# big integer multiply

links

* <http://www.intel.com/content/dam/www/public/us/en/documents/white-papers/ia-large-integer-arithmetic-paper.pdf>
* <https://lists.libre-soc.org/pipermail/libre-soc-dev/2022-April/004700.html>
* <https://news.ycombinator.com/item?id=21151646>

## Variant 1

Row-based multiply using temporary vector. Simple implementation
of Knuth M: <https://git.libre-soc.org/?p=libreriscv.git;a=blob;f=openpower/sv/bitmanip/mulmnu.c;hb=HEAD>

```
      for (i = 0; i < m; i++) {
         unsigned product = u[i]*v[j] + w[i + j];
         phi[i] = product>>16;
         plo[i] = product;
      }
      for (i = 0; i < m; i++) {
         t = (phi[i]<<16) | plo[i] + k;
         w[i + j] = t;          // (I.e., t & 0xFFFF).
         k = t >> 16;
      }
```

**maddx RT, RA, RB, RC** (RS=RT+VL for SVP64, RS=RT+1 for scalar)

    prod[0:127] = (RA) * (RB)
    sum[0:127] = EXTZ(RC) + prod
    RT <- sum[64:127]
    RS <- sum[0:63]

**addxd RT, RA, RB** (RS=RB+VL for SVP64, RS=RB+1 for scalar)

    cat[0:127] = (RS) || (RB)
    sum[0:127] = cat + EXTZ(RA)
    RA = sum[0:63]
    RT = sum[64:127]

These two combine as, simply:

    # assume VL=8, therefore RS starts at r8.v
    # q           : r16
    # multiplier  : r17
    # multiplicand: r20.v
    # carry       : r18
    li r18, 0
    sv.maddx r0.v, r16, r17, r20.v
    # here, RS=RB+VL, therefore again RS starts at r8.v
    sv.addxd r0.v, r18, r0.v

## Variant 2

```
      for (i = 0; i < m; i++) {
         unsigned product = u[i]*v[j] + k;
         k = product>>16;
         plo[i] = product; // & 0xffff
      }
      k = 0;
      for (i = 0; i < m; i++) {
         t = plo[i] + w[i + j] + k;
         w[i + j] = t;          // (I.e., t & 0xFFFF).
         k = t >> 16; // carry: should only be 1 bit
      }
```

**maddx RT, RA, RB, RC**

    prod[0:127] = (RA) * (RB)
    sum[0:127] = EXTZ(RC) + prod
    RT <- sum[64:127]
    RC <- sum[0:63]

# big integer division

links

* <https://skanthak.homepage.t-online.de/division.html>
* <https://news.ycombinator.com/item?id=26562819>
* <https://gmplib.org/~tege/division-paper.pdf>
* <https://github.com/Richard-Mace/huge-integer-class/blob/master/HugeInt.cpp> nice-looking well-commented c++ implementation
* <https://lists.libre-soc.org/pipermail/libre-soc-dev/2022-April/004739.html>
* <https://github.com/bcoin-org/libtorsion/blob/master/src/mpi.c#L2872>

the most efficient division algorithm is probably Knuth's Algorithm D (with modifications from the exercises section of his book) which is O(n^2) and uses 2N-by-N-bit div/rem

an oversimplified version of the knuth algorithm d with 32-bit words is:
(TODO find original: <https://raw.githubusercontent.com/hcs0/Hackers-Delight/master/divmnu64.c.txt>

```
void div(uint32_t *n, uint32_t *d, uint32_t* q, int n_bytes, int d_bytes) {
    // assumes d[0] != 0, also n, d, and q have their most-significant-word in index 0
    int q_bytes = n_bytes - d_bytes;
    for(int i = 0; i < q_bytes / sizeof(n[0]); i++) {
        // calculate guess for quotient word
        q[i] = (((uint64_t)n[i] << 32) + n[i + 1]) / d[0];
        // n -= q[i] * d
        uint32_t carry = 0, carry2 = 0;
        for(int j = d_bytes / sizeof(d[0]) - 1; j >= 0; j--) {
            uint64_t v = (uint64_t)q[i] * d[j] + carry;
            carry = v >> 32;
            v = (uint32_t)v;
            v = n[i + j] - v + carry2;
            carry2 = v >> 32; // either ~0 or 0
            n[i + j] = v;
        }
        // fixup if carry2 != 0
    }
    // now remainder is in n and quotient is in q
}
```

The key loop may be implemented with a 4-in, 2-out mul-twin-add
(which is too much):

```
On Sat, Apr 16, 2022, 22:06 Jacob Lifshay <programmerjake@gmail.com> wrote:
and a mrsubcarry (the one actually needed by bigint division):

    # for big_c - big_a * word_b
    result <- RC + ~(RA * RB) + CARRY # wrong, needs further thought
    CARRY <- HIGH_HALF(result)
    RT <- LOW_HALF(result)

turns out, after some checking with 4-bit words, afaict the correct
algorithm for mrsubcarry is:

    # for big_c - big_a * word_b
    result <- RC + ~(RA * RB) + CARRY
    result_high <- HIGH_HALF(result)
    if CARRY <= 1 then # unsigned comparison
        result_high <- result_high + 1
    end
    CARRY <- result_high
    RT <- LOW_HALF(result)

afaict, that'll make the following algorithm work:

so the inner loop in the bigint division algorithm would end up being
(assuming n, d, and q all fit in registers):

    li r3, 1 # carry in for subtraction
    mtspr CARRY, r3 # init carry spr
    setvl loop_count
    sv.mrsubcarry rn.v, rd.v, rq.s, rn.v
```

This algorithm may be morphed into a pair of Vector operations by temporary
storage of the products.

```
      uint32_t borrow = 0;
      for(int i = 0; i <= n; i++) {
         uint32_t vn_i = i < n ? vn[i] : 0;
         uint64_t value = un[i + j] - (uint64_t)qhat * vn_i;
         plo[i] = value & 0xffffffffLL;
         phi[i] = value >> 32;
      }
      for(int i = 0; i <= n; i++) {
         uint64_t value = (((uint64_t)phi[i]<<32) | plo[i]) - borrow;
         borrow = ~(value >> 32)+1; // -(uint32_t)(value >> 32);
         un[i + j] = (uint32_t)value;
      }
      bool need_fixup = borrow != 0;
```


Transformation of 4-in, 2-out into a pair of operations:

* 3-in, 2-out `msubx RT, RA, RB, RC` producing {RT,RS} where RS=RT+VL
* 3-in, 2-out `subxd RT, RA, RB` a hidden RS=RT+VL as input, RA dual

<img src="/openpower/sv/weirdmuladd.jpg" width=800 />

A trick used in the DCT and FFT twin-butterfly instructions,
originally borrowed from `lq` and LD/ST-with-update, is to
have a second hidden (implicit) destination register, RS.
RS is calculated as RT+VL, where all scalar operations
assume VL=1.  With `sv.msubx` *creating* a pair of Vector
results, `sv.weirdaddx` correspondingly has to pick the
pairs up, containing the split lo-hi 128-bit products,
in order to carry on the algorithm.

**msubx RT, RA, RB, RC** (RS=RT+VL for SVP64, RS=RT+1 for scalar)

    prod[0:127] = (RA) * (RB)
    sub[0:127] = EXTZ(RC) - prod
    RT <- sub[64:127]
    RS <- sub[0:63]

**subxd RT, RA, RB** (RS=RB+VL for SVP64, RS=RB+1 for scalar)

    cat[0:127] = (RS) || (RB)
    sum[0:127] = cat - EXTS(RA)
    RA = ~sum[0:63] + 1
    RT = sum[64:127]

These two combine as, simply:

    # assume VL=8, therefore RS starts at r8.v
    # q       : r16
    # dividend: r17
    # divisor : r20.v
    # carry   : r18
    li r18, 0
    sv.msubx r0.v, r16, r17, r20.v
    # here, RS=RB+VL, therefore again RS starts at r8.v
    sv.subxd r0.v, r18, r0.v

As a result, a big-integer subtract and multiply may be carried out
in only 3 instructions, one of which is setting a scalar integer to
zero.

An Rc=1 variant tests not against RT but RA, which allows detection
of a fixup in Knuth Algorithm D: the condition where RA is not zero.