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8-bit number to binary string (gynvael.coldwind.pl)
158 points by Audiophilip on Nov 5, 2015 | hide | past | favorite | 49 comments

The big constants used in the formula

    void to_bin(unsigned char c, char *out) {
      *(unsigned long long*)out = 3472328296227680304ULL +
        (((c * 9241421688590303745ULL) / 128) & 72340172838076673ULL);
are much more obvious when they remain in hex:

0x3030303030303030 -- the byte array of '0's encoded as ASCII

0x8040201008040201 -- the magic

0x101010101010101 -- only 0 or 1 to be added to the appropriate array element

Therefore, more readable:

    void to_bin(unsigned char c, char *out) {
      /* endian dependent, works on x86 */
      *(unsigned long long*)out = 0x3030303030303030ULL +
        (((c * 0x8040201008040201ULL) >> 7) & 0x0101010101010101ULL);

The function is of course endian dependent (I'd add a comment about that fact in it) and the explanation of the quoted function on the page starts at "If you shift the the constant by 7 bits to the right." The page is a result of an edit where the older function (that ends with + c / 128;) was explained first.

Still, nice.

I wish I could do this in JavaScript.

You can but you have to use jquery.

Towards the bottom of the article it says:

the last thing I did was changing the constants to decimal - they look way more scary this way ;>

because, you know, programming should be innately difficult and confusing so we can laugh at others who understand less than we do.

Never underestimate how many times code (any code) posted online will be copy/pasted throughout the world everywhere from school projects to production systems.

Cool hacks seem to be lightening rods for the humorless. There are entire contests dedicated to obfuscating code! This was a neat, unintuitive solution that he revealed in steps with a surprise ending. As an essay it worked perfectly.

The whole thing is whimsy. You'd never want to use this approach at all in a production system. Using decimal constants instead of hex is the least of the sins here.

There should be room in programming for whimsy. This stuff is fun, and play is healthy. If somebody takes this thing seriously and copies the code into a production system, that's their own fault, not the author's.

> You'd never want to use this approach at all in a production system

Actually it's not less useful than the code the compilers you depend on use all the time. They really depend on such "bit twiddles." Guy Steele agrees too, read his foreword to the "Hacker's Delight," a whole book about such functions:


The difference being that the compiler knows what sort of bit-twiddling it should be doing and what sort of bit-twiddling it should not be doing.

I would not use this code without being very thoughtful and intentional about where it was being used.

> I would not use this code without being very thoughtful and intentional about where it was being used.

Sure, definitely not for everybody and every occasion. It's a platform dependent code. See my comment to cnvogel and look at his implementation for an example.

My job is basically fixing the code and processes of corporate programmers who copy/paste examples from websites into production systems. There's little understanding of how the code works and almost no reading of related documentation. If I compiles, I ships.

It's like if bridge designers didn't really know mechanics and just copy/pasted minecraft designs into the real world. Sure, it's not the fault of minecraft designers, but we can help kill fewer people by making better examples.

I'm sorry you have that job. Or, maybe congratulations if you're the consultant who comes in and makes bucket loads of money fixing other people's crap.

Either way, expecting People Who Know What They're Doing to shut up about it so that we can prevent People Who Don't Know What They're Doing from misunderstanding is shameful. The linked code wasn't published in a textbook, nor was it submitted as a Stack Overflow answer. You have no right to expect people to sanitize their writings just in case they see it.

Sorry if I sound prickly, but this sentiment makes my blood boil.

I'm sorry but this is a false analogy [1].

You're suggesting that something that doesn't happen (mechanical engineers + minecraft) is analogous to something that does happen (corporate programmers + code poetry).

Are you then suggesting that we outlaw minecraft because there may be a brain dead mechanical engineer alive?

What you're suggesting is akin to either outlawing essays like this or only allowing "Professional Engineers" (i.e. certification required) to write code?

The only solution is for the latter since the former is typical of a professional engineer seeking intellectual stimulation and should be applauded.

1. https://en.wikipedia.org/wiki/Argument_from_analogy#False_an...

as scott adams says: I agree with your analysis of your hallucination.

"Professional Engineers" (i.e. certification required) to write code?

Do we let self taught basement dwellers design bridges crossed by millions of people and 100 story buildings?

yikes dude. go to bed, have a good rest, and i hope that you wake up in a better mood tomorrow.

I don't think that analogy really works. If bridge designers were doing this, there would be massive outcry and we'd put a stop to it. We allow it to happen with (terrible) programmers only because the harm they do is pretty limited, and nobody's getting killed because of it.

there would be massive outcry and we'd put a stop to it.

Massive security branches? Massive fraud? Ineffective nationalized cybercommands?

Just because we can't see the world burning with our eyes doesn't mean lead in the air isn't giving us brain damage.

These are all because of people copy/pasting code from the internet?

There are probably 100x more servers sitting in server farms running websites (and burning power) than need to be because companies would rather hire 10 cheap Ruby or PHP programmers than one "expensive" Erlang/Elixer programmer. So there is some harm to bad programmers, even for non-critical websites.

I think the analogy holds. Software doesn't kill people ...until it does: https://en.m.wikipedia.org/wiki/Therac-25

Software obviously kills people sometimes. But I'm pretty sure that people who blindly copy/paste code from the internet into production systems don't.

This code was already difficult and confusing and the author obviously pokes fun at his own code by decimalizing the constants. No need to accuse him of elitist or immoral behaviour.

It's a good and common practice to keep the constants in the sanest form and not obfuscate them, see the collections of such functions, like the already quoted


or like


By keeping the decimal constants he diminishes the chance to be attributed.

Sure, sure, in most contexts code as communication medium is laudable. Here, the game is to make people accustomed to understanding go "the hell just happened?" when it works. I like it.

Yeah but that does not apply to jokes.

You mean the whole article is a joke just because the article ends with "the last thing I did was changing the constants to decimal - they look way more scary this way"?

I see it as a presentation of an efficient specialized function, just like the other bit hacks are (some already linked, one more, a whole book, here: http://www.hackersdelight.org/ ).

If it's is to impress the uninitiated, such are more probable to not know the hex codes too. For them is the decimal number the one less scary than the hex one. For others, converting back to hex is just a tiny annoyance, nothing more.

But it's not efficient. The author even explicitly says so at the beginning:

> (Disclaimer: It's not faster. It's not better. It's not production-quality code. It's just is and it was fun to make ;>)

It's a joke, a joke with some context, that you tell other people who know the context. Would you accuse people of intentionally obfuscating things when they riff on "the door is ajar" joke?

No, on modern 64-bit processors it's definitely more efficient and faster than the loop variant. It's measurable. The author's disclaimer that it isn't faster is actually valid only for old or small processors.

See cnvogel's attempt to measure it:


I also haven't accused anybody anything here.

You just won't give up will you?

Do you post similar rants to mechanical engineers building rube goldberg machines and electrical engineers building impractical extremely high voltage transformers?

Can you imagine that there are people who actually program on such a low level that they actually care about that kind of functions, constants and the number of T states that the resulting assembly code takes? I'm one of those. If these aspects aren't relevant to you please allow me to address them.

The article author actually engineered an effective function for spreading out bits to bytes. I invite you to read the comments here that are actually mine and what's in them.

Awesome. I'm always amazed by bitmagic like this. Today I saw one on stackoverflow and spent a good deal of time failing to understand it:


And of course I'm reminded of the fast inverse square root:


Why isn't this faster than the "standard" loop based version? The disclaimer at the start says it's not, but I would have thought that 4 64bit math operations would be much faster than a loop with 8 steps and comparisons and so on.

First and foremost: YES, I KNOW! This is completely senseless microbenchmarking and micro-optimization. I ONLY LOOK AT THIS FOR FUN.

That being said. The 64bit multiplication is faster, and also constant in time. I've measured this using the timestamp counter which doesn't necessarily count actual CPU clock cycles, but possibly an integer multiple of it (seems to output always numbers divisable by 8 for me).

    100 loop:01100100(152) 64bit:01100100(48)
    101 loop:01100101(128) 64bit:01100101(48)
    102 loop:01100110(160) 64bit:01100110(48)
    103 loop:01100111(128) 64bit:01100111(48)
    104 loop:01101000(192) 64bit:01101000(48)
    105 loop:01101001(88) 64bit:01101001(48)
    106 loop:01101010(152) 64bit:01101010(48)
    107 loop:01101011(128) 64bit:01101011(48)
    108 loop:01101100(160) 64bit:01101100(48)

That's on an Intel(R) Core(TM) i5 CPU M 450 @ 2.40GHz and the cycles include the rdtsc, a few movs, and the call to the respective function.


I couldn't resist so I tried a table look-up (16 elements) version which was about twice as fast as the multiply version.

Intel Core i5-3210M @ 2.50GHz

Funny, I read about the first part of the sentence 'I tried a table look-up' and thought immediately WHAT ABOUT THE CACHE?

Then I continued '16 elements', oh, OK then, next time I'll try to read the whole sentence first.

Thanks for measuring.

From your github source:

    /* anyone have a big-endian machine at hand to test this? */
    #define MULT  0x8040201008040201ULL
    #define MULT  0x0102040810204080ULL
Even without the machine I believe that such a big endian implementation can't work. Hint: you can try it on the little endian machine too, the resulting bits must be appropriate, and I believe they wouldn't. That's the beauty of the magic constants, their construction isn't so easy as it appears to be.

You are right, the carry bits will clobber the neighboring bytes, I think.

> the carry bits will clobber

Actually, not the carry bits, the "content" bits. The 8-bit content must appear on more places there for the magic to work.

It is surely faster on modern 64-bit CPU's, and surely not faster on the 8-bit ones.

Rather than add 0x3030303030303030, you could OR with it because you know there will never be a carry. This may save cycles on some architectures.

This is an example of SIMD Within A Register. There are some more neat examples here: http://aggregate.org/MAGIC/

For Python

    >>> f1=lambda c: __import__('struct').pack('<Q', 3472328296227680304  + (((c * 9241421688590303745) / 128) & 72340172838076673))
    >>> f1(1)
    >>> f1(2)
    >>> f1(5)
    >>> f2=lambda c: struct.pack('<Q', (((c * 0x8040201008040201) / 128) & 0x0101010101010101))
    >>> f2(1)
    >>> f2(2)
    >>> f2(101)
Use "<" for little-endian for x86

Here's a fancier, slightly less-magic, version. How to compile is left as an exercise for the reader (hint: Haswell or newer, requires BMI2 instruction set):

    void toBinary(uint8_t byte) {
        uint64_t zeroes = *(uint64_t *)"00000000";
        uint64_t resultStr =
            zeroes | __builtin_bswap64(_pdep_u64(byte, 0x0101010101010101ULL));
        printf("hello: %.*s\n", 8, (char *)&resultStr);

This really reminds me of the old demoscene diskmag Hugi article about adding 16 bit colors (without MMX).


> x86 is little endian - that's why it's backwards

This can be read the wrong way, although it is correct: the author is referring to the final byte sequence and not the input bit sequence. Endianness applies to bytes, not bits.

Nice little problem. Thanks for sharing!

Cool but I want to say I've seen similar, google turns up: http://www.asmcommunity.net/forums/topic/?id=28498

I recommend reading over this: http://0x80.pl/articles/convert-to-bin.html

btw all this SWAR stuff should be aligned before applying.

Be careful with alignment. On architectures where it matters, the caller may pass in a "char* out" which is not suitably aligned, and doing an "unsigned long long" store into that address may fault.

Feature request: add a parameter that specifies the base into which the function rewrites the number.

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