Fast inverse square root explained

Fast inverse square root, sometimes referred to as or by the hexadecimal constant , is an algorithm that estimates $\backslash frac$, the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number

x in IEEE 754 floating-point format. The algorithm is best known for its implementation in 1999 in Quake III Arena, a first-person shooter video game heavily based on 3D graphics. With subsequent hardware advancements, especially the x86 SSE instruction rsqrtss, this algorithm is not generally the best choice for modern computers, though it remains an interesting historical example.

The algorithm accepts a 32-bit floating-point number as the input and stores a halved value for later use. Then, treating the bits representing the floating-point number as a 32-bit integer, a logical shift right by one bit is performed and the result subtracted from the number, which is a floating-point representation of an approximation of $\backslash sqrt$. This results in the first approximation of the inverse square root of the input. Treating the bits again as a floating-point number, it runs one iteration of Newton's method, yielding a more precise approximation.

History

William Kahan and K.C. Ng at Berkeley wrote an unpublished paper in May 1986 describing how to calculate the square root using bit-fiddling techniques followed by Newton iterations. In the late 1980s, Cleve Moler at Ardent Computer learned about this technique and passed it along to his coworker Greg Walsh. Greg Walsh devised the now-famous constant and fast inverse square root algorithm. Gary Tarolli was consulting for Kubota, the company funding Ardent at the time, and likely brought the algorithm to 3dfx Interactive circa 1994.

Jim Blinn demonstrated a simple approximation of the inverse square root in a 1997 column for IEEE Computer Graphics and Applications. Reverse engineering of other contemporary 3D video games uncovered a variation of the algorithm in Activision's 1997 Interstate '76.^[1]

Quake III Arena, a first-person shooter video game, was released in 1999 by id Software and used the algorithm. Brian Hook may have brought the algorithm from 3dfx to id Software. A discussion of the code appeared on the Chinese developer forum CSDN in 2000, and Usenet and the gamedev.net forum spread the code widely in 2002 and 2003. Speculation arose as to who wrote the algorithm and how the constant was derived; some guessed John Carmack. Quake IIIs full source code was released at QuakeCon 2005, but provided no answers. The authorship question was answered in 2006 when Greg Walsh contacted Beyond3D as their speculation gained popularity on Slashdot.

In 2007 the algorithm was implemented in some dedicated hardware vertex shaders using field-programmable gate arrays (FPGA).^[2]

Motivation

The inverse square root of a floating point number is used in digital signal processing to normalize a vector, scaling it to length 1 to produce a unit vector. For example, computer graphics programs use inverse square roots to compute angles of incidence and reflection for lighting and shading. 3D graphics programs must perform millions of these calculations every second to simulate lighting. When the code was developed in the early 1990s, most floating point processing power lagged the speed of integer processing. This was troublesome for 3D graphics programs before the advent of specialized hardware to handle transform and lighting. Computation of square roots usually depends upon many division operations, which for floating point numbers are computationally expensive. The fast inverse square generates a good approximation with only one division step.

The length of the vector is determined by calculating its Euclidean norm: the square root of the sum of squares of the vector components. When each component of the vector is divided by that length, the new vector will be a unit vector pointing in the same direction. In a 3D graphics program, all vectors are in three-dimensional space, so

\boldsymbol v would be a vector (v _{1,} v _{2,} v _{3)} .\|\boldsymbol{v}\| = \sqrt{v ^{2 }} _{3} is the Euclidean norm of the vector.\boldsymbol{\hat{v}} = (\boldsymbol{v)/() }is the normalized (unit) vector. Substituting $\backslash |\backslash boldsymbol\; v\backslash |$, the equation can also be written as:\boldsymbol{\hat{v}} = (\boldsymbol{v)/() }\sqrtwhich relates the unit vector to the inverse square root of the distance components. The inverse square root can be used to compute \boldsymbol{\hat{v}} because this equation is equivalent to\boldsymbol{\hat{v}} = \boldsymbol{v} (1)/(\sqrt{v) ^{2 }} ^{3} where the fraction term is the inverse square root of v ^{2} _{3} .

At the time, floating-point division was generally expensive compared to multiplication; the fast inverse square root algorithm bypassed the division step, giving it its performance advantage.

Overview of the code

The following code is the fast inverse square root implementation from Quake III Arena, stripped of C preprocessor directives, but including the exact original comment text:float Q_rsqrt(float number)

At the time, the general method to compute the inverse square root was to calculate an approximation for $\backslash frac$, then revise that approximation via another method until it came within an acceptable error range of the actual result. Common software methods in the early 1990s drew approximations from a lookup table. The key of the fast inverse square root was to directly compute an approximation by utilizing the structure of floating-point numbers, proving faster than table lookups. The algorithm was approximately four times faster than computing the square root with another method and calculating the reciprocal via floating-point division. The algorithm was designed with the IEEE 754-1985 32-bit floating-point specification in mind, but investigation from Chris Lomont showed that it could be implemented in other floating-point specifications.

The advantages in speed offered by the fast inverse square root trick came from treating the 32-bit floating-point word^[3] as an integer, then subtracting it from a "magic" constant, . This integer subtraction and bit shift results in a bit pattern which, when re-defined as a floating-point number, is a rough approximation for the inverse square root of the number. One iteration of Newton's method is performed to gain some accuracy, and the code is finished. The algorithm generates reasonably accurate results using a unique first approximation for Newton's method; however, it is much slower and less accurate than using the SSE instruction rsqrtss on x86 processors also released in 1999.

Worked example

As an example, the number

x =0 . 15625 can be used to calculate $\backslash frac\; \backslash approx\; 2.52982$. The first steps of the algorithm are illustrated below:

0011_1110_0010_0000_0000_0000_0000_0000 Bit pattern of both x and i 0001_1111_0001_0000_0000_0000_0000_0000 Shift right one position: (i >> 1) 0101_1111_0011_0111_0101_1001_1101_1111 The magic number 0x5F3759DF 0100_0000_0010_0111_0101_1001_1101_1111 The result of 0x5F3759DF - (i >> 1)

Interpreting as IEEE 32-bit representation:

0_01111100_01000000000000000000000 1.25 × 2⁻³ 0_00111110_00100000000000000000000 1.125 × 2⁻⁶⁵ 0_10111110_01101110101100111011111 1.432430... × 2⁶³ 0_10000000_01001110101100111011111 1.307430... × 2¹

Reinterpreting this last bit pattern as a floating point number gives the approximation

y =2 . 61486 , which has an error of about 3.4%. After one single iteration of Newton's method, the final result is y =2 . 52549 , an error of only 0.17%.

Avoiding undefined behavior

According to the C standard, reinterpreting a floating point value as an integer by casting then dereferencing the pointer to it is not valid (undefined behavior). Another way would be to place the floating point value in an anonymous union containing an additional 32-bit unsigned integer member, and accesses to that integer provides a bit level view of the contents of the floating point value. However, type punning through a union is also undefined behavior in C++.

1. include // uint32_t

float Q_rsqrt(float number)

In modern C++, the recommended method for implementing this function's casts is through C++20's std::bit_cast. This also allows the function to work in constexpr context:

import std;

constexpr std::float32_t Q_rsqrt(std::float32_t number) noexcept

Algorithm

The algorithm computes $\backslash frac$ by performing the following steps:

1. Alias the argument

x to an integer as a way to compute an approximation of the binary logarithm $\backslash log\_(x)$

1. Use this approximation to compute an approximation of $\backslash log\_\backslash left(\backslash frac\backslash right)\; =\; -\backslash frac\; \backslash log\_(x)$
2. Alias back to a float, as a way to compute an approximation of the base-2 exponential
3. Refine the approximation using a single iteration of Newton's method.

Floating-point representation

See main article: Single-precision floating-point format.

Since this algorithm relies heavily on the bit-level representation of single-precision floating-point numbers, a short overview of this representation is provided here. To encode a non-zero real number

x as a single precision float, the first step is to write x as a normalized binary number:\begin{align} x & = \pm 1 .b _{1b} _{2b} _{3\ldots} x 2 ^{e _{x}} \end{align}

where the exponent $e\_x$ is an integer, and $1.b\_1b\_2b\_3\backslash ldots$ is the binary representation of the significand. Since the single bit before the point in the significand is always 1, it does not need be stored. The equation can be rewritten as:

\begin{align} x & = \pm 2 ^{e _{x}} ( 1 + m _{x) \end{align}}

where $m\_x$ means $0.b\_1b\_2b\_3\backslash ldots$, so

• $M\_x\; =\; m\_x\; \backslash times\; L$, where $L\; =\; 2^$^[5] (23 bits)

These fields are then packed, left to right, into a 32-bit container.

As an example, consider again the number

x = 0 . 15625 = 0 . 00101 _{2} . Normalizing x yields:

$x\; =\; +2^(1\; +\; 0.25)$

and thus, the three unsigned integer fields are:

S = 0 E = -3 + 127 = 124 = 0111 1100 _{2} M = 0 . 25 x 2 ^{23} = 2 097 152 = 0010 0000 0000 0000 0000 0000 _{2}

these fields are packed as shown in the figure below:

Aliasing to an integer as an approximate logarithm

If $\backslash frac$ were to be calculated without a computer or a calculator, a table of logarithms would be useful, together with the identity $\backslash log\_b\backslash left(\backslash frac\backslash right)\; =\; \backslash log\_b\backslash left(x^\backslash right)\; =\; -\backslash frac\; \backslash log\_b(x)$, which is valid for every base

b . The fast inverse square root is based on this identity, and on the fact that aliasing a float32 to an integer gives a rough approximation of its logarithm. Here is how:

If

x is a positive normal number:x = 2 ^{e _{x}} ( 1 + m _{x)}

then

log _{2(x)} = e _{x} + log _{2(1} + m _{x)}

and since

m _{x} \in [ 0 , 1 ) , the logarithm on the right-hand side can be approximated bylog _{2(1} + m _{x)} ≈ m _{x} + \sigma

where

\sigma is a free parameter used to tune the approximation. For example, \sigma = 0 yields exact results at both ends of the interval, while $\backslash sigma\; =\; \backslash frac\; -\; \backslash frac\; \backslash approx\; 0.0430357$ yields the optimal approximation (the best in the sense of the uniform norm of the error). However, this value is not used by the algorithm as it does not take subsequent steps into account.

Thus there is the approximation

log _{2(x)} ≈ e _{x} + m _{x} + \sigma.

Interpreting the floating-point bit-pattern of

x as an integer I _{x} yields^[6] \begin{align} I _{x} & = E _{x} L + M _{x\\} & = L (e _{x} + B + m _{x)\\} & = L (e _{x} + m _{x} + \sigma + B - \sigma)\\ & ≈ L log _{2(x)} + L (B - \sigma). \end{align}

It then appears that

I _{x} is a scaled and shifted piecewise-linear approximation of log _{2(x)} , as illustrated in the figure on the right. In other words, log _{2(x)} is approximated bylog _{2(x)} ≈ (I _{x})/(L) - (B - \sigma).

First approximation of the result

The calculation of $y=\backslash frac$ is based on the identity

log _{2(y)} = - \tfrac{ 1 }{ 2 } log _{2(x)}

Using the approximation of the logarithm above, applied to both

x and y , the above equation gives:(I _{y})/(L) - (B - \sigma) ≈ - (1)/(\left( 2) (I _{x})/(L) - (B - \sigma)\right) Thus, an approximation of I _{y} is:I _{y} ≈ \tfrac{ 3 }{ 2 } L (B - \sigma) - \tfrac{ 1 }{ 2 } I _{x}

which is written in the code as

i = 0x5f3759df - (i >> 1);

The first term above is the magic number

\tfrac{ 3 }{ 2 } L (B - \sigma) = 0x5F3759DF

from which it can be inferred that

\sigma ≈ 0 . 0450466 . The second term, (1)/(2) I _{x} , is calculated by shifting the bits of I _{x} one position to the right.

Newton's method

See main article: Newton's method. With

y as the inverse square root, f(y) = (1)/(y ^{2}) - x =0 . The approximation yielded by the earlier steps can be refined by using a root-finding method, a method that finds the zero of a function. The algorithm uses Newton's method: if there is an approximation, y _{n} for y , then a better approximation y _{n+1} can be calculated by taking y _{n} - (f(y _{n)})/(f'(y _{n)}) , where f'(y _{n)} is the derivative of f(y) at y _{n} . For the above f(y) ,y _{n+1} = (y _{n} \left( 3- xy ^{2 \right)} _{n})/(2) where f(y) = (1)/(y ^{2}) - x and f'(y) =- (2)/(y ^{3}) .

Treating

y as a floating-point number, y = y*(threehalfs - x/2*y*y); is equivalent toy _{n+1} = y _{n} \left ( (32- (x)/(2))/(y) ^{2 \right} ^{n} ) = (y _{n} \left( 3- xy ^{2 \right)} _{n})/(2) . By repeating this step, using the output of the function (y _{n+1} ) as the input of the next iteration, the algorithm causes y to converge to the inverse square root. For the purposes of the Quake III engine, only one iteration was used. A second iteration remained in the code but was commented out.

Accuracy

As noted above, the approximation is very accurate. The single graph on the right plots the error of the function (that is, the error of the approximation after it has been improved by running one iteration of Newton's method), for inputs starting at 0.01, where the standard library gives 10.0 as a result, and InvSqrt gives 9.982522, making the relative difference 0.0017478, or 0.175% of the true value, 10. The absolute error only drops from then on, and the relative error stays within the same bounds across all orders of magnitude.

Subsequent improvements

Magic number

It is not known precisely how the exact value for the magic number was determined. Chris Lomont developed a function to minimize approximation error by choosing the magic number

R over a range. He first computed the optimal constant for the linear approximation step as, close to, but this new constant gave slightly less accuracy after one iteration of Newton's method. Lomont then searched for a constant optimal even after one and two Newton iterations and found, which is more accurate than the original at every iteration stage. He concluded by asking whether the exact value of the original constant was chosen through derivation or trial and error. Lomont said that the magic number for 64-bit IEEE754 size type double is, but it was later shown by Matthew Robertson to be exactly .

Jan Kadlec reduced the relative error by a further factor of 2.7 by adjusting the constants in the single Newton's method iteration as well,^[7] arriving after an exhaustive search atconv.i = 0x5F1FFFF9 - (conv.i >> 1);conv.f *= 0.703952253f * (2.38924456f - x * conv.f * conv.f);return conv.f;

A complete mathematical analysis for determining the magic number is now available for single-precision floating-point numbers.^[8]

Zero finding

Intermediate to the use of one vs. two iterations of Newton's method in terms of speed and accuracy is a single iteration of Halley's method. In this case, Halley's method is equivalent to applying Newton's method with the starting formula

f(y) = (1)/(y ^{1/2}) - xy ^{3/2} = 0 . The update step is theny _{n+1} = y _{n} - (f(y _{n)})/(f'(y _{n)}) = y _{n} \left( (3 + xy ^{2} _{n})/(1 + 3 xy ^{2} _{n}) \right), where the implementation should calculate xy ^{2} _{n} only once, via a temporary variable.

Obsolescence

Subsequent additions by hardware manufacturers have made this algorithm redundant for the most part. For example, on x86, Intel introduced the SSE instruction rsqrtss in 1999. In a 2009 benchmark on the Intel Core 2, this instruction took 0.85ns per float compared to 3.54ns for the fast inverse square root algorithm, and had less error.

Some low-cost embedded systems do not have specialized square root instructions. However, manufacturers of these systems usually provide trigonometric and other math libraries, based on algorithms such as CORDIC.

See also

• • Magic number

References

Bibliography

• Blinn. Jim. Floating Point Tricks. IEEE Computer Graphics & Applications. July 1997. 17. 4. 10.1109/38.595279. 80.
• Book: Blinn , Jim . Jim Blinn's Corner: Notation, notation notation. Morgan Kaufmann. 2003. 1-55860-860-5.
• Book: Eberly , David . 3D Game Engine Design. Morgan Kaufmann. 2001. 978-1-55860-593-0. registration.
• Goldberg. David. What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys. 1991. 23. 1. 5–48. 10.1145/103162.103163. 222008826.
• Book: Hardy. Godfrey. A Course of Pure Mathematics. 1908. Cambridge University Press. Cambridge, UK. 0-521-72055-9.
• Book: Hennessey , John . . Patterson, David A. . Computer Organization and Design. registration. Morgan Kaufmann Publishers. San Francisco, CA. 1998. 2nd. 978-1-55860-491-9.
• Web site: Fast Inverse Square Root. Lomont. Chris. February 2003. 2009-02-13.
• Web site: The Mathematics Behind the Fast Inverse Square Root Function Code. McEniry. Charles. August 2007. https://web.archive.org/web/20150511044204/http://www.daxia.com/bibis/upload/406Fast_Inverse_Square_Root.pdf. 2015-05-11.
• . Middendorf . Lars . Mühlbauer . Felix . Umlauf . George . Bodba . Christophe . June 1, 2007 . Embedded Vertex Shader in FPGA . IFIP TC10 Working Conference:International Embedded Systems Symposium (IESS) . Embedded System Design: Topics, Techniques and Trends . Rettberg, Achin . et al. . Springer . . 978-0-387-72257-3 . 10.1007/978-0-387-72258-0_14 . live . https://web.archive.org/web/20190501235316/https://link.springer.com/content/pdf/10.1007/978-0-387-72258-0_14.pdf . 2019-05-01. free .

• Web site: https://web.archive.org/web/20090215020337/http://www.hackszine.com/blog/archive/2008/12/quakes_fast_inverse_square_roo.html. Quake's fast inverse square root. Striegel. Jason. 2008-12-04. Hackszine. O'Reilly Media. 2009-02-15. 2013-01-07.
  • Moroz. Leonid V.. Walczyk. Cezary J.. Hrynchyshyn. Andriy. Holimath. Vijay. Cieslinski. Jan L.. Fast calculation of inverse square root with the use of magic constant analytical approach. Applied Mathematics and Computation. Elsevier Science Inc.. January 2018. 316. C. 10.1016/j.amc.2017.08.025. 245–255. 1603.04483. 7494112.

Further reading

• Kushner. David. August 2002. The wizardry of Id. IEEE Spectrum. 39. 8. 42–47. 10.1109/MSPEC.2002.1021943.

External links

• 0x5f3759df, further investigations into accuracy and generalizability of the algorithm by Christian Plesner Hansen
• Origin of Quake3's Fast InvSqrt
• Quake III Arena source code (archived in Software Heritage)
• Implementation of InvSqrt in DESMOS
• "Fast Inverse Square Root — A Quake III Algorithm" (YouTube)

Notes and References

1. Web site: Peelar . Shane . 1 June 2021 . Fast reciprocal square root... in 1997?! .
2. Book: Zafar . Saad . Adapa . Raviteja . 2014 International Conference on Advances in Electrical Engineering (ICAEE) . Hardware architecture design and mapping of 'Fast Inverse Square Root' algorithm . January 2014 . https://ieeexplore.ieee.org/document/6838433 . 1–4 . 10.1109/ICAEE.2014.6838433. 978-1-4799-3543-7 . 2005623 .
3. Use of the type long reduces the portability of this code on modern systems. For the code to execute properly, [[sizeof]](long) must be 4 bytes, otherwise negative outputs may result. Under many modern 64-bit systems, [[sizeof]](long) is 8 bytes. The more portable replacement is int32_t.
4. E _{x} should be in the range [ 1 , 254 ] for x to be representable as a normal number.
5. The only real numbers that can be represented exactly as floating point are those for which $M\_x$ is an integer. Other numbers can only be represented approximately by rounding them to the nearest exactly representable number.
6. Since x is positive, S _{x} = 0 .
7. Web site: Kadlec . Jan . 2010 . Řrřlog::Improving the fast inverse square root . 2020-12-14 . personal blog . 2018-07-09 . https://web.archive.org/web/20180709021629/http://rrrola.wz.cz/inv_sqrt.html . bot: unknown .
8. Muller. Jean-Michel. December 2020. Elementary Functions and Approximate Computing. Proceedings of the IEEE. 108. 12. 2146. 10.1109/JPROC.2020.2991885. 219047769 . 0018-9219.