# 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 }
```