# Operators and functions¶

## Arithmetic operators¶

xtensor provides overloads of traditional arithmetic operators for xexpression objects:

• unary operator+

• unary operator-

• operator+

• operator-

• operator*

• operator/

• operator%

All these operators are element-wise operators and apply the lazy broadcasting rules explained in a previous section.

#incude "xtensor/xarray.hpp"

xt::xarray<int> a = {{1, 2}, {3, 4}};
xt::xarray<int> b = {1, 2};

xt::xarray<int> res = 2 * (a + b);
// => res = {{4, 8}, {8, 12}}


## Logical operators¶

xtensor also provides overloads of the logical operators:

• operator!

• operator||

• operator&&

Like arithmetic operators, these logical operators are element-wise operators and apply the lazy broadcasting rules. In addition to these element-wise logical operators, xtensor provides two reducing boolean functions:

• any(E&& e) returns true if any of e elements is truthy, false otherwise.

• all(E&& e) returns true if all elements of e are truthy, false otherwise.

and an element-wise ternary function (similar to the : ? ternary operator):

• where(E&& b, E1&& e1, E2&& e2) returns an xexpression whose elements are those of e1 when corresponding elements of b are truthy, and those of e2 otherwise.

#include "xtensor/xarray.hpp"

xt::xarray<bool> b = { false, true, true, false };
xt::xarray<int> a1 = { 1,   2,  3,  4 };
xt::xarray<int> a2 = { 11, 12, 13, 14 };

xt::xarray<int> res = xt::where(b, a1, a2);
// => res = { 11, 2, 3, 14 }


Unlike in numpy.where, xt::where takes full advantage of the lazyness of xtensor.

## Comparison operators¶

xtensor provides overloads of the inequality operators:

• operator<

• operator<=

• operator>

• operator>=

These overloads of inequality operators are quite different from the standard C++ inequality operators: they are element-wise operators returning boolean xexpression:

#include "xtensor/xarray.hpp"

xt::xarray<int> a1 = {  1, 12,  3, 14 };
xt::xarray<int> a2 = { 11,  2, 13, 4  };
xt::xarray<bool> comp = a1 < a2;
// => comp = { true, false, true, false }


However, equality operators are similar to the traditional ones in C++:

• operator==(const E1& e1, const E2& e2) returns true if e1 and e2 hold the same elements.

• operator!=(const E1& e1, const E2& e2) returns true if e1 and e2 don’t hold the same elements.

Element-wise equality comparison can be achieved through the xt::equal function.

#include "xtensor/xarray.hpp"

xt::xarray<int> a1 = {  1,  2, 3, 4};
xt::xarray<int> a2 = { 11, 12, 3, 4};

bool res = (a1 == a2);
// => res = false

xt::xarray<bool> re = xt::equal(a1, a2);
// => re = { false, false, true, true }


## Bitwise operators¶

xtensor also contains the following bitwise operators:

• Bitwise and: operator&

• Bitwise or: operator|

• Bitwise xor: operator^

• Bitwise not: operator~

• Bitwise left/right shift: left_shift, right_shift

## Mathematical functions¶

xtensor provides overloads for many of the standard mathematical functions:

• basic functions: abs, remainder, fma, …

• exponential functions: exp, expm1, log, log1p, …

• power functions: pow, sqrt, cbrt, …

• trigonometric functions: sin, cos, tan, …

• hyperbolic functions: sinh, cosh, tanh, …

• Error and gamma functions: erf, erfc, tgamma, lgamma, ….

• Nearest integer floating point operations: ceil, floor, trunc, …

See the API reference for a comprehensive list of available functions. Like operators, the mathematical functions are element-wise functions and apply the lazy broadcasting rules.

## Casting¶

xtensor will implicitly promote and/or cast tensor expression elements as needed, which suffices for most use-cases. But explicit casting can be performed via cast, which performs an element-wise static_cast.

#include "xtensor/xarray.hpp"

xt::xarray<int> a = { 3, 5, 7 };

auto res = a / 2;
// => res = { 1, 2, 3 }

auto res2 = xt::cast<double>(a) / 2;
// => res2 = { 1.5, 2.5, 3.5 }


## Reducers¶

xtensor provides reducers, that is, means for accumulating values of tensor expressions over prescribed axes. The return value of a reducer is an xexpression with the same shape as the input expression, with the specified axes removed.

#include "xtensor/xarray.hpp"
#include "xtensor/xmath.hpp"

xt::xarray<double> a = xt::ones<double>({3, 2, 4, 6, 5});
xt::xarray<double> res = xt::sum(a, {1, 3});
// => res.shape() = { 3, 4, 5 };
// => res(0, 0, 0) = 12


You can also call the reduce generator with your own reducing function:

#include "xtensor/xarray.hpp"
#include "xtensor/xreducer.hpp"

xt::xarray<double> arr = some_init_function({3, 2, 4, 6, 5});
xt::xarray<double> res = xt::reduce([](double a, double b) { return a*a + b*b; },
arr,
{1, 3});


The reduce generator also accepts a xreducer_functors object, a tuple of three functions (one for reducing, one for initialization and one for merging). A generator is provided to build the xreducer_functors object, the last function can be omitted:

#include "xtensor/xarray.hpp"
#include "xtensor/xreducer.hpp"

xt::xarray<double> arr = some_init_function({3, 2, 4, 6, 5});
xt::xarray<double> res = xt::reduce(xt::make_xreducer_functor([](double a, double b) { return a*a + b*b; },
[](double a) { return a * 2; })
arr,
{1, 3});


If no axes are provided, the reduction is performed over all the axes, and the result is a 0-D expression. Since xtensor’s expressions are lazy evaluated, you need to explicitely call the access operator to trigger the evaluation and get the result:

#include "xtensor/xarray.hpp"
#include "xtensor/xreducer.hpp"

xt::xarray<double> arr = some_init_function({3, 2, 4, 6, 5});
double res = xt::reduce([](double a, double b) { return a*a + b*b; }, arr)();


## Accumulators¶

Similar to reducers, xtensor provides accumulators which are used to implement cumulative functions such as cumsum or cumprod. Accumulators can currently only work on a single axis. Additionally, the accumulators are not lazy and do not return an xexpression, but rather an evaluated xarray or xtensor.

#include "xtensor/xarray.hpp"
#include "xtensor/xmath.hpp"

xt::xarray<double> a = xt::ones<double>({5, 8, 3});
xt::xarray<double> res = xt::cumsum(a, 1);
// => res.shape() = {5, 8, 3};
// => res(0, 0, 0) = 1
// => res(0, 7, 0) = 8


You can also call the accumumulate generator with your own accumulating function. For example, the implementation of cumsum is as follows:

#include "xtensor/xarray.hpp"
#include "xtensor/xaccumulator.hpp"

xt::xarray<double> arr = some_init_function({5, 5, 5});
xt::xarray<double> res = xt::accumulate([](double a, double b) { return a + b; },
arr,
1);


## Evaluation strategy¶

Generally, xtensor implements a lazy execution model, but under certain circumstances, a greedy execution model with immediate execution can be favorable. For example, reusing (and recomputing) the same values of a reducer over and over again if you use them in a loop can cost a lot of CPU cycles. Additionally, greedy execution can benefit from SIMD acceleration over reduction axes and is faster when the entire result needs to be computed.

Therefore, xtensor allows to select an evaluation_strategy. Currently, two evaluation strategies are implemented: evaluation_strategy::immediate and evaluation_strategy::lazy. When immediate evaluation is selected, the return value is not an xexpression, but an in-memory datastructure such as a xarray or xtensor (depending on the input values).

Choosing an evaluation_strategy is straightforward. For reducers:

#include "xtensor/xarray.hpp"
#include "xtensor/xreducer.hpp"

xt::xarray<double> a = xt::ones<double>({3, 2, 4, 6, 5});
auto res = xt::sum(a, {1, 3}, xt::evaluation_strategy::immediate);
// or select the default:
// auto res = xt::sum(a, {1, 3}, xt::evaluation_strategy::lazy);


Note: for accumulators, only the immediate evaluation strategy is currently implemented.

## Universal functions and vectorization¶

xtensor provides utilities to vectorize any scalar function (taking multiple scalar arguments) into a function that will perform on xexpression s, applying the lazy broadcasting rules which we described in a previous section. These functions are called xfunction s. They are xtensor’s counterpart to numpy’s universal functions.

Actually, all arithmetic and logical operators, inequality operator and mathematical functions we described before are xfunction s.

The following snippet shows how to vectorize a scalar function taking two arguments:

#include "xtensor/xarray.hpp"
#include "xtensor/xvectorize.hpp"

int f(int a, int b)
{
return a + 2 * b;
}

auto vecf = xt::vectorize(f);
xt::xarray<int> a = { 11, 12, 13 };
xt::xarray<int> b = {  1,  2,  3 };
xt::xarray<int> res = vecf(a, b);
// => res = { 13, 16, 19 }