== "Zfh" and "Zfhmin" Extensions for Half-Precision Floating-Point, Version 1.0

This chapter describes the Zfh standard extension for 16-bit
half-precision binary floating-point instructions compliant with the
IEEE 754-2008 arithmetic standard. The Zfh extension depends on the
single-precision floating-point extension, F. The NaN-boxing scheme
described in <<nanboxing>> is extended to allow a
half-precision value to be NaN-boxed inside a single-precision value
(which may be recursively NaN-boxed inside a double- or quad-precision
value when the D or Q extension is present).

[NOTE]
====
This extension primarily provides instructions that consume
half-precision operands and produce half-precision results. However, it
is also common to compute on half-precision data using higher
intermediate precision. Although this extension provides explicit
conversion instructions that suffice to implement that pattern, future
extensions might further accelerate such computation with additional
instructions that implicitly widen their operands—e.g.,
halflatexmath:[$\times$]halflatexmath:[$+$]singlelatexmath:[$\rightarrow$]single—or
implicitly narrow their results—e.g.,
halflatexmath:[$+$]singlelatexmath:[$\rightarrow$]half.
====
=== Half-Precision Load and Store Instructions

New 16-bit variants of LOAD-FP and STORE-FP instructions are added,
encoded with a new value for the funct3 width field.

include::images/wavedrom/sp-load-store.edn[]
[[sp-load-store]]
//.Half-precision load and store instructions

FLH and FSH are only guaranteed to execute atomically if the effective
address is naturally aligned.

FLH and FSH do not modify the bits being transferred; in particular, the
payloads of non-canonical NaNs are preserved. FLH NaN-boxes the result
written to _rd_, whereas FSH ignores all but the lower 16 bits in _rs2_.

=== Half-Precision Computational Instructions

A new supported format is added to the format field of most
instructions, as shown in <<tab:fpextfmth>>.

[[tab:fpextfmth]]
.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
|===

The half-precision floating-point computational instructions are defined
analogously to their single-precision counterparts, but operate on
half-precision operands and produce half-precision results.

include::images/wavedrom/spfloat-zfh.edn[]

include::images/wavedrom/spfloat2-zfh.edn[]

=== Half-Precision Conversion and Move Instructions

New floating-point-to-integer and integer-to-floating-point conversion
instructions are added. These instructions are defined analogously to
the single-precision-to-integer and integer-to-single-precision
conversion instructions. FCVT.W.H or FCVT.L.H converts a half-precision
floating-point number to a signed 32-bit or 64-bit integer,
respectively. FCVT.H.W or FCVT.H.L converts a 32-bit or 64-bit signed
integer, respectively, into a half-precision floating-point number.
FCVT.WU.H, FCVT.LU.H, FCVT.H.WU, and FCVT.H.LU variants convert to or
from unsigned integer values. FCVT.L[U].H and FCVT.H.L[U] are RV64-only
instructions.

include::images/wavedrom/half-prec-conv-and-mv.edn[]
[[half-prec-conv-and-mv]]

New floating-point-to-floating-point conversion instructions are added.
These instructions are defined analogously to the double-precision
floating-point-to-floating-point conversion instructions. FCVT.S.H or
FCVT.H.S converts a half-precision floating-point number to a
single-precision floating-point number, or vice-versa, respectively. If
the D extension is present, FCVT.D.H or FCVT.H.D converts a
half-precision floating-point number to a double-precision
floating-point number, or vice-versa, respectively. If the Q extension
is present, FCVT.Q.H or FCVT.H.Q converts a half-precision
floating-point number to a quad-precision floating-point number, or
vice-versa, respectively.

include::images/wavedrom/half-prec-flpt-to-flpt-conv.edn[]
[[half-prec-flpt-to-flpt-conv]]

Floating-point to floating-point sign-injection instructions, FSGNJ.H,
FSGNJN.H, and FSGNJX.H are defined analogously to the single-precision
sign-injection instruction.

include::images/wavedrom/flt-to-flt-sgn-inj-instr.edn[]
[[flt-to-flt-sgn-inj-instr]]

Instructions are provided to move bit patterns between the
floating-point and integer registers. FMV.X.H moves the half-precision
value in floating-point register _rs1_ to a representation in IEEE
754-2008 standard encoding in integer register _rd_, filling the upper
XLEN-16 bits with copies of the floating-point number's sign bit.

FMV.H.X moves the half-precision value encoded in IEEE 754-2008 standard
encoding from the lower 16 bits of integer register _rs1_ to the
floating-point register _rd_, NaN-boxing the result.

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

include::images/wavedrom/flt-pt-to-int-move.edn[]
[[flt-pt-to-int-move]]

=== Half-Precision Floating-Point Compare Instructions

The half-precision floating-point compare instructions are defined
analogously to their single-precision counterparts, but operate on
half-precision operands.

include::images/wavedrom/half-pr-flt-pt-compare.edn[]
[[half-pr-flt-pt-compare]]

=== Half-Precision Floating-Point Classify Instruction

The half-precision floating-point classify instruction, FCLASS.H, is
defined analogously to its single-precision counterpart, but operates on
half-precision operands.

include::images/wavedrom/half-pr-flt-pt-class.edn[]
[[half-pr-flt-class]]

=== "Zfhmin" Standard Extension for Minimal Half-Precision Floating-Point

This section describes the Zfhmin standard extension, which provides
minimal support for 16-bit half-precision binary floating-point
instructions. The Zfhmin extension is a subset of the Zfh extension,
consisting only of data transfer and conversion instructions. Like Zfh,
the Zfhmin extension depends on the single-precision floating-point
extension, F. The expectation is that Zfhmin software primarily uses the
half-precision format for storage, performing most computation in higher
precision.

The Zfhmin extension includes the following instructions from the Zfh
extension: FLH, FSH, FMV.X.H, FMV.H.X, FCVT.S.H, and FCVT.H.S. If the D
extension is present, the FCVT.D.H and FCVT.H.D instructions are also
included. If the Q extension is present, the FCVT.Q.H and FCVT.H.Q
instructions are additionally included.

[NOTE]
====
Zfhmin does not include the FSGNJ.H instruction, because it suffices to
instead use the FSGNJ.S instruction to move half-precision values
between floating-point registers.

Half-precision addition, subtraction, multiplication, division, and
square-root operations can be faithfully emulated by converting the
half-precision operands to single-precision, performing the operation
using single-precision arithmetic, then converting back to
half-precision. cite:[roux:hal-01091186] Performing half-precision fused multiply-addition using
this method incurs a 1-ulp error on some inputs for the RNE and RMM
rounding modes.

Conversion from 8- or 16-bit integers to half-precision can be emulated
by first converting to single-precision, then converting to
half-precision. Conversion from 32-bit integer can be emulated by first
converting to double-precision. If the D extension is not present and a
1-ulp error under RNE or RMM is tolerable, 32-bit integers can be first
converted to single-precision instead. The same remark applies to
conversions from 64-bit integers without the Q extension.
====
