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[[!tag standards]]

# REMAP <a name="remap" />

* <https://bugs.libre-soc.org/show_bug.cgi?id=143> matrix multiply
* <https://bugs.libre-soc.org/show_bug.cgi?id=867> add svindex
* <https://bugs.libre-soc.org/show_bug.cgi?id=885> svindex in simulator
* <https://bugs.libre-soc.org/show_bug.cgi?id=911> offset svshape option
* <https://bugs.libre-soc.org/show_bug.cgi?id=864> parallel reduction
* <https://bugs.libre-soc.org/show_bug.cgi?id=930> DCT/FFT "strides"
* see [[sv/remap/appendix]] for examples and usage
* see [[sv/propagation]] for a future way to apply REMAP
* [[remap/discussion]]

REMAP is an advanced form of Vector "Structure Packing" that
provides hardware-level support for commonly-used *nested* loop patterns.
For more general reordering an Indexed REMAP mode is available.

REMAP allows the usual vector loop `0..VL-1` to be "reshaped" (re-mapped)
from a linear form to a 2D or 3D transposed form, or "offset" to permit
arbitrary access to elements (when elwidth overrides are used),
independently on each Vector src or dest
register.

The initial primary motivation of REMAP was for Matrix Multiplication, reordering of sequential
data in-place: in-place DCT and FFT were easily justified given the
high usage in Computer Science.
Four SPRs are provided which may be applied to any GPR, FPR or CR Field
so that for example a single FMAC may be
used in a single loop to perform 5x3 times 3x4 Matrix multiplication,
generating 60 FMACs *without needing explicit assembler unrolling*.
Additional uses include regular "Structure Packing"
such as RGB pixel data extraction and reforming.

REMAP, like all of SV, is abstracted out, meaning that unlike traditional
Vector ISAs which would typically only have a limited set of instructions
that can be structure-packed (LD/ST and Move operations
being the most common), REMAP may be applied to
literally any instruction: CRs, Arithmetic, Logical, LD/ST, anything.

When SUBVL is greater than 1 a given group of Subvector
elements are kept together: effectively the group becomes the
element, and with REMAP applying to elements
(not sub-elements) each group is REMAPed together.
Swizzle *can* however be applied to the same
instruction as REMAP, providing re-sequencing of
Subvector elements which REMAP cannot. Also as explained in [[sv/mv.swizzle]], [[sv/mv.vec]] and the [[svp64/appendix]], Pack and Unpack Mode bits
can extend down into Sub-vector elements to influence vec2/vec3/vec4
sequential reordering, but even here, REMAP is not extended down to
the actual sub-vector elements themselves.

In its general form, REMAP is quite expensive to set up, and on some
implementations may introduce
latency, so should realistically be used only where it is worthwhile.
Given that most other ISAs require full loop-unrolling for Matrix,
DCT and FFT, savings are still anticipated.
Commonly-used patterns such as Matrix Multiply, DCT and FFT have
helper instruction options which make REMAP easier to use.

There are four types of REMAP:

* **Matrix**, also known as 2D and 3D reshaping, can perform in-place
  Matrix transpose and rotate. The Shapes are set up for an "Outer Product"
  Matrix Multiply.
* **FFT/DCT**, with full triple-loop in-place support: limited to
  Power-2 RADIX
* **Indexing**, for any general-purpose reordering, also includes
  limited 2D reshaping.
* **Parallel Reduction**, for scheduling a sequence of operations
  in a Deterministic fashion, in a way that may be parallelised,
  to reduce a Vector down to a single value.

Best implemented on top of a Multi-Issue Out-of-Order Micro-architecture,
REMAP Schedules are 100% Deterministic **including Indexing** and are
designed to be incorporated in between the Decode and Issue phases,
directly into Register Hazard Management

As long as the SVSHAPE SPRs
are not written to directly, Hardware may treat REMAP as 100%
Deterministic: all REMAP Management instructions take static
operands (no dynamic register operands)
with the exception of Indexed Mode, and even then
Architectural State is permitted to assume that the Indices
are cacheable from the point at which the `svindex` instruction
is executed.

Parallel Reduction is unusual in that it requires a full vector array
of results (not a scalar) and uses the rest of the result Vector for
the purposes of storing intermediary calculations.  As these intermediary
results are Deterministically computed they may be useful.
Additionally, because the intermediate results are always written out
it is possible to service Precise Interrupts without affecting latency
(a common limitation of Vector ISAs).

# Basic principle

* normal vector element read/write of operands would be sequential
  (0 1 2 3 ....)
* this is not appropriate for (e.g.) Matrix multiply which requires
  accessing elements in alternative sequences (0 3 6 1 4 7 ...)
* normal Vector ISAs use either Indexed-MV or Indexed-LD/ST to "cope"
  with this.  both are expensive (copy large vectors, spill through memory)
  and very few Packed SIMD ISAs cope with non-Power-2.
* REMAP **redefines** the order of access according to set
  (Deterministic) "Schedules".
* Matrix Schedules are not at all restricted to power-of-two boundaries
  making it unnecessary to have for example specialised 3x4 transpose
  instructions of other Vector ISAs.

Only the most commonly-used algorithms in computer science have REMAP
support, due to the high cost in both the ISA and in hardware.  For
arbitrary remapping the `Indexed` REMAP may be used.

# Example Usage

* `svshape` to set the type of reordering to be applied to an
  otherwise usual `0..VL-1` hardware for-loop
* `svremap` to set which registers a given reordering is to apply to
  (RA, RT etc)
* `sv.{instruction}` where any Vectorised register marked by `svremap`
  will have its ordering REMAPPED according to the schedule set
  by `svshape`.

The following illustrative example multiplies a 3x4 and a 5x3
matrix to create
a 5x4 result:

```
    svshape 5, 4, 3, 0, 0           # Outer Product
    svremap 15, 1, 2, 3, 0, 0, 0, 0
    sv.fmadds *0, *32, *64, *0
```

* svshape sets up the four SVSHAPE SPRS for a Matrix Schedule
* svremap activates four out of five registers RA RB RC RT RS (15)
* svremap requests:
  - RA to use SVSHAPE1
  - RB to use SVSHAPE2
  - RC to use SVSHAPE3
  - RT to use SVSHAPE0
  - RS Remapping to not be activated
* sv.fmadds has Vectors at RT=0, RA=32, RB=64, RC=0
* With REMAP being active each register's element index is
  *independently* transformed using the specified SHAPEs.

Thus the Vector Loop is arranged such that the use of
the multiply-and-accumulate instruction executes precisely the required
Schedule to perform an in-place in-registers Outer Product
Matrix Multiply with no
need to perform additional Transpose or register copy instructions.
The example above may be executed as a unit test and demo,
[here](https://git.libre-soc.org/?p=openpower-isa.git;a=blob;f=src/openpower/decoder/isa/test_caller_svp64_matrix.py;h=c15479db9a36055166b6b023c7495f9ca3637333;hb=a17a252e474d5d5bf34026c25a19682e3f2015c3#l94)
     
# REMAP types

This section summarises the motivation for each REMAP Schedule
and briefly goes over their characteristics and limitations.
Further details on the Deterministic Precise-Interruptible algorithms
used in these Schedules is found in the [[sv/remap/appendix]].

## Matrix (1D/2D/3D shaping)

Matrix Multiplication is a huge part of High-Performance Compute,
and 3D.
In many PackedSIMD as well as Scalable Vector ISAs, non-power-of-two
Matrix sizes are a serious challenge. PackedSIMD ISAs, in order to
cope with for example 3x4 Matrices, recommend rolling data-repetition and loop-unrolling.
Aside from the cost of the load on the L1 I-Cache, the trick only
works if one of the dimensions X or Y are power-two. Prime Numbers
(5x7, 3x5) become deeply problematic to unroll.

Even traditional Scalable Vector ISAs have issues with Matrices, often
having to perform data Transpose by pushing out through Memory and back,
or computing Transposition Indices (costly) then copying to another
Vector (costly).

Matrix REMAP was thus designed to solve these issues by providing Hardware
Assisted
"Schedules" that can view what would otherwise be limited to a strictly
linear Vector as instead being 2D (even 3D) *in-place* reordered.
With both Transposition and non-power-two being supported the issues
faced by other ISAs are mitigated.

Limitations of Matrix REMAP are that the Vector Length (VL) is currently
restricted to 127: up to 127 FMAs (or other operation)
may be performed in total.
Also given that it is in-registers only at present some care has to be
taken on regfile resource utilisation. However it is perfectly possible
to utilise Matrix REMAP to perform the three inner-most "kernel" loops of
the usual 6-level large Matrix Multiply, without the usual difficulties
associated with SIMD.

Also the `svshape` instruction only provides access to part of the
Matrix REMAP capability. Rotation and mirroring need to be done by
programming the SVSHAPE SPRs directly, which can take a lot more
instructions.

## FFT/DCT Triple Loop

DCT and FFT are some of the most astonishingly used algorithms in
Computer Science.  Radar, Audio, Video, R.F. Baseband and dozens more.  At least
two DSPs, TMS320 and Hexagon, have VLIW instructions specially tailored
to FFT.

An in-depth analysis showed that it is possible to do in-place in-register
DCT and FFT as long as twin-result "butterfly" instructions are provided.
These can be found in the [[openpower/isa/svfparith]] page if performing
IEEE754 FP transforms. *(For fixed-point transforms, equivalent 3-in 2-out
integer operations would be required)*. These "butterfly" instructions
avoid the need for a temporary register because the two array positions
being overwritten will be "in-flight" in any In-Order or Out-of-Order
micro-architecture.

DCT and FFT Schedules are currently limited to RADIX2 sizes and do not
accept predicate masks.  Given that it is common to perform recursive
convolutions combining smaller Power-2 DCT/FFT to create larger DCT/FFTs
in practice the RADIX2 limit is not a problem.  A Bluestein convolution
to compute arbitrary length is demonstrated by
[Project Nayuki](https://www.nayuki.io/res/free-small-fft-in-multiple-languages/fft.py)

## Indexed

The purpose of Indexing is to provide a generalised version of
Vector ISA "Permute" instructions, such as VSX `vperm`.  The
Indexing is abstracted out and may be applied to much more
than an element move/copy, and is not limited for example
to the number of bytes that can fit into a VSX register.
Indexing may be applied to LD/ST (even on Indexed LD/ST
instructions such as `sv.lbzx`), arithmetic operations,
extsw: there is no artificial limit.

The only major caveat is that the registers to be used as
Indices must not be modified by any instruction after Indexed Mode
is established, and neither must MAXVL be altered. Additionally,
no register used as an Index may exceed MAXVL-1.

Failure to observe
these conditions results in `UNDEFINED` behaviour.
These conditions allow a Read-After-Write (RAW) Hazard to be created on
the entire range of Indices to be subsequently used, but a corresponding
Write-After-Read Hazard by any instruction that modifies the Indices
**does not have to be created**. Given the large number of registers
involved in Indexing this is a huge resource saving and reduction
in micro-architectural complexity. MAXVL is likewise
included in the RAW Hazards because it is involved in calculating
how many registers are to be considered Indices.

With these Hazard Mitigations in place, high-performance implementations
may read-cache the Indices from the point where a given `svindex` instruction
is called (or SVSHAPE SPRs - and MAXVL- directly altered).

The original motivation for Indexed REMAP was to mitigate the need to add
an expensive `mv.x` to the Scalar ISA, which was likely to be rejected as
a stand-alone instruction.  Usually a Vector ISA would add a non-conflicting
variant (as in VSX `vperm`) but it is common to need to permute by source,
with the risk of conflict, that has to be resolved, for example, in AVX-512
with `conflictd`.

Indexed REMAP on the other hand **does not prevent conflicts** (overlapping
destinations), which on a superficial analysis may be perceived to be a
problem, until it is recalled that, firstly, Simple-V is designed specifically
to require Program Order to be respected, and that Matrix, DCT and FFT
all *already* critically depend on overlapping Reads/Writes: Matrix
uses overlapping registers as accumulators.  Thus the Register Hazard
Management needed by Indexed REMAP *has* to be in place anyway.

The cost compared to Matrix and other REMAPs (and Pack/Unpack) is
clearly that of the additional reading of the GPRs to be used as Indices,
plus the setup cost associated with creating those same Indices.
If any Deterministic REMAP can cover the required task, clearly it
is adviseable to use it instead.

*Programmer's note: some algorithms may require skipping of Indices exceeding
VL-1, not MAXVL-1. This may be achieved programmatically by performing
an `sv.cmp *BF,*RA,RB` where RA is the same GPRs used in the Indexed REMAP,
and RB contains the value of VL returned from `setvl`. The resultant
CR Fields may then be used as Predicate Masks to exclude those operations
with an Index exceeding VL-1.*

## Parallel Reduction

Vector Reduce Mode issues a deterministic tree-reduction schedule to the underlying micro-architecture.  Like Scalar reduction, the "Scalar Base"
(Power ISA v3.0B) operation is leveraged, unmodified, to give the
*appearance* and *effect* of Reduction.

In Horizontal-First Mode, Vector-result reduction **requires**
the destination to be a Vector, which will be used to store
intermediary results.

Given that the tree-reduction schedule is deterministic,
Interrupts and exceptions
can therefore also be precise.  The final result will be in the first
non-predicate-masked-out destination element, but due again to
the deterministic schedule programmers may find uses for the intermediate
results.

When Rc=1 a corresponding Vector of co-resultant CRs is also
created.  No special action is taken: the result and its CR Field
are stored "as usual" exactly as all other SVP64 Rc=1 operations.

Note that the Schedule only makes sense on top of certain instructions:
X-Form with a Register Profile of `RT,RA,RB` is fine because two sources
and the destination are all the same type.  Like Scalar
Reduction, nothing is prohibited:
the results of execution on an unsuitable instruction may simply
not make sense. With care, even 3-input instructions (madd, fmadd, ternlogi) 
may be used.

Critical to note regarding use of Parallel-Reduction REMAP is that,
exactly as with all REMAP Modes, the `svshape` instruction *requests*
a certain Vector Length (number of elements to reduce) and then
sets VL and MAXVL at the number of **operations** needed to be
carried out.  Thus, equally as importantly, like Matrix REMAP
the total number of operations
is restricted to 127.  Any Parallel-Reduction requiring more operations
will need to be done manually in batches (hierarchical
recursive Reduction).

Also important to note is that the Deterministic Schedule is arranged
so that some implementations *may* parallelise it (as long as doing so
respects Program Order and Register Hazards).  Performance (speed)
of any given
implementation is neither strictly defined or guaranteed.  As with
the Vulkan(tm) Specification, strict compliance is paramount whilst
performance is at the discretion of Implementors.

**Parallel-Reduction with Predication**

To avoid breaking the strict RISC-paradigm, keeping the Issue-Schedule
completely separate from the actual element-level (scalar) operations,
Move operations are **not** included in the Schedule.  This means that
the Schedule leaves the final (scalar) result in the first-non-masked 
element of the Vector used.  With the predicate mask being dynamic
(but deterministic) this result could be anywhere.

If that result is needed to be moved to a (single) scalar register
then a follow-up `sv.mv/sm=predicate rt, *ra` instruction will be
needed to get it, where the predicate is the exact same predicate used
in the prior Parallel-Reduction instruction.

* If there was only a single
  bit in the predicate then the result will not have moved or been altered
  from the source vector prior to the Reduction
* If there was more than one bit the result will be in the
  first element with a predicate bit set.

In either case the result is in the element with the first bit set in
the predicate mask.

For *some* implementations
the vector-to-scalar copy may be a slow operation, as may the Predicated
Parallel Reduction itself.
It may be better to perform a pre-copy
of the values, compressing them (VREDUCE-style) into a contiguous block,
which will guarantee that the result goes into the very first element
of the destination vector, in which case clearly no follow-up
vector-to-scalar MV operation is needed.

**Usage conditions**

The simplest usage is to perform an overwrite, specifying all three
register operands the same.

    svshape parallelreduce, 6
    sv.add *8, *8, *8

The Reduction Schedule will issue the Parallel Tree Reduction spanning
registers 8 through 13, by adjusting the offsets to RT, RA and RB as
necessary (see "Parallel Reduction algorithm" in a later section).

A non-overwrite is possible as well but just as with the overwrite
version, only those destination elements necessary for storing
intermediary computations will be written to: the remaining elements
will **not** be overwritten and will **not** be zero'd.

    svshape parallelreduce, 6
    sv.add *0, *8, *8

However it is critical to note that if the source and destination are
not the same then the trick of using a follow-up vector-scalar MV will
not work.

## Sub-Vector Horizontal Reduction

Note that when SVM is clear and SUBVL!=1 a Parallel Reduction is performed
on all first Subvector elements, followed by another separate independent
Parallel Reduction on all the second Subvector elements and so on.

    for selectsubelement in (x,y,z,w):
       parallelreduce(0..VL-1, selectsubelement)

By contrast, when SVM is set and SUBVL!=1, a Horizontal
Subvector mode is enabled, applying the Parallel Reduction
Algorithm to the Subvector Elements. The Parallel Reduction
is independently applied VL times, to each group of Subvector
elements. Bear in mind that predication is never applied down
into individual Subvector elements, but will be applied
to select whether the *entire* Parallel Reduction on each
group is performed or not.

      for (i = 0; i < VL; i++)
        if (predval & 1<<i) # predication
           el = element[i]
           parallelreduction([el.x, el.y, el.z, el.w])

Note that as this is a Parallel Reduction, for best results
it should be an overwrite operation, where the result for
the Horizontal Reduction of each Subvector will be in the
first Subvector element.
Also note that use of Rc=1 is `UNDEFINED` behaviour.

In essence what is happening here is that Structure Packing is being
combined with Parallel Reduction.  If the Subvector elements may be
laid out as a 2D matrix, with the Subvector elements on rows,
and Parallel Reduction is applied per row, then if `SVM` is **clear**
the Matrix is transposed (like Pack/Unpack)
before still applying the Parallel Reduction to the **row**.

# Determining Register Hazards

For high-performance (Multi-Issue, Out-of-Order) systems it is critical
to be able to statically determine the extent of Vectors in order to
allocate pre-emptive Hazard protection.  The next task is to eliminate
masked-out elements using predicate bits, freeing up the associated
Hazards.

For non-REMAP situations `VL` is sufficient to ascertain early
Hazard coverage, and with SVSTATE being a high priority cached
quantity at the same level of MSR and PC this is not a problem.

The problems come when REMAP is enabled.  Indexed REMAP must instead
use `MAXVL` as the earliest (simplest)
batch-level Hazard Reservation indicator,
but Matrix, FFT and Parallel Reduction must all use completely different
schemes.  The reason is that VL is used to step through the total
number of *operations*, not the number of registers.  The "Saving Grace"
is that all of the REMAP Schedules are Deterministic.

Advance-notice Parallel computation and subsequent cacheing
of all of these complex Deterministic REMAP Schedules is
*strongly recommended*, thus allowing clear and precise multi-issue
batched Hazard coverage to be deployed, *even for Indexed Mode*.
This is only possible for Indexed due to the strict guidelines
given to Programmers.

In short, there exists solutions to the problem of Hazard Management,
with varying degrees of refinement possible at correspondingly
increasing levels of complexity in hardware.

# REMAP area of SVSTATE

The following bits of the SVSTATE SPR are used for REMAP:

|32.33|34.35|36.37|38.39|40.41| 42.46 | 62 |
| --  | --  | --  | --  | --  | ----- | ------ |
|mi0  |mi1  |mi2  |mo0  |mo1  | SVme  | RMpst    |

mi0-2 and mo0-1 each select SVSHAPE0-3 to apply to a given register.
mi0-2 apply to RA, RB, RC respectively, as input registers, and
likewise mo0-1 apply to output registers (RT/FRT, RS/FRS) respectively.
SVme is 5 bits (one for each of mi0-2/mo0-1) and indicates whether the
SVSHAPE is actively applied or not.

* bit 0 of SVme indicates if mi0 is applied to RA / FRA
* bit 1 of SVme indicates if mi1 is applied to RB / FRB
* bit 2 of SVme indicates if mi2 is applied to RC / FRC
* bit 3 of SVme indicates if mo0 is applied to RT / FRT
* bit 4 of SVme indicates if mo1 is applied to Effective Address / FRS / RS
  (LD/ST-with-update has an implicit 2nd write register, RA)

# svremap instruction <a name="svremap"> </a>

There is also a corresponding SVRM-Form for the svremap
instruction which matches the above SPR:

    svremap SVme,mi0,mi1,mi2,mo0,mo2,pst

|0     |6     |11  |13   |15   |17   |19   |21    | 22.25 |26..31 |
| --   | --   | -- | --  | --  | --  | --  | --   | ----  | ----- |
| PO   | SVme |mi0 | mi1 | mi2 | mo0 | mo1 | pst  | rsvd  | XO    |

# SHAPE Remapping SPRs

There are four "shape" SPRs, SHAPE0-3, 32-bits in each,
which have the same format.  

Shape is 32-bits.  When SHAPE is set entirely to zeros, remapping is
disabled: the register's elements are a linear (1D) vector.

|31.30|29..28 |27..24| 23..21 | 20..18  | 17..12  |11..6 |5..0  | Mode  |
|---- |------ |------| ------ | ------- | ------- |----- |----- | ----- |
|0b00 |skip   |offset| invxyz | permute | zdimsz  |ydimsz|xdimsz|Matrix |
|0b00 |elwidth|offset|sk1/invxy|0b110/0b111|SVGPR|ydimsz|xdimsz|Indexed|
|0b01 |submode|offset| invxyz | submode2| zdimsz  |mode  |xdimsz|DCT/FFT|
|0b10 |submode|offset| invxyz | rsvd    | rsvd    |rsvd  |xdimsz|Preduce|
|0b11 |       |      |        |         |         |      |      |rsvd   |

mode sets different behaviours (straight matrix multiply, FFT, DCT).

* **mode=0b00** sets straight Matrix Mode
* **mode=0b00** with permute=0b110 or 0b111 sets Indexed Mode
* **mode=0b01** sets "FFT/DCT" mode and activates submodes
* **mode=0b10** sets "Parallel Reduction" Schedules.

## Parallel Reduction Mode

Creates the Schedules for Parallel Tree Reduction.

* **submode=0b00** selects the left operand index
* **submode=0b01** selects the right operand index

* When bit 0 of `invxyz` is set, the order of the indices
  in the inner for-loop are reversed. This has the side-effect
  of placing the final reduced result in the last-predicated element.
  It also has the indirect side-effect of swapping the source
  registers: Left-operand index numbers will always exceed
  Right-operand indices.
  When clear, the reduced result will be in the first-predicated
  element, and Left-operand indices will always be *less* than
  Right-operand ones.
* When bit 1 of `invxyz` is set, the order of the outer loop
  step is inverted: stepping begins at the nearest power-of two
  to half of the vector length and reduces by half each time.
  When clear the step will begin at 2 and double on each
  inner loop.

## FFT/DCT mode

submode2=0 is for FFT. For FFT submode the following schedules may be 
selected:

* **submode=0b00** selects the ``j`` offset of the innermost for-loop
  of Tukey-Cooley
* **submode=0b10** selects the ``j+halfsize`` offset of the innermost for-loop
  of Tukey-Cooley
* **submode=0b11** selects the ``k`` of exptable (which coefficient)

When submode2 is 1 or 2, for DCT inner butterfly submode the following
schedules may be selected.  When submode2 is 1, additional bit-reversing
is also performed.

* **submode=0b00** selects the ``j`` offset of the innermost for-loop,
    in-place
* **submode=0b010** selects the ``j+halfsize`` offset of the innermost for-loop,
  in reverse-order, in-place
* **submode=0b10** selects the ``ci`` count of the innermost for-loop,
  useful for calculating the cosine coefficient
* **submode=0b11** selects the ``size`` offset of the outermost for-loop,
  useful for the cosine coefficient ``cos(ci + 0.5) * pi / size``

When submode2 is 3 or 4, for DCT outer butterfly submode the following
schedules may be selected.  When submode is 3, additional bit-reversing
is also performed.

* **submode=0b00** selects the ``j`` offset of the innermost for-loop,
* **submode=0b01** selects the ``j+1`` offset of the innermost for-loop,

`zdimsz` is used as an in-place "Stride", particularly useful for
column-based in-place DCT/FFT.

## Matrix Mode

In Matrix Mode, skip allows dimensions to be skipped from being included
in the resultant output index.  this allows sequences to be repeated:
```0 0 0 1 1 1 2 2 2 ...``` or in the case of skip=0b11 this results in
modulo ```0 1 2 0 1 2 ...```

* **skip=0b00** indicates no dimensions to be skipped
* **skip=0b01** sets "skip 1st dimension"
* **skip=0b10** sets "skip 2nd dimension"
* **skip=0b11** sets "skip 3rd dimension"

invxyz will invert the start index of each of x, y or z. If invxyz[0] is
zero then x-dimensional counting begins from 0 and increments, otherwise
it begins from xdimsz-1 and iterates down to zero. Likewise for y and z.

offset will have the effect of offsetting the result by ```offset``` elements:

    for i in 0..VL-1:
        GPR(RT + remap(i) + SVSHAPE.offset) = ....

this appears redundant because the register RT could simply be changed by a compiler, until element width overrides are introduced.  also
bear in mind that unlike a static compiler SVSHAPE.offset may
be set dynamically at runtime.

xdimsz, ydimsz and zdimsz are offset by 1, such that a value of 0 indicates
that the array dimensionality for that dimension is 1. any dimension
not intended to be used must have its value set to 0 (dimensionality
of 1).  A value of xdimsz=2 would indicate that in the first dimension
there are 3 elements in the array.  For example, to create a 2D array
X,Y of dimensionality X=3 and Y=2, set xdimsz=2, ydimsz=1 and zdimsz=0

The format of the array is therefore as follows:

    array[xdimsz+1][ydimsz+1][zdimsz+1]

However whilst illustrative of the dimensionality, that does not take the
"permute" setting into account.  "permute" may be any one of six values
(0-5, with values of 6 and 7 indicating "Indexed" Mode).  The table
below shows how the permutation dimensionality order works:

| permute | order | array format             |
| ------- | ----- | ------------------------ |
| 000     | 0,1,2 | (xdim+1)(ydim+1)(zdim+1) |
| 001     | 0,2,1 | (xdim+1)(zdim+1)(ydim+1) |
| 010     | 1,0,2 | (ydim+1)(xdim+1)(zdim+1) |
| 011     | 1,2,0 | (ydim+1)(zdim+1)(xdim+1) |
| 100     | 2,0,1 | (zdim+1)(xdim+1)(ydim+1) |
| 101     | 2,1,0 | (zdim+1)(ydim+1)(xdim+1) |
| 110     | 0,1   | Indexed (xdim+1)(ydim+1) |
| 111     | 1,0   | Indexed (ydim+1)(xdim+1) |

In other words, the "permute" option changes the order in which
nested for-loops over the array would be done.  See executable
python reference code for further details.

*Note: permute=0b110 and permute=0b111 enable Indexed REMAP Mode,
described below*

With all these options it is possible to support in-place transpose,
in-place rotate, Matrix Multiply and Convolutions, without being
limited to Power-of-Two dimension sizes.

## Indexed Mode

Indexed Mode activates reading of the element indices from the GPR
and includes optional limited 2D reordering.
In its simplest form (without elwidth overrides or other modes):

```
def index_remap(i):
    return GPR((SVSHAPE.SVGPR<<1)+i) + SVSHAPE.offset

for i in 0..VL-1:
    element_result = ....
    GPR(RT + indexed_remap(i)) = element_result
```

With element-width overrides included, and using the pseudocode
from the SVP64 [[sv/svp64/appendix#elwidth]] elwidth section
this becomes:

```
def index_remap(i):
    svreg = SVSHAPE.SVGPR << 1
    srcwid = elwid_to_bitwidth(SVSHAPE.elwid)
    offs = SVSHAPE.offset
    return get_polymorphed_reg(svreg, srcwid, i) + offs

for i in 0..VL-1:
    element_result = ....
    rt_idx = indexed_remap(i)
    set_polymorphed_reg(RT, destwid, rt_idx, element_result)
```

Matrix-style reordering still applies to the indices, except limited
to up to 2 Dimensions (X,Y). Ordering is therefore limited to (X,Y) or
(Y,X). Only one dimension may optionally be skipped. Inversion of either
X or Y or both is possible. Pseudocode for Indexed Mode (including elwidth
overrides) may be written in terms of Matrix Mode, specifically
purposed to ensure that the 3rd dimension (Z) has no effect:

```
def index_remap(ISHAPE, i):
    MSHAPE.skip   = 0b0 || ISHAPE.sk1
    MSHAPE.invxyz = 0b0 || ISHAPE.invxy
    MSHAPE.xdimsz = ISHAPE.xdimsz
    MSHAPE.ydimsz = ISHAPE.ydimsz
    MSHAPE.zdimsz = 0 # disabled
    if ISHAPE.permute = 0b110 # 0,1
       MSHAPE.permute = 0b000 # 0,1,2
    if ISHAPE.permute = 0b111 # 1,0
       MSHAPE.permute = 0b010 # 1,0,2
    el_idx = remap_matrix(MSHAPE, i)
    svreg = ISHAPE.SVGPR << 1
    srcwid = elwid_to_bitwidth(ISHAPE.elwid)
    offs = ISHAPE.offset
    return get_polymorphed_reg(svreg, srcwid, el_idx) + offs
```

The most important observation above is that the Matrix-style
remapping occurs first and the Index lookup second.  Thus it
becomes possible to perform in-place Transpose of Indices which
may have been costly to set up or costly to duplicate
(waste register file space).

# svshape instruction  <a name="svshape"> </a>

`svshape` is a convenience instruction that reduces instruction
count for common usage patterns, particularly Matrix, DCT and FFT. It sets up
(overwrites) all required SVSHAPE SPRs and also modifies SVSTATE
including VL and MAXVL. Using `svshape` therefore does not also
require `setvl`.

Form: SVM-Form SV "Matrix" Form (see [[isatables/fields.text]])

    svshape SVxd,SVyd,SVzd,SVRM,vf

| 0.5|6.10  |11.15  |16..20 | 21..24 | 25 | 26..31|  name    |
| -- | --   | ---   | ----- | ------ | -- | ------| -------- |
|OPCD| SVxd | SVyd  | SVzd  | SVRM   | vf | XO    | svshape  |

Fields:

* **SVxd** - SV REMAP "xdim"
* **SVyd** - SV REMAP "ydim"
* **SVzd** - SV REMAP "zdim"
* **SVRM** - SV REMAP Mode (0b00000 for Matrix, 0b00001 for FFT etc.)
* **vf** - sets "Vertical-First" mode
* **XO** - standard 6-bit XO field

*Note: SVxd, SVyz and SVzd are all stored "off-by-one".  In the assembler
mnemonic the values `1-32` are stored in binary as `0b00000..0b11111`*

| SVRM   | Remap Mode description |
| --     | --              |
| 0b0000 | Matrix 1/2/3D    |
| 0b0001 | FFT Butterfly   |
| 0b0010 | reserved |
| 0b0011 | DCT Outer butterfly  |
| 0b0100 | DCT Inner butterfly, on-the-fly (Vertical-First Mode) |
| 0b0101 | DCT COS table index generation |
| 0b0110 | DCT half-swap   |
| 0b0111 | Parallel Reduction |
| 0b1000 | reserved for svshape2 |
| 0b1001 | reserved for svshape2 |
| 0b1010 | reserved |
| 0b1011 | iDCT Outer butterfly  |
| 0b1100 | iDCT Inner butterfly, on-the-fly (Vertical-First Mode) |
| 0b1101 | iDCT COS table index generation |
| 0b1110 | iDCT half-swap   |
| 0b1111 | FFT half-swap   |

Examples showing how all of these Modes operate exists in the online
[SVP64 unit tests](https://git.libre-soc.org/?p=openpower-isa.git;a=tree;f=src/openpower/decoder/isa;hb=HEAD)
and the full pseudocode setting up all SPRs
is in the [[openpower/isa/simplev]] page.

In Indexed Mode, there are only 5 bits available to specify the GPR
to use, out of 128 GPRs (7 bit numbering).  Therefore, only the top
5 bits are given in the `SVxd` field: the bottom two implicit bits
will be zero (`SVxd || 0b00`).

`svshape` has *limited applicability* due to being a 32-bit instruction.
The full capability of SVSHAPE SPRs may be accessed by directly writing
to SVSHAPE0-3 with `mtspr`. Circumstances include Matrices with dimensions
larger than 32, and in-place Transpose.  Potentially a future v3.1 Prefixed
instruction, `psvshape`, may extend the capability here.

# svindex instruction  <a name="svindex"> </a>

`svindex` is a convenience instruction that reduces instruction
count for Indexed REMAP Mode. It sets up
(overwrites) all required SVSHAPE SPRs and can modify the REMAP
SPR as well.  The relevant SPRs *may* be directly programmed with
`mtspr` however it is laborious to do so: svindex saves instructions
covering much of Indexed REMAP capability.

Form: SVI-Form SV "Indexed" Form (see [[isatables/fields.text]])

    svindex SVG,rmm,SVd,ew,yx,mr,sk

| 0.5|6.10 |11.15  |16.20 | 21..25      | 26..31|  name    | Form |
| -- | --  | ---   | ---- | ----------- | ------| -------- | ---- |
|OPCD| SVG | rmm   | SVd  | ew/yx/mm/sk | XO    | svindex | SVI-Form |

Fields:

* **SVd** - SV REMAP x/y dim 
* **rmm** - REMAP mask: sets remap mi0-2/mo0-1 and SVSHAPEs,
  controlled by mm
* **ew** - sets element width override on the Indices
* **SVG** - GPR SVG<<2 to be used for Indexing
* **yx** - 2D reordering to be used if yx=1
* **mm** - mask mode. determines how `rmm` is interpreted.
* **sk** - Dimension skipping enabled
* **XO** - standard 6-bit XO field

*Note: SVd, like SVxd, SVyz and SVzd of `svshape`, are all stored
"off-by-one".  In the assembler
mnemonic the values `1-32` are stored in binary as `0b00000..0b11111`*.

*Note: when `yx=1,sk=0` the second dimension is calculated as
`CEIL(MAXVL/SVd)`*.

When `mm=0`:

* `rmm`, like REMAP.SVme, has bit 0 
  correspond to mi0, bit 1 to mi1, bit 2 to mi2,
  bit 3 to mo0 and bit 4 to mi1
* all SVSHAPEs and the REMAP parts of SVSHAPE are first reset (initialised to zero)
* for each bit set in the 5-bit `rmm`, in order, the first
  as-yet-unset SVSHAPE will be updated
  with the other operands in the instruction, and the REMAP
  SPR set.
* If all 5 bits of `rmm` are set then both mi0 and mo1 use SVSHAPE0.
* SVSTATE persistence bit is cleared
* No other alterations to SVSTATE are carried out

Example 1: if rmm=0b00110 then SVSHAPE0 and SVSHAPE1 are set up,
and the REMAP SPR set so that mi1 uses SVSHAPE0 and mi2
uses mi2.  REMAP.SVme is also set to 0b00110, REMAP.mi1=0
(SVSHAPE0) and REMAP.mi2=1 (SVSHAPE1)

Example 2: if rmm=0b10001 then again SVSHAPE0 and SVSHAPE1
are set up, but the REMAP SPR is set so that mi0 uses SVSHAPE0
and mo1 uses SVSHAPE1. REMAP.SVme=0b10001, REMAP.mi0=0, REMAP.mo1=1

Rough algorithmic form:

    marray = [mi0, mi1, mi2, mo0, mo1]
    idx = 0
    for bit = 0 to 4:
        if not rmm[bit]: continue
        setup(SVSHAPE[idx])
        SVSTATE{marray[bit]} = idx
        idx = (idx+1) modulo 4

When `mm=1`:

* bits 0-2 (MSB0 numbering) of `rmm` indicate an index selecting mi0-mo1
* bits 3-4 (MSB0 numbering) of `rmm` indicate which SVSHAPE 0-3 shall
  be updated
* only the selected SVSHAPE is overwritten
* only the relevant bits in the REMAP area of SVSTATE are updated
* REMAP persistence bit is set.

Example 1: if `rmm`=0b01110 then bits 0-2 (MSB0) are 0b011 and
bits 3-4 are 0b10. thus, mo0 is selected and SVSHAPE2
to be updated. REMAP.SVme[3] will be set high and REMAP.mo0
set to 2 (SVSHAPE2).

Example 2: if `rmm`=0b10011 then bits 0-2 (MSB0) are 0b100 and
bits 3-4 are 0b11.  thus, mo1 is selected and SVSHAPE3
to be updated. REMAP.SVme[4] will be set high and REMAP.mo1
set to 3 (SVSHAPE3).

Rough algorithmic form:

    marray = [mi0, mi1, mi2, mo0, mo1]
    bit = rmm[0:2]
    idx = rmm[3:4]
    setup(SVSHAPE[idx])
    SVSTATE{marray[bit]} = idx
    SVSTATE.pst = 1

In essence, `mm=0` is intended for use to set as much of the
REMAP State SPRs as practical with a single instruction,
whilst `mm=1` is intended to be a little more refined.

**Usage guidelines**

* **Disable 2D mapping**: to only perform Indexing without
 reordering use `SVd=1,sk=0,yx=0` (or set SVd to a value larger
 or equal to VL)
* **Modulo 1D mapping**: to perform Indexing cycling through the
 first N Indices use `SVd=N,sk=0,yx=0` where `VL>N`. There is
 no requirement to set VL equal to a multiple of N.
* **Modulo 2D transposed**: `SVd=M,sk=0,yx=1`, sets
 `xdim=M,ydim=CEIL(MAXVL/M)`.

Beyond these mappings it becomes necessary to write directly to
the SVSTATE SPRs manually.

# svshape2 (offset) <a name="svshape2"> </a>

`svshape2` is an additional convenience instruction that prioritises
setting `SVSHAPE.offset`. Its primary purpose is for use when
element-width overrides are used. It has identical capabilities to `svindex` and
in terms of both options (skip, etc.) and ability to activate REMAP
(rmm, mask mode) but unlike `svindex` it does not set GPR REMAP,
only a 1D or 2D `svshape`, and
unlike `svshape` it can set an arbirrary `SVSHAPE.offset` immediate.

One of the limitations of Simple-V is that Vector elements start on the boundary
of the Scalar regfile, which is fine when element-width overrides are not
needed. If the starting point of a Vector with smaller elwidths must begin
in the middle of a register, normally there would be no way to do so except
through LD/ST.  `SVSHAPE.offset` caters for this scenario and `svshape2`is
makes it easier.

    svshape2 offs,yx,rmm,SVd,sk,mm

| 0.5|6..9|10|11.15  |16..20 | 21..25 | 25 | 26..31|  name    |
| -- |----|--| ---   | ----- | ------ | -- | ------| -------- |
|OPCD|offs|yx| rmm   | SVd   | 100/mm | sk | XO    | svshape  |

* **offs** (4 bits) - unsigned offset
* **yx** (1 bit) - swap XY to YX
* **SVd** dimension size
* **rmm** REMAP mask
* **mm** mask mode
* **sk** (1 bit) skips 1st dimension if set

Dimensions are calculated exactly as `svindex`. `rmm` and
`mm` are as per `svindex`.  

*Programmer's Note: offsets for `svshape2` may be specified in the range
0-15. Given that the principle of Simple-V is to fit on top of
byte-addressable register files and that GPR and FPR are 64-bit (8 bytes)
it should be clear that the offset may, when `elwidth=8`, begin an
element-level operation starting element zero at any arbitrary byte.
On cursory examination attempting to go beyond the range 0-7 seems
unnecessary given that the **next GPR or FPR** is an
alias for an offset in the range 8-15.  Thus by simply increasing
the starting Vector point of the operation to the next register it
can be seen that the offset of 0-7 would be sufficient.  Unfortunately
however some operations are EXTRA2-encoded it is **not possible**
to increase the GPR/FPR register number by one, because EXTRA2-encoding
of GPR/FPR Vector numbers are restricted to even numbering.
For CR Fields the EXTRA2 encoding is even more sparse.
The additional offset range (8-15) helps overcome these limitations.*

*Hardware Implementor's note: with the offsets only being immediates
and with register numbering being entirely immediate as well it is
possible to correctly compute Register Hazards without requiring
reading the contents of any SPRs.  If however there are
instructions that have directly written to the SVSTATE or SVSHAPE
SPRs and those instructions are still in-flight then this position
is clearly **invalid**.*

# TODO

* investigate https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6879380/#!po=19.6429
in https://bugs.libre-soc.org/show_bug.cgi?id=653
* UTF-8 <https://bugs.libre-soc.org/show_bug.cgi?id=794>
* Triangular REMAP
* Cross-Product REMAP (actually, skew Matrix: https://en.m.wikipedia.org/wiki/Skew-symmetric_matrix)
* Convolution REMAP