the 16-bit half-precision format. For more about the evolution of SoftFloat releases, see `SoftFloat-history.html`.

The functional interface of SoftFloat Release 3 and later differs in many details from the releases that came before. For specifics of these differences, see section 9 below, Changes from SoftFloat Release 2.

## 2. Limitations

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

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 64-bit integers. Earlier releases of SoftFloat included implementations of 32-bit single-precision and 64-bit double-precision floating-point that did not require 64-bit integers, but this option is not supported starting with Release 3. Since 1999, ISO standards for C have mandated compiler support for 64-bit integers. A compiler conforming to the 1999 C Standard or later is recommended but not strictly required.

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 Release 3e.

## 3. Acknowledgments and License

The SoftFloat package was written by me, John R. Hauser. Release 3 of SoftFloat was a completely new implementation supplanting earlier releases. The project to create Release 3 (now through 3e) 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:

> 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 Release 3e as well as to each source file individually.

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:

1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.

2. 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.

3. 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.

## 4. Types and Functions

The types and functions of SoftFloat are declared in header file `softfloat.h`.

### 4.1. Boolean and Integer Types

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

### 4.2. Floating-Point Types

The `softfloat.h` header defines five floating-point types:

> `float16_t` | 16-bit half-precision binary format
> `float32_t` | 32-bit single-precision binary format
> `float64_t` | 64-bit double-precision binary format
> `extFloat80_t` | 80-bit double-extended-precision binary format (old Intel or Motorola format)
> `float128_t` | 128-bit quadruple-precision binary format

The non-extended types are each exactly the size specified: 16 bits for `float16_t`, 32 bits for `float32_t`, 64 bits for `float64_t`, and 128 bits for `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 80-bit double-extended-precision floating-point values in memory. This structure is the same size as type `extFloat80_t` and contains at least these two fields (not necessarily in this order):

>     uint16_t signExp;
>     uint64_t signif;

Field `signExp` contains the sign and exponent of the floating-point value, with the sign in the most significant bit (bit 15) and the encoded exponent in the other 15 bits. Field `signif` is the complete 64-bit significand of the floating-point value. (In the usual encoding for 80-bit extended floating-point, the leading 1 bit of normalized numbers is not implicit but is stored in the most significant bit of the significand.)

### 4.3. Supported Floating-Point Functions

SoftFloat implements these arithmetic operations for its floating-point types:

- conversions between any two floating-point formats;
- for each floating-point format, conversions to and from signed and unsigned 32-bit and 64-bit integers;
- for each format, the usual addition, subtraction, multiplication, division, and square root operations;
- for each format except `extFloat80_t`, the fused multiply-add operation defined by the IEEE Standard;
- for each format, the floating-point remainder operation defined by the IEEE Standard;
- for each format, a “round to integer” operation that rounds to the nearest integer value in the same format; and
- comparisons between two values in the same floating-point format.

The following operations required by the 2008 IEEE Floating-Point Standard are not supported in SoftFloat Release 3e:

- nextUp, nextDown, minNum, maxNum, minNumMag, maxNumMag, scaleB, and logB;
- conversions between floating-point formats and decimal or hexadecimal character sequences;
- all “quiet-computation” operations (copy, negate, abs, and copySign, which all involve only simple copying and/or manipulation of the floating-point sign bit); and
- all “non-computational” operations other than isSignaling (which is supported).

### 4.4. Non-canonical Representations in `extFloat80_t`

Because the 80-bit double-extended-precision format, `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 of 0. For finite values outside the subnormal range, the canonical representation always has an encoded exponent that is nonzero and a leading significand bit of 1.

For an infinity or NaN, the leading significand bit is similarly expected to be 1. An infinity or NaN with a leading significand bit of 0 is again considered non-canonical. Hence, altogether, to be canonical, a value of type `extFloat80_t` must have a leading significand bit of 1, unless the value is subnormal or zero, in which case the leading significand bit and the encoded exponent must both be zero.

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.

### 4.5. Conventions for Passing Arguments and Results

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

> `float64_t f64_add( float64_t, float64_t );`

The story is more complex when function inputs and outputs are 80-bit and 128-bit 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 128-bit floating-point values, SoftFloat supplies this function:

> `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 128-bit inputs and outputs are “in memory”, pointed to by pointer arguments.

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 32 bits or smaller, only the pass-by-pointer versions of functions (e.g., `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 64 bits or larger, both function versions (`f128M_add` and `f128_add`) are provided, because the cost of passing by value is then more reasonable. Applications that must be portable across 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.

## 5. Reserved Names

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.

## 6. Mode Variables

The following global variables control rounding mode, underflow detection, and the 80-bit extended format’s rounding precision:

> `softfloat_roundingMode`
> `softfloat_detectTininess`
> `extF80_roundingPrecision`

These mode variables are covered in the next several subsections. For some SoftFloat ports, these variables may be per-thread (declared `thread_local`), meaning that different execution threads have their own separate copies of the variables.

### 6.1. Rounding Mode

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

> `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

Variable `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 to 1. Rounding to odd is also known as jamming.

### 6.2. Underflow Detection

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

> `softfloat_tininess_beforeRounding`
> `softfloat_tininess_afterRounding`

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.

### 6.3. Rounding Precision for the 80-Bit Extended Format

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:

> `extF80_add`
> `extF80_sub`
> `extF80_mul`
> `extF80_div`
> `extF80_sqrt`

When `extF80_roundingPrecision` is set to its default value of 80, these operations are rounded to the full precision of the 80-bit double-extended-precision format, like occurs for other formats. Setting `extF80_roundingPrecision` to 32 or to 64 causes the operations listed to be rounded to 32-bit precision (equivalent to `float32_t`) or to 64-bit precision (equivalent 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`.

## 7. Exceptions and Exception Flags

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

> `softfloat_flag_inexact`
> `softfloat_flag_underflow`
> `softfloat_flag_overflow`
> `softfloat_flag_infinite`
> `softfloat_flag_invalid`

Variable `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.

## 8. Function Details

In this section, `<float>` appears in function names as a substitute for one of these abbreviations:

> `f16` | indicates `float16_t`, passed by value
> `f32` | indicates `float32_t`, passed by value
> `f64` | indicates `float64_t`, passed by value
> `extF80M` | indicates `extFloat80_t`, passed indirectly via pointers
> `extF80` | indicates `extFloat80_t`, passed by value
> `f128M` | indicates `float128_t`, passed indirectly via pointers
> `f128` | indicates `float128_t`, passed by value

The circumstances under which values of floating-point types `extFloat80_t` and `float128_t` may be passed either by value or indirectly via pointers was discussed earlier in section 4.5, Conventions for Passing Arguments and Results.

### 8.1. Conversions from Integer to Floating-Point

All conversions from a 32-bit or 64-bit integer, signed or unsigned, to a floating-point format are supported. Functions performing these conversions have these names:

> `ui32_to_<float>`
> `ui64_to_<float>`
> `i32_to_<float>`
> `i64_to_<float>`

Conversions from 32-bit integers to 64-bit double-precision and larger formats are always exact, and likewise conversions from 64-bit integers to 80-bit double-extended-precision and 128-bit quadruple-precision are also always exact.

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 );

### 8.2. Conversions from Floating-Point to Integer

Conversions from a floating-point format to a 32-bit or 64-bit integer, signed or unsigned, are supported with these functions:

> `<float>_to_ui32`
> `<float>_to_ui64`
> `<float>_to_i32`
> `<float>_to_i64`

The functions have signatures as follows, depending on whether the floating-point input is passed by value or via pointers:

>     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 as C require that conversions to integers be rounded toward zero, the following functions are provided for improved speed and convenience:

> `<float>_to_ui32_r_minMag`
> `<float>_to_ui64_r_minMag`
> `<float>_to_i32_r_minMag`
> `<float>_to_i64_r_minMag`

These functions round only toward zero (to minimum magnitude). The signatures for these functions are the same as above without the redundant `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 );

### 8.3. Conversions Among Floating-Point Types

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.

### 8.4. Basic Arithmetic Functions

The following basic arithmetic functions are provided:

> `<float>_add`
> `<float>_sub`
> `<float>_mul`
> `<float>_div`
> `<float>_sqrt`

Each floating-point operation takes two operands, except for `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:

>     float64_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 );

When floating-point values are passed indirectly through pointers, arguments `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 80-bit double-extended-precision (`extFloat80_t`) functions is affected by variable `extF80_roundingPrecision`, as explained earlier in section 6.3, Rounding Precision for the 80-Bit Extended Format.

### 8.5. Fused Multiply-Add Functions

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 80-bit double-extended-precision format, `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:

>     float64_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
>      );

The functions compute (`a` × `b`) + `c` with a single rounding. When floating-point values are passed indirectly through pointers, arguments `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.

### 8.6. Remainder Functions

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:

>     float64_t f64_rem( float64_t a, float64_t b );

>     void
>      f128M_rem(
>          const float128_t *aPtr, const float128_t *bPtr, float128_t *destPtr );

When floating-point values are passed indirectly through pointers, arguments `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`, where n is the integer closest to `a` ÷ `b`. If `a` ÷ `b` is exactly halfway between two integers, n is the even integer closest to `a` ÷ `b`. The IEEE Standard’s remainder operation is always exact and so requires no rounding.

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.

### 8.7. Round-to-Integer Functions

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:

>     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
>      );

When floating-point values are passed indirectly through pointers, `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.

### 8.8. Comparison Functions

For each format, the following floating-point comparison functions are provided:

> `<float>_eq`
> `<float>_le`
> `<float>_lt`

Each comparison takes two operands of the same type and returns a Boolean. The abbreviation `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:

> `<float>_eq_signaling`
> `<float>_le_quiet`
> `<float>_lt_quiet`

The `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.

### 8.9. Signaling NaN Test Functions

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 );

### 8.10. Raise-Exception Function

SoftFloat provides a single function for raising floating-point exceptions:

>     void softfloat_raiseFlags( uint_fast8_t exceptions );

The `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.

## 9. Changes from SoftFloat Release 2

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

### 9.1. Name Changes

The most obvious and pervasive difference compared to Release 2 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:

> 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, Release 3 adopted the following new abbreviations for function names:

> used in names in Release 2:`` | used in names in Release 3:
> `int32` | `i32`
> `int64` | `i64`
> `float32` | `f32`
> `float64` | `f64`
> `floatx80` | `extF80`
> `float128` | `f128`

Thus, for example, the function to add two 32-bit floating-point numbers, previously called `float32_add` in Release 2, is now `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)

### 9.2. Changes to Function Arguments

Besides simple name changes, some operations were given a different interface in Release 3 than they had in Release 2:

- Since Release 3, integer arguments and results of functions have standard types from header `<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 Release 3 and later pass Booleans as standard type `bool` from `<stdbool.h>`, whereas previously these were again passed as a port-specific type (usually `int`).

- As explained earlier in section 4.5, Conventions for Passing Arguments and Results, SoftFloat functions in Release 3 and later may pass 80-bit and 128-bit 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 Release 2, floating-point arguments and results were always passed by value, regardless of their size.

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

### 9.3. Added Capabilities

With Release 3, some new features have been added that were not present in Release 2:

- 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 Release 2.

  (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.)

- As of Release 3b, 16-bit half-precision, `float16_t`, is supported.

- Functions have been added for converting between the floating-point types and unsigned integers. Release 2 supported only signed integers, not unsigned.

- Fused multiply-add functions have been added for all floating-point formats except 80-bit double-extended-precision, `extFloat80_t`.

- New rounding modes are supported: `softfloat_round_near_maxMag` (round to nearest, with ties to maximum magnitude, away from zero), and, as of Release 3c, optional `softfloat_round_odd` (round to odd, also known as jamming).

### 9.4. Better Compatibility with the C Language

Release 3 of SoftFloat was written to conform better to the ISO C Standard’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.

### 9.5. New Organization as a Library

Starting with Release 3, 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.

### 9.6. Optimization Gains (and Losses)

Individual SoftFloat functions have been variously improved in Release 3 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 80-bit and 128-bit formats, `extFloat80_t` and `float128_t`, code size has also generally been reduced.

However, because Release 2 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.

## 10. Future Directions

The following improvements are anticipated for future releases of SoftFloat:

- more functions from the 2008 version of the IEEE Floating-Point Standard;
- consistent, defined behavior for non-canonical representations of extended format `extFloat80_t` (discussed in section 4.4, Non-canonical Representations in `extFloat80_t`).

## 11. Contact Information

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`.