# Introduction

Although best used with SVP64 REMAP these instructions may be used in a Scalar-only context to save considerably on DCT, DFT and FFT processing. Whilst some hardware implementations may not necessarily implement them efficiently (slower Micro-coding) savings still come from the reduction in temporary registers as well as instruction count.

# Rationale for Twin Butterfly Integer DCT Instruction(s)

The number of general-purpose uses for DCT is huge. The number of instructions needed instead of these Twin-Butterfly instructions is also huge (eight) and given that it is extremely common to explicitly loop-unroll them quantity hundreds to thousands of instructions are dismayingly common (for all ISAs).

The goal is to implement instructions that calculate the expression:

``````    fdct_round_shift((a +/- b) * c)
``````

For the single-coefficient butterfly instruction, and:

``````    fdct_round_shift(a * c1  +/- b * c2)
``````

For the double-coefficient butterfly instruction.

In a 32-bit context `fdct_round_shift` is defined as `ROUND_POWER_OF_TWO(x, 14)`

``````    #define ROUND_POWER_OF_TWO(value, n) \
(((value) + (1 << ((n)-1))) >> (n))
``````

These instructions are at the core of ALL FDCT calculations in many major video codecs, including -but not limited to- VP8/VP9, AV1, etc. ARM includes special instructions to optimize these operations, although they are limited in precision: `vqrdmulhq_s16`/`vqrdmulhq_s32`.

The suggestion is to have a single instruction to calculate both values `((a + b) * c) >> N`, and `((a - b) * c) >> N`. The instruction will run in accumulate mode, so in order to calculate the 2-coeff version one would just have to call the same instruction with different order a, b and a different constant c.

``````    #include <stdint.h>
#define ROUND_POWER_OF_TWO(value, n) \
(((value) + (1 << ((n)-1))) >> (n))
void twin_int(int16_t *t, int16_t x0, int16_t x1, int16_t cospi_16_64) {
t = ROUND_POWER_OF_TWO((x0 + x1) * cospi_16_64, 14);
t = ROUND_POWER_OF_TWO((x0 - x1) * cospi_16_64, 14);
}
``````

8 instructions are required - replaced by just the one (maddsubrs):

``````    add 9,5,4
subf 5,5,4
mullw 9,9,6
mullw 5,5,6
srawi 9,9,14
srawi 5,5,14
``````

\newpage{}

## Integer Butterfly Multiply Add/Sub FFT/DCT

Add the following to Book I Section 3.3.9.1

A-Form

``````    |0     |6     |11      |16     |21      |26    |31 |
| PO   |  RT  |   RA   |   RB  |   SH   |   XO |Rc |
``````

Pseudo-code:

``````    n <- SH
sum <- (RT || RT) + (RA || RA)
diff <- (RT || RT) - (RA || RA)
prod1 <- MULS(RB, sum)
prod2 <- MULS(RB, diff)
if n = 0 then
prod1_lo <- prod1[XLEN+1:(XLEN*2)]
prod2_lo <- prod2[XLEN+1:(XLEN*2)]
RT <- prod1_lo
RS <- prod2_lo
else
round <- *(XLEN*2 + 1)
round[XLEN*2 - n + 1] <- 1
prod1 <- prod1 + round
prod2 <- prod2 + round
res1 <- prod1[XLEN - n + 1:XLEN*2 - n]
res2 <- prod2[XLEN - n + 1:XLEN*2 - n]
RT <- res1
RS <- res2
``````

Similar to `RTp`, this instruction produces an implicit result, `RS`, which under Scalar circumstances is defined as `RT+1`. For SVP64 if `RT` is a Vector, `RS` begins immediately after the Vector `RT` where the length of `RT` is set by `SVSTATE.MAXVL` (Max Vector Length).

Special Registers Altered:

``````    None
``````

# [DRAFT] Integer Butterfly Multiply Add and Round Shift FFT/DCT

A-Form

Pseudo-code:

``````    n <- SH
prod <- MULS(RB, RA)
if n = 0 then
prod_lo <- prod[XLEN:(XLEN*2) - 1]
RT <- (RT) + prod_lo
else
res[0:XLEN*2-1] <- (EXTSXL((RT), 1) || (RT)) + prod
round <- *XLEN*2
round[XLEN*2 - n] <- 1
res <- res + round
RT <- res[XLEN - n:XLEN*2 - n -1]
``````

Special Registers Altered:

``````None
``````

# [DRAFT] Integer Butterfly Multiply Sub and Round Shift FFT/DCT

A-Form

• msubrs RT,RA,RB,SH

Pseudo-code:

``````    n <- SH
prod <- MULS(RB, RA)
if n = 0 then
prod_lo <- prod[XLEN:(XLEN*2) - 1]
RT <- (RT) - prod_lo
else
res[0:XLEN*2-1] <- (EXTSXL((RT), 1) || (RT)) - prod
round <- *XLEN*2
round[XLEN*2 - n] <- 1
res <- res + round
RT <- res[XLEN - n:XLEN*2 - n -1]
``````

Special Registers Altered:

``````None
``````

This pair of instructions is supposed to be used in complement to the maddsubrs to produce the double-coefficient butterfly instruction. In order for that to work, instead of passing c2 as coefficient, we have to pass c2-c1 instead.

In essence, we are calculating the quantity `a * c1 +/- b * c1` first, with `maddsubrs` without shifting (so `SH=0`) and then we add/sub `b * (c2-c1)` from the previous `RT`, and then do the shifting.

In the following example, assume `a` in `R1`, `b` in `R10`, `c1` in `R11` and `c2 - c1` in `R12`. The first instruction will put `a * c1 + b * c1` in `R1` (`RT`), `a * c1 - b * c1` in `RS` (here, `RS = RT +1`, so `R2`). Then, `maddrs` will add `b * (c2 - c1)` to `R1` (`RT`), and `msubrs` will subtract it from `R2` (`RS`), and then round shift right both quantities 14 bits:

``````    maddsubrs 1,10,0,11
msubrs 2,10,12,14
``````

In scalar code, that would take ~16 instructions for both operations.

\newpage{}

# Twin Butterfly Floating-Point DCT and FFT Instruction(s)

Add the following to Book I Section 4.6.6.3

## Floating-Point Twin Multiply-Add DCT [Single]

X-Form

``````    |0     |6     |11      |16     |21      |31 |
| PO   |  FRT |  FRA   |  FRB  |   XO   |Rc |
``````

Pseudo-code:

``````    FRS <- FPADD32(FRT, FRB)
sub <- FPSUB32(FRT, FRB)
FRT <- FPMUL32(FRA, sub)
``````

The two IEEE754-FP32 operations

``````    FRS <- [(FRT) + (FRB)]
FRT <- [(FRT) - (FRB)] * (FRA)
``````

are simultaneously performed.

The Floating-Point operand in register FRT is added to the floating-point operand in register FRB and the result stored in FRS.

Using the exact same operand input register values from FRT and FRB that were used to create FRS, the Floating-Point operand in register FRB is subtracted from the floating-point operand in register FRT and the result then rounded before being multiplied by FRA to create an intermediate result that is stored in FRT.

The add into FRS is treated exactly as `fadds`. The creation of the result FRT is not the same as that of `fmsubs`, but is instead as if `fsubs` were performed first followed by `fmuls`. The creation of FRS and FRT are treated as parallel independent operations which occur at the same time.

Note that if Rc=1 an Illegal Instruction is raised. Rc=1 is `RESERVED`

Similar to `FRTp`, this instruction produces an implicit result, `FRS`, which under Scalar circumstances is defined as `FRT+1`. For SVP64 if `FRT` is a Vector, `FRS` begins immediately after the Vector `FRT` where the length of `FRT` is set by `SVSTATE.MAXVL` (Max Vector Length).

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI VXIMZ
``````

X-Form

``````    |0     |6     |11      |16     |21      |31 |
| PO   |  FRT |  FRA   |  FRB  |   XO   |Rc |
``````

Pseudo-code:

``````    FRS <- FPMULADD32(FRT, FRA, FRB, -1, 1)
FRT <- FPMULADD32(FRT, FRA, FRB, 1, 1)
``````

The two operations

``````    FRS <- -([(FRT) * (FRA)] - (FRB))
FRT <-   [(FRT) * (FRA)] + (FRB)
``````

are performed.

The floating-point operand in register FRT is multiplied by the floating-point operand in register FRA. The floating-point operand in register FRB is added to this intermediate result, and the intermediate stored in FRS.

Using the exact same values of FRT, FRT and FRB as used to create FRS, the floating-point operand in register FRT is multiplied by the floating-point operand in register FRA. The floating-point operand in register FRB is subtracted from this intermediate result, and the intermediate stored in FRT.

FRT is created as if a `fmadds` operation had been performed. FRS is created as if a `fnmsubs` operation had simultaneously been performed with the exact same register operands, in parallel, independently, at exactly the same time.

Note that if Rc=1 an Illegal Instruction is raised. Rc=1 is `RESERVED`

Similar to `FRTp`, this instruction produces an implicit result, `FRS`, which under Scalar circumstances is defined as `FRT+1`. For SVP64 if `FRT` is a Vector, `FRS` begins immediately after the Vector `FRT` where the length of `FRT` is set by `SVSTATE.MAXVL` (Max Vector Length).

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI VXIMZ
``````

X-Form

``````    |0     |6     |11      |16     |21      |31 |
| PO   |  FRT |  FRA   |  FRB  |   XO   |Rc |
``````

Pseudo-code:

``````    FRS <- FPADD64(FRT, FRB)
sub <- FPSUB64(FRT, FRB)
FRT <- FPMUL64(FRA, sub)
``````

The two IEEE754-FP64 operations

``````    FRS <- [(FRT) + (FRB)]
FRT <- [(FRT) - (FRB)] * (FRA)
``````

are simultaneously performed.

The Floating-Point operand in register FRT is added to the floating-point operand in register FRB and the result stored in FRS.

Using the exact same operand input register values from FRT and FRB that were used to create FRS, the Floating-Point operand in register FRB is subtracted from the floating-point operand in register FRT and the result then rounded before being multiplied by FRA to create an intermediate result that is stored in FRT.

The add into FRS is treated exactly as `fadd`. The creation of the result FRT is not the same as that of `fmsub`, but is instead as if `fsub` were performed first followed by `fmuls. The creation of FRS and FRT are treated as parallel independent operations which occur at the same time.

Note that if Rc=1 an Illegal Instruction is raised. Rc=1 is `RESERVED`

Similar to `FRTp`, this instruction produces an implicit result, `FRS`, which under Scalar circumstances is defined as `FRT+1`. For SVP64 if `FRT` is a Vector, `FRS` begins immediately after the Vector `FRT` where the length of `FRT` is set by `SVSTATE.MAXVL` (Max Vector Length).

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI VXIMZ
``````

X-Form

``````    |0     |6     |11      |16     |21      |31 |
| PO   |  FRT |  FRA   |  FRB  |   XO   |Rc |
``````

Pseudo-code:

``````    FRS <- FPMULADD64(FRT, FRA, FRB, -1, 1)
FRT <- FPMULADD64(FRT, FRA, FRB, 1, 1)
``````

The two operations

``````    FRS <- -([(FRT) * (FRA)] - (FRB))
FRT <-   [(FRT) * (FRA)] + (FRB)
``````

are performed.

The floating-point operand in register FRT is multiplied by the floating-point operand in register FRA. The float- ing-point operand in register FRB is added to this intermediate result, and the intermediate stored in FRS.

Using the exact same values of FRT, FRT and FRB as used to create FRS, the floating-point operand in register FRT is multiplied by the floating-point operand in register FRA. The float- ing-point operand in register FRB is subtracted from this intermediate result, and the intermediate stored in FRT.

FRT is created as if a `fmadd` operation had been performed. FRS is created as if a `fnmsub` operation had simultaneously been performed with the exact same register operands, in parallel, independently, at exactly the same time.

Note that if Rc=1 an Illegal Instruction is raised. Rc=1 is `RESERVED`

Similar to `FRTp`, this instruction produces an implicit result, `FRS`, which under Scalar circumstances is defined as `FRT+1`. For SVP64 if `FRT` is a Vector, `FRS` begins immediately after the Vector `FRT` where the length of `FRT` is set by `SVSTATE.MAXVL` (Max Vector Length).

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI VXIMZ
``````

A-Form

``````    |0     |6     |11      |16     |21      |26    |31 |
| PO   | FRT  |  FRA   |  FRB  |     /  |   XO |Rc |
``````

Pseudo-code:

``````    FRT <- FPADD32(FRA, FRB)
FRS <- FPSUB32(FRB, FRA)
``````

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI
``````

A-Form

``````    |0     |6     |11      |16     |21      |26    |31 |
| PO   | FRT  |  FRA   |  FRB  |     /  |   XO |Rc |
``````

Pseudo-code:

``````    FRT <- FPADD64(FRA, FRB)
FRS <- FPSUB64(FRB, FRA)
``````

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI
``````

## Floating-Point Subtract FFT/DCT [Single]

A-Form

``````    |0     |6     |11      |16     |21      |26    |31 |
| PO   | FRT  |  FRA   |  FRB  |     /  |   XO |Rc |
``````
• ffsubs FRT,FRA,FRB (Rc=0)

Pseudo-code:

``````    FRT <- FPSUB32(FRB, FRA)
``````

Special Registers Altered:

``````    FPRF FR FI
FX OX UX XX
VXSNAN VXISI
``````

## Floating-Point Subtract FFT/DCT [Double]

A-Form

``````    |0     |6     |11      |16     |21      |26    |31 |
| PO   | FRT  |  FRA   |  FRB  |     /  |   XO |Rc |
``````
• ffsub FRT,FRA,FRB (Rc=0)

Pseudo-code:

``````    FRT <- FPSUB64(FRB, FRA)
``````    FPRF FR FI