2. Getting Started¶

After installation, you should have a folder examples in your main pySecDec directory. Here we describe a few of the examples available in the examples directory. A full list of examples is given in List of Examples.

2.1. A Simple Example¶

We first show how to compute a simple dimensionally regulated integral:

$\int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (x+y)^{-2+\epsilon}.$

To run the example change to the easy directory and run the commands:

$ python3 generate_easy.py
$ make -C easy
$ python3 integrate_easy.py

Additional build options are discussed in the next section. This will evaluate and print the result of the integral:

Numerical Result: + ((1.00000000000000022e+00,0.00000000000000000e+00) +/- (5.65352153979095401e-17,0.00000000000000000e+00))*eps^-1 + ((3.06852819440053548e-01,0.00000000000000000e+00) +/- (1.18502493127591741e-15,0.00000000000000000e+00)) + O(eps)
Analytic Result: + (1.000000)*eps^-1 + (0.306853) + O(eps)

The file generate_easy.py defines the integral and calls pySecDec to perform the sector decomposition. When run it produces the directory easy which contains the code required to numerically evaluate the integral. The make command builds this code and produces a library. The file integrate_easy.py loads the integral library and evaluates the integral. The user is encouraged to copy and adapt these files to evaluate their own integrals.

Note

If the user is interested in evaluating a loop integral there are many convenience functions that make this much easier. Please see Evaluating a Loop Integral for more details.

In generate_easy.py we first import make_package, a function which can decompose, subtract and expand regulated integrals and write a C++ package to evaluate them. To define our integral we give it a name which will be used as the name of the output directory and C++ namespace. The integration_variables are declared along with a list of the name of the regulators. We must specify a list of the requested_orders to which pySecDec should expand our integral in each regulator. Here we specify requested_orders = [0] which instructs make_package to expand the integral up to and including $\mathcal{O}(\epsilon)$ . Next, we declare the polynomials_to_decompose, here sympy syntax should be used.

#!/usr/bin/env python3

from pySecDec import make_package

if __name__ == "__main__":

    make_package(
        name = 'easy',
        integration_variables = ['x','y'],
        regulators = ['eps'],

        requested_orders = [0],
        polynomials_to_decompose = ['(x+y)^(-2+eps)']
    )

Once the C++ library has been written and built we run integrate_easy.py. Here the library is loaded using IntegralLibrary. Calling the instance of IntegralLibrary with easy_integral() numerically evaluates the integral and returns the result.

#!/usr/bin/env python3

from pySecDec.integral_interface import IntegralLibrary
from math import log

if __name__ == "__main__":

    # load c++ library
    easy = IntegralLibrary('easy/easy_pylink.so')

    # integrate
    _, _, result = easy()

    # print result
    print('Numerical Result:' + result)
    print('Analytic Result:' + ' + (%f)*eps^-1 + (%f) + O(eps)' % (1.0,1.0-log(2.0)))

2.2. Evaluating a Loop Integral¶

A simple example of the evaluation of a loop integral with pySecDec is box1L. This example computes a one-loop box with one off-shell leg (with off-shellness s1) and one internal massive line (with mass squared msq), it is shown in Fig. 2.1.

Diagrammatic representation of `box1L` — Fig. 2.1 Diagrammatic representation of box1L¶

To run the example change to the box1L directory and run the commands:

$ python3 generate_box1L.py
$ make -C box1L
$ python3 integrate_box1L.py

This will print the result of the integral evaluated with Mandelstam invariants s=4.0, t=-0.75 and s1=1.25, msq=1.0:

eps^-2: -0.142868356275422825 - 1.63596224151119965e-6*I +/- ( 0.00118022544307414272 + 0.000210769456586696187*I )
eps^-1: 0.639405625715768089 + 1.34277036689902802e-6*I +/- ( 0.00650722394065588166 + 0.000971496627153705891*I )
eps^0 : -0.425514350373418893 + 1.86892487760861536*I +/- ( 0.00706834403694714484 + 0.0186497890361357298*I )

The file generate_box1L.py defines the loop integral and calls pySecDec to perform the sector decomposition. When run it produces the directory box1L which contains the code required to numerically evaluate the integral. The make command builds this code and produces a library. The file integrate_box1L.py loads the integral library and evaluates the integral for a specified numerical point.

The content of the python files is described in detail in the following sections. The user is encouraged to copy and adapt these files to evaluate their own loop integrals.

2.2.1. Defining a Loop Integral¶

To explain the input format, let us look at generate_box1L.py from the one-loop box example. The first line reads

import pySecDec as psd

This line specifies that the module pySecDec should be imported with the alias psd. The functions loop_package and loop_integral from this module will be needed shortly.

The following part contains the definition of the loop integral li:

li = psd.LoopIntegralFromGraph(
    # Give adjacency list and indicate whether the propagator
    # connecting the numbered vertices is massive or massless
    # in the first entry of each list item.
    internal_lines = [['m',[1,2]], ['0',[2,3]], ['0',[3,4]], ['0',[4,1]]],
    # List the names of the external momenta and the labels
    # of the vertecies they are attached to.
    external_lines = [['p1',1], ['p2',2], ['p3',3], ['p4',4]],
    # Define the kinematics and the names of the kinematic
    # invariants.
    replacement_rules = [
        ('p1*p1', 's1'),
        ('p2*p2', '0'),
        ('p3*p3', '0'),
        ('p4*p4', '0'),
        ('p3*p2', 't/2'),
        ('p1*p2', 's/2-s1/2'),
        ('p1*p4', 't/2-s1/2'),
        ('p2*p4', 's1/2-t/2-s/2'),
        ('p3*p4', 's/2'),
        ('m**2', 'msq')
    ]
)

Here the class LoopIntegralFromGraph is used to Feynman parametrize the loop integral given the adjacency list. Alternatively, the class LoopIntegralFromPropagators can be used to construct the Feynman integral given the momentum representation.

The symbols for the kinematic invariants and the masses also need to be given as an ordered list. The ordering is important as the numerical values assigned to these list elements at the numerical evaluation stage should have the same order.

Mandelstam_symbols = ['s','t','s1']
mass_symbols = ['msq']

Next, the function loop_package is called. It will create a folder called box1L. It performs the algebraic sector decomposition steps and writes a package containing the C++ code for the numerical evaluation. The argument requested_orders specifies the order in the regulator to which the integral should be expanded. For a complete list of possible options see loop_package.

psd.loop_package(
    name = 'box1L',

    loop_integral = li,

    real_parameters = Mandelstam_symbols + mass_symbols,

    # the highest order of the final epsilon expansion -->
    # change this value to whatever you think is appropriate
    requested_orders = [0],

    # the method to be used for the sector decomposition
    # valid values are ``iterative`` or ``geometric`` or ``geometric_ku``
    decomposition_method = 'geometric'
)

2.2.2. Building the integration Library¶

After running the python script generate_box1L.py the folder box1L is created and should contain the following files and subdirectories

Makefile       box1L.hpp           integrate_box1L.cpp  disteval/
Makefile.conf  box1L.pdf           box1L_data/          pylink/
README         integral_names.txt  box1L_integral/      src/

In the folder box1L, typing

$ make

will create the static library box1L_integral/libbox1L_integral.a and the shared library box1L_pylink.so which can be linked to external programs. The make command can also be run in parallel by using the -j option. The number of threads each instance of tform uses can be set via the environment variable FORMTHREADS.

New in version 1.4: The environment variable FORMOPT sets FORM’s code optimization level. If not set, the value that was passed to make_package or loop_package is used.

To build the dynamic library libbox1L.so set dynamic as build target:

$ make dynamic

New in version 1.6: To build the disteval library (which will consist of multiple files in box1L/disteval/ directory) set disteval as build target:

$ make disteval

Note that disteval libraries are designed with a focus on optimization for modern processors by making use of AVX2 and FMA instruction sets. For CPUs that support these, best performance is achieved by using the newest compiler available on the system (chosen via the CXX variable), and by enabling the support of AVX2 and FMA (via the CXXFLAGS variable). For example:

$ make disteval CXX="g++-12" CXXFLAGS="-mavx2 -mfma"

To build the libraries with NVidia C Compiler (NVCC) for GPU support, type

$ make SECDEC_WITH_CUDA_FLAGS="-arch=sm_XX"

where sm_XX must be replaced by the target NVidia GPU architechtures; see the arch option of NVCC. The SECDEC_WITH_CUDA_FLAGS variable, which enables GPU code compilation, contains flags which are passed to NVCC during code compilation and linking. Multiple GPU architectures may be specified as described in the NVCC manual, for example SECDEC_WITH_CUDA_FLAGS="-gencode arch=compute_XX,code=sm_XX -gencode arch=compute_YY,code=sm_YY" where XX and YY are the target GPU architectures. The script examples/easy/print-cuda-arch.sh can be used to obtain the compute architecture of your current machine.

To evaluate the integral numerically a program can call one of these libraries. How to do this interactively or via a python script is explained in the section Python Interface. Alternatively, a C++ program can be produced as explained in the section C++ Interface.

2.2.3. Python Interface¶

To evaluate the integral for a given numerical point we can use integrate_box1L.py. First it imports the necessary python packages and loads the C++ library.

from pySecDec.integral_interface import IntegralLibrary
import sympy as sp

# load c++ library
box1L = IntegralLibrary('box1L/box1L_pylink.so')

Next, an integrator is configured for the numerical integration. The full list of available integrators and their options is given in integral_interface.

# choose integrator
box1L.use_Vegas(flags=2) # ``flags=2``: verbose --> see Cuba manual

If you want to use GPUs, change to the CudaQmc integrator. For example, to run on all available GPUs and CPU cores using the Korobov transform with weight 3, change the above lines to

# choose integrator
box1L.use_Qmc(transform='Korobov3')

Calling the box1L library numerically evaluates the integral. Note that the order of the real parameters must match that specified in generate_box1L.py. A list of possible settings for the library, in particular details of how to set the contour deformation parameters, is given in IntegralLibrary. To change the accuracy settings of the integration, the most important parameters are epsrel, epsabs and maxeval, which can be added to the integrator argument list:

# choose integrator
box1L.use_Vegas(flags=2,epsrel=0.01, epsabs=1e-07, maxeval=1000000)

# integrate
str_integral_without_prefactor, str_prefactor, str_integral_with_prefactor = box1L(real_parameters=[4.0, -0.75, 1.25, 1.0])

In case of a sign check error (sign_check_error), the arguments number_of_presamples, deformation_parameters_maximum, and deformation_parameters_minimum as described in IntegralLibrary can be used to modify the contour. At this point the string str_integral_with_prefactor contains the full result of the integral and can be manipulated as required. In the integrate_box1L.py an example is shown how to parse the expression with sympy and access individual orders of the regulator.

Note

Instead of parsing the result, it can simply be printed with the line print(str_integral_with_prefactor).

# convert complex numbers from c++ to sympy notation
str_integral_with_prefactor = str_integral_with_prefactor.replace(',','+I*')
str_prefactor = str_prefactor.replace(',','+I*')
str_integral_without_prefactor = str_integral_without_prefactor.replace(',','+I*')

# convert result to sympy expressions
integral_with_prefactor = sp.sympify(str_integral_with_prefactor.replace('+/-','*value+error*'))
integral_with_prefactor_err = sp.sympify(str_integral_with_prefactor.replace('+/-','*value+error*'))
prefactor = sp.sympify(str_prefactor)
integral_without_prefactor = sp.sympify(str_integral_without_prefactor.replace('+/-','*value+error*'))
integral_without_prefactor_err = sp.sympify(str_integral_without_prefactor.replace('+/-','*value+error*'))

# examples how to access individual orders
print('Numerical Result')
print('eps^-2:', integral_with_prefactor.coeff('eps',-2).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-2).coeff('error'), ')')
print('eps^-1:', integral_with_prefactor.coeff('eps',-1).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',-1).coeff('error'), ')')
print('eps^0 :', integral_with_prefactor.coeff('eps',0).coeff('value'), '+/- (', integral_with_prefactor_err.coeff('eps',0).coeff('error'), ')')

An example of how to loop over several kinematic points is shown in the example integrate_box1L_multiple_points.py.

2.2.4. Command-line interface with disteval¶

New in version 1.6.

The disteval library, once built, can be used directly from the command line via the pySecDec.disteval Python module:

$ python3 -m pySecDec.disteval box1L/disteval/box1L.json s=4.0 t=-0.75 s1=1.25 msq=1.0
[...]
[
  (
    +eps^-2*(-1.4285714285714279e-01+9.0159338621354360e-18j)
    +eps^-2*(+3.4562234592930473e-17+1.4290950719747641e-17j)*plusminus
    +eps^-1*(+6.3843370937935406e-01+2.5048341561937569e-10j)
    +eps^-1*(+4.4293092326873179e-10+4.6245608965315405e-10j)*plusminus
    +eps^0*(-4.2634981062934296e-01+1.8664974523210687e+00j)
    +eps^0*(+5.5826851229628189e-06+4.8099553795389634e-06j)*plusminus
  )
]

Note that the output is a list of expressions; here a list of a single item. This is because as we shall see in Evaluating a Weighted Sum of Integrals, a single library can produce multiple resulting expressions.

The general usage of the command-line interface is:

$ python3 -m pySecDec.disteval integrand.json [options] <var>=value ...

The evaluation can be controlled via the provided command-line options:

--epsabs=<number>: stop if this absolute precision is reached (default: 1e-10);
--epsrel=<number>: stop if this relative precision is reached (default: 1e-4);
--timeout=<number>: stop after at most this many seconds (defaul: inf);
--points=<number>: use this initial Quasi-Monte-Carlo lattice size (default: 1e4);
--presamples=<number>: use this many points for presampling (default: 1e4);
--shifts=<number>: use this many lattice shifts per integral (default: 32);
--coefficients=<path>: use coefficients from this directory;
--format=<path>: output the result in this format (sympy, mathematica, or json; default: sympy).

This list of options can also be obtained from within the command line by running:

$ python3 -m pySecDec.disteval --help

2.2.5. Python interface with disteval¶

Similarly to Python Interface, disteval libraries can be used from Python, this time via the DistevalLibrary class. An example of this usage be found in integrate_box1L_disteval.py. The example starts by importing the necessary packages and loading the library:

from pySecDec.integral_interface import DistevalLibrary
import sympy as sp

box1L = DistevalLibrary('box1L/disteval/box1L.json')

Then, calling the box1L library to perform the evaluation at the given parameter values:

# integrate
str_result = box1L(parameters={"s": 4.0, "t": -0.75, "s1": 1.25, "msq": 1.0}, verbose=False)

And finally, converting the result to a sympy object and printing it:

# convert result to sympy expressions
result = sp.sympify(str_result)
value = result[0].subs({"plusminus": 0})
error = result[0].coeff("plusminus")

# examples how to access individual orders
print('Numerical Result')
print('eps^-2:', value.coeff('eps',-2), '+/- (', error.coeff('eps',-2), ')')
print('eps^-1:', value.coeff('eps',-1), '+/- (', error.coeff('eps',-1), ')')
print('eps^0 :', value.coeff('eps',0), '+/- (', error.coeff('eps',0), ')')

2.2.6. C++ Interface¶

Usually it is easier to obtain a numerical result using the Python Interface. However, the library can also be used directly from C++. Inside the generated box1L folder the file integrate_box1L.cpp demonstrates this.

After the lines parsing the input parameters, an secdecutil::Integrator is constructed and its parameters are set:

// Set up Integrator
secdecutil::integrators::Qmc<
                            box1L::integrand_return_t,
                                box1L::maximal_number_of_integration_variables,
                                integrators::transforms::Korobov<3>::type,
                                box1L::user_integrand_t
                            > integrator;
integrator.verbosity = 1;

The amplitude is constructed via a call to name::make_amplitudes() and packed into a name::handler_t.

// Construct the amplitudes
std::vector<box1L::nested_series_t<box1L::sum_t>> unwrapped_amplitudes =
    box1L::make_amplitudes(real_parameters, complex_parameters, "box1L_coefficients", integrator);

// Pack amplitudes into handler
box1L::handler_t<box1L::amplitudes_t> amplitudes
(
    unwrapped_amplitudes,
    integrator.epsrel, integrator.epsabs
    // further optional arguments: maxeval, mineval, maxincreasefac, min_epsrel, min_epsabs, max_epsrel, max_epsabs
);
amplitudes.verbose = true;

If desired, the contour deformation can be adjusted via additional arguments to name::handler_t.

2.3. Evaluating a Weighted Sum of Integrals¶

New in version 1.5.

Let us examine example easy_sum, which demonstrates how two weighted sums of dimensionally regulated integrals can be evaluated. The example computes the following two weighted sums:

$& 2 s\ I_1 + 3 s\ I_2, \\ & \frac{s}{2 \epsilon}\ I_1 + \frac{s \epsilon}{3}\ I_2,$

where

$I_1 & = \int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (x+y)^{-2+\epsilon}, \\ I_2 & = \int_0^1 \mathrm{d} x \int_0^1 \mathrm{d} y \ (2x+3y)^{-1+\epsilon}.$

First, we import the necessary python packages and open the if __name__ == "__main__" guard, as required by multiprocessing.

#!/usr/bin/env python3
import pySecDec as psd

if __name__ == "__main__":

The common arguments for the integrals are collected in the common_args dictionary.

common_args = {
    'real_parameters': ['s'],
    'regulators': ['eps'],
    'requested_orders': [0]
}

Next, the coefficients of the integrals for each weighted sum are specified. Each coefficient is specified as a string with an arbitrary arithmetic (i.e. rational) expression. Coefficients can also be specified as instances of the Coefficient class. Coefficients can depend on the regulators, the sum_package function will automatically determine the correct orders to which the coefficients and integrals should be expanded in order to obtain the requested_orders.

coefficients = {
    "sum1" : [
        '2*s',       # easy1
        '3*s'        # easy2
    ],
    "sum2" : [
        's/(2*eps)', # easy1
        's*eps/3'    # easy2
    ]
}

The integrals are specified using the MakePackage wrapper function (which has the same arguments as make_package), for loop integrals the LoopPackage wrapper may be used (it has the same arguments as loop_package).

integrals = [
    psd.MakePackage('easy1',
        integration_variables = ['x','y'],
        polynomials_to_decompose = ['(x+y)^(-2+eps)'],
        **common_args),
    psd.MakePackage('easy2',
        integration_variables = ['x','y'],
        polynomials_to_decompose = ['(2*x+3*y)^(-1+eps)'],
        **common_args)
]

Finally, the list of integrals and coefficients are passed to sum_package. This will generate a C++ library which efficiently evaluates both weighted sums of integrals, sharing the results of the integrals between the different sums.

# generate code sum of (int * coeff)
psd.sum_package('easy_sum', integrals,
    coefficients = coefficients, **common_args)

The generated C++ library can be compiled and called via the python and/or C++ interface as described above.

2.4. Using Expansion By Regions (Generic Integral)¶

New in version 1.5.

The example make_regions_ebr provides a simple introduction to the expansion by regions functionality within pySecDec. For a more detailed discussion of expansion by regions see our paper [PSD21].

The necessary packages are loaded and the if __name__ == "__main__" guard is opened.

#!/usr/bin/env python3
import pySecDec as psd

if __name__ == "__main__":

Expansion by regions is applied to a generic integral using the make_regions function.

regions_generators = psd.make_regions(
    name = 'make_regions_ebr',
    integration_variables = ['x'],
    regulators = ['delta'],
    requested_orders = [0],
    smallness_parameter = 't',
    polynomials_to_decompose = ['(x)**(delta)','(t + x + x**2)**(-1)'],
    expansion_by_regions_order = 0,
    real_parameters = ['t'],
    complex_parameters = [],
    decomposition_method = 'geometric_infinity_no_primary',
    polytope_from_sum_of=[1]
)

The output of make_regions can be passed to sum_package in order to generate a C++ library suitable for evaluating the expanded integral.

psd.sum_package(
    'make_regions_ebr',
    regions_generators,
    regulators = ['delta'],
    requested_orders = [0],
    real_parameters = ['t']
)

The generated C++ library can be compiled and called via the python and/or C++ interface as described above.

2.5. Using Expansion By Regions (Loop Integral)¶

New in version 1.5.

The example generate_box1L_ebr demonstrates how expansion by regions can be applied to loop integrals within pySecDec by applying it to the 1-loop box integral as described in Section 4.2 of [Mis18]. For a more detailed discussion of expansion by regions see our paper [PSD21].

First, the necessary packages are loaded and the if __name__ == "__main__" guard is opened.

#!/usr/bin/env python3

import pySecDec as psd

# This example is the one-loop box example in Go Mishima's paper arXiv:1812.04373

if __name__ == "__main__":

The loop integral can be constructed via the convenience functions in loop_integral, here we use LoopintegralFromGraph. Note that powerlist=["1+n1","1+n1/2","1+n1/3","1+n1/5"], here n1 is an extra regulator required to regulate the singularities which appear when expanding this loop integral. We use the “trick” of introducing only a single regulator divided by different prime numbers for each power, rather than unique regulators for each propagator (though this is also supported by pySecDec). Poles in the extra regulator n1 may appear in individual regions but are expected to cancel when all regions are summed.

# here we define the Feynman diagram
li = psd.LoopIntegralFromGraph(
    internal_lines = [['mt',[3,1]],['mt',[1,2]],['mt',[2,4]],['mt',[4,3]]],
    external_lines = [['p1',1],['p2',2],['p3',3],['p4',4]],
    powerlist=["1+n1","1+n1/2","1+n1/3","1+n1/5"],
    regulators=["eps","n1"],
    # renames the parameters to get the same polynomials as in 1812.04373
    Feynman_parameters=["x%i" % i for i in range(1,5)],

replacement_rules = [
    # note that in those relations all momenta are incoming
    # general relations:
    ('p1*p1', 'm1sq'),
    ('p2*p2', 'm2sq'),
    ('p3*p3', 'm3sq'),
    ('p4*p4', 'm4sq'),
    ('p1*p2', 's/2-(m1sq+m2sq)/2'),
    ('p1*p3', 't/2-(m1sq+m3sq)/2'),
    ('p1*p4', 'u/2-(m1sq+m4sq)/2'),
    ('p2*p3', 'u/2-(m2sq+m3sq)/2'),
    ('p2*p4', 't/2-(m2sq+m4sq)/2'),
    ('p3*p4', 's/2-(m3sq+m4sq)/2'),
    ('u', '(m1sq+m2sq+m3sq+m4sq)-s-t'),
    # relations for our specific case:
    ('mt**2', 'mtsq'),
    ('m1sq',0),
    ('m2sq',0),
    ('m3sq','mHsq'),
    ('m4sq','mHsq'),
    ('mHsq', 0),
])

Expansion by regions is applied to a loop integral using the loop_regions function. We expand around a small mass mtsq.

# find the regions
terms = psd.loop_regions(
    name = "box1L_ebr",
    loop_integral=li,
    smallness_parameter = "mtsq",
    expansion_by_regions_order=0)

The output of loop_regions can be passed to sum_package in order to generate a C++ library suitable for evaluating the expanded integral.

# write the code to sum up the regions
psd.sum_package(
    "box1L_ebr",
    terms,
    li.regulators,
    requested_orders = [0,0],
    real_parameters = ['s','t','u','mtsq'],
    complex_parameters = [])

The generated C++ library can be compiled and called via the python and/or C++ interface as described above.

2.6. List of Examples¶

Here we list the available examples. For more details regarding each example see [PSD17], [PSD18] and [PSD21].

easy:	a simple parametric integral, described in Section 2.1
box1L:	a simple 1-loop, 4-point, 4-propagator integral, described in Section 2.2
triangle2L:	a 2-loop, 3-point, 6-propagator diagram, also known as P126
box2L_numerator:	a massless planar on-shell 2-loop, 4-point, 7-propagator box with a numerator, either defined as an inverse propagator `box2L_invprop.py` or in terms of contracted Lorentz vectors `box2L_contracted_tensor.py`
pentabox_fin:	a 2-loop, 5-point, 8-propagator diagram, evaluated in $6-2 \epsilon$ dimensions where it is finite
triangle3L:	a 2-loop, 3-point, 7-propagator integral, demonstrates that the symmetry finder can significantly reduce the number of sectors
formfactor4L:	a single-scale 4-loop 3-point integral in $6-2 \epsilon$ dimensions
bubble6L:	a single-scale 6-loop 2-point integral, evaluated at a Euclidean phase-space point
elliptic2L_euclidean:	an integral known to contain elliptic functions, evaluated at a Euclidean phase-space point
elliptic2L_physical:	an integral known to contain elliptic functions, evaluated at a physical phase-space point
banana_3mass:	a 3-loop 2-point integral with three different internal masses known to contain hyperelliptic functions, evaluated at a physical phase-space point
hyperelliptic:	a 2-loop 4-point nonplanar integral known to contain hyperelliptic functions, evaluated at a physical phase-space point
triangle2L_split:	a 2-loop, 3-point, 6-propagator integral without a Euclidean region due to special kinematics
Nbox2L_split:	three 2-loop, 4-point, 5-propagator integrals that need `split=True` due to special kinematics
hypergeo5F4:	a general dimensionally regulated parameter integral
hz2L_nonplanar:	a 2-loop, 4-point, 7-propagator integral with internal and external masses
box1L_ebr:	uses expansion by regions to expand a 1-loop box with a small internal mass, this integral is also considered in Section 4.2 of [Mis18]
bubble1L_ebr:	uses expansion by regions to expand a 1-loop, 2-point integral in various limits, demonstrates the use of an additional regulator as described in [PSD21]
bubble1L_dotted_ebr:	uses expansion by regions to expand a 1-loop, 2-point integral, demonstrates the $t$ and $z$ methods described in [PSD21]
bubble2L_largem_ebr:	uses expansion by regions to expand a 1-loop, 2-point integral with a large mass
bubble2L_smallm_ebr:	uses expansion by regions to expand a 1-loop, 2-point integral with a small mass
formfactor1L_ebr:	uses expansion by regions to compute various 1-loop, 3-point form factor integrals from the literature, demonstrates the use of `add_monomial_regulator_power` to introduce an additional regulator as described in [PSD21]
triangle2L_ebr:	uses expansion by regions to compute a 2-loop, 3-point integral with a large mass
make_regions_ebr:	uses expansion by regions to compute a simple generic integral with a small parameter
easy_sum:	calculates the sum of two integrals with different coefficients, demonstrates the use of `sum_package`
yyyy1L:	calculates a 1-loop 4-photon helicity amplitude, demonstrates the use of `sum_package`
two_regulators:	an integral involving poles in two different regulators.
userdefined_cpp:	a collection of examples demonstrating how to combine polynomials to be decomposed with other user-defined functions
regions:	prints a list of the regions obtained by applying expansion by regions to formfactor1L_massless
region_tools:	demonstrates the standalone usage of suggested_extra_regulator_exponent, extra_regulator_constraints and find_regions