# Basics¶

The remainder of this document provides a basic overview of the Enoki library. All code snippets assume that the following lines are present:

#include <iostream>
#include <enoki/array.h>

/* Don't forget to include the 'enoki' namespace */
using namespace enoki;


Note

The next few sections will introduce the basics of Enoki in the context of vectorization for CPUs. Later sections will discuss vectorization via GPU arrays and Automatic differentiation. It is recommended that you read the following sections even if you are mainly interested in the latter two topics.

## Static arrays¶

The most important data structure in this library is enoki::Array, a generic container that stores a fixed-size array of an arbitrary data type. This is somewhat similar to the standard template library class std::array. The main distinction between the two is that enoki::Array forwards all C++ operators (and other standard mathematical functions) to the contained elements.

For instance, the snippet

using StrArray = Array<std::string, 2>;

StrArray x("Hello ", "How are "),
y("world!", "you?");

// Prints: "[Hello world!,  How are you?]"
std::cout << x + y << std::endl;


is valid because Array::operator+() can carry out the addition by invoking std::string::operator+() (this would not compile if std::array was used, since it has no such semantics).

Enoki arrays also have support for a convenient feature that is commonly known as broadcasting—in this simple example, broadcasting is triggered if we add a raw string corresponding to an array of dimension zero. Fancier applications of broadcasting are discussed later.

// Prints: "[Hello you,  How are you]"
std::cout << x + "you" << std::endl;


The real use case of Enoki arrays, however, involves arrays of integer or floating point values, for which arithmetic operations can often be reduced to fast vectorized instructions provided by current processor architectures.

The library ships with partial template overloads that become active when the Type and Size parameters supplied to the enoki::Array<Type, Size> template correspond to combinations that are natively supported by the targeted hardware. For instance, the template overloads for single precision arrays look as follows:

Altogether, Enoki currently supports the ARM NEON, SSE4.2, AVX, AVX2, and AVX512 instruction sets and vectorizes arithmetic involving single and double precision floating point values as well as signed and unsigned 32-bit and 64-bit integers.

It is worth pointing out that that enoki::Array does not require Size to exactly match what is supported by the hardware to benefit from vectorization. Enoki relies on template metaprogramming techniques to ensure optimal code generation even in such challenging situations. For instance, arithmetic operations involving a hypothetical Array<float, 27> type will generate one AVX512 instruction [1], one AVX instruction, and one 4-wide SSE instruction that leaves the last entry unused.

A perhaps more sensible use of this feature is to instantiate packed arrays with a Size that is an integer multiple of what is supported natively as a way of aggressively unrolling the underlying computations. If Size is not specified, the implementation chooses a reasonable value (4 on machines with SSE/NEON, 8 on machines with AVX, and 16 on AVX512).

## Initializing, reading, and writing data¶

Arrays can be initialized by broadcasting a scalar value, or by specifying the values of individual entries.

/* Initialize all entries with a constant */
MyFloat f1(1.f);

/* Initialize the entries individually */
MyFloat f2(1.f, 2.f, 3.f, 4.f);


The enoki namespace also contains a large number of global functions that create or manipulate Enoki arrays in various ways. One example is the enoki::load() function, which is the method of choice to initialize an array with data that is currently stored in memory:

float *mem = /* ... pointer to floating point data ... */;
MyFloat f3;

/* Load entries of 'f3' from 'mem' */
f3 = load<MyFloat>(mem);           /* if known that 'mem' is aligned */
f3 = load_unaligned<MyFloat>(mem); /* otherwise */


Both enoki::load() and enoki::load_unaligned() are template functions that load an array of the specified type (MyFloat in this case) from a given address in memory. The first indicates that the memory address is aligned to a multiple of alignof(MyFloat), which is equal to 16 bytes in this example. It is a good idea to align data and use aligned versions of operations, since this reduces the number of cache lines that must be accessed.

Warning

Performing an aligned load from an unaligned memory address will cause a general protection fault that immediately terminates the application.

Note

It is generally desirable to use high-level Enoki template functions like enoki::load() whenever possible since they are designed to produce the most efficient instruction sequences for the specified target architecture. In this case, we could also have written

f3 = MyFloat(mem[0], mem[1], mem[2], mem[3]);


which is functionally equivalent—however, it is not guaranteed that the compiler will be able to exploit the equivalence to generate optimal code in this case.

An analogous pair of operations stores the contents of arrays in memory:

/* Store entries of 'f3' to 'mem' */
store(mem, f3);                    /* if known that 'mem' is aligned */
store_unaligned(mem, f3)           /* otherwise */


Note that load() and load_unaligned() require the target type as a template parameter, while the stores do not, since it can be inferred from the provided value.

Gather, scatter, and scatter-add operations are also supported using a similar pattern:

/* 32 and 64 bit integers are supported as indices for scatter/gather operations */
Array<int, 4> idx(1, 2, 3, 4);

/* Gather f3 from mem -- this is equivalent to
setting f3[i] = mem[idx[i]] (i = 0, 1, ..) */
f3 = gather<MyFloat>(mem, idx);

/* Scatter f3 to mem -- this is equivalent to
setting mem[idx[i]] = f3[i] (i = 0, 1, ..) */
scatter(mem, f3, idx);

/* Atomic scatter-add of f3 to mem -- this is equivalent to
setting mem[idx[i]] += f3[i] (i = 0, 1, ..). The operation
is atomic in the sense that it gives the correct results
if 'idx' refers to the same index twice. */


Finally, the following initialization methods also exist:

/* Efficient way to create an array of any type filled with zero entries */
f1 = zero<MyFloat>();

/* Initialize entries with index sequence 0, 1, 2, ... */
f1 = arange<MyFloat>();

/* Initialize entries with a linearly increasing sequence with endpoints 0 and 1 */
f1 = linspace<MyFloat>(0.f, 1.f);


## Element access¶

The components of Array can be accessed via operator[]. If you find yourself using this much, your code is likely not making good use of the vector units.

f2[2] = 1.f;


Alternatively, the functions x(), y(), z(), and w() can be used to access the first four components. The following line is equivalent to the one above.

f2.z() = 1.f;


Components of a vector can be efficiently reordered using the following syntax:

// f1: [0, 10, 20, 30]
f2 = shuffle<0, 2, 1, 3>(f1);
// f2: [0, 20, 10, 30]


Finally, Enoki provides an overloaded operator<<(std::ostream&, ...) stream insertion operator to facilitate the inspection of array contents:

/* The line below prints: [1, 2, 3, 4] */
std::cout << MyFloat(1.f, 2.f, 3.f, 4.f) << std::endl;


## Vertical operations¶

Enoki provides the following vertical operations. The word vertical implies that they are independently applied to all array elements.

/* Basic arithmetic operations*/
f1 *= (f2 + 1.f) / (f2 - 1.f);

/* Basic math library functions */
f2 = ceil(f1); f2 = floor(f1); f2 = round(f1); f2 = trunc(f1);
f2 = abs(f1);  f2 = sqrt(f1); f2 = cbrt(f1); f2 = sign(f1);
f2 = min(f1, f2); f2 = max(f1, f2);

f1 = fmadd(f1, f2, f3); /* f1 * f2 + f3 */
f1 = fmsub(f1, f2, f3); /* f1 * f2 - f3 */
f1 = fnmadd(f1, f2, f3); /* -f1 * f2 + f3 */
f1 = fnmsub(f1, f2, f3); /* -f1 * f2 - f3 */

/* Efficient reciprocal and reciprocal square root */
f1 = rcp(f1); f1 = rsqrt(f1);

/* Trigonometric and inverse trigonometric functions */
f2 = sin(f1);   f2 = cos(f1);    f2 = tan(f1);
f2 = csc(f1);   f2 = sec(f1);    f2 = cot(f1);
f2 = asin(f1);  f2 = acos(f1);   f2 = atan(f2);
f2 = atan2(f1, f2);
auto [s, c] = sincos(f1);

/* Hyperbolic and inverse hyperbolic functions */
f2 = sinh(f1);  f2 = cosh(f1);  f2 = tanh(f1);
f2 = csch(f1);  f2 = sech(f1);  f2 = coth(f1);
f2 = asinh(f1); f2 = acosh(f1); f2 = atanh(f2);
auto [sh, ch] = sincosh(f1);

/* Exponential function, natural logarithm, power function */
f2 = exp(f1);   f2 = log(f1);   f2 = pow(f1, f2);

/* Exponent/mantissa manipulation */
f1 = ldexp(f1, f2);
auto [mant, exp] = frexp(f1);

/* Special functions */
f2 = erf(f1); f2 = erfinv(f1); f2 = erfi(f1);
f2 = i0e(f1); f2 = dawson(f1);

f1 = comp_ellint_1(f1);     f1 = ellint_1(f1, f2);
f1 = comp_ellint_2(f1);     f1 = ellint_2(f1, f2);
f1 = comp_ellint_2(f1, f2); f1 = ellint_3(f1, f2, f3);

/* Bit shifts and rotations (only for integer arrays) */
i1 = sl<3>(i1);   i1 = sr<3>(i1);   /* Shift by a compile-time constant ("immediate") */
i1 = i1 >> i2;    i1 = i1 << i2;    /* Element-wise shift by a variable amount */
i1 = rol<3>(i1);  i1 = ror<3>(i1);  /* Rotate by a compile-time constant ("immediate") */
i1 = rol(i1, i2); i1 = ror(i1, i2); /* Element-wise rotation by a variable amount */

/* Trailing/leading zero count, population count (only for integer arrays) */
i1 = lzcnt(i1);  i1 = tzcnt(i1);  i1 = popcnt(i1);


### Casting¶

A cast is another type of vertical operation. Enoki performs efficient conversion between any pair of types using native conversion instructions whenever possible:

using Source = Array<int64_t, 32>;
using Target = Array<double, 32>;

Source source = ...;
Target target(source);


## Horizontal operations¶

In contrast to the above vertical operations, the following horizontal operations consider the entries of a packed array jointly and return a scalar.

Depending on the size of the array, these are implemented using between $$log_2(N)$$ and $$N-1$$ vertical reduction operations and shuffles. Horizontal operations should generally be avoided since they don’t fully utilize the hardware vector units (ways of avoiding them are discussed later).

/* Horizontal sum, equivalent to f1[0] + f1[1] + f1[2] + f1[3] */
float s0 = hsum(f1);

/* Horizontal product, equivalent to f1[0] * f1[1] * f1[2] * f1[3] */
float s1 = hprod(f1);

/* Horizontal minimum, equivalent to std::min({ f1[0], f1[1], f1[2], f1[3] }) */
float s2 = hmin(f1);

/* Horizontal maximum, equivalent to std::max({ f1[0], f1[1], f1[2], f1[3] }) */
float s3 = hmax(f1);

/* Horizontal mean , equivalent to (f1[0] + f1[1] + f1[2] + f1[3]) / 4.f */
float s4 = hmean(f1);


The following linear algebra primitives are also realized in terms of horizontal operations:

/* Dot product of two arrays */
float dp = dot(f1, f2);

/* For convenience: absolute value of the dot product */

/* Squared 2-norm of a vector */
float sqn = squared_norm(f1);

/* 2-norm of a vector */
float nrm = norm(f1);


Comparisons involving Enoki types are generally applied component-wise and produce a mask representing the outcome of the comparison. The internal representation of a mask is an implementation detail that varies widely from architecture to architecture – an overview is given in the section on Architectural differences handled by Enoki.

Masks enable powerful branchless logic in combination with a range of other bit-level operations. The following snippets show some example usage of mask types:

auto mask = f1 > 1.f;

/* Bit-level AND operation: Zero out entries where the comparison was false */


Masks can be combined in various ways

mask ^= (f1 > cos(f2)) | ~(f2 <= f1);


The following range tests also generate masks

mask = isnan(f1);    /* Per-component NaN test */
mask = isinf(f1);    /* Per-component +/- infinity test */
mask = isfinite(f1); /* Per-component test for finite values */


Note

Using the -ffast-math compiler option may break detection of NaN values, and so is typically not recommended.

Enoki provides a number of helpful trait classes to access array-related types. For instance, enoki::mask_t determines the mask type associated with an array, which permits replacing the auto statement above.

using FloatMask = mask_t<MyFloat>;


A comprehensive list of type traits is available in the reference. Similar to the horizontal operations for addition and multiplication involving arithmetic arrays, mask arrays also provide a set of horizontal operations:

/* Do *all* entries have a mask value corresponding to 'true'? */

/* Do *some* entries have a mask value corresponding to 'true'? */

/* Do *none* of the entries have a mask value corresponding to 'true'? */

/* Count *how many* entries have a mask value corresponding to 'true'? */


Note

The special case of the equality and inequality comparison operators: following the principle of least surprise, enoki::operator==() and enoki::operator!=() return a boolean value (i.e. they internally perform a horizontal reduction). Vertical comparison operators named eq() and neq() are also available. The following pairs of operations are thus equivalent:

MyFloat f1 = ..., f2 = ...;

bool b1 = (f1 == f2);
bool b2 = all(eq(f1, f2));

bool b3 = (f1 != f2);
bool b4 = any(neq(f1, f2));


One of the most useful bit-level operation is select() which chooses between two arguments using a mask. This is extremely useful for writing branch-free code. Argument order matches the C ternary operator, i.e. condition ? true_value : false_value maps to select(condition, true_value, false_value).

f1 = select(f1 < 0.f, f1, f2);

/* The above select() statement is equivalent to the following less efficient expression */
f1 = ((f1 < 0.f) & f1) | (~(f1 < 0.f) & f2);


Enoki also provides a special masked assignment operator, which updates entries of an array matching the given mask:

f1[f1 > 0.f] = f2;
f1[f1 < 0.f] += 1.f;


Compared to select(), a masked update may generate slightly more efficient code on some platforms. Apart from this, the two approaches can be used interchangeably. An alternative syntax involving the function enoki::masked() also exists:

masked(f1, f1 > 0.f) = f2;
masked(f1, f1 < 0.f) += 1.f;


This is functionally equivalent to the previous example. The enoki::masked() syntax exists because it extends to cases where f1 is scalar, i.e. not an Enoki array. Using Enoki functions with scalar arguments will be discussed later.

## The special case of 3D arrays¶

Because information of dimension 3 occurs frequently (spatial coordinates, color information, …) and generally also benefits very slightly from vectorization, Enoki represents 3-vectors in packed arrays of size 4, leaving the last component unused. Any vertical operations are applied to the entire array including the fourth component, while horizontal operations skip the last component.

An efficient cross product operation realized using shuffles is available for 3-vectors:

f1 = cross(f1, f2);


Generally, a better way to work with 3D data while achieving much greater instruction level parallelism is via nested arrays and the Structure of Arrays (SoA) approach discussed next.

Footnotes

 [1] Different combinations are used when not all of these instruction sets are available.