# Universal functions¶

Standard mathematical functions can be calculated on any scalar, scalar-valued iterable (ranges, lists, tuples containing numbers), and on `ndarray`s without having to change the call signature. In all cases the functions return a new `ndarray` of typecode `float` (since these functions usually generate float values, anyway). The only exceptions to this rule are the `exp`, and `sqrt` functions, which, if `ULAB_SUPPORTS_COMPLEX` is set to 1 in ulab.h, can return complex arrays, depending on the argument. All functions execute faster with `ndarray` arguments than with iterables, because the values of the input vector can be extracted faster.

At present, the following functions are supported (starred functions can operate on, or can return complex arrays):

`acos`, `acosh`, `arctan2`, `around`, `asin`, `asinh`, `atan`, `arctan2`, `atanh`, `ceil`, `cos`, `degrees`, `exp*`, `expm1`, `floor`, `log`, `log10`, `log2`, `radians`, `sin`, `sinh`, `sqrt*`, `tan`, `tanh`.

These functions are applied element-wise to the arguments, thus, e.g., the exponential of a matrix cannot be calculated in this way, only the exponential of the matrix entries.

```# code to be run in micropython

from ulab import numpy as np

a = range(9)
b = np.array(a)

# works with ranges, lists, tuples etc.
print('a:\t', a)
print('exp(a):\t', np.exp(a))

# with 1D arrays
print('\n=============\nb:\n', b)
print('exp(b):\n', np.exp(b))

# as well as with matrices
c = np.array(range(9)).reshape((3, 3))
print('\n=============\nc:\n', c)
print('exp(c):\n', np.exp(c))
```
```a:   range(0, 9)
exp(a):      array([1.0, 2.718281828459045, 7.38905609893065, 20.08553692318767, 54.59815003314424, 148.4131591025766, 403.4287934927351, 1096.633158428459, 2980.957987041728], dtype=float64)

=============
b:
array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float64)
exp(b):
array([1.0, 2.718281828459045, 7.38905609893065, 20.08553692318767, 54.59815003314424, 148.4131591025766, 403.4287934927351, 1096.633158428459, 2980.957987041728], dtype=float64)

=============
c:
array([[0.0, 1.0, 2.0],
[3.0, 4.0, 5.0],
[6.0, 7.0, 8.0]], dtype=float64)
exp(c):
array([[1.0, 2.718281828459045, 7.38905609893065],
[20.08553692318767, 54.59815003314424, 148.4131591025766],
[403.4287934927351, 1096.633158428459, 2980.957987041728]], dtype=float64)
```

## Computation expenses¶

The overhead for calculating with micropython iterables is quite significant: for the 1000 samples below, the difference is more than 800 microseconds, because internally the function has to create the `ndarray` for the output, has to fetch the iterable’s items of unknown type, and then convert them to floats. All these steps are skipped for `ndarray`s, because these pieces of information are already known.

Doing the same with `list` comprehension requires 30 times more time than with the `ndarray`, which would become even more, if we converted the resulting list to an `ndarray`.

```# code to be run in micropython

from ulab import numpy as np
import math

a = *1000
b = np.array(a)

@timeit
def timed_vector(iterable):
return np.exp(iterable)

@timeit
def timed_list(iterable):
return [math.exp(i) for i in iterable]

print('iterating over ndarray in ulab')
timed_vector(b)

print('\niterating over list in ulab')
timed_vector(a)

print('\niterating over list in python')
timed_list(a)
```
```iterating over ndarray in ulab
execution time:  441  us

iterating over list in ulab
execution time:  1266  us

iterating over list in python
execution time:  11379  us
```

## arctan2¶

The two-argument inverse tangent function is also part of the `vector` sub-module. The function implements broadcasting as discussed in the section on `ndarray`s. Scalars (`micropython` integers or floats) are also allowed.

```# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2.2, 33.33, 444.444])
print('a:\n', a)
print('\narctan2(a, 1.0)\n', np.arctan2(a, 1.0))
print('\narctan2(1.0, a)\n', np.arctan2(1.0, a))
print('\narctan2(a, a): \n', np.arctan2(a, a))
```
```a:
array([1.0, 2.2, 33.33, 444.444], dtype=float64)

arctan2(a, 1.0)
array([0.7853981633974483, 1.14416883366802, 1.5408023243361, 1.568546328341769], dtype=float64)

arctan2(1.0, a)
array([0.7853981633974483, 0.426627493126876, 0.02999400245879636, 0.002249998453127392], dtype=float64)

arctan2(a, a):
array([0.7853981633974483, 0.7853981633974483, 0.7853981633974483, 0.7853981633974483], dtype=float64)
```

## around¶

`numpy`’s `around` function can also be found in the `vector` sub-module. The function implements the `decimals` keyword argument with default value `0`. The first argument must be an `ndarray`. If this is not the case, the function raises a `TypeError` exception. Note that `numpy` accepts general iterables. The `out` keyword argument known from `numpy` is not accepted. The function always returns an ndarray of type `mp_float_t`.

```# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2.2, 33.33, 444.444])
print('a:\t\t', a)
print('\ndecimals = 0\t', np.around(a, decimals=0))
print('\ndecimals = 1\t', np.around(a, decimals=1))
print('\ndecimals = -1\t', np.around(a, decimals=-1))
```
```a:           array([1.0, 2.2, 33.33, 444.444], dtype=float64)

decimals = 0         array([1.0, 2.0, 33.0, 444.0], dtype=float64)

decimals = 1         array([1.0, 2.2, 33.3, 444.4], dtype=float64)

decimals = -1        array([0.0, 0.0, 30.0, 440.0], dtype=float64)
```

## exp¶

If `ULAB_SUPPORTS_COMPLEX` is set to 1 in ulab.h, the exponential function can also take complex arrays as its argument, in which case the return value is also complex.

```# code to be run in micropython

from ulab import numpy as np

a = np.array([1, 2, 3])
print('a:\t\t', a)
print('exp(a):\t\t', np.exp(a))
print()

b = np.array([1+1j, 2+2j, 3+3j], dtype=np.complex)
print('b:\t\t', b)
print('exp(b):\t\t', np.exp(b))
```
```a:           array([1.0, 2.0, 3.0], dtype=float64)
exp(a):              array([2.718281828459045, 7.38905609893065, 20.08553692318767], dtype=float64)

b:           array([1.0+1.0j, 2.0+2.0j, 3.0+3.0j], dtype=complex)
exp(b):              array([1.468693939915885+2.287355287178842j, -3.074932320639359+6.71884969742825j, -19.88453084414699+2.834471132487004j], dtype=complex)
```

## sqrt¶

If `ULAB_SUPPORTS_COMPLEX` is set to 1 in ulab.h, the exponential function can also take complex arrays as its argument, in which case the return value is also complex. If the input is real, but the results might be complex, the user is supposed to specify the output `dtype` in the function call. Otherwise, the square roots of negative numbers will result in `NaN`.

```# code to be run in micropython

from ulab import numpy as np

a = np.array([1, -1])
print('a:\t\t', a)
print('sqrt(a):\t\t', np.sqrt(a))
print('sqrt(a):\t\t', np.sqrt(a, dtype=np.complex))
```
```a:           array([1.0, -1.0], dtype=float64)
sqrt(a):             array([1.0, nan], dtype=float64)
sqrt(a):             array([1.0+0.0j, 0.0+1.0j], dtype=complex)
```

## Vectorising generic python functions¶

The examples above use factory functions. In fact, they are nothing but the vectorised versions of the standard mathematical functions. User-defined `python` functions can also be vectorised by help of `vectorize`. This function takes a positional argument, namely, the `python` function that you want to vectorise, and a non-mandatory keyword argument, `otypes`, which determines the `dtype` of the output array. The `otypes` must be `None` (default), or any of the `dtypes` defined in `ulab`. With `None`, the output is automatically turned into a float array.

The return value of `vectorize` is a `micropython` object that can be called as a standard function, but which now accepts either a scalar, an `ndarray`, or a generic `micropython` iterable as its sole argument. Note that the function that is to be vectorised must have a single argument.

```# code to be run in micropython

from ulab import numpy as np

def f(x):
return x*x

vf = np.vectorize(f)

# calling with a scalar
print('{:20}'.format('f on a scalar: '), vf(44.0))

# calling with an ndarray
a = np.array([1, 2, 3, 4])
print('{:20}'.format('f on an ndarray: '), vf(a))

# calling with a list
print('{:20}'.format('f on a list: '), vf([2, 3, 4]))
```
```f on a scalar:       array([1936.0], dtype=float64)
f on an ndarray:     array([1.0, 4.0, 9.0, 16.0], dtype=float64)
f on a list:         array([4.0, 9.0, 16.0], dtype=float64)
```

As mentioned, the `dtype` of the resulting `ndarray` can be specified via the `otypes` keyword. The value is bound to the function object that `vectorize` returns, therefore, if the same function is to be vectorised with different output types, then for each type a new function object must be created.

```# code to be run in micropython

from ulab import numpy as np

l = [1, 2, 3, 4]
def f(x):
return x*x

vf1 = np.vectorize(f, otypes=np.uint8)
vf2 = np.vectorize(f, otypes=np.float)

print('{:20}'.format('output is uint8: '), vf1(l))
print('{:20}'.format('output is float: '), vf2(l))
```
```output is uint8:     array([1, 4, 9, 16], dtype=uint8)
output is float:     array([1.0, 4.0, 9.0, 16.0], dtype=float64)
```

The `otypes` keyword argument cannot be used for type coercion: if the function evaluates to a float, but `otypes` would dictate an integer type, an exception will be raised:

```# code to be run in micropython

from ulab import numpy as np

int_list = [1, 2, 3, 4]
float_list = [1.0, 2.0, 3.0, 4.0]
def f(x):
return x*x

vf = np.vectorize(f, otypes=np.uint8)

print('{:20}'.format('integer list: '), vf(int_list))
# this will raise a TypeError exception
print(vf(float_list))
```
```integer list:        array([1, 4, 9, 16], dtype=uint8)

Traceback (most recent call last):
File "/dev/shm/micropython.py", line 14, in <module>
TypeError: can't convert float to int
```

### Benchmarks¶

It should be pointed out that the `vectorize` function produces the pseudo-vectorised version of the `python` function that is fed into it, i.e., on the C level, the same `python` function is called, with the all-encompassing `mp_obj_t` type arguments, and all that happens is that the `for` loop in `[f(i) for i in iterable]` runs purely in C. Since type checking and type conversion in `f()` is expensive, the speed-up is not so spectacular as when iterating over an `ndarray` with a factory function: a gain of approximately 30% can be expected, when a native `python` type (e.g., `list`) is returned by the function, and this becomes around 50% (a factor of 2), if conversion to an `ndarray` is also counted.

The following code snippet calculates the square of a 1000 numbers with the vectorised function (which returns an `ndarray`), with `list` comprehension, and with `list` comprehension followed by conversion to an `ndarray`. For comparison, the execution time is measured also for the case, when the square is calculated entirely in `ulab`.

```# code to be run in micropython

from ulab import numpy as np

def f(x):
return x*x

vf = np.vectorize(f)

@timeit
def timed_vectorised_square(iterable):
return vf(iterable)

@timeit
def timed_python_square(iterable):
return [f(i) for i in iterable]

@timeit
def timed_ndarray_square(iterable):
return np.array([f(i) for i in iterable])

@timeit
def timed_ulab_square(ndarray):
return ndarray**2

print('vectorised function')
squares = timed_vectorised_square(range(1000))

print('\nlist comprehension')
squares = timed_python_square(range(1000))

print('\nlist comprehension + ndarray conversion')
squares = timed_ndarray_square(range(1000))

print('\nsquaring an ndarray entirely in ulab')
a = np.array(range(1000))
squares = timed_ulab_square(a)
```
```vectorised function
execution time:  7237  us

list comprehension
execution time:  10248  us

list comprehension + ndarray conversion
execution time:  12562  us

squaring an ndarray entirely in ulab
execution time:  560  us
```

From the comparisons above, it is obvious that `python` functions should only be vectorised, when the same effect cannot be gotten in `ulab` only. However, although the time savings are not significant, there is still a good reason for caring about vectorised functions. Namely, user-defined `python` functions become universal, i.e., they can accept generic iterables as well as `ndarray`s as their arguments. A vectorised function is still a one-liner, resulting in transparent and elegant code.

A final comment on this subject: the `f(x)` that we defined is a generic `python` function. This means that it is not required that it just crunches some numbers. It has to return a number object, but it can still access the hardware in the meantime. So, e.g.,

```led = pyb.LED(2)

def f(x):
if x < 100:
led.toggle()
return x*x
```

is perfectly valid code.