:stem: latexmath

[[single-float]]
== "F" Extension for Single-Precision Floating-Point, Version 2.2

This chapter describes the standard instruction-set extension for
single-precision floating-point, which is named "F" and adds
single-precision floating-point computational instructions compliant
with the IEEE 754-2008 arithmetic standard cite:[ieee754-2008]. The F extension depends on the "Zicsr" extension for control and status register access.

=== F Register State

The F extension adds 32 floating-point registers, `f0-f31`, each 32
bits wide, and a floating-point control and status register `fcsr`,
which contains the operating mode and exception status of the
floating-point unit. This additional state is shown in
<<fprs>>. We use the term FLEN to describe the width of
the floating-point registers in the RISC-V ISA, and FLEN=32 for the F
single-precision floating-point extension. Most floating-point
instructions operate on values in the floating-point register file.
Floating-point load and store instructions transfer floating-point
values between registers and memory. Instructions to transfer values to and from the integer register file are also provided.

[TIP]
====
We considered a unified register file for both integer and
floating-point values as this simplifies software register allocation
and calling conventions, and reduces total user state. However, a split
organization increases the total number of registers accessible with a
given instruction width, simplifies provision of enough regfile ports
for wide superscalar issue, supports decoupled floating-point-unit
architectures, and simplifies use of internal floating-point encoding
techniques. Compiler support and calling conventions for split register
file architectures are well understood, and using dirty bits on
floating-point register file state can reduce context-switch overhead.
====

[[fprs]]
.RISC-V standard F extension single-precision floating-point state
[cols="<,^,>",options="header",width="50%",align="center",grid="rows"]
|===
| [.small]#FLEN-1#| >| [.small]#0#
3+^| [.small]#f0#
3+^| [.small]#f1#
3+^| [.small]#f2#
3+^| [.small]#f3#
3+^| [.small]#f4#
3+^| [.small]#f5#
3+^| [.small]#f6#
3+^| [.small]#f7#
3+^| [.small]#f8#
3+^| [.small]#f9#
3+^| [.small]#f10#
3+^| [.small]#f11#
3+^| [.small]#f12#
3+^| [.small]#f13#
3+^| [.small]#f14#
3+^| [.small]#f15#
3+^| [.small]#f16#
3+^| [.small]#f17#
3+^| [.small]#f18#
3+^| [.small]#f19#
3+^| [.small]#f20#
3+^| [.small]#f21#
3+^| [.small]#f22#
3+^| [.small]#f23#
3+^| [.small]#f24#
3+^| [.small]#f25#
3+^| [.small]#f26#
3+^| [.small]#f27#
3+^| [.small]#f28#
3+^| [.small]#f29#
3+^| [.small]#f30#
3+^| [.small]#f31#
3+^| [.small]#FLEN#
| [.small]#31#| >| [.small]#0#
3+^|  [.small]#fcsr#
3+^| [.small]#32#
|===

=== Floating-Point Control and Status Register

The floating-point control and status register, `fcsr`, is a RISC-V
control and status register (CSR). It is a 32-bit read/write register
that selects the dynamic rounding mode for floating-point arithmetic
operations and holds the accrued exception flags, as shown in <<fcsr>>.

[[fcsr, Floating-Point Control and Status Register]]
.Floating-point control and status register
include::images/wavedrom/float-csr.adoc[]

The `fcsr` register can be read and written with the FRCSR and FSCSR
instructions, which are assembler pseudoinstructions built on the
underlying CSR access instructions. FRCSR reads `fcsr` by copying it
into integer register _rd_. FSCSR swaps the value in `fcsr` by copying
the original value into integer register _rd_, and then writing a new
value obtained from integer register _rs1_ into `fcsr`.

The fields within the `fcsr` can also be accessed individually through
different CSR addresses, and separate assembler pseudoinstructions are defined
for these accesses. The FRRM instruction reads the Rounding Mode field `frm`
(`fcsr` bits 7--5) and copies it into the least-significant three bits of
integer register _rd_, with zero in all other bits. FSRM swaps the value in
`frm` by copying the original value into integer register _rd_, and then
writing a new value obtained from the three least-significant bits of integer
register _rs1_ into `frm`. FRFLAGS and FSFLAGS are defined analogously for the
Accrued Exception Flags field `fflags` (`fcsr` bits 4--0).

Bits 31--8 of the `fcsr` are reserved for other standard extensions. If
these extensions are not present, implementations shall ignore writes to
these bits and supply a zero value when read. Standard software should
preserve the contents of these bits.

Floating-point operations use either a static rounding mode encoded in
the instruction, or a dynamic rounding mode held in `frm`. Rounding
modes are encoded as shown in <<rm>>. A value of 111 in the
instruction's _rm_ field selects the dynamic rounding mode held in
`frm`. The behavior of floating-point instructions that depend on
rounding mode when executed with a reserved rounding mode is _reserved_, including both static reserved rounding modes (101-110) and dynamic reserved rounding modes (101-111). Some instructions, including widening conversions, have the _rm_ field but are nevertheless mathematically unaffected by the rounding mode; software should set their _rm_ field to 
RNE (000) but implementations must treat the _rm_ field as usual (in
particular, with regard to decoding legal vs. reserved encodings).

[[rm]]
.Rounding mode encoding.
[%autowidth,float="center",align="center",cols="^,^,<",options="header"]
|===
|Rounding Mode |Mnemonic |Meaning
|000 |RNE |Round to Nearest, ties to Even
|001 |RTZ |Round towards Zero
|010 |RDN |Round Down (towards latexmath:[$-\infty$])
|011 |RUP |Round Up (towards latexmath:[$+\infty$])
|100 |RMM |Round to Nearest, ties to Max Magnitude
|101 | |_Reserved for future use._
|110 | |_Reserved for future use._
|111 |DYN |In instruction's _rm_ field, selects dynamic rounding mode; In Rounding Mode register, _reserved_.
|===

[NOTE]
====
The C99 language standard effectively mandates the provision of a
dynamic rounding mode register. In typical implementations, writes to
the dynamic rounding mode CSR state will serialize the pipeline. Static
rounding modes are used to implement specialized arithmetic operations
that often have to switch frequently between different rounding modes.

The ratified version of the F spec mandated that an illegal-instruction
exception was raised when an instruction was executed with a reserved
dynamic rounding mode. This has been weakened to reserved, which matches
the behavior of static rounding-mode instructions. Raising an
illegal-instruction exception is still valid behavior when encountering a
reserved encoding, so implementations compatible with the ratified spec
are compatible with the weakened spec.
====

The accrued exception flags indicate the exception conditions that have
arisen on any floating-point arithmetic instruction since the field was
last reset by software, as shown in <<bitdef>>. The base
RISC-V ISA does not support generating a trap on the setting of a
floating-point exception flag.
(((floating-point, exception flag)))

[[bitdef]]
.Accrued exception flag encoding.
[%autowidth,float="center",align="center",cols="^,<",options="header",]
|===
|Flag Mnemonic |Flag Meaning
|NV |Invalid Operation
|DZ |Divide by Zero
|OF |Overflow
|UF |Underflow
|NX |Inexact
|===

[NOTE]
====
As allowed by the standard, we do not support traps on floating-point
exceptions in the F extension, but instead require explicit checks of
the flags in software. We considered adding branches controlled directly
by the contents of the floating-point accrued exception flags, but
ultimately chose to omit these instructions to keep the ISA simple.
====

=== NaN Generation and Propagation

Except when otherwise stated, if the result of a floating-point
operation is NaN, it is the canonical NaN. The canonical NaN has a
positive sign and all significand bits clear except the MSB, a.k.a. the
quiet bit. For single-precision floating-point, this corresponds to the pattern `0x7fc00000`.
(((NaN, generation)))
(((NaN, propagation)))

[TIP]
====
We considered propagating NaN payloads, as is recommended by the
standard, but this decision would have increased hardware cost.
Moreover, since this feature is optional in the standard, it cannot be
used in portable code.

Implementors are free to provide a NaN payload propagation scheme as a
nonstandard extension enabled by a nonstandard operating mode. However, the canonical NaN scheme described above must always be supported and should be the default mode.
====
'''
[NOTE]
====
We require implementations to return the standard-mandated default
values in the case of exceptional conditions, without any further
intervention on the part of user-level software (unlike the Alpha ISA
floating-point trap barriers). We believe full hardware handling of
exceptional cases will become more common, and so wish to avoid
complicating the user-level ISA to optimize other approaches.
Implementations can always trap to machine-mode software handlers to
provide exceptional default values.
====

=== Subnormal Arithmetic

Operations on subnormal numbers are handled in accordance with the IEEE 754-2008 standard.
(((operations, subnormal)))

In the parlance of the IEEE standard, tininess is detected after
rounding.
(((tininess, handling)))

[NOTE]
====
Detecting tininess after rounding results in fewer spurious underflow
signals.
====

=== Single-Precision Load and Store Instructions

Floating-point loads and stores use the same base+offset addressing mode as the integer base ISAs, with a base address in register _rs1_ and a 12-bit signed byte offset. The FLW instruction loads a single-precision floating-point value from memory into floating-point register _rd_. FSW stores a single-precision value from floating-point register _rs2_ to memory.

include::images/wavedrom/sp-load-store-2.adoc[]
[[sp-ldst]]
//.SP load and store

FLW and FSW are only guaranteed to execute atomically if the effective
address is naturally aligned.

FLW and FSW do not modify the bits being transferred; in particular, the payloads of non-canonical NaNs are preserved.

As described in <<ldst>>, the execution environment defines whether misaligned floating-point loads and stores are handled invisibly or raise a contained or fatal trap.

[[single-float-compute]]
=== Single-Precision Floating-Point Computational Instructions

Floating-point arithmetic instructions with one or two source operands
use the R-type format with the OP-FP major opcode. FADD.S and FMUL.S
perform single-precision floating-point addition and multiplication
respectively, between _rs1_ and _rs2_. FSUB.S performs the
single-precision floating-point subtraction of _rs2_ from _rs1_. FDIV.S performs the single-precision floating-point division of _rs1_ by _rs2_. FSQRT.S computes the square root of _rs1_. In each case, the result is written to _rd_.

The 2-bit floating-point format field _fmt_ is encoded as shown in
<<fmt>>. It is set to _S_ (00) for all instructions in the F extension.

[[fmt]]
.Format field encoding
[%autowidth,float="center",align="center",cols="^,^,<",options="header",]
|===
|_fmt_ field |Mnemonic |Meaning
|00 |S |32-bit single-precision
|01 |D |64-bit double-precision
|10 |H |16-bit half-precision
|11 |Q |128-bit quad-precision
|===

All floating-point operations that perform rounding can select the
rounding mode using the _rm_ field with the encoding shown in
<<rm>>.

Floating-point minimum-number and maximum-number instructions FMIN.S and FMAX.S write, respectively, the smaller or larger of _rs1_ and _rs2_ to _rd_. For the purposes of these instructions only, the value
latexmath:[$-0.0$] is considered to be less than the value
latexmath:[$+0.0$]. If both inputs are NaNs, the result is the canonical NaN. If only one operand is a NaN, the result is the non-NaN operand.
Signaling NaN inputs set the invalid operation exception flag, even when the result is not NaN.

[NOTE]
====
Note that in version 2.2 of the F extension, the FMIN.S and FMAX.S
instructions were amended to implement the proposed IEEE 754-201x
minimumNumber and maximumNumber operations, rather than the IEEE
754-2008 minNum and maxNum operations. These operations differ in their
handling of signaling NaNs.
====

include::images/wavedrom/spfloat.adoc[]
[[spfloat]]
//.Single-Precision Floating-Point Computational Instructions
(((floating point, fused multiply-add)))

Floating-point fused multiply-add instructions require a new standard
instruction format. R4-type instructions specify three source registers (_rs1_, _rs2_, and _rs3_) and a destination register (_rd_). This format is only used by the floating-point fused multiply-add instructions.

FMADD.S multiplies the values in _rs1_ and _rs2_, adds the value in
_rs3_, and writes the final result to _rd_. FMADD.S computes
_(rs1latexmath:[$\times$]rs2)latexmath:[$\+$]rs3_.

FMSUB.S multiplies the values in _rs1_ and _rs2_, subtracts the value in _rs3_, and writes the final result to _rd_. FMSUB.S computes
_(rs1latexmath:[$\times$]rs2)latexmath:[$\-$]rs3_.

FNMSUB.S multiplies the values in _rs1_ and _rs2_, negates the product, adds the value in _rs3_, and writes the final result to _rd_. FNMSUB.S computes _-(rs1latexmath:[$\times$]rs2)latexmath:[$\+$]rs3_.

FNMADD.S multiplies the values in _rs1_ and _rs2_, negates the product, subtracts the value in _rs3_, and writes the final result to _rd_. FNMADD.S computes _-(rs1latexmath:[$\times$]rs2)latexmath:[$\-$]rs3_.

[NOTE]
====
The FNMSUB and FNMADD instructions are counterintuitively named, owing
to the naming of the corresponding instructions in MIPS-IV. The MIPS
instructions were defined to negate the sum, rather than negating the
product as the RISC-V instructions do, so the naming scheme was more
rational at the time. The two definitions differ with respect to
signed-zero results. The RISC-V definition matches the behavior of the
x86 and ARM fused multiply-add instructions, but unfortunately the
RISC-V FNMSUB and FNMADD instruction names are swapped compared to x86
and ARM.
====

include::images/wavedrom/spfloat2.adoc[]
[[fnmaddsub]]
//.F[N]MADD/F[N]MSUB instructions

[NOTE]
====
The fused multiply-add (FMA) instructions consume a large part of the
32-bit instruction encoding space. Some alternatives considered were to
restrict FMA to only use dynamic rounding modes, but static rounding
modes are useful in code that exploits the lack of product rounding.
Another alternative would have been to use rd to provide rs3, but this
would require additional move instructions in some common sequences. The current design still leaves a large portion of the 32-bit encoding space open while avoiding having FMA be non-orthogonal.
====

The fused multiply-add instructions must set the invalid operation
exception flag when the multiplicands are latexmath:[$\infty$] and zero, even when the addend is a quiet NaN.

[NOTE]
====
The IEEE 754-2008 standard permits, but does not require, raising the
invalid exception for the operation latexmath:[$\infty\times 0\ +$]qNaN.
====

=== Single-Precision Floating-Point Conversion and Move Instructions

Floating-point-to-integer and integer-to-floating-point conversion
instructions are encoded in the OP-FP major opcode space. FCVT.W.S or
FCVT.L.S converts a floating-point number in floating-point register
_rs1_ to a signed 32-bit or 64-bit integer, respectively, in integer
register _rd_. FCVT.S.W or FCVT.S.L converts a 32-bit or 64-bit signed
integer, respectively, in integer register _rs1_ into a floating-point
number in floating-point register _rd_. FCVT.WU.S, FCVT.LU.S, FCVT.S.WU,
and FCVT.S.LU variants convert to or from unsigned integer values. For
XLENlatexmath:[$>32$], FCVT.W[U].S sign-extends the 32-bit result to the
destination register width. FCVT.L[U].S and FCVT.S.L[U] are RV64-only
instructions. If the rounded result is not representable in the
destination format, it is clipped to the nearest value and the invalid
flag is set. <<int_conv>> gives the range of valid inputs
for FCVT._int_.S and the behavior for invalid inputs.
(((floating-point, conversion)))

All floating-point to integer and integer to floating-point conversion
instructions round according to the _rm_ field. A floating-point
register can be initialized to floating-point positive zero using
FCVT.S.W _rd_, `x0`, which will never set any exception flags.

[[int_conv]]
.Domains of float-to-integer conversions and behavior for invalid inputs
[%autowidth,float="center",align="center",cols="<,>,>,>,>",options="header",]
|===
| |FCVT.W.S |FCVT.WU.S |FCVT.L.S |FCVT.LU.S
|Minimum valid input (after rounding) |latexmath:[$-2^{31}$] |0
|latexmath:[$-2^{63}$] |0

|Maximum valid input (after rounding) |latexmath:[$2^{31}-1$]
|latexmath:[$2^{32}-1$] |latexmath:[$2^{63}-1$] |latexmath:[$2^{64}-1$]

|Output for out-of-range negative input |latexmath:[$-2^{31}$] |0
|latexmath:[$-2^{63}$] |0

|Output for latexmath:[$-\infty$] |latexmath:[$-2^{31}$] |0
|latexmath:[$-2^{63}$] |0

|Output for out-of-range positive input |latexmath:[$2^{31}-1$]
|latexmath:[$2^{32}-1$] |latexmath:[$2^{63}-1$] |latexmath:[$2^{64}-1$]

|Output for latexmath:[$+\infty$] or NaN |latexmath:[$2^{31}-1$]
|latexmath:[$2^{32}-1$] |latexmath:[$2^{63}-1$] |latexmath:[$2^{64}-1$]
|===

All floating-point conversion instructions set the Inexact exception
flag if the rounded result differs from the operand value and the
Invalid exception flag is not set.

include::images/wavedrom/spfloat-cn-cmp.adoc[]
[[fcvt]]
//.SP float convert and move

Floating-point to floating-point sign-injection instructions, FSGNJ.S,
FSGNJN.S, and FSGNJX.S, produce a result that takes all bits except the
sign bit from _rs1_. For FSGNJ, the result's sign bit is _rs2_'s sign
bit; for FSGNJN, the result's sign bit is the opposite of _rs2_'s sign
bit; and for FSGNJX, the sign bit is the XOR of the sign bits of _rs1_
and _rs2_. Sign-injection instructions do not set floating-point
exception flags, nor do they canonicalize NaNs. Note, FSGNJ.S _rx, ry,
ry_ moves _ry_ to _rx_ (assembler pseudoinstruction FMV.S _rx, ry_);
FSGNJN.S _rx, ry, ry_ moves the negation of _ry_ to _rx_ (assembler
pseudoinstruction FNEG.S _rx, ry_); and FSGNJX.S _rx, ry, ry_ moves the absolute value of _ry_ to _rx_ (assembler pseudoinstruction FABS.S _rx,
ry_).

include::images/wavedrom/spfloat-sign-inj.adoc[]
[[inj]]

[NOTE]
====
The sign-injection instructions provide floating-point MV, ABS, and NEG, as well as supporting a few other operations, including the IEEE
copySign operation and sign manipulation in transcendental math function libraries. Although MV, ABS, and NEG only need a single register operand, whereas FSGNJ instructions need two, it is unlikely most microarchitectures would add optimizations to benefit from the reduced number of register reads for these relatively infrequent instructions. Even in this case, a microarchitecture can simply detect when both source registers are the same for FSGNJ instructions and only read a single copy.
====

Instructions are provided to move bit patterns between the
floating-point and integer registers. FMV.X.W moves the single-precision value in floating-point register _rs1_ represented in IEEE 754-2008 encoding to the lower 32 bits of integer register _rd_. The bits are not modified in the transfer, and in particular, the payloads of non-canonical NaNs are preserved. For RV64, the higher 32 bits of the destination register are filled with copies of the floating-point number's sign bit.

FMV.W.X moves the single-precision value encoded in IEEE 754-2008
standard encoding from the lower 32 bits of integer register _rs1_ to
the floating-point register _rd_. The bits are not modified in the
transfer, and in particular, the payloads of non-canonical NaNs are
preserved.

[NOTE]
====
The FMV.W.X and FMV.X.W instructions were previously called FMV.S.X and FMV.X.S. The use of W is more consistent with their semantics as an instruction that moves 32 bits without interpreting them. This became clearer after defining NaN-boxing. To avoid disturbing existing code, both the W and S versions will be supported by tools.
====

include::images/wavedrom/spfloat-mv.adoc[]
[[spfloat-mv]]
//.SP floating point move

[TIP]
====
The base floating-point ISA was defined so as to allow implementations
to employ an internal recoding of the floating-point format in registers to simplify handling of subnormal values and possibly to reduce functional unit latency. To this end, the F extension avoids
representing integer values in the floating-point registers by defining conversion and comparison operations that read and write the integer register file directly. This also removes many of the common cases where explicit moves between integer and floating-point registers are required, reducing instruction count and critical paths for common mixed-format code sequences.
====

=== Single-Precision Floating-Point Compare Instructions

Floating-point compare instructions (FEQ.S, FLT.S, FLE.S) perform the
specified comparison between floating-point registers
(latexmath:[$\mbox{\em rs1}
= \mbox{\em rs2}$], latexmath:[$\mbox{\em rs1} < \mbox{\em rs2}$],
latexmath:[$\mbox{\em rs1} \leq
\mbox{\em rs2}$]) writing 1 to the integer register _rd_ if the
condition holds, and 0 otherwise.

FLT.S and FLE.S perform what the IEEE 754-2008 standard refers to as
_signaling_ comparisons: that is, they set the invalid operation
exception flag if either input is NaN. FEQ.S performs a _quiet_
comparison: it only sets the invalid operation exception flag if either input is a signaling NaN. For all three instructions, the result is 0 if either operand is NaN.

include::images/wavedrom/spfloat-comp.adoc[]
[[spfloat-comp]]
//.SP floating point compare

[NOTE]
====
The F extension provides a latexmath:[$\leq$] comparison, whereas the
base ISAs provide a latexmath:[$\geq$] branch comparison. Because
latexmath:[$\leq$] can be synthesized from latexmath:[$\geq$] and
vice-versa, there is no performance implication to this inconsistency,
but it is nevertheless an unfortunate incongruity in the ISA.
====

=== Single-Precision Floating-Point Classify Instruction

The FCLASS.S instruction examines the value in floating-point register
_rs1_ and writes to integer register _rd_ a 10-bit mask that indicates
the class of the floating-point number. The format of the mask is
described in <<fclass>>. The corresponding bit in _rd_ will
be set if the property is true and clear otherwise. All other bits in
_rd_ are cleared. Note that exactly one bit in _rd_ will be set.
FCLASS.S does not set the floating-point exception flags.
(((floating-point, classification)))

include::images/wavedrom/spfloat-classify.adoc[]
[[spfloat-classify]]
//.SP floating point classify

[[fclass]]
.Format of result of FCLASS instruction.
[%autowidth,float="center",align="center",cols="^,<",options="header",]
|===
|_rd_ bit |Meaning
|0 |_rs1_ is latexmath:[$-\infty$].
|1 |_rs1_ is a negative normal number.
|2 |_rs1_ is a negative subnormal number.
|3 |_rs1_ is latexmath:[$-0$].
|4 |_rs1_ is latexmath:[$+0$].
|5 |_rs1_ is a positive subnormal number.
|6 |_rs1_ is a positive normal number.
|7 |_rs1_ is latexmath:[$+\infty$].
|8 |_rs1_ is a signaling NaN.
|9 |_rs1_ is a quiet NaN.
|===