Introduction
In 2003, Alex Fr wrote an excellent article that explains how to perform SIMD (single instruction, multiple data) processing with Intel's Streaming SIMD Extensions (SSE). SSE is a set of instructions supported by Intel processors that perform highspeed operations on large chunks of data.
In 2008, Intel introduced a new set of highperformance instructions called Advanced Vector Extensions (AVX). They perform many of the same operations as SSE instructions, but operate on larger chunks of data at higher speed. Recently, Intel has released additional instructions in the AVX2 and AVX512 sets. This article focuses on accessing AVX and AVX2 instructions through special C functions called intrinsic functions.
Instead of presenting the entire set of AVX/AVX2 intrinsics, this article focuses on math computation. In particular, the goal is to multiply complex numbers. To perform this operation with AVX/AVX2, three types of intrinsics are needed:
 Initialization intrinsics
 Arithmetic intrinsics
 Permute/shuffle intrinsics
This article discusses the intrinsics in each category and explains how they're used in code. The end of the article shows how to integrate these intrinsics to multiply complex numbers.
It's important to understand the difference between a processor instruction and an intrinsic function. An AVX instruction is an assembly command that performs an indivisible operation. For example, the AVX instruction vaddps
adds two operands and places the result in a third.
To perform the operation in C/C++, the intrinsic function _mm256_add_ps()
maps directly to vaddps
, combining the performance of assembly with the convenience of a highlevel function. An intrinsic function doesn't necessarily map to a single instruction, but AVX/AVX2 intrinsics provide reliably high performance compared to other C/C++ functions.
1. Preliminary Requirements
To understand the content of this article, you need a basic familiarity with C and SIMD processing. To execute the code, you need a CPU that supports AVX or AVX/AVX2. Here are the CPUs that support AVX:
 Intel's Sandy Bridge/Sandy Bridge E/Ivy Bridge/Ivy Bridge E
 Intel's Haswell/Haswell E/Broadwell/Broadwell E
 AMD's Bulldozer/Piledriver/Steamroller/Excavator
Every CPU that supports AVX2 also supports AVX. Here are the devices:
 Intel's Haswell/Haswell E/Broadwell/Broadwell E
 AMD's Excavator
Most of the functions discussed in this article are provided by AVX. But a few are AVX2specific. To distinguish between them, I've preceded the names of AVX2 intrinsics with (2)
in the tables throughout this article.
2. Overview of Vector Programming
AVX instructions improve an application's performance by processing large chunks of values at the same time instead of processing the values individually. These chunks of values are called vectors, and AVX vectors can contain up to 256 bits of data. Common AVX vectors contain four double
s (4 x 64 bits = 256), eight float
s (8 x 32 bits = 256), or eight int
s (8 x 32 bits = 256).
An example will demonstrate the power of AVX/AVX2 processing. Suppose a function needs to multiply eight float
s of one array by eight float
s of a second array and add the result to a third array. Without vectors, the function might look like this:
multiply_and_add(const float* a, const float* b, const float* c, float* d) {
for(int i=0; i<8; i++) {
d[i] = a[i] * b[i];
d[i] = d[i] + c[i];
}
}
Here's what the function looks like with AVX2:
__m256 multiply_and_add(__m256 a, __m256 b, __m256 c) {
return _mm256_fmadd_ps(a, b, c);
}
This AVX2 intrinsic function _mm256_fmadd_ps
processes twentyfour float
s, but it doesn't map to a single instruction. Instead, it executes three instructions: vfmadd132ps
, vfmadd213ps
, and vfmadd231ps
. Despite this, it executes quickly and it's much faster than looping through the individual elements.
Despite the power of Intel's intrinsics, they make many programmers nervous. This is usually for two reasons. First, the data types have strange names like __m256
. Second, the functions have strange names like _mm256_fmadd_ps
. Therefore, before I discuss the intrinsic functions in detail, I want to discuss Intel's data types and naming conventions.
3. Fundamentals of AVX Programming
Most of this article focuses on the mathrelated intrinsic functions provided by AVX and AVX2. But before looking at the functions, it's important to understand three points:
 Data types
 Function naming conventions
 Compiling AVX applications
This section touches on each of these points and provides a simple application that subtracts one vector from another.
3.1 Data Types
A few intrinsics accept traditional data types like int
s or float
s, but most operate on data types that are specific to AVX and AVX2. There are six main vector types and Table 1 lists each of them.
Table 1: AVX/AVX2 Data Types
Data Type  Description 
__m128  128bit vector containing 4 float s 
__m128d  128bit vector containing 2 double s 
__m128i  128bit vector containing integers 
__m256  256bit vector containing 8 float s 
__m256d  256bit vector containing 4 double sā 
__m256i  256bit vector containing integers 
Each type starts with two underscores, an m
, and the width of the vector in bits. AVX512 supports 512bit vector types that start with _m512
, but AVX/AVX2 vectors don't go beyond 256 bits.
If a vector type ends in d
, it contains double
s, and if it doesn't have a suffix, it contains float
s. It might look like _m128i
and _m256i
vectors must contain int
s, but this isn't the case. An integer vector type can contain any type of integer, from char
s to short
s to unsigned long long
s. That is, an _m256i
may contain 32 char
s, 16 short
s, 8 int
s, or 4 long
s. These integers can be signed or unsigned.
3.2 Function Naming Conventions
The names of AVX/AVX2 intrinsics can be confusing at first, but the naming convention is really straightforward. Once you understand it, you'll be able to judge approximately what a function does by looking at its name. A generic AVX/AVX2 intrinsic function is given as follows:
_mm<bit_width>_<name>_<data_type>
The parts of this format are given as follows:
<bit_width>
identifies the size of the vector returned by the function. For 128bit vectors, this is empty. For 256bit vectors, this is set to 256
. <name>
describes the operation performed by the intrinsic <data_type>
identifies the data type of the function's primary arguments
This last part, <data_type>
, is a little complicated. It identifies the content of the input values, and can be set to any of the following values:
ps
 vectors contain float
s (ps
stands for packed singleprecision) pd
 vectors contain double
s (pd
stands for packed doubleprecision) epi8/epi16/epi32/epi64
 vectors contain 8bit/16bit/32bit/64bit signed integers epu8/epu16/epu32/epu64
 vectors contain 8bit/16bit/32bit/64bit unsigned integers si128
/si256
 unspecified 128bit vector or 256bit vector m128/m128i/m128d/m256/m256i/m256d
 identifies input vector types when they're different than the type of the returned vector
As an example, consider _mm256_srlv_epi64
. Even if you don't know what srlv
means, the _mm256
prefix tells you the function returns a 256bit vector and the _epi64
tells you that the arguments contain 64bit signed integers.
As a second example, consider _mm_testnzc_ps
. The _mm
implies that the function returns a 128bit vector. The _ps
at the end implies that the argument vectors contain float
s.
AVX data types start with two underscores and an m
. Functions start with an underscore and two m
s. I tend to get this confused, so I came up with a way to remember the difference: a data type represents memory and a function represents a multimedia operation. That's the best I can do.
3.3 Building AVX Applications
To build an application that uses AVX intrinsics, you don't need to link any libraries. But you need to include the immintrin.h header file. This header includes other headers that map AVX/AVX2 functions to instructions.
The code in hello_avx.c shows what a basic AVX application looks like:
#include <immintrin.h>
#include <stdio.h>
int main() {
__m256 evens = _mm256_set_ps(2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0);
__m256 odds = _mm256_set_ps(1.0, 3.0, 5.0, 7.0, 9.0, 11.0, 13.0, 15.0);
__m256 result = _mm256_sub_ps(evens, odds);
float* f = (float*)&result;
printf("%f %f %f %f %f %f %f %f\n",
f[0], f[1], f[2], f[3], f[4], f[5], f[6], f[7]);
return 0;
}
To build the application, you need to tell the compiler that the architecture supports AVX. The flag depends on the compiler, and gcc requires the mavx
flag. Therefore, the hello_avx.c source file can be compiled with the following command:
gcc mavx o hello_avx hello_avx.c
In this example, all the functions start with _mm256
and end with _ps
, so I hope it's clear that all the operations involve 256bit vectors containing float
s. I also hope it's clear that each element in the resulting vector equals 1.0. If you run the application, you'll see that this is the case.
4. Initialization Intrinsics
Before you can operate on AVX vectors, you need to fill the vectors with data. Therefore, the first set of intrinsics discussed in this article initialize vectors with data. There are two ways of doing this: initializing vectors with scalar values and initializing vectors with data loaded from memory.
4.1 Initialization with Scalar Values
AVX provides intrinsic functions that combine one or more values into a 256bit vector. Table 2 lists their names and provides a description of each.
There are similar intrinsics that initialize 128bit vectors, but those are provided by SSE, not AVX. The only difference in the functions' names is that _mm256_
is replaced with _mm_
.
Table 2: Initialization Intrinsics
Function  Description 
_mm256_setzero_ps/pd  Returns a floatingpoint vector filled with zeros 
_mm256_setzero_si256  Returns an integer vector whose bytes
are set to zero 
_mm256_set1_ps/pd  Fill a vector with a floatingpoint value 
_mm256_set1_epi8/epi16
_mm256_set1_epi32/epi64  Fill a vector with an integer 
_mm256_set_ps/pd  Initialize a vector with eight floats (ps)
or four doubles (pd)

_mm256_set_epi8/epi16
_mm256_set_epi32/epi64
 Initialize a vector with integers 
_mm256_set_m128/m128d/
_mm256_set_m128i  Initialize a 256bit vector with
two 128bit vectors

_mm256_setr_ps/pd  Initialize a vector with eight floats (ps)
or four doubles (pd) in reverse order 
_mm256_setr_epi8/epi16
_mm256_setr_epi32/epi64  Initialize a vector with integers in reverse order 
The first functions in the table are the easiest to understand. _m256_setzero_ps
returns an __m256
vector containing eight float
s set to zero. Similarly, _m256_setzero_si256
returns an __m256i
vector whose bytes are set to zero. As an example, the following line of code creates a 256bit vector that contains four double
s set to zero:
_m256d dbl_vector = _m256_setzero_pd();
The functions whose names contain set1
accept a single value and repeat it throughout the vector. For example, the following line of code creates an __m256i
whose sixteen short values are set to 47:
_m256i short_vector = _m256_set1_pd();
The rest of the functions in Table 2 contain _set_
or _setr_
. These functions accept a series of values, one for each element of the vector. These values are placed in the returned vector, and the order important to understand. The following function call returns a vector containing eight ints whose values range from 1 to 8:
_m256i int_vector = _m256_set_epi32(1, 2, 3, 4, 5, 6, 7, 8);
You might expect the values to be stored in the order in which they're given. But Intel's architecture is littleendian, so the leastsignificant value (8), is stored first and the mostsignificant value (1) is stored last.
You can verify this by casting int_vector
to an int
pointer and printing the stored values. This is shown with the following code:
__m256i int_vector = _mm256_set_epi32(1, 2, 3, 4, 5, 6, 7, 8);
int *ptr = (int*)&int_vector;
printf("%d %d %d %d %d %d %d %d\n", ptr[0], ptr[1], ptr[2], ptr[3], ptr[4], ptr[5], ptr[6], ptr[7]);
> 8 7 6 5 4 3 2 1
If you want the values to be stored in the given order, you can create vectors with one of the _setr_
functions, where r
presumably stands for reverse. The following code shows how this works:
__m256i int_vector = _mm256_setr_epi32(1, 2, 3, 4, 5, 6, 7, 8);
int *ptr = (int*)&int_vector;
printf("%d %d %d %d %d %d %d %d\n", ptr[0], ptr[1], ptr[2], ptr[3], ptr[4], ptr[5], ptr[6], ptr[7]);
> 1 2 3 4 5 6 7 8
It's interesting to note that neither AVX nor AVX2 provide intrinsics that initialize vectors with unsigned integers. However, they provide functions that operate on vectors with unsigned integers.
4.2 Loading Data from Memory
A common usage of AVX/AVX2 is to load data from memory into vectors, process the vectors, and store the results back to memory. The first step is accomplished with the intrinsic functions listed in Table 3. The last two functions are preceded with (2)
because they're provided by AVX2, not AVX.
Table 3: Vector Load Intrinsics
Data Type  Description 
_mm256_load_ps/pd  Loads a floatingpoint vector from an
aligned memory address 
_mm256_load_si256  Loads an integer vector from an aligned
memory address 
_mm256_loadu_ps/pd  Loads a floatingpoint vector from an
unaligned memory address 
_mm256_loadu_si256  Loads an integer vector from an unaligned
memory address 
_mm_maskload_ps/pd
_mm256_maskload_ps/pd  Load portions of a 128bit/256bit
floatingpoint vector according to a mask 
(2)_mm_maskload_epi32/64
(2)_mm256_maskload_epi32/64  Load portions of a 128bit/256bit
integer vector according to a mask 
When loading data into vectors, memory alignment becomes particularly important. Each _mm256_load_*
intrinsic accepts a memory address that must be aligned on a 32byte boundary. That is, the address must be divisible by 32. The following code shows how this can be used in practice:
float* aligned_floats = (float*)aligned_alloc(32, 64 * sizeof(float));
... Initialize data ...
__m256 vec = _mm256_load_ps(aligned_floats);
Any attempt to load unaligned data with _m256_load_*
produces a segmentation fault. If the data isn't aligned at a 32bit boundary, the _m256_loadu_*
functions should be used instead. This is shown with the following code:
float* unaligned_floats = (float*)malloc(64 * sizeof(float));
... Initialize data ...
__m256 vec = _mm256_loadu_ps(unaligned_floats);
Suppose you want to process a float
array using AVX vectors, but the length of the array is 11, which isn't divisible by 8. In this case, the last five float
s of the second __m256
vector need to be set to zero so they don't affect the computation. This selective loading can be accomplished with the _maskload_
functions at the bottom of Table 3.
Each _maskload_
function accepts two arguments: a memory address and an integer vector with the same number of elements as the returned vector. For each element in the integer vector whose highest bit is one, the corresponding element in the returned vector is read from memory. If the highest bit in the integer vector is zero, the corresponding element in the returned vector is set to zero.
An example will clarify how these functions are used. The code in mask_load.c reads eight int
s into a vector, and the last three should be set to zero. The function to use is _mm256_maskload_epi32
, and its second argument should be a __m256i
mask vector. This mask vector contains five int
s whose highest bit equals 1 and three int
s whose highest bit is zero. Here's what the code looks like:
#include <immintrin.h>
#include <stdio.h>
int main() {
int i;
int int_array[8] = {100, 200, 300, 400, 500, 600, 700, 800};
__m256i mask = _mm256_setr_epi32(20, 72, 48, 9, 100, 3, 5, 8);
__m256i result = _mm256_maskload_epi32(int_array, mask);
int* res = (int*)&result;
printf("%d %d %d %d %d %d %d %d\n",
res[0], res[1], res[2], res[3], res[4], res[5], res[6], res[7]);
return 0;
}
If you run this application on a system that supports AVX2, it will print the following result:
100 200 300 400 500 0 0 0
There are three points I'd like to mention:
 The code sets the mask vector's content with a
_setr_
function instead of _set_
because it orders the vector's elements as they're delivered to the function.  The highest bit of a negative integer is always 1. This is why the mask vector contains five negative numbers and three positive numbers.
 The
_mm256_maskload_epi32
function is provided by AVX2, not AVX. Therefore, to compile this code with gcc, the mavx2
flag must be used instead of mavx
.
In addition to the functions listed in Table 3, AVX2 provides gather functions that load indexed data from memory.
5. Arithmetic Intrinsics
Math is the primary reason for AVX's existence, and the fundamental operations are addition, subtraction, multiplication, and division. This section presents the intrinsics that perform these operations, and also looks at the new fused multiplyandadd functions provided by AVX2.
5.1 Addition and Subtraction
Table 4 lists the AVX/AVX2 intrinsics that perform addition and subtraction. Most of them operate on vectors containing integers because of the concern with saturation.
Table 4: Addition and Subtraction Intrinsics
Data Type  Description 
_mm256_add_ps/pd  Add two floatingpoint vectors 
_mm256_sub_ps/pd  Subtract two floatingpoint vectors 
(2)_mm256_add_epi8/16/32/64  Add two integer vectors 
(2)_mm236_sub_epi8/16/32/64  Subtract two integer vectors 
(2)_mm256_adds_epi8/16
(2)_mm256_adds_epu8/16  Add two integer vectors with saturation 
(2)_mm256_subs_epi8/16
(2)_mm256_subs_epu8/16  Subtract two integer vectors with saturation

_mm256_hadd_ps/pd  Add two floatingpoint vectors horizontally 
_mm256_hsub_ps/pd  Subtract two floatingpoint vectors horizontally 
(2)_mm256_hadd_epi16/32  Add two integer vectors horizontally 
(2)_mm256_hsub_epi16/32  Subtract two integer vectors horizontally 
(2)_mm256_hadds_epi16  Add two vectors containing shorts horizontally with saturation 
(2)_mm256_hsubs_epi16  Subtract two vectors containing shorts horizontally with saturation 
_mm256_addsub_ps/pd  Add and subtract two floatingpoint vectors 
When adding or subtracting integer vectors, it's important to see the difference between the _add_
/_sub_
functions and the _adds_
/_subs_
functions. The extra s
stands for saturation, which is produced when the result requires more memory than the vector can store. Functions that take saturation into account clamp the result to the minimum/maximum value that can be stored. Functions without saturation ignore the memory issue when saturation occurs.
For example, suppose a vector contains signed bytes, so the maximum value for each element is 127 (0x7F). If an operation adds 98 to 85, the mathematical sum is 183 (0xB7).
 If the values are added with
_mm256_add_epi8
, the saturation will be ignored and the stored result will be 73 (0xB7).  If the values are added with
_mm256_adds_epi8
, the result will be clamped to the maximum value, 127 (0x7F).
As another example, consider two vectors that contain signed short integers. The minimum value is 32,768. If you compute 18,000  19,000, the mathematical result is 37,000 (0xFFFF6F78 as a 32bit integer).
 If the values are subtracted with
_mm256_sub_epi16
, the saturation will be ignored and the stored result will be 28,536 (0x6F78).  If the values are subtracted with
_mm256_subs_epi16
, the result will be clamped to the minimum value, 32,768 (0x8000).
The _hadd_
/_hsub_
functions perform addition and subtraction horizontally. That is, instead of adding or subtracting elements of different vectors, they add or subtract adjacent elements within each vector. The results are stored in an interleaved fashion. Figure 1 shows how this works for _mm256_hadd_pd
, which horizontally adds double
vectors A and B:
Figure 1: Horizontal Addition of Two Vectors
This may seem strange to add and subtract elements horizontally, but these operations are helpful when multiplying complex numbers. This is explained later in this article.
The last function in Table 4, _mm256_addsub_ps/pd
, alternately subtracts and adds elements of two floatingpoint vectors. That is, even elements are subtracted and odd elements are added. For example, if vec_a
contains (0.1, 0.2, 0.3, 0.4) and vec_b
contains (0.5, 0.6, 0.7, 0.8), _mm256_addsub_pd(vec_a, vec_b)
equals (0.4, 0.8, 0.4, 1.2).
5.2 Multiplication and Division
Table 5 lists the AVX/AVX2 intrinsics that perform multiplication and division. As with addition and subtraction, there are special intrinsics for operating on integers.
Table 5: Multiplication and Division Intrinsics
Data Type  Description 
_mm256_mul_ps/pd  Multiply two floatingpoint vectors 
(2)_mm256_mul_epi32/
(2)_mm256_mul_epu32  Multiply the lowest four elements of vectors containing 32bit integers 
(2)_mm256_mullo_epi16/32  Multiply integers and store low halves 
(2)_mm256_mulhi_epi16/
(2)_mm256_mulhi_epu16  Multiply integers and store high halves

(2)_mm256_mulhrs_epi16  Multiply 16bit elements to form 32bit elements 
_mm256_div_ps/pd  Divide two floatingpoint vectors 
If two Nbit numbers are multiplied on a computer, the result can take up 2N bits. For this reason, only the four low elements of the _mm256_mul_epi32
and _mm256_mul_epu32
intrinsics are multiplied together, and the result is a vector containing four long integers. Figure 2 shows how this works:
Figure 2: Multiplying Low Elements of Integer Vectors
The _mullo_
functions are similar to the integer _mul_
functions, but instead of multiplying low elements, they multiply every element of both vectors store only the low half of each product. Figure 3 shows what this looks like:
Figure 3: Multiplying Integers and Storing Low Halves
The _mm256_mulhi_epi16
and _mm256_mulhi_epu16
intrinsics are similar, but they store the high halves of the integer products.
5.3 Fused Multiply and Add (FMA)
As mentioned earlier, the result of multiplying two Nbit numbers can occupy 2N bits. Therefore, when you multiply two floatingpoint values, a and b, the result is really round(a * b), where round(x) returns the floatingpoint value closest to x. This loss of accuracy increases as further operations are performed.
AVX2 provides instructions that fuse multiplication and addition together. That is, instead of returning round(round(a * b) + c), they return round(a * b + c). As a result, these instructions provide greater speed and accuracy than performing multiplication and addition separately.
Table 6 lists the FMA intrinsics provided by AVX2 and includes a description of each. Every instruction in the table accepts three input vectors, and I've referred to them as a, b, and c.
Table 6: FMA Intrinsics
Data Type  Description 
(2)_mm_fmadd_ps/pd/
(2)_mm256_fmadd_ps/pd  Multiply two vectors and add the product to a third (res = a * b + c) 
(2)_mm_fmsub_ps/pd/
(2)_mm256_fmsub_ps/pd  Multiply two vectors and subtract a vector from the product (res = a * b  c)

(2)_mm_fmadd_ss/sd  Multiply and add the lowest element in the vectors (res[0] = a[0] * b[0] + c[0]) 
(2)_mm_fmsub_ss/sd  Multiply and subtract the lowest element in the vectors (res[0] = a[0] * b[0]  c[0]) 
(2)_mm_fnmadd_ps/pd
(2)_mm256_fnmadd_ps/pd  Multiply two vectors and add the negated product to a third (res = (a * b) + c)

(2)_mm_fnmsub_ps/pd/
(2)_mm256_fnmsub_ps/pd  Multiply two vectors and add the negated product to a third (res = (a * b)  c) 
(2)_mm_fnmadd_ss/sd  Multiply the two lowest elements and add the negated product to the lowest element of the third vector (res[0] = (a[0] * b[0]) + c[0]) 
(2)_mm_fnmsub_ss/sd  Multiply the lowest elements and subtract the lowest element of the third vector from the negated product (res[0] = (a[0] * b[0])  c[0]) 
(2)_mm_fmaddsub_ps/pd/
(2)_mm256_fmaddsub_ps/pd  Multiply two vectors and alternately add and subtract from the product (res = a * b  c) 
(2)_mm_fmsubadd_ps/pd/
(2)_mmf256_fmsubadd_ps/pd  Multiply two vectors and alternately subtract and add from the product (res = a * b  c) 
If an intrinsic's name ends with _ps
or _pd
, every element of the input vectors is included in the operation. If an intrinsic's name ends with _ss
or _sd
, only the lowest element is included. The rest of the elements in the output vector are set equal to the elements of the first input vector. For example, suppose vec_a
= (1.0, 2.0), vec_b
= (5.0, 10.0), and vec_c
= (7.0, 14.0). In this case, _mm_fmadd_sd(vec_a, vec_b, vec_c)
returns (12.0, 2.0) because (1.0 * 5.0) + 7.0 = 12.0 and 2.0 is the second element of vec_a
.
It's important to see the difference between the _fmadd_
/_fmsub_
and _fnmadd_
/_fnmsub_
intrinsics. The latter functions negate the product of the first two input vectors before adding or subtracting the third input vector.
The _fmaddsub_
and _fmsubadd_
intrinsics alternate between adding and subtracting elements of the third vector. The _fmaddsub_
intrinsics add odd elements and subtract even elements. The _fmsubadd_
intrinsics subtract odd elements and add even elements. The code in fmatest.c shows how the _mm256_fmaddsub_pd
intrinsic can be used in practice.
#include <immintrin.h>
#include <stdio.h>
int main() {
__m256d veca = _mm256_setr_pd(6.0, 6.0, 6.0, 6.0);
__m256d vecb = _mm256_setr_pd(2.0, 2.0, 2.0, 2.0);
__m256d vecc = _mm256_setr_pd(7.0, 7.0, 7.0, 7.0);
__m256d result = _mm256_fmaddsub_pd(veca, vecb, vecc);
double* res = (double*)&result;
printf("%lf %lf %lf %lf\n", res[0], res[1], res[2], res[3]);
return 0;
}
When this code is compiled and executed on a processor that supports AVX2, the printed results are as follows:
5.000000 19.000000 5.000000 19.000000
The FMA instructions are provided by AVX2, so you might think the mavx2
flag is needed for building the application with gcc. But I've found that the mfma
flag is required instead. Otherwise, I get strange compile errors.
6. Permuting and Shuffling
Many applications must rearrange vector elements to ensure that operations are performed properly. AVX/AVX2 provides a number of intrinsic functions for this purpose, and two major categories are the _permute_
functions and the _shuffle_
functions. This section presents both types of intrinsics.
6.1 Permuting
AVX provides functions that return a vector containing the rearranged elements of a vector. Table 7 lists these permute functions and provides a description of each.
Table 7: Permute Intrinsics
Data Type  Description 
_mm_permute_ps/pd/
_mm256_permute_ps/pd  Select elements from the input vector based on an 8bit control value 
(2)_mm256_permute4x64_pd/
(2)_mm256_permute4x64_epi64  Select 64bit elements from the input vector based on an 8bit control value 
_mm256_permute2f128_ps/pd  Select 128bit chunks from two input vectors based on an 8bit control value

_mm256_permute2f128_si256  Select 128bit chunks from two input vectors based on an 8bit control value 
_mm_permutevar_ps/pd
_mm256_permutevar_ps/pd  Select elements from the input vector based on bits in an integer vector

(2)_mm256_permutevar8x32_ps /
(2)_mm256_permutevar8x32_epi32  Select 32bit elements (float s and int s) using indices in an integer vector 
The _permute_
intrinsics accept two arguments: an input vector and an 8bit control value. The bits of the control value determine which of the input vector's elements is inserted into the output. For _mm256_permute_ps
, each pair of control bits determines an upper and lower output element by selecting one of the upper or lower elements in the input vector. This is complicated, so I hope Figure 4 will make the operation a little clearer:
Figure 4: Operation of the Permute Intrinsic Function
As shown in the figure, values of the input vector may be repeated multiple times in the output. Other input values may not be selected at all.
In _mm256_permute_pd
, the low four bits of the control value select between adjacent pairs of double
s. _mm256_permute4x4_pd
is similar, but uses all of the control bits to select which 64bit element is placed in the output. In the _permute2f128_
intrinsics, the control value selects 128bit chunks from the two input vectors instead of selecting elements from one input vector.
The _permutevar_
intrinsics perform the same operation as the _permute_
intrinsics. But instead of using 8bit control values to select elements, they rely on integer vectors with the same size as the input vector. For example, the input vector of _mm256_permute_ps
is a _mm256
, so the integer vector is an _mm256i
. The high bits of the integer vector perform selection in the same way as the bits of the 8bit control values of the _permute_
intrinsics.
6.2 Shuffling
Like the _permute_
intrinsics, the _shuffle_
intrinsics select elements from one or two input vectors and place them in the output vector. Table 8 lists the functions and provides a description of each.
Table 8: Shuffle Intrinsics
Data Type  Description 
_mm256_shuffle_ps/pd  Select floatingpoint elements according to an 8bit value

_mm256_shuffle_epi8/
_mm256_shuffle_epi32  Select integer elements according to an
8bit value 
(2)_mm256_shufflelo_epi16/
(2)_mm256_shufflehi_epi16  Select 128bit chunks from two input vectors based on an 8bit control value

All _shuffle_
intrinsics operate on 256bit vectors. In each case, the last argument is an 8bit value that determines which input elements should be placed in the output vector.
For _mm256_shuffle_ps
, only the high four bits of the control value are used. If the input vectors contain int
s or float
s, all the control bits are used. For _mm256_shuffle_ps
, the first two pairs of bits select elements from the first vector and the second two pairs of bits select elements from the second vector. Figure 5 illustrates how this works:
Figure 5: Operation of the Shuffle Intrinsic Function
To shuffle 16bit values, AVX2 provides _mm256_shufflelo_epi16
and _mm256_shufflehi_epi16
. As with _mm256_shuffle_ps
, the control value is split into four pairs of bits that select from eight elements. But for _mm256_shufflelo_epi16
, the eight elements are taken from the eight low 16bit values. For _mm256_shufflehi_epi16
, the eight elements are taken from the eight high 16bit values.
7. Complex Multiplication
Complex multiplication is a timeconsuming operation that must be performed repeatedly in signal processing applications. I won't go deeply into the theory, but every complex number can be expressed as a + bi, where a and b are floatingpoint values and i is the squareroot of 1. a is called the real part and b is called the imaginary part. If (a + bi) and (c + di) are multiplied, the product equals (ac  bd) + (ad + bc)i.
Complex numbers can be stored in interleaved fashion, which means each real part is followed by the imaginary part. Suppose vec1 is a __m256d
that stores two complex numbers, (a + bi) and (x + yi), and vec2
is a __m256d
that stores (c + di) and (z + wi). Figure 6 illustrates how the values are stored. As shown, the prod
vector stores the two products: (ac  bd) + (ad + bc)i and (xz  yw) + (xw + yz)i.
Figure 6: Complex Multiplication Using Vectors
I don't know the fastest way to compute complex products with AVX/AVX2. But I've devised a method that has given me good results. It consists of five steps:
 Multiply
vec1
and vec2
and store the result in vec3
.  Switch the real/imaginary values of
vec2
.  Negate the imaginary values of
vec2
.  Multiply
vec1
and vec2
and store the result in vec4
.  Use horizontal subtraction on
vec3
and vec4
to produce the answer in vec1
.
The code in complex_mult.c shows how this can be performed using AVX intrinsics:
#include <immintrin.h>
#include <stdio.h>
int main() {
__m256d vec1 = _mm256_setr_pd(4.0, 5.0, 13.0, 6.0);
__m256d vec2 = _mm256_setr_pd(9.0, 3.0, 6.0, 7.0);
__m256d neg = _mm256_setr_pd(1.0, 1.0, 1.0, 1.0);
__m256d vec3 = _mm256_mul_pd(vec1, vec2);
vec2 = _mm256_permute_pd(vec2, 0x5);
vec2 = _mm256_mul_pd(vec2, neg);
__m256d vec4 = _mm256_mul_pd(vec1, vec2);
vec1 = _mm256_hsub_pd(vec3, vec4);
double* res = (double*)&vec1;
printf("%lf %lf %lf %lf\n", res[0], res[1], res[2], res[3]);
return 0;
}
This code operates on double
vectors, but the method can easily be extended to support float
vectors.
Using the Code
The AVX_examples.zip archive contains the four source files mentioned in this article. I haven't provided any makefiles, but the code can be compiled with the following commands:
gcc mavx o hello_avx hello_avx.c
gcc mavx2 o mask_load mask_load.c
gcc mfma o fmatest fmatest.c
gcc mavx o complex_mult complex_mult.c
Of course, the applications will only execute properly if the processor supports AVX or AVX/AVX2.
Points of Interest
Many developers may avoid learning AVX/AVX2 in the hope that the compiler will perform autovectorization. Autovectorization is a great feature, but if you understand the intrinsics, you can rearrange your algorithm to take better advantage of SIMD processing. By inserting AVX/AVX2 intrinsics, I have dramatically improved the processing speed of my signal processing applications.
History
2/20  Fixed formatting and image links
4/2  Fixed a couple typographical errors