Basic mathematical operations involving numbers with uncertainties only require a simple import:

```
>>> from uncertainties import ufloat
```

The `ufloat()` function creates numbers with uncertainties. Existing
calculation code can usually run with no or little modification and
automatically produce results with uncertainties.

The `uncertainties` module contains other features, which can be
made accessible through

```
>>> import uncertainties
```

The `uncertainties` package also contains sub-modules for
*advanced mathematical functions*, and
*arrays and matrices*.

Numbers with uncertainties can be input either numerically, or through one of many string representations, so that files containing numbers with uncertainties can easily be parsed. Thus, x = 0.20±0.01 can be expressed in many convenient ways, including:

```
>>> x = ufloat(0.20, 0.01) # x = 0.20+/-0.01
```

```
>>> from uncertainties import ufloat_fromstr
>>> x = ufloat_fromstr("0.20+/-0.01")
>>> x = ufloat_fromstr("(2+/-0.1)e-01") # Factored exponent
>>> x = ufloat_fromstr("0.20(1)") # Short-hand notation
>>> x = ufloat_fromstr("20(1)e-2") # Exponent notation
>>> x = ufloat_fromstr(u"0.20±0.01") # Pretty-print form
>>> x = ufloat_fromstr("0.20") # Automatic uncertainty of +/-1 on last digit
```

Each number created this way is an **independent (random) variable**
(for details, see the *Technical Guide*).

More information can be obtained with `pydoc uncertainties.ufloat`
and `pydoc uncertainties.ufloat_fromstr` (“20(1)×10^{-2}” is
also recognized, etc.).

Calculations can be performed directly, as with regular real numbers:

```
>>> square = x**2
>>> print square
0.040+/-0.004
```

Besides being able to apply basic mathematical operations to numbers
with uncertainty, this package provides generalizations of **most of
the functions from the standard** `math` **module**. These
mathematical functions are found in the `uncertainties.umath`
module:

```
>>> from uncertainties.umath import * # Imports sin(), etc.
>>> sin(x**2)
0.03998933418663417+/-0.003996800426643912
```

The list of available mathematical functions can be obtained with the
`pydoc uncertainties.umath` command.

It is possible to put numbers with uncertainties in NumPy arrays and matrices:

```
>>> arr = numpy.array([ufloat(1, 0.01), ufloat(2, 0.1)])
>>> 2*arr
[2.0+/-0.02 4.0+/-0.2]
>>> print arr.sum()
3.00+/-0.10
```

Thus, usual operations on NumPy arrays can be performed transparently even when these arrays contain numbers with uncertainties.

*More complex operations on NumPy arrays and matrices* can be
performed through the dedicated `uncertainties.unumpy` module.

Numbers with uncertainties can be printed conveniently:

```
>>> print x
0.200+/-0.010
```

The resulting form can generally be parsed back with
`ufloat_fromstr()` (except for the LaTeX form).

The nominal value and the uncertainty always have the **same
precision**: this makes it easier to compare them.

More **control over the format** can be obtained (in Python 2.6+)
through the usual `format()` method of strings:

```
>>> print 'Result = {:10.2f}'.format(x)
Result = 0.20+/- 0.01
```

(Python 2.6 requires `'{0:10.2f}'` instead, with the usual explicit
index. In Python 2.5 and earlier versions, `str.format()` is not
available, but one can use the `format()` method of numbers with
uncertainties instead: `'Result = %s' % x.format('10.2f')`.)

**All the float format specifications** are accepted, except those
with the `n` format type. In particular, a fill character, an
alignment option, a sign or zero option, a width, or the `%` format
type are all supported.

When `uncertainties` must **choose the number of significant
digits on the uncertainty**, it is defined with the Particle Data
Group
rounding rules (these rules keep the number of digits small, while
preventing the uncertainty from being displayed with a large relative
error):

```
>>> print 'Automatic number of digits on the uncertainty: {}'.format(x)
Automatic number of digits on the uncertainty: 0.200+/-0.010
>>> print x
0.200+/-0.010
```

It is possible to **control the number of significant digits of the
uncertainty** by adding the precision modifier `u` after the
precision (and before any valid float format type like `f`, `e`,
the empty format type, etc.):

```
>>> print '1 significant digit on the uncertainty: {:.1u}'.format(x)
1 significant digit on the uncertainty: 0.20+/-0.01
>>> print '3 significant digits on the uncertainty: {:.3u}'.format(x)
3 significant digits on the uncertainty: 0.2000+/-0.0100
>>> print '1 significant digit, exponent notation: {:.1ue}'.format(x)
1 significant digit, exponent notation: (2.0+/-0.1)e-01
>>> print '1 significant digit, percentage: {:.1u%}'.format(x)
1 significant digit, percentage: (20+/-1)%
```

The usual **float formats with a precision** retain their original
meaning (e.g. `.2e` uses two digits after the decimal point): code
that works with floats produces similar results when running with
numbers with uncertainties.

A **common exponent** is automatically calculated if an exponent is
needed for the larger of the nominal value (in absolute value) and the
uncertainty (the rule is the same as for floats). The exponent is
generally **factored**, for increased legibility:

```
>>> print x*1e7
(2.00+/-0.10)e+06
```

When a *format width* is used, the common exponent is not factored:

```
>>> print 'Result = {:10.1e}'.format(x*1e-10)
Result = 2.0e-11+/- 0.1e-11
```

(Using a (minimal) width of 1 is thus a way of forcing exponents to not be factored.) Thanks to this feature, each part (nominal value and standard deviation) is correctly aligned across multiple lines, while the relative magnitude of the error can still be readily estimated thanks to the common exponent.

Formatting options can be added at the end of the format string: `S`
for the **shorthand notation**, `L` for a **LaTeX** output, `P`
for **pretty-printing**:

```
>>> print '{:+.1uS}'.format(x) # Sign, 1 digit for the uncertainty, shorthand
+0.20(1)
>>> print '{:L}'.format(x*1e7) # Automatic exponent form, LaTeX
\left(2.00 \pm 0.10\right) \times 10^{6}
```

The pretty-printing mode uses “±” and superscript exponents: the
default output is such that `print '{:.2e}'.format(x)` yields
“(2.00+/-0.10)e-01”, whereas the pretty-printing mode in `print
u'{:.2eP}'.format(x)` yields “(2.00±0.10)×10^{-1}”. Note that
the pretty-printing mode implies using Unicode format strings
(`u'…'` in Python 2, but simply `'…'` in Python 3).

These formatting options can be combined (when meaningful).

An uncertainty which is *exactly* **zero** is always formatted as an
integer:

```
>>> print ufloat(3.1415, 0)
3.1415+/-0
>>> print ufloat(3.1415e10, 0)
(3.1415+/-0)e+10
>>> print ufloat(3.1415, 0.0005)
3.1415+/-0.0005
>>> print '{:.2f}'.format(ufloat(3.14, 0.001))
3.14+/-0.00
>>> print '{:.2f}'.format(ufloat(3.14, 0.00))
3.14+/-0
```

**All the digits** of a number with uncertainty are given in its
representation:

```
>>> y = ufloat(1.23456789012345, 0.123456789)
>>> print y
1.23+/-0.12
>>> print repr(y)
1.23456789012345+/-0.123456789
>>> y
1.23456789012345+/-0.123456789
```

**More information** on formatting can be obtained with `pydoc
uncertainties.UFloat.__format__` (customization of the LaTeX output,
etc.).

It is sometimes useful to have a **consistent formatting** across
multiple parts of a program. Python’s string.Formatter class
allows one to do just that. Here is how it can be used to consistently
use the shorthand notation for numbers with uncertainties:

```
class ShorthandFormatter(string.Formatter):
def format_field(self, value, format_spec):
if isinstance(value, uncertainties.UFloat):
return value.format(format_spec+'S') # Shorthand option added
# Special formatting for other types can be added here (floats, etc.)
else:
# Usual formatting:
return super(ShorthandFormatter, self).format_field(
value, format_spec)
frmtr = ShorthandFormatter()
print frmtr.format("Result = {0:.1u}", x) # 1-digit uncertainty
```

prints with the shorthand notation: `Result = 0.20(1)`.

The nominal value and the uncertainty (standard deviation) can also be accessed independently:

```
>>> print square
0.040+/-0.004
>>> print square.nominal_value
0.04
>>> print square.n # Abbreviation
0.04
>>> print square.std_dev
0.004
>>> print square.s # Abbreviation
0.004
```

The various contributions to an uncertainty can be obtained through
the `error_components()` method, which maps the **independent
variables a quantity depends on** to their **contribution to the total
uncertainty**. According to the *linear error propagation theory* implemented in `uncertainties`, the sum of the
squares of these contributions is the squared uncertainty.

The individual contributions to the uncertainty are more easily usable
when the variables are **tagged**:

```
>>> u = ufloat(1, 0.1, "u variable") # Tag
>>> v = ufloat(10, 0.1, "v variable")
>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.223606797749979
>>> for (var, error) in sum_value.error_components().items():
... print "{}: {}".format(var.tag, error)
...
u variable: 0.1
v variable: 0.2
```

The variance (i.e. squared uncertainty) of the result
(`sum_value`) is the quadratic sum of these independent
uncertainties, as it should be (`0.1**2 + 0.2**2`).

The tags *do not have to be distinct*. For instance, *multiple* random
variables can be tagged as `"systematic"`, and their contribution to
the total uncertainty of `result` can simply be obtained as:

```
>>> syst_error = math.sqrt(sum( # Error from *all* systematic errors
... error**2
... for (var, error) in result.error_components().items()
... if var.tag == "systematic"))
```

The remaining contribution to the uncertainty is:

```
>>> other_error = math.sqrt(result.std_dev**2 - syst_error**2)
```

The variance of `result` is in fact simply the quadratic sum of
these two errors, since the variables from
`result.error_components()` are independent.

Comparison operators behave in a natural way:

```
>>> print x
0.200+/-0.010
>>> y = x + 0.0001
>>> y
0.2001+/-0.01
>>> y > x
True
>>> y > 0
True
```

One important concept to keep in mind is that `ufloat()` creates a
random variable, so that two numbers with the same nominal value and
standard deviation are generally different:

```
>>> y = ufloat(1, 0.1)
>>> z = ufloat(1, 0.1)
>>> print y
1.00+/-0.10
>>> print z
1.00+/-0.10
>>> y == y
True
>>> y == z
False
```

In physical terms, two rods of the same nominal length and uncertainty
on their length are generally of different sizes: `y` is different
from `z`.

More detailed information on the semantics of comparison operators for
numbers with uncertainties can be found in the *Technical Guide*.

The covariance matrix between various variables or calculated quantities can be simply obtained:

```
>>> sum_value = u+2*v
>>> cov_matrix = uncertainties.covariance_matrix([u, v, sum_value])
```

has value

```
[[0.01, 0.0, 0.01],
[0.0, 0.01, 0.02],
[0.01, 0.02, 0.05]]
```

In this matrix, the zero covariances indicate that `u` and `v` are
independent from each other; the last column shows that `sum_value`
does depend on these variables. The `uncertainties` package
keeps track at all times of all correlations between quantities
(variables and functions):

```
>>> sum_value - (u+2*v)
0.0+/-0
```

Reciprocally, **correlated variables can be created** transparently,
provided that the NumPy package is available.

Correlated variables can be obtained through the *covariance* matrix:

```
>>> (u2, v2, sum2) = uncertainties.correlated_values([1, 10, 21], cov_matrix)
```

creates three new variables with the listed nominal values, and the given covariance matrix:

```
>>> sum_value
21.0+/-0.223606797749979
>>> sum2
21.0+/-0.223606797749979
>>> sum2 - (u2+2*v2)
0.0+/-3.83371856862256e-09
```

The theoretical value of the last expression is exactly zero, like for
`sum - (u+2*v)`, but numerical errors yield a small uncertainty
(3e-9 is indeed very small compared to the uncertainty on `sum2`:
correlations should in fact cancel the uncertainty on `sum2`).

The covariance matrix is the desired one:

```
>>> uncertainties.covariance_matrix([u2, v2, sum2])
```

reproduces the original covariance matrix `cov_matrix` (up to
rounding errors).

Alternatively, correlated values can be defined through a
*correlation* matrix (the correlation matrix is the covariance matrix
normalized with individual standard deviations; it has ones on its
diagonal), along with a list of nominal values and standard deviations:

```
>>> (u3, v3, sum3) = uncertainties.correlated_values_norm(
... [(1, 0.1), (10, 0.1), (21, 0.22360679774997899)], corr_matrix)
>>> print u3
1.00+/-0.10
```

The three returned numbers with uncertainties have the correct
uncertainties and correlations (`corr_matrix` can be recovered
through `correlation_matrix()`).

This package allows **code which is not meant to be used with numbers
with uncertainties to handle them anyway**. This is for instance
useful when calling external functions (which are out of the user’s
control), including functions written in C or Fortran. Similarly,
**functions that do not have a simple analytical form** can be
automatically wrapped so as to also work with arguments that contain
uncertainties.

It is thus possible to take a function `f()` *that returns a
single float*, and to automatically generalize it so that it also
works with numbers with uncertainties:

```
>>> wrapped_f = uncertainties.wrap(f)
```

The new function `wrapped_f()` *accepts numbers with uncertainties*
as arguments *wherever a Python float is used* for `f()`.
`wrapped_f()` returns the same values as `f()`, but with
uncertainties.

With a simple wrapping call like above, uncertainties in the function
result are automatically calculated numerically. **Analytical
uncertainty calculations can be performed** if derivatives are
provided to `wrap()`.

More details are available in the documentation string of `wrap()`
(accessible through the `pydoc` command, or Python’s `help()`
shell function).

It is sometimes useful to modify the error on certain parameters so as
to study its impact on a final result. With this package, the
**uncertainty of a variable can be changed** on the fly:

```
>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.223606797749979
>>> prev_uncert = u.std_dev
>>> u.std_dev = 10
>>> sum_value
21.0+/-10.00199980003999
>>> u.std_dev = prev_uncert
```

The relevant concept is that `sum_value` does depend on the
variables `u` and `v`: the `uncertainties` package keeps
track of this fact, as detailed in the *Technical Guide*, and uncertainties can thus be updated at any time.

When manipulating ensembles of numbers, *some* of which contain
uncertainties while others are simple floats, it can be useful to
access the **nominal value and uncertainty of all numbers in a uniform
manner**. This is what the `nominal_value()` and
`std_dev()` functions do:

```
>>> print uncertainties.nominal_value(x)
0.2
>>> print uncertainties.std_dev(x)
0.01
>>> uncertainties.nominal_value(3)
3
>>> uncertainties.std_dev(3)
0.0
```

Finally, a utility method is provided that directly yields the
standard score
(number of standard deviations) between a number and a result with
uncertainty: with `x` equal to 0.20±0.01,

```
>>> x.std_score(0.17)
-3.0
```

Since the application of *linear error propagation theory* involves the calculation of **derivatives**, this
package automatically performs such calculations; users can thus
easily get the derivative of an expression with respect to any of its
variables:

```
>>> u = ufloat(1, 0.1)
>>> v = ufloat(10, 0.1)
>>> sum_value = u+2*v
>>> sum_value.derivatives[u]
1.0
>>> sum_value.derivatives[v]
2.0
```

These values are obtained with a *fast differentiation algorithm*.

The capabilities of the `uncertainties` package in terms of array
handling are detailed in *Uncertainties in arrays*.

Details about the theory behind this package and implementation
information are given in the
*Technical Guide*.