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Origin of Quake3's Fast InvSqrt()
geo writes "Beyond3D.com's Ryszard Sommefeldt dons his seersucker hunting jacket and meerschaum pipe to take on his secret identity as graphics code sleuth extraordinaire. In today's thrilling installment, the origins of one of the more famous snippets of graphics code in recent years is under the microscope — Quake3's Fast InvSqrt(), which has been known to cause strong geeks to go wobbly in the knees while contemplating its simple beauty and power." From the article: ""
"English motherfucker, do you speak it?" Anyone care to explain what that function does?
Only the State obtains its revenue by coercion. - Murray Rothbard
The linked site seems to be down (gee, you think it might be slashdotted?), but this paper seems to be covering the same topic.
See what I've been reading.
int i = (int)x;
Then C will simply convert the float value into an integer value (throwing away fractional part). But this isn't what we want. We want to operate on the bits of an IEEE floating point value directly, and integers are the best way to do that.
So first, we lie to the compiler by telling it we have a pointer to an int:
(int *) &f
And then we deference the pointer to get it into an operable int:
i = *(int *) &f
Note what's important here is to keep the compiler from modifying any part of the original 32-bit value.
Schwab
Editor, A1-AAA AmeriCaptions
Crap, lets try that again, with the link this time ;).Here you go.
Oh no... it's the future.
(int) x would convert the floating point value to an integer (truncation, basically).
*(int*) &x treats the bits as an integer, with no behind the scenes conversion to an actual int value.
Introduction
Note!
This article is a republishing of something I had up on my personal website a year or so ago before I joined Beyond3D, which is itself the culmination of an investigation started in April 2004. So if timeframes appear a little wonky, it's entirely on purpose! One for the geeks, enjoy.
Origin of Quake3's Fast InvSqrt()
To most folks the following bit of C code, found in a few places in the recently released Quake3 source code, won't mean much. To the Beyond3D crowd it might ring a bell or two. It might even make some sense.
InvSqrt()
Finding the inverse square root of a number has many applications in 3D graphics, not least of all the normalisation of 3D vectors. Without something like the nrm instruction in a modern fragment processor where you can get normalisation of an fp16 3-channel vector for free on certain NVIDIA hardware if you're (or the compiler is!) careful, or if you need to do it outside of a shader program for whatever reason, inverse square root is your friend. Most of you will know that you can calculate a square root using Newton-Raphson iteration and essentially that's what the code above does, but with a twist.
How the code works
The magic of the code, even if you can't follow it, stands out as the i = 0x5f3759df - (i>>1); line. Simplified, Newton-Raphson is an approximation that starts off with a guess and refines it with iteration. Taking advantage of the nature of 32-bit x86 processors, i, an integer, is initially set to the value of the floating point number you want to take the inverse square of, using an integer cast. i is then set to 0x5f3759df, minus itself shifted one bit to the right. The right shift drops the least significant bit of i, essentially halving it.
Using the integer cast of the seeded value, i is reused and the initial guess for Newton is calculated using the magic seed value minus a free divide by 2 courtesy of the CPU.
But why that constant to start the guessing game? Chris Lomont wrote a paper analysing it while at Purdue in 2003. He'd seen the code on the gamedev.net forums and that's probably also where DemoCoder saw it before commenting in the first NV40 Doom3 thread on B3D. Chris's analysis for his paper explains it for those interested in the base math behind the implementation. Suffice to say the constant used to start the Newton iteration is a very clever one. The paper's summary wonders who wrote it and whether they got there by guessing or derivation.
So who did write it? John Carmack?
While discussing NV40's render path in the Doom3 engine as mentioned previously, the code was brought up and attributed to John Carmack; and he's the obvious choice since it appears in the source for one of his engines. Michael Abrash was mooted as a possible author too. Michael stands up here as x86 assembly optimiser extraordinaire, author of the legendary Zen of Assembly Language and Zen of Graphics Programming tomes, and employee of id during Quake's development where he worked alongside Carmack on optimising Quake's software renderer for the CPUs around at the time.
Asking John whether it was him or Michael returned a "not quite".
-----Original Message-----
From: John Carmack
Sent: 26 April 2004 19:51
Subject: Re: Origin of fast approximated inverse square root
At 06:38 PM 4/26/2004 +0100, you wrote:
>Hi John,
>
>There's a discussion on Beyond3D.com's forums about who the author of
>the following is:
>
>float InvSqrt (float x){
> float xhalf = 0.5f*x;
> int i = *(int*)
> i = 0x5f3759df - (i>>1);
> x = *(float*)
> x = x*(1.5f - xhalf*x*x);
> return x;
>}
>
>Is that something we can attribute to you? Analysis shows it to be
>extremely clever in its method and supposedly from the Q3 source.
>Most people say it's your work, a few say it's Michael Abrash's. Do
>you know who's responsible, possibly with a history of sorts?
Not me,
I'll take a swing at this one. It's because the author doesn't want the value of x, but the integer representation of the value at x's memory address.
If x is 3.14159, (2) will result in i==3, whereas (1) will result in whatever the 4-byte IEEE-754 representation of 3.14159 is (0x40490FD0, if Google is correct). By using (1), the author is able to use integer bitwise opeartions (>>) to perform "free" floating point operations. When i is sent back into floating point form via:
x = *(float*)
x now contains the value of the integer operation:
i = 0x5f3759df - (i >> 1);
which was presumably faster than an identical floating point operation. It's a nifty little solution, especially with regard to the selection of the magic number.
Seriously, try looking away from the genius who obviously wrote it.
- There is no single comment which would make reading and understanding what happens here much easier!
- Introduction of a magic number with no explanation whatsoever
- Magic pointer arithmetics without demystification
- Portability? Abuse of a single processor architecture, without warning that this would not work on non-x86
I know it is good code. But it is simply bad code!First off, this function calculates 1.0/sqrt(x), not sqrt(x). InvSqrt is a particularily nasty function because both the divide and the square root stall the floating point pipeline on IA32 processors. As a result, instead of shooting out one result per cycle that the pipelining normally allows, the processor will stall for 32 cycles for the divide after it has stalled for the 43 cycles for the square root(P4). This is a big hit to realtime performance and it also prevents 76 multiplies from getting done while the pipeline is stalled. Secondly, IA32 processors are super scalar and have multiple integer units which can do portions of this calculation in parallel. This algorithm is brilliant because it uses the integer units for a portion of the most difficult part of the calculation and the remaining floating point multiplies only take about 6 clock cycles on the FPU. The difference in clock cycles you are counting is likely because the routine as written will be implemented as a function call and the stack push overhead will eat you alive. If this is implemented inline, it's about 6 times as good as simply calling the processor's assembly instructions for root and divide in sequence with the penalty that it isn't as accurate. It is virtually impossible to beat sqrt on IA-32 but 1.0/sqrt can be computed faster with newton raphson iteration in one fell swoop than by coposition of the operations. I've worked several years implementing similar optimizations in the reference implementation of ISO/IEC 18026, a standard for digital map conversion. Most of the routines that had optimizations like this added to them saw at least 30% speed improvements. This is a bit of a soft number because many things were reordered to make the pipeline fill better but in general, a complicated function especially of trig fucntions that can be computed in one iteration of well designed newton-raphson will be much faster than the coposition of the CPU's implementation of the component functions. In short, don't write off careful numerics they can provide great sped improvements, just don't use them in code that people will want to understand later if you don't document exactly what you did and why.
Holy crap--you forgot the link but the mods followed your instructions anyway.
Mods: I want +5, Funny for this. No, no, wait: +5, Informative. No, wait, anyone can google something and be "informative." I want a +5, Interesting.
Thanks.
Dear Slashdot: next time you want to mess with the site, add a rich-text editor for comments.
I believe you meant to say x^(-1/2)
Too bad the people modding you up don't have math degrees. =P
:(){
Here's an old version of one of my webpages:
c ities.com/SiliconValley/9498/sqroot.html
http://web.archive.org/web/19990210111728/www.geo
And here's an updated version of the same page:
http://www.azillionmonkeys.com/qed/sqroot.html
It isn't an exact rendering of the code in question, but it explains enough for any skilled hacker to 1) understand what's going on and 2) modify the code to create the resulting code that's in the Quake 3 source. Furthermore this web page has existed since about 1997 (archive.org doesn't go back that far for some reason.)
Now *IF* in fact the code origin comes from someone who took ideas from my site, I should point out that *I* am not the originator of the idea either (though I did write the relevant code). Bruce Holloway (who I credit on the page) was the first person to point out this technique to me at around the 1997 timeframe (prior to this, I created my own method which is similar, but not really as fast). (Vesa Karvonen informed by of the technique (through a code snippet with no explanation) at roughly the same time as well.) It was a technique well known to hard core 3D accelerator and CPU implementors, and follows an intentional design idea from the IEEE-754 specification.
Prof. William Kahan, one of the key people who specified the IEEE-754 standard (the standard for floating point the many CPUs use, starting with Intel's 8087 coprocessor) apparently presented this idea, and is the source for where Bruce Holloway got the idea. The IEEE-754 standard came out around the 1982 time frame. Though, its very likely that these ideas originate from even earlier in computing history.
It starts by taking a guess at the right answer, and then improving the guess until it's accurate enough to use.
The first step depends heavily on the fact that a floating point number on a computer is represented as a significand (aka mantissa) and an exponent (a power of two). For the moment, consider taking just the square root of X instead of its inverse. You could separate out the exponent part of the floating point number, divide it by two, and then put the result back together with the original significand, and have a reasonable starting point.
From there, you could improve your guesses to get a better approximation. The simplest version of that would be like a high-low game -- you split the difference between the current guess and the previous guess, and then add or subtract that depending on whether your previous guess was high or low. Eventually, you'll get arbitrarily close to the correct answer.
This can take quite a few iterations to get to the right answer though. To improve that, Newton-Raphson looks at the curve of the function you're working with, and projects a line tangent to the curve at the point of the current guess. Where that line crosses the origin gives you the next guess. That's probably a lot easier to understand from picture.
In this case, we're looking for the inverse square root, which changes the curve, but not the basic idea. As a general rule, the closer your first guess, the fewer iterations you need to get some particular level of accuracy. That's the point of the:
While the originator of this constant is unknown, and some of it is rather obscure, the basic idea of most of it is fairly simple: we start by shifting the original number right a bit. This divides both the mantissa and the exponent part by two, with the possibility that IF the exponent was odd, it shifts a bit from the exponent into the mantissa. The subtraction from the magic number then does a couple of things. For one thing, if a bit from the exponent was shifted into the mantissa, it removes it. The rest of the subtraction is trickier. If memory serves, it's based on the harmonic mean of the difference between sqrt(x) and (x/2) for every possible floating point number of the size you're using.
This is where the fact that it's 1/sqrt(x) instead of sqrt(x) means a lot: 1/sqrt(x) is a curve, but it's a fairly flat curve -- much flatter than sqrt(x). The result is that we can approximate a point on the curve fairly accurately with a line. In this case, it's really two lines, which gets it a bit closer still.
From there, the number has had a bit of extra tweaking done -- it doesn't actually give the most accurate first guess, but its errors are often enough in the opposite direction from those you get in the Newton-Raphson iteration steps that it gives slightly more accurate final results.
The universe is a figment of its own imagination.
The trick of this function is to take the 32 bits of data that are really a float, but process it as if it's an integer. So you take that cumbersome number 21 as a float, then BAM! presto, turn it directly to an integer not through type conversion, but by simply treating those same 32 bits as if they were representing an integer all along.
Let's use the number 21 as an example in the function call.
The binary representation of 21 as a float is 01000001 10101000 00000000 00000000 (broken into 8-bit words for clarity). The function then goes to create a new integer i, whose value is also 01000001 10101000 00000000 00000000 (which happens to be 1101529088 in decimal). The magical line of the code, i = 0x5f3759df - (i>>1), takes that integer i, shifts its bits one to the right (turning our 01000001 10101000 00000000 00000000 into 00100000 11010100 00000000 00000000, or 550764544 in decimal), then subtracts it (still doing integer math here) from 0x5f3759df (which is 01011111 00110111 01011001 11011111 or 1597463007 in decimal), and winds up with 00111110 01100011 01011001 11011111 (or 1046698463 in decimal).
Now, for its next trick, it takes that wonky integer 1046698463, and turns it back into a floating point number, by the same trick used above, i.e. simply by looking at those same 32 bits, and pretending they're a float, not an int. The binary representation of 1046698463, 00111110 01100011 01011001 11011111, is the same as 0.22202251851558685 in float.
From here on out, it's all floating math. Apply the Newton-Rhapson method (thats the next line), we get x = 0.22202251851558685 * (1.5 - ( (21*.5) * 0.22202251851558685^2 )) = 0.218117811. We return this value at the closing of the function. As it turns out, the inverse square root of 21 is 0.21821789... (thanks Google calc). So, I have no idea WHY the Float to Int to Float trick works, but it works very well.
Whew!
The algorithm is simple Newton-Raphson -- make a good initial guess, then iterate making the guess better. I think Newton-Raphson on 1/sqrt picks up 5-6 bits each try in the line "x = x*(1.5f - xhalf*x*x)". It can be repeated to get a more accurate result each time it's repeated.
The problem with Newton-Raphson is making a good first guess--otherwise, you need an extra iteration or two. And that's what the magic number is doing, making a good first guess.
So let's work out what a good first guess would look like for 1/sqrt(f), to see where this code came from.
Floating Point numbers are stored with a mantissa and an exponent: f = mantissa * (2 ^ exponent), where exponent is 8-bits wide and the mantissa is 23-bits wide.
Let's take an example: 1/sqrt(16) would have f = 1.0 * (2 ^ 4). We want the result 0.25 which is f = 1.0 * ( 2 ^ -2).
So our first guess should take our exponent, negate it, and cut it in half. (Try more examples to see that this works--it's basically the definition of 1/sqrt(f)). We'll ignore the mantissa--if we can just get within a factor of 2 of the answer in one step, we're doing pretty well.
Unfortunately, the exponent is stored in FP numbers in an offset format. In memory, The mantissa is in the low 23 bits, and the most-significant bit is the sign (which will be 0 if we're taking roots). For now, let's just assume we have our exponents as 8-bit values, to work out what we need to do with the +127 offset.
We want new_actual_exp = -(actual_exp)/2. But in memory, exp = (actual_exp + 127). Or, actual_exp = exp - 127.
Substituting gives (new_exp - 127) = -(exp - 127)/2. Simplify this to: new_exp = 127 - (exp - 127)/2 => new_exp = 3*127/2 - (exp / 2).
Now the exponent is shifted 23 places in memory, so let's write out our code (and ignore the mantissa completely for now...): rewriting as hex: Well, first, it's arguable whether it should be 0x5f000000 or 0x5f400000 (The "4" is actually in the mantissa). I'm guessing resolving that dilemma led to the original author discovering that choosing a particular pattern of bits in the mantissa can help make the initial guess even more accurate, leading to the 0x5f3759df constant.
I haven't worked it out, but Chris Lomont http://www.lomont.org/Math/Papers/2003/InvSqrt.pd