next-pow2-wasm v1.0.0

next-pow2-wasm

Next power of 2 functions in WASM, eg. for aligning pointers

Usage

This module includes WebAssembly functions for rounding a number to the next N = 2 ** n, which is often used for aligning words to their natural boundaries.

The formula in C is (ptr + (N - 1)) & (-N), where ptr is the number to align from and N is a power of 2.

In WebAssembly, to align to the next i32 (4 bytes), the routine would look as follows:

(func $i32.align_i32
  (param $ptr i32) (result i32)

  (i32.and (i32.add (local.get $ptr) (i32.const 3))
           (i32.const -4)))

A more generalised form is a bit more complicated:

(func $i32.next_pow2
  (param $base i32)
  (param $power i32)
  (result i32)

  ;; This block makes sure $base \in [n, n + 1)
  (i32.add
    (local.get $base)
    (i32.sub (local.get $power) (i32.const 1))) ;; N - 1

  ;; This block negates $power, ie. -N
  (i32.sub (i32.const 0) (local.get $power))

  ;; Clear lowest bits, ie. round down to $power
  (i32.and))

WebAssembly does not have a i32.neg operation, so instead we must do 0 - N.

Fixing memory alignment

Why

Unaligned memory access can have serious performance implications on some processor architectures, and even though WASM abstracts away the details of the actual processor, it cannot patch unaligned access for you.

Further, if you try to wrap a data structure as a multi-byte TypedArray from Javascript, as is common for returning complex objects from WASM, the addressing needs to be a multiple of the byte-width of that type, eg. Float64Arrays need to start at a multiple of 8, or an Error will be thrown. Take the example of creating a view to the WASM linear memory eg. new Float64Array(wasm.memory.buffer, 12, 5), which says "create a view to the doubles/Numbers in the ArrayBuffer wasm.memory.buffer starting from byte-offset 12 and for the next 5 doubles (40 bytes)". This will fail as an offset of 12 bytes is not a multiple of 8. Therefore you you need to round to the next multiple of 8 in your WASM code before you write your data.

The Linux Kernel documentation on UNALIGNED MEMORY ACCESSES has further details on the implications of unaligned access:

Why unaligned access is bad
===========================

The effects of performing an unaligned memory access vary from architecture
to architecture. It would be easy to write a whole document on the differences
here; a summary of the common scenarios is presented below:

 - Some architectures are able to perform unaligned memory accesses
   transparently, but there is usually a significant performance cost.
 - Some architectures raise processor exceptions when unaligned accesses
   happen. The exception handler is able to correct the unaligned access,
   at significant cost to performance.
 - Some architectures raise processor exceptions when unaligned accesses
   happen, but the exceptions do not contain enough information for the
   unaligned access to be corrected.
 - Some architectures are not capable of unaligned memory access, but will
   silently perform a different memory access to the one that was requested,
   resulting in a subtle code bug that is hard to detect!

It should be obvious from the above that if your code causes unaligned
memory accesses to happen, your code will not work correctly on certain
platforms and will cause performance problems on others.

Specialised for memory access

i8

Single bytes are always naturally aligned, as that is the smallest atom in WASM linear memory.

i16

Round to the next multiple of 2 (eg. i32.load16_*):

(local i32 $unaligned)
(local i32 $aligned)

(set_local $unaligned (i32.const 3))

(set_local $aligned (i32.and (i32.const -2)
                             (i32.add (i32.const 1)
                                      (get_local $unaligned)))))

i32/f32

Round to the next multiple of 4 (eg. i32, f32):

(local i32 $unaligned)
(local i32 $aligned)

(set_local $unaligned (i32.const 3))

(set_local $aligned (i32.and (i32.const -4)
                             (i32.add (i32.const 3)
                                      (get_local $unaligned)))))

i64/f64

Round to the next multiple of 8 (eg. i64, f64):

(local i32 $unaligned)
(local i32 $aligned)

(set_local $unaligned (i32.const 13))

(set_local $aligned (i32.and (i32.const -8)
                             (i32.add (i32.const 7)
                                      (get_local $unaligned)))))

How it works

The general formula is: (x + (N - 1)) & (-N), where N must be a power of 2, eg. 2, 4, 8, 16, 32, …

The trick is that adding N - 1 will push the unaligned number into the interval between two multiples of [kN, (k + 1)N) (end exclusive), eg. [0, 8), [8, 16), [16, 24), [24, 32) etc. All we need now is to round down to the closest multiple of N. Since N is a power of 2 (8 = 2^3), we can exploit the fact that negative numbers are represented as two's compliment, meaning that while +8 has leading zeros, -8 will have leading ones, ie. +8 = 00001000 and -8 = 11111000. This means we can use -8 as a mask to only keep bits that will keep the number a multiple of 8 (N = 8 in this example). Clearing lower three bits here yield a number that's a multiple of 8. Check for yourself here:

00010110                = 22
│││││││└───── b8 * 1    = 0
││││││└────── b7 * 2    = 2
│││││└─────── b6 * 4    = 4
││││└──────── b5 * 8    = 0
│││└───────── b4 * 16   = 16
││└────────── b3 * 32   = 0
│└─────────── b2 * 64   = 0
└──────────── b1 * 128  = 0

See the worked example below rounding the value 15 to the next multiple of 8:

  00001111 = 15
+ 00000111 = +7
---------------
  00010110 = 22 = 15 + 7    note that 22 is between [16, 24)

- 00001000 = +8
---------------
  11111000 = -8 = -(+8)     two's complement, note the leading 1's

  11111000 = -8
& 00010110 = 22
---------------
  00010000 = 16             recall that `1 & 0 = 0` and `1 & 1 = 1`

License

ISC