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[[!tag standards]]

# REMAP <a name="remap" />

see [[sv/propagation]] because it is currently the only way to apply
REMAP.  

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, independently on each Vector src or dest
register.

Their primary use is for Matrix Multiplication, reordering of sequential
data in-place.  Four SPRs are provided so that a single FMAC may be
used in a single loop to perform 4x4 times 4x4 Matrix multiplication,
generating 64 FMACs.  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 typically), REMAP may be applied to
literally any instruction: CRs, Arithmetic, Logical, LD/ST, anything.

Note that REMAP does not apply to sub-vector elements: that is what
swizzle is for.  Swizzle *can* however be applied to the same instruction
as REMAP.

REMAP is quite expensive to set up, and on some implementations introduce
latency, so should realistically be used only where it is worthwhile

# Principle

* normal vector element read/write as 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)
* REMAP **redefines** the order of access according to set "Schedules"

Only the most commonly-used algorithms in computer science have REMAP
support, due to the high cost in both the ISA and in hardware.

# REMAP SPR

| 0  | 2  | 4  | 6  | 8  | 10.14 | 15..23 |
| -- | -- | -- | -- | -- | ----- | ------ |
|mi0 |mi1 |mi2 |mo0 |mo1 | SVme  | rsv    |

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 (FRT, FRS respectively).
SVme is 5 bits, and indicates indicate 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
  (LD/ST-with-update has an implicit 2nd write register, RA)

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

    |0     |6     |11  |13   |15   |17   |19   |21    | 22    |26     |31 |
    | PO   | SVme |mi0 | mi1 | mi2 | mo0 | mo1 | pst  | rsvd | XO    | / |

# SHAPE 1D/2D/3D vector-matrix remapping SPRs

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

[[!inline raw="yes" pages="openpower/sv/shape_table_format" ]]

The algorithm below shows how REMAP works more clearly, and may be
executed as a python program:

```
[[!inline quick="yes" raw="yes" pages="openpower/sv/remap.py" ]]
```

An easier-to-read version (using python iterators) shows the loop nesting:

```
[[!inline quick="yes" raw="yes" pages="openpower/sv/remapyield.py" ]]
```

Each element index from the for-loop `0..VL-1`
is run through the above algorithm to work out the **actual** element
index, instead.  Given that there are four possible SHAPE entries, up to
four separate registers in any given operation may be simultaneously
remapped:

    function op_add(rd, rs1, rs2) # add not VADD!
      ...
      ...
      for (i = 0; i < VL; i++)
        xSTATE.srcoffs = i # save context
        if (predval & 1<<i) # predication uses intregs
           ireg[rd+remap1(id)] <= ireg[rs1+remap2(irs1)] +
                                  ireg[rs2+remap3(irs2)];
           if (!int_vec[rd ].isvector) break;
        if (int_vec[rd ].isvector)  { id += 1; }
        if (int_vec[rs1].isvector)  { irs1 += 1; }
        if (int_vec[rs2].isvector)  { irs2 += 1; }

By changing remappings, 2D matrices may be transposed "in-place" for one
operation, followed by setting a different permutation order without
having to move the values in the registers to or from memory.

Note that:

* Over-running the register file clearly has to be detected and
  an illegal instruction exception thrown
* When non-default elwidths are set, the exact same algorithm still
  applies (i.e. it offsets *polymorphic* elements *within* registers rather 
  than entire registers).
* If permute option 000 is utilised, the actual order of the
  reindexing does not change.  However, modulo MVL still occurs
  which will result in repeated operations (use with caution).
* If two or more dimensions are set to zero, the actual order does not change!
* The above algorithm is pseudo-code **only**.  Actual implementations
  will need to take into account the fact that the element for-looping
  must be **re-entrant**, due to the possibility of exceptions occurring.
  See SVSTATE SPR, which records the current element index.
  Continuing after return from an interrupt may introduce latency
  due to re-computation of the remapped offsets.
* Twin-predicated operations require **two** separate and distinct
  element offsets.  The above pseudo-code algorithm will be applied
  separately and independently to each, should each of the two
  operands be remapped.  *This even includes unit-strided LD/ST*
  and other operations
  in that category, where in that case it will be the **offset** that is
  remapped.
* Offset is especially useful, on its own, for accessing elements
  within the middle of a register.  Without offsets, it is necessary
  to either use a predicated MV, skipping the first elements, or
  performing a LOAD/STORE cycle to memory.
  With offsets, the data does not have to be moved.
* Setting the total elements (xdim+1) times (ydim+1) times (zdim+1) to
  less than MVL is **perfectly legal**, albeit very obscure.  It permits
  entries to be regularly presented to operands **more than once**, thus
  allowing the same underlying registers to act as an accumulator of
  multiple vector or matrix operations, for example.
* Note especially that Program Order **must** still be respected
  even when overlaps occur that read or write the same register
  elements *including polymorphic ones*

Clearly here some considerable care needs to be taken as the remapping
could hypothetically create arithmetic operations that target the
exact same underlying registers, resulting in data corruption due to
pipeline overlaps.  Out-of-order / Superscalar micro-architectures with
register-renaming will have an easier time dealing with this than
DSP-style SIMD micro-architectures.

## svstate instruction

Please note: this is **not** intended for production.  It sets up
(overwrites) all required SVSHAPE SPRs and indicates that the
*next instruction* shall have those REMAP shapes applied to it,
assuming that instruction is of the form FRT,FRA,FRC,FRB.

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

| 0.5|6.10  |11.15  |16..20 | 21..25 | 25 | 26..30 |31|  name    |
| -- | --   | ---   | ----- | ------ | -- | ------ |--| -------- |
|OPCD| SVxd | SVyd  | SVzd  | SVRM   | vf | XO     |/ | svstate  |


Fields:

* **SVxd** - SV REMAP "xdim"
* **SVyd** - SV REMAP "ydim"
* **SVMM** - SV REMAP Mode (0b00000 for Matrix, 0b00001 for FFT)
* **vf** - sets "Vertical-First" mode
* **XO** - standard 5-bit XO field

# 4x4 Matrix to vec4 Multiply Example

The following settings will allow a 4x4 matrix (starting at f8), expressed
as a sequence of 16 numbers first by row then by column, to be multiplied
by a vector of length 4 (starting at f0), using a single FMAC instruction.

* SHAPE0: xdim=4, ydim=4, permute=yx, applied to f0
* SHAPE1: xdim=4, ydim=1, permute=xy, applied to f4
* VL=16, f4=vec, f0=vec, f8=vec
* FMAC f4, f0, f8, f4

The permutation on SHAPE0 will use f0 as a vec4 source. On the first
four iterations through the hardware loop, the REMAPed index will not
increment. On the second four, the index will increase by one. Likewise
on each subsequent group of four.

The permutation on SHAPE1 will increment f4 continuously cycling through
f4-f7 every four iterations of the hardware loop.

At the same time, VL will, because there is no SHAPE on f8, increment
straight sequentially through the 16 values f8-f23 in the Matrix. The
equivalent sequence thus is issued:

    fmac f4, f0, f8, f4
    fmac f5, f0, f9, f5
    fmac f6, f0, f10, f6
    fmac f7, f0, f11, f7
    fmac f4, f1, f12, f4
    fmac f5, f1, f13, f5
    fmac f6, f1, f14, f6
    fmac f7, f1, f15, f7
    fmac f4, f2, f16, f4
    fmac f5, f2, f17, f5
    fmac f6, f2, f18, f6
    fmac f7, f2, f19, f7
    fmac f4, f3, f20, f4
    fmac f5, f3, f21, f5
    fmac f6, f3, f22, f6
    fmac f7, f3, f23, f7

The only other instruction required is to ensure that f4-f7 are
initialised (usually to zero).

It should be clear that a 4x4 by 4x4 Matrix Multiply, being effectively
the same technique applied to four independent vectors, can be done by
setting VL=64, using an extra dimension on the SHAPE0 and SHAPE1 SPRs,
and applying a rotating 1D SHAPE SPR of xdim=16 to f8 in order to get
it to apply four times to compute the four columns worth of vectors.

# Warshall transitive closure algorithm

TODO move to [[sv/remap/discussion]] page, copied from here
http://lists.libre-soc.org/pipermail/libre-soc-dev/2021-July/003286.html

with thanks to Hendrik.

> Just a note:  interpreting + as 'or', and * as 'and',
> operating on Boolean matrices, 
> and having result, X, and Y be the exact same matrix,
> updated while being used,
> gives the traditional Warshall transitive-closure
> algorithm, if the loops are nested exactly in thie order.

this can be done with the ternary instruction which has
an in-place triple boolean input:

    RT = RT | (RA & RB)

and also has a CR Field variant of the same

notes from conversations:

> > for y in y_r:
> >  for x in x_r:
> >    for z in z_r:
> >      result[y][x] +=
> >         a[y][z] *
> >         b[z][x]

> This nesting of loops works for matrix multiply, but not for transitive
> closure. 

> > it can be done:
> >
> >   for z in z_r:
> >    for y in y_r:
> >     for x in x_r:
> >       result[y][x] +=
> >          a[y][z] *
> >          b[z][x]
>
> And this ordering of loops *does* work for transitive closure, when a,
> b, and result are the very same matrix, updated while being used.
>
> By the way, I believe there is a graph algorithm that does the
> transitive closure thing, but instead of using boolean, "and", and "or",
> they use real numbers, addition, and minimum.  I think that one computes
> shortest paths between vertices.
>
> By the time the z'th iteration of the z loop begins, the algorithm has
> already peocessed paths that go through vertices numbered < z, and it
> adds paths that go through vertices numbered z.
>
> For this to work, the outer loop has to be the one on teh subscript that
> bridges a and b (which in this case are teh same matrix, of course).

# SUBVL Remap

Remapping even of SUBVL (vec2/3/4) elements is permitted, as if the
sub-vectir elements were simply part of the main VL loop.  This is the
*complete opposite* of predication which **only** applies to the whole
vec2/3/4.  In pseudocode this would be:

      for (i = 0; i < VL; i++)
        if (predval & 1<<i) # apply to VL not SUBVL
          for (j = 0; j < SUBVL; j++)
             id = i*SUBVL + j # not, "id=i".
             ireg[RT+remap1(id)] ...

The reason for allowing SUBVL Remaps is that some regular patterns using
Swizzle which would otherwise require multiple explicit instructions
with 12 bit swizzles encoded in them may be efficently encoded with Remap
instead.  Not however that Swizzle is *still permitted to be applied*.

An example where SUBVL Remap is appropriate is the Rijndael MixColumns
stage:

<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/76/AES-MixColumns.svg/600px-AES-MixColumns.svg.png" width="400px" />

Assuming that the bytes are stored `a00 a01 a02 a03 a10 .. a33`
a 2D REMAP allows:

* the column bytes (as a vec4) to be iterated over as an inner loop,
  progressing vertically (`a00 a10 a20 a30`)
* the columns themselves to be iterated as an outer loop
* a 32 bit `GF(256)` Matrix Multiply on the vec4 to be performed.

This entirely in-place without special 128-bit opcodes.  Below is
the pseudocode for [[!wikipedia Rijndael MixColumns]]

```
void gmix_column(unsigned char *r) {
    unsigned char a[4];
    unsigned char b[4];
    unsigned char c;
    unsigned char h;
    // no swizzle here but still SUBVL.Remap
    // can be done as vec4 byte-level
    // elwidth overrides though.
    for (c = 0; c < 4; c++) {
        a[c] = r[c];
        h = (unsigned char)((signed char)r[c] >> 7);
        b[c] = r[c] << 1;
        b[c] ^= 0x1B & h; /* Rijndael's Galois field */
    }
    // SUBVL.Remap still needed here
    // bytelevel elwidth overrides and vec4
    // These may then each be 4x 8bit bit Swizzled
    // r0.vec4 = b.vec4
    // r0.vec4 ^= a.vec4.WXYZ
    // r0.vec4 ^= a.vec4.ZWXY
    // r0.vec4 ^= b.vec4.YZWX ^ a.vec4.YZWX
    r[0] = b[0] ^ a[3] ^ a[2] ^ b[1] ^ a[1];
    r[1] = b[1] ^ a[0] ^ a[3] ^ b[2] ^ a[2];
    r[2] = b[2] ^ a[1] ^ a[0] ^ b[3] ^ a[3]; 
    r[3] = b[3] ^ a[2] ^ a[1] ^ b[0] ^ a[0];
}
```

With the assumption made by the above code that the column bytes have
already been turned around (vertical rather than horizontal) SUBVL.REMAP
may transparently fill that role, in-place, without a complex byte-level
mv operation.

The application of the swizzles allows the remapped vec4 a, b and r
variables to perform four straight linear 32 bit XOR operations where a
scalar processor would be required to perform 16 byte-level individual
operations.  Given wide enough SIMD backends in hardware these 3 bit
XORs may be done as single-cycle operations across the entire 128 bit
Rijndael Matrix.

The other alternative is to simply perform the actual 4x4 GF(256) Matrix
Multiply using the MDS Matrix.

# 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