The Aggregate Magic Algorithms (2015)

It's a bit of a trope that mathematicians consider all knowledge to fall into two groups: 'unknown' and 'obvious/trivial'.

These "filler" expressions are typical in mathematical demonstrations (as an indication that step won't be explained). But sometimes it's not obvious

> Clearly, floor of base 2 log of x is (WORDBITS-lzc(x)).

Yes. Because the 1st bit indicates the MSB of the number and the way binary works

If you have only one bit set, then WORDBITS-lzc(x) is log2(x). If you have less significant bits enabled that won't be enough to bump the floor to the next number

I dunno, I guess it's alright to use these phrases for things that are truly trivial and direct. But plenty of people use these phrases to introduce things that are definitely not obvious. It's quite annoying and I suspect it's either ingrained in habit, or it's just the author wanting to "show off", sort of like "wow, this is so obvious to me, isn't it to you?", nevermind that they spent a great deal of time to understand it when they first came across it.

Grrr x)

I come from a firmware development background so this kind of bit manipulation tricks are familiar to me. That equation makes sense after a few seconds looking at it but I can't always run a proof in my mind to be 100% certain.

I think anyone who has tried to teach or explain something will see why this happens. You have to make some assumptions about what readers know/can figure out otherwise you would literally be explaining everything starting from binary numbers every time.

The "clearly" stuff is just shorthand for "I assume you know this because its foundational to understanding the thing were talking about now / or I'm too lazy to explain this now / or this is so far off-topic that this isnt the place / etc. etc" . Nothing more - it can be read as arrogance on the authors part but I think thats an uncharitable interpretation.

>> Clearly, floor of base 2 log of x is (WORDBITS-lzc(x)).

> Uh, is it?

Of course, that is intuitively obvious to even the most casual observer. ;)

http://www.urbandictionary.com/define.php?term=intuitively%2...

It's poor form to describe items as clear, obvious, trivial, 'of course', easy to see, and so on. To look too much into it: if the item is clearly​ understood then there's no need to identify it as clear. Clearly it's clear. It bothers me too.

It is... if you're the kind of person who does bit twiddling. And for everyone else it's gobbledygook.

Basically, you're using a binary representation, so everything is in powers of two. Let's just talk integers. 2^n is just the nth bit set and a bunch of zeros. Every number between 2^n and 2^n+1 has the nth digit set and some lower digits. The log base 2 is between n and n+1. So the floor is n. So, biggest digit set=floor of log base 2.

Once you've grasped that principle, the exact details are, in fact, clear. :)

From the article How to Read Mathematics [0], that kind of statement "is a signal to the reader that what's involved here is perhaps tedious and even difficult, but involves no deep insights."

[0] http://web.stonehill.edu/compsci/History_Math/math-read.htm

HN discussion: https://news.ycombinator.com/item?id=14232977

A clarification: this isn't to say such tricks are never used in a non-toy environment. There is, for example, this search algorithm I made for a serious purpose:

https://stackoverflow.com/q/40141793/1763356

The bit math is milder, but it still comes in handy.

Probably not. These low level tricks are supposed to be hidden away inside functions, and any good compiler ought to produce similar machine code in the majority of these examples.

> Do these actually run faster than the code produced by a good compiler? Benchmarks?

Some of these are absolutely suspect. Now, these are all architecture specific: my results below are for gcc, targeting the Intel chip in the MBP in my lap. (Of course, some of the article's code samples assume certain things about an architecture, and are not portable.)

For example,

> Absolute Value of a Float

> IEEE floating point uses an explicit sign bit, so the absolute value can be taken by a bitwise AND with the complement of the sign bit. For IA32 32-bit, the sign bit is an int value of 0x80000000, for IA32 64-bit, the sign bit is the long long int value 0x8000000000000000. Of course, if you prefer to use just int values, the IA32 64-bit sign bit is 0x80000000 at an int address offset of +1. For example:

  double x;

  /* make x = abs(x) */
  *(((int *) &x) + 1) &= 0x7fffffff;

First, this is undefined behavior. The portable way to do this is fabs(), which is part of the standard library. For me, calling fabs() compiles to a single instruction:

  andps   LCPI1_0(%rip), %xmm0

where LCPI1_0 is

  .quad   9223372036854775807     ## 0x7fffffffffffffff
  .quad   9223372036854775807     ## 0x7fffffffffffffff

fabs() compiled down to essentially the same bitwise &, in assembly, that the article is doing. The article's "advice" also gets compiled to exactly the same instruction & set of constants, which I actually find way more interesting.

> Integer Constant Multiply

> Given an integer value x and an integer or floating point value y, the value of xy can be computed efficiently using a sequence derived from the binary value of x. For example, if x is 5 (4 + 1):*

  y2 = y + y;
  y4 = y2 + y2;
  result = y + y4;

> In the special case that y is an integer, this can be done with shifts:

  y4 = (y << 2);
  result = y + y4;

Compilers will do this for you, today, automatically. In the example case of multiplying by 5, gcc will do better than the article's code, by using lea cleverly:

  leal    (%rdi,%rdi,4), %eax

gcc also compiles the article's shift-then-add strategy to exactly the same leal.

The article mentions this optimization again, later,

> Shift-and-Add Optimization

> Rather obviously, if an integer multiply can be implemented by a shift-and-add sequence, then a shift-and-add sequence can be implemented by multiplying by the appropriate constant... with some speedup on processors like the AMD Athlon. Unfortunately, GCC seems to believe constant multiplies should always be converted into shift-and-add sequences, so there is a problem in using this optimization in C source code.

The above leal instruction was the output of GCC, so this is obviously wrong. One can maybe say the article is old, but I'm pretty sure GCC has had the "use leal to do constant multiplies" optimization for over a decade now, but then, perhaps the article is that old. If so, the advice would appear to no longer apply.

Write it the obvious, readable way first. If you decide to optimize at the level this article is discussing, you should be reading the assembly to see if your optimizations will have any effect.

But not all of them are bad. The "is this unsigned integer a power of 2" code is correct, I believe, and if you understand binary, not that hard to understand.

In some cases I suspect it depends a lot on the hardware. I did a bit of research into the "Integer minimum" algorithm. Given the following straightforward implementation:

    int min(int x, int y) { return y < x ? y : x; }

..most compilers will produce something very much like the following for x86-64 (from Compiler Explorer https://gcc.godbolt.org using -Ofast -march=haswell):

    min(int, int):
        cmp     esi, edi
        mov     eax, edi
        cmovle  eax, esi
        ret

Compare that to the generated instructions for the algorithm listed in the "Integer minimum or maximum" section (x+(((y-x)>>(WORDBITS-1))&(y-x))):

    min(int, int):
        sub     esi, edi
        mov     eax, esi
        sar     eax, 31
        and     eax, esi
        add     eax, edi
        ret

It's not immediately obvious to me which one is likely to be faster (I'm not an expert on x86-64 assembly).