On Wednesday, 18 September 2019 at 18:42:32 UTC, Joseph Rushton Wakeling wrote:

>On Saturday, 4 May 2019 at 19:08:10 UTC, Murilo wrote:

>BigInt takes too long. I need something as fast as the primitive types.

What hardware are you planning on running your programs on? I'm not sure how good a speed you can guarantee without native hardware support.

Hello together.

You can also run D code with 64-bit integers on 32-Bit architectures, so where should be problems to have 128-bit on 32-Bit architectures or particularly 64-bit architecture?

In 32-Bit ARM you would do 128-Bit arithmetics like this:

```
c = a + b // r0..r3 = a, r4..r7 = b, r8..r11 = c
ADD r8, r0, r4
ADC r9, r1, r5
ADC r10, r2, r6
ADC r11, r3, r7
c = a - b
SUB r8, r0, r4
SBC r9, r1, r5
SBC r10, r2, r6
SBC r11, r3, r7
//this would work for any multiple of 32-Bit arithmetics and even for any
//multiple of 8-Bit arithmetics since ARM has zero/sign extend instructions for 8-bit
//negation, logical operations, shifts and rotations are similarly easy to implement for 128-bit
c = a * b // result depends on the type of a, b, c actually but assume ucent
UMULL r9, r8, r0, r4 @ c[0..64] = a[0..32] * b[0..32]
UMULL r11, r10, r0, r6 @ c[64..128] = a[0..32] * b[64..96]
UMLAL r11, r10, r1, r5 @ c[64..128] += a[32..64] * b[32..64]
UMLAL r11, r10, r2, r4 @ c[64..128] += a[64..96] * b[0..32]
MLA r11, r0, r7 @ c[96..128] += a[0..32] * b[96..128]
MLA r11, r1, r6 @ c[96..128] += a[32..64] * b[64..96]
MLA r11, r2, r5 @ c[96..128] += a[64..96] * b[32..64]
MLA r11, r0, r7 @ c[96..128] += a[96..128] * b[0..32]
UMLAL r10, r9, r0, r5 // c[32..96] += a[0..32] * b[32..64]
ADC r11, r11, #0 // consider overflow
UMLAL r10, r9, r1, r4 // c[32..96] += a[32..64] * b[0..32]
ADC r11, r11, #0 // consider overflow
```

You maybe can even reduce the multiplication code by one instruction with a smarter solution. Well, I almost suggested using godbolt.org to check, what Assembly code is generated by C# when using 128-Bit but I see that C# is not supported there (but D is :-) ).

Only the division is slightly more complicated. You'd probably inverse the divisor and multiply it to the divident `c = a * (1/b)`

(and only calculating the upper 128-bit of the 256-bit product).

If b is 32-Bit then

```
2^n * 1/b = floor(0xFF..FF/b) + (0xFF..FF % b +1) * 1/b
<=> //using x = floor(0xFF..FF/b), y = 0xFF..FF % b + 1
q = a/b = a/2^n * (x + y/2^n * (x + y/2^n * ( ... )))
//for n = 32 (step size) calculate UQ128 q as follows
q = x << 96
q += y * x << 64
q += y² * x << 32
q += y³ * x + (y⁴ * x >> 32) + (y⁵ * x >> 64) ... //until nothing changes
q = (q * a)[128..256] //getting the upper 128-bit of 256-bit result
1/b = x * (1 + y/2^n + (y/2^n)² + (y/2^n)³ + ...)
= x * ((y/2^n)^-m - 1) / ((y/2^n)^-1 - 1)
//q = r0..r3
UDIV(x, 0xFF..FF, divisor) //x = floor(0xFF..FF / b)
MLS(y, x, divisor, x) //= y = x - x * divisor = 0xFF..FF % b
ADD(y, y, 1)
... //multiplications and additions
```

This algorithm is simple but has Worst-case execution time O(n) where n is the bit length which are a lot of multiplications. The result of 1/b is not perfectly accurate since it divides 0.FF..FF as divident and not 1.0 and the bigger y, the slower is this algorithm. But as I look at the worst case, I notice optimization potential:

```
q = x * ((1 << 32) + y) * ((1 << 64) + (y *= y) ) // 32-bit x 32-bit x 64-bit
q += (q * (y *= y) ) >> 128 // upper 128-bit of 128-bit x 128-bit
q += (q * (y *= y) ) >> 256 // upper 128-bit of 128-bit x 256-bit
... //another 3 times in the worst case
// the algorithm can stop in between, if y >> 2^n has become = 0
q = (q * a)[128..256]
```

I basically (re)found the Goldschmidt Division. The individual multiplications of x and y can be optimized because register parts of x and y can be entirely zero so that multiplications need only 32x32 bit multiplication in the best case.

For divisors of more than 32-bit, one can try to enlarge the rest of the division.

```
b' // highest non-zero word in the 128-bit integer b
x = floor(0xFF..FF / b') << ((3 - n)*32) // n is the Least-significant-first index of word b' in b
y = 0xFF..FF % b' + 1 // y now can be up to 127 bit large in the first step!
q = x // x is ucent, three words of it are zero
q += (q * y) >> 128 // upper 128 bit of 128x128 multiplication
q += (q * (y *= y)) >> 256 // upper 128 bit of 128x256 multiplication
...
```

The only difference is that the first value of y can become quite large already and x is already 128-bit where 3 words of it are zero.

A division of 128-bit by a constant is very easy, just a constant multiplication of the precalculated inverse and taking the upper 128-bits of the result.

The only reason I see for not implementing it is low priority and calculation with reals is sufficient in most cases and faster (at least the register pressure will go down significantly). The only disadvantage of reals is the limited precision but except for extremely high precision applications and cryptographic-related things, I don't know a need for 128-bit (at least there is only few performance gain of using 128-bit for parallel operations). Oftentimes you can replace 128-bit arithmetics in cryptographic and multimedia with SIMD instructions which are used for parallel arithmetics in processors.

We actually need to make us aware of what 128-bit actually means! You can store any timestamp you likely ever would need already in a 80-Bit integer (about the order of Femtoseconds in a year if I remember right, so just 90-Bit would give you Femtoseconds in one millenium). 128-bit numbers can store integers up to 300*10¹² !! This is said by physicists to be far more than the number of atoms in our universe, so most of the values which can be stored in 128-Bit are not even numbers anymore in the original sense.

Probably they are waiting for 128-bit architectures but hm...

You can translate the upper code into a template for your ASM language (like AMD64) and there you go.