#### integer floats with remainder theory - copyright RS

The relevance of integer floats is that we can do 2 things:Float on Integer instruction sets at half resolution 32Bit Int > 16Bit.16Bit > 24Bit.8Bit > 28Bit.4BitRemainder theorem is the capacity to convert back and forth with data.

RAM/Memory and hard drive storage are major components & we also need to consider compression & color formats like DOT5

Machine learning & Server improvement: Files attached please utilise.

https://is.gd/ProcessorLasso

https://is.gd/3DMLSorcerer

### Integer_Float Remainder op Form 1:(c)RS

One being the integer and the remainder being the floating component....

thusly we need two integers per float for example 2 32bit integers will make one single float instruction....

integer A : Remainder B

A + B = float

(A + B) x (A²+B²)

= float C dislocating A and B by a certain number of places = a float that travels as the integer.

Expansion data sets:

A1 : B1

A2 : B2

Ar : Br

F1 : Bf1

F2 : Bf2

Fr : Bfr

A : Integer

F : Float

r : Remainder

The data set expansion can be infinite and the expansion of the data set doubles the precision,

With the remainder... infinite computation = infinite precision.

Not only that but the computation can be executed as an Integer or as a float or indeed with both.

Relevance is that on computers there are a lot of integer registers; Float also..

Also the data can be compressed in ram without using larger buffer widths.

copyright Rupert Summerskill

#### COP-Roll : (c)Rupert S

#### ROLL Operation Syntax : RS :(Integer & Float)

Processing Cache (displacement) Operation Roll Arithmetic Maths : For

Multiplication, Division, Addition & Subtraction : P-COR-SAM

*

Addressable by Compiler update, Firmware update, CPU/GPU Rules

Firmware, Bios, Operating System & Program/Machine Learning.

Machine Learning will considerably enhance Cache & Processor routine

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*

In a single loop a multiply of a float point precision of under 1 for example 0.00001 requires that:

In Integer float :

Multiply of a sum such as 15.05 * 3 is 2 operations:

(15 x 3) + ((roll 0.05 left 2 places)*3) = R=(5 x 3) + 45

In other words : 2 storage values R remainder (the float component) & the number,

However multiplication of a float such as 0.01 is a division in one example & a multiply roll in another,

Roll is a memory operation in CPU terms & is a single processor loop push

In all operations where division is banned we have to decide whether the operation is multiples or division of base value 10 or 1,10,100>,

Such an operation can be carried out by addition or subtraction or roll, Values such as 200* ,

Require multiple additions under the multiply is banned principle.

Multiple sets of memory arrays in a series parallel is the equivalent of multiplication through addition,

Subtraction through addition requires inverting the power phase of a single component array.

Thus we are able to addition and subtraction all sums ? traditional math solves have done this before,

Roll operations are our fast way to multiply;

However arrays of addition & subtraction are a (logical fast loop)..

Full Operation in a single cycle, Because there is no sideways roll.

However direct memory displacement between 010100 & 101000 can use a cache to displace a 1,

Such an arrangement such as a 4 digit displacement cache to roll the operation on memory transfer.

Displace on operation (cycle 2) does minimize operations.

Having that cache further up the binary pipeline does reduce the number of roll cache modifier buffers that we need,

However the time we save & the time we lose & the CPU space we lose or gain.. depends specifically how limited the Roll Cache is.

### Integer_Float Remainder op Form 2:(c)RS

32Bit (2x16Bit) is the most logical for 32Bit registers

64Bit (2x32Bit) is the most logical for 32Bit registers

Byte Swap operation

Byte Inversion operation

For example DWord: 8

2 x DWord: 8 Bit Integer & 8 Bit 4 roll places & 4 Bit Value.

Displacing the value 4 bits in 8 makes the value an integer,

Alternatively Adaptive maths adds 0 as for example multiplication & removes it afterwards..

The usage of adaptation takes the second DWord & effectively makes it an accurate remainder.

In that example i believe one less operation is needed in the 16Bit example,

Operation example 2 uses an embedded multiply x 10 & divide after (to get resulting float)

32Bit memory space: 2x 16Bit Value, 1 Integer 16Bit & 1 0. value,

That can effectively be displaced 16 decimal places

The maths required as displayed above require inverting Multiply & Division,

For Mul & Div Ops on remainder; However does not when used finally:

In the FLOAT Unit FPU for large precision maths

This allows fully Integer CPU to do Float maths and store them as integer..

Both allowing fully the use of all registers & also storage as purely Integer_Float,

It also allows Full cache usage for SiMD,AVX & Vector Units.

Byte Inversion simply allows Byte Swap & Inversion to fully realise performance improvements..

& Also Byte Inversion maths.

SiMD,AVX,Vector : ByteSwap,Invert,Mul,Div etcetera Ergo Float compatible & Acceleration

Float : High Precision finalisation .. Lower Frequency = More potential

Integer + Byte Functions : Pure Acceleration with minimal loss Core Function utilisation

This is all algebra; Categorically.

(c) Rupert S https://science.n-helix.com

Optimisation & Use:

https://science.n-helix.com/2021/02/multi-operation-maths.html

https://science.n-helix.com/2019/05/compiler-optimisation.html

https://science.n-helix.com/2020/01/float-hlsl-spir-v-compiler-role.html

Compiler books & reading : https://science.n-helix.com/2017/04/boinc.html

https://science.n-helix.com/2019/06/kernel.html

https://science.n-helix.com/2019/06/vulkan-stack.html

http://science.n-helix.com/2018/09/hpc-pack-install-guide.html

https://science.n-helix.com/2019/05/compiler-optimisation.html

https://science.n-helix.com/2020/01/float-hlsl-spir-v-compiler-role.html

Compiler books & reading : https://science.n-helix.com/2017/04/boinc.html

https://science.n-helix.com/2019/06/kernel.html

https://science.n-helix.com/2019/06/vulkan-stack.html

http://science.n-helix.com/2018/09/hpc-pack-install-guide.html

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