# NOTE

This section is under revision (and is optional)

# REMAP CSR

There is one 32-bit CSR which may be used to indicate which registers, if used in any operation, must be "reshaped" (re-mapped) from a linear form to a 2D or 3D transposed form, or "offset" to permit arbitrary access to elements within a register.

Their primary use is for Matrix Multiplication, reordering of sequential data in-place. Three CSRs are provided so that a single FMAC may be used in a single loop to perform 4x4 times 4x4 Matrix multiplication, generating 64 FMACs.

The 32-bit REMAP CSR may reshape up to 3 registers:

29..28 | 27..26 | 25..24 | 23 | 22..16 | 15 | 14..8 | 7 | 6..0 |
---|---|---|---|---|---|---|---|---|

shape2 | shape1 | shape0 | 0 | regidx2 | 0 | regidx1 | 0 | regidx0 |

regidx0-2 refer not to the Register CSR CAM entry but to the underlying
*real* register (see regidx, the value) and consequently is 7-bits wide.
When set to zero (referring to x0), clearly reshaping x0 is pointless,
so is used to indicate "disabled".
shape0-2 refers to one of three SHAPE CSRs. A value of 0x3 is reserved.
Bits 7, 15, 23, 30 and 31 are also reserved, and must be set to zero.

It is anticipated that these specialist CSRs not be very often used. Unlike the CSR Register and Predication tables, the REMAP CSRs use the full 7-bit regidx so that they can be set once and left alone, whilst the CSR Register entries pointing to them are disabled, instead.

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

There are three "shape" CSRs, SHAPE0, SHAPE1, SHAPE2, 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..24 | 23..21 | 20..18 | 17..12 | 11..6 | 5..0 |
---|---|---|---|---|---|---|

applydim | offset | invxyz | permute | zdimsz | ydimsz | xdimsz |

applydim will set to zero the dimensions less than this. applydim=0 applies all three. applydim=1 applies y and z. applydim=2 applys only z. applydim=3 is reserved.

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 equivalent to the sequential element loop to appear to run for offset (additional) iterations prior to actually generating output.

xdimsz, ydimsz and zdimsz are offset by 1, such that a value of 0 indicates that the array dimensionality for that dimension is 1. A value of xdimsz=2 would indicate that in the first dimension there are 3 elements in the array. The format of the array is therefore as follows:

```
array[xdim+1][ydim+1][zdim+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 being reserved, and not legal). 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) |

In other words, the "permute" option changes the order in which nested for-loops over the array would be done.

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

```
xdim = 3
ydim = 4
zdim = 1
lims = [xdim, ydim, zdim]
idxs = [0,0,0] # starting indices
order = [0,1,2] # experiment with different permutations, here
offset = 2 # experiment with different offset, here
VL = xdim * ydim * zdim # multiply (or add) to this to get "cycling"
applydim = 0
invxyz = [0,0,0]
# run for offset iterations before actually starting
for idx in range(offset):
for i in range(3):
idxs[order[i]] = idxs[order[i]] + 1
if (idxs[order[i]] != lims[order[i]]):
break
idxs[order[i]] = 0
break_count = 0
for idx in range(VL):
ix = [0] * 3
for i in range(3):
if i >= applydim:
ix[i] = idxs[i]
if invxyz[i]:
ix[i] = lims[i] - 1 - ix[i]
new_idx = ix[0] + ix[1] * xdim + ix[2] * xdim * ydim
print new_idx,
break_count += 1
if break_count == lims[order[0]]:
print
break_count = 0
for i in range(3):
idxs[order[i]] = idxs[order[i]] + 1
if (idxs[order[i]] != lims[order[i]]):
break
idxs[order[i]] = 0
```

Here, it is assumed that this algorithm be run within all pseudo-code
throughout this document where a (parallelism) for-loop would normally
run from 0 to VL-1 to refer to contiguous register
elements; instead, where REMAP indicates to do so, the element index
is run through the above algorithm to work out the **actual** element
index, instead. Given that there are three possible SHAPE entries, up to
three 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+remap(id)] <= ireg[rs1+remap(irs1)] +
ireg[rs2+remap(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. Also, the reason for having REMAP separate from the three SHAPE CSRs is so that in a chain of matrix multiplications and additions, for example, the SHAPE CSRs need only be set up once; only the REMAP CSR need be changed to target different registers.

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 elements
*within*registers rather than entire registers). - If permute option 000 is utilised, the actual order of the reindexing does not change!
- 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 MSTATE CSR, which records the current element index. - 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 C.LDSP*and other operations in that category, where in that case it will be the**offset**that is remapped (see Compressed Stack LOAD/STORE section). - 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.

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.

# 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 CSRs, and applying a rotating 1D SHAPE CSR of xdim=16 to f8 in order to get it to apply four times to compute the four columns worth of vectors.