Stan Math Library  4.9.0 Automatic Differentiation
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This is meant as a basic guide for writing and maintaining functions in the stan-dev/math repository. Stan Math is the automatic differentiation (autodiff) library behind the Stan language, and the math functions exposed at the Stan language level are implemented here in C++.

In the course of the Math library's existence, C++ has changed substantially. Math was originally written before C++11. It currently targets C++14. In the near future it will transition to C++17. With this in mind, there are many different ways to write Math functions. This guide tries to document best practices, conventions which not all functions in Math follow, but should be followed for new code to keep the code from getting unwieldy (the old patterns will be updated eventually).

The title contains "Current State" to emphasize that if any information here is out of date or any advice does not work, it should be reported as a bug (issue tracker link).

This document builds on the information in the Stan Math Library. While some parts of the paper are out of date now it is very useful for understanding the patterns used in the math library.

# Preliminary resources

Before contributing to Stan Math it's recommended you become familiar with C++. While this guide attempts to cover the edges and odd things we do in the math library, we recommend the resources below for getting started on autodiff in C++.

Books and Papers:

Talks:

Blog Posts and Articles:

A generally good resource for making minimal examples for bugs is godbolt.org. The godbolt link here (backup gist) contains a function for printing C++ types and has Eigen and boost headers included.

The example function section is an amalgamation of the common_pitfalls section. It's recommended you skim over the common pitfalls before going over the examples below.

# Code structure

The Stan Math library is spit into 4 main source folders that hold functions, type traits, and classes.

• prim: General Scalar, Matrix, and std::vector<T> types
• rev: Specializations for reverse mode automatic differentiation
• fwd: Specializations for forward mode automatic differentiation.
• mix: Sources to allow mixes of forward and reverse mode.
• opencl: Sources for doing reverse mode automatic differentiation on GPUs.

Within each of those folders you will find any one of the following folders

• core: Base implementations of custom scalars or backend setup.
• Ex: in prim this is operators for complex numbers and the setup for threading, rev's core is the scalar and its base operators for reverse mode and the stack allocator, and fwd has the scalar for forward mode autodiff and its operators.
• err: Functions that perform a check and if true throw an error.
• fun: The math functions exposed to the Stan language.
• functor: Functions that take in other functions and data as input such as reduce_sum()
• meta: Type traits and helper functions used internally.

Any function callable from the math library, excluding those in the internal namespace, should be compatible with Stan's reverse and forward mode autodiff types. Any new function introduced to the Math library is expected to support higher-order automatic differentiation. Though exceptions to supporting higher-order autodiff have been made in the past such as the differential equations solvers only supporting reverse mode autodiff.

You can contribute a new function by writing the function in the prim/fun folder. The function will need to accept arithmetic, reverse mode (stan::math::var ), and forward mode (stan::math::fvar) scalar types and should be tested using the expect_ad() testing framework. The rev, fwd, mix, and opencl folders are used for adding specialization for:

1. rev: reverse-mode autodiff specializations
2. fwd: forward-mode autodiff specializations
3. mix: mixed-mode autodiff specializations
4. opencl: GPU (reverse-mode?) autodiff specializations

# Adding a simple example function

A generic function that takes in an Eigen vector and performs a dot product on itself could be written in prim/fun as a for loop accumulating squared values of the original vector. In the example directly below, the lines with comments (1), (2), (3), and (4) are explained in more detail as they are Stan Math specific components of the function required to allow the function to work inside of the math library.

// stan/math/prim/fun/dot_self.hpp
template <typename EigVec, require_eigen_vector_t<EigVec>* = nullptr> // (1)
inline value_type_t<EigVec> dot_self(const EigVec& x) {
const auto& x_ref = to_ref(x); // (2)
value_type_t<EigVec> sum_x = 0.0; // (3)
for (int i = 0; i < x.size(); ++i) {
sum_x += x_ref.coeff(i) * x_ref.coeff(i); // (4)
}
return sum_x;
}

For a real implementation of this function, we would want to use the member method of the input Eigen matrix x.squaredNorm(), but unrolling dot_self() gives a nice example with a few subtleties we need to take into account.

### (1) Using Stan's require type traits to specialize overloads

The Stan math library requires that all functions are able to accept Eigen expression templates, hence the general EigVec template above. Though the math library wants general template parameters for the functions in prim, functions should be restricted to a subset of types. For example, without the require in the above function, it would be perfectly valid to pass in an Eigen::MatrixXd, but that would not be good! In order to restrict the types that can go into a function, the math library uses the require type traits to detect which function a particular type should use. This is very similar conceptually to C++20 requires.

To avoid document duplication please see the require_meta_doc for a lengthier discussion on the requires template types. The guide there explains the different styles and patterns of requires that we use throughout the math library as well as in this example. require_meta_doc has subsections which will show you the requires we have already set up for types we commonly see.

We want to restrict dot_self() to only accept Eigen types which have one row or one column at compile time, which for us is require_eigen_vector_t. The C++ convention for require types looks odd at first, as it sets the require template parameter to be a pointer with a default value of nullptr.

template <typename EigVec, require_eigen_vector_t<EigVec>* = nullptr>

One C++ trick is that any non-type template parameter of type void* with a default of nullptr will be ignored by the compiler. Under the hood, if any of the requires are successful they will return back a type void. So in the case we are passing a type that our requires accepts we get back a void* = nullptr which is safely ignored by the compiler. In the case that the type does not satisfy the require then the function is removed from the set of possible functions the caller could use via SFINAE. This scheme leads to a very nice pattern for writing generic templates for functions while also being able to restrict the set of types that a function can be used for.

### (2) Ensure Eigen matrices are only evaluated once

Before accessing individual coefficients of an Eigen type, use stan::math::to_ref to make sure accessing the coefficients will not cause a compiler error or force the coefficient to be computed multiple times. The function stan::math::to_ref is a smart helper function that will evaluate expressions that perform computations, but will not evaluate expressions that are only a view or slice of a matrix or vector.

In the Stan Math library, we allow functions to accept Eigen expressions. This allows expressions to be accumulated across multiple function calls and can often give much more optimized code. For instance, in the example below the additions that go into subtract() will be combined together to form one large computation instead of many smaller ones.

The type created on the right hand side of the equals is an Eigen expression that looks like the following pseudotype of the real generated expression.

When the expression is assigned to an actual matrix it will execute code that performs the matrix multiply of B and C and then runs only one loop for the elementwise addition and subtraction.

Eigen::MatrixXd tmp = B * C;
Eigen::MatrixXd x(tmp.rows(), tmp.cols());
for (Eigen::Index i = 0; i < x.size(); ++i) {
x(i) = A(i) + tmp(i) - D(i) + E(i);
}

Using lazily evaluated expressions allows Eigen to avoid redundant copies, reads and writes to our data. However, this comes at the cost of more complicated template traits and patterns as well as being careful around handling inputs to functions.

In (3), when we access the coefficients of x in the loop, if its type is similar to the expression above we can get incorrect results as Eigen does not guarantee any safety of results when performing coefficient level access on an expression type that transforms its inputs. So stan::math::to_ref looks at its input type, and if the input type is an Eigen expression that performs a calculation then it returns as if we called .eval() on our input, otherwise, it returns the same type as x.

Another example where stan::math::to_ref is required is the example below, which you can copy and paste into godbolt.org to try yourself.

#include <Eigen/Dense>
#include <string_view>
#include <iostream>
template <typename T>
double dot_self(const T& A) {
double sum_a = 0;
for (int i = 0; i < A.size(); ++i) {
sum_a += A.coeff(i) * A.coeff(i);
}
return sum_a;
}
int main() {
Eigen::MatrixXd A_mat = Eigen::MatrixXd::Random(5, 5);
Eigen::VectorXd A_vec = Eigen::VectorXd::Random(5);
// This works fine
//double b2 = dot_self((A_mat * A_vec).eval());
// This fails
double b2 = dot_self(A_mat * A_vec);
std::cout << b2;
}
auto dot_self(const T &a)
Returns squared norm of a vector or matrix.
Definition dot_self.hpp:21

Pasting the above into Godbolt will show a compilation error! That's because Eigen does not allow coefficient levels access to product functions. Instead, that expression needs to be evaluated before it is accessed elementwise. Alternatively, consider the following example, where we pass in an expression for elementwise multiplication.

int main() {
Eigen::VectorXd A_vec1 = Eigen::VectorXd::Random(5);
Eigen::VectorXd A_vec2 = Eigen::VectorXd::Random(5);
double b2 = dot_self(A_vec1.array() * A_vec2.array());
std::cout << b2;
}

The code above will compile and execute, but in the body of the dot_self function, when we call A.coeff(i) twice in the for loop we will end up evaluating the expression twice.

Suppose the input to dot_self is instead a segment of an existing vector such as the below example.

Eigen::Matrix<var, Dynamic, 1> A = Eigen::VectorXd::Random(20);
var b = dot_self(A.segment(1, 5));

When calling A.segment(1, 5) Eigen will create an instance of an expression class called Eigen::Block<Eigen::VectorXd> that holds a view into the original vector. An expression that is only a view into an object is safe to read coefficient-wise. In this case, stan::math::to_ref knows this and as long as you use const auto& as the object type returned from to_ref(x), then to_ref(x) will deduce the correct type Block instead of casting to an evaluated vector. We use const auto& here even when the output will store an object as C++'s lifetime rules allow for this.

‍The lifetime of a temporary object may be extended by binding to a const lvalue reference or to an rvalue reference (since C++11), see reference initialization for details.

If we used auto here and if the return type of stan::math::to_ref is an Eigen::MatrixXd then it would make a copy. But with const auto& we do not make a copy as the lifetime of the object referenced by the const auto& will be the same as if we had done auto.

### (3) Using value_type_t and other type querying type traits

The type trait stan::value_type_t is one of several type traits we use in the library to query information about types. stan::value_type_t will return the inner type of a container, so value_type_t<Eigen::Matrix<double, -1, -1>> will return double, value_type_t<std::vector<std::vector<double>>> will return a std::vector<double> and value_type_t<double> will simply return a double.

See the docs for stan::value_type, stan::scalar_type and stan::base_type for more information on these.

### (4) Accessing Eigen matrices though <tt>.coeff()</tt> and <tt>.coeffRef()</tt>

Eigen performs bounds checks by default when using [] or () to access elements. However .coeff() and .coeffRef() do not. Because Stan programs perform bounds checking at a higher level it's safe to remove the bounds checks here. Note that .coeff() should be used when accessing values of an Eigen types and .coeffRef() should be used when assigning values of an Eigen type.

### Testing

From there you can use the unit test suite for automatic differentiation to verify your code produces the correct gradients and higher order derivatives and your done!

// Example tests in test/unit/math/mix/fun/dot_self_test.cpp
TEST(MathMixMatFun, dotSelf) {
auto f = [](const auto& y) { return stan::math::dot_self(y); };
Eigen::VectorXd x0(0);
Eigen::VectorXd x1(1);
x1 << 2;
Eigen::VectorXd x2(2);
x2 << 2, 3;
Eigen::VectorXd x3(3);
x3 << 2, 3, 4;
for (const auto& a : std::vector<Eigen::VectorXd>{x0, x1, x2, x3}) {
}
}
(Expert) Numerical traits for algorithmic differentiation variables.
STL namespace.

That's it, your done! Make a PR, add some docs, have a cup of tea all is well.

But, while your function will work for all of our types, it probably won't be as fast as it could be. Indeed, since the above is using Stan's basic scalar reverse mode, for a vector of size N there will be N individual callback functions on the reverse pass callback stack. This is not the worst, Stan's base scalar operations are as efficient as they can be, but by writing a specialization for reverse mode we can have one callback with much more efficient memory access. The next section may seem a bit advanced, but for those who enjoy adventure, let's talk about how to make this function go faster.

# Specializing for reverse mode

The following material depends on understanding Stan Math's automatic differentiation, which is described in the Preliminary resources

### Using type traits to expose Our new function

First we need to stop our current function from compiling for reverse mode autodiff. This can be done using the requires type traits in Stan Math. The only thing that has changed in the function above and below is the additional template parameter for the requirement on the function.

template <typename EigVec, require_eigen_vector_t<EigVec>* = nullptr,
require_not_st_var<EigVec>* = nullptr>
inline double dot_self(const EigVec& x) {
auto x_ref = to_ref(x);
var sum = 0.0;
for (int i = 0; i < x.size(); ++i) {
sum += x_ref.coeff(i) * x_ref.coeff(i);
}
return sum;
}

The require type trait in the above stops the function from working specifically with Eigen vectors whose inner Scalar type is a stan::math::var type. Now we can write the signature of our function that will go into the stan/math/rev/fun/ folder

template <typename EigVec, require_eigen_vector_vt<is_var, EigVec>* = nullptr>
inline double dot_self(const EigVec& x) {
// ...
}

The above signature is defined to only accept Eigen vectors whose stan::value_type is stan::math::var .

## Working in reverse mode

So now let's write our reverse mode specialization in rev/fun. In the below I've placed numbered comments to indicate places in the function which use patterns specific to reverse mode autodiff in Stan Math that will be discussed further.

template <typename EigVec, require_eigen_vt<is_var, EigVec>* = nullptr>
inline var dot_self(const EigVec& v) {
// (1) put v into our arena
arena_t<EigVec> arena_v(v);
// (2) calculate forward pass using
// (3) the .val() method for matrices of var types
var res = dot_self(arena_v.val());
// (4) Place a callback for the reverse pass on the callback stack.
reverse_pass_callback([res, arena_v]() mutable {
});
return res;
}

There's a lot going on here particular to Stan math and so we will talk about each special bit.

## (1) Arena memory

The type arena_t<T> and the function stan::math::to_arena takes an object and places it on Stan's arena allocator.

As discussed in Thinking About Automatic Differentiation in Fun New Ways reverse mode automatic differentiation comes down to.

1. Calculating a function z = f(x, y)‘s output (forward pass)
2. Storing the input and output into a memory arena
3. Adding a callback to a callback stack that calculates the adjoint of the function

This process allows us to call stan::math::grad , call each of the callbacks in the callback stack, and accumulate the gradients starting from the final outputs back to the inputs. (reverse pass)

Reverse mode autodiff in Stan Math requires a huge number of temporaries and intermediates to be computed and saved for the reverse pass. There are so many allocations that the overhead of malloc() becomes noticeable. To avoid this, Math provides its own memory arena. The assumptions of the Math arena are:

1. Objects allocated in the arena are TriviallyCopyable and TriviallyDestructable. These objects do not need their destructors called. Although there are ways to destruct something allocated in the arena, it is not automatic.
2. All objects allocated on the arena have the same lifetime.
3. Memory allocated from the arena is available and valid until stan::math::recover_memory is called.

The arena memory pattern fits nicely into automatic differentiation for MCMC as we will most commonly be calling gradients with data of the same size over and over again.

Functions that use reverse mode autodiff are allowed to save what they need to the arena, and then the objects saved here will be available during the reverse pass.

There is a utility for saving objects into the arena that do need their destructors called. This is discussed under stan::math::make_chainable_ptr . Arena types are helper functions and types for easily saving variables to the arena.

stan::arena_t defines, for a given type T, the type of a lightweight object pointing to memory in the arena that can store a T. Memory for stan::arena_t objects is only recovered when stan::math::recover_memory is called. No destructors are called for objects stored with stan::arena_t.

Copying a variable x to the arena can be done with either stan::math::to_arena while declaring the type of the object with auto or creating the type directly with the stan::arena_t type trait. Using auto and stan::math::to_arena is the preferred form. Usually when you see code that directly uses stan::arena_t it is because the code was written before stan::math::to_arena existed.

template <typename T>
auto myfunc(const T& x) {
auto arena_x = to_arena(x);
//...
}
template <typename T>
auto myfunc(const T& x) {
arena_t<T> arena_x = x;
// ...
}

## (2) Calculating the forward pass

Because of our previous requires template we can now call dot_self() with the values of the matrix. The compiler will then know to call the previous definition of dot_self written to work with double scalar types.

## (3) Accessing the values and adjoints of matrices.

The calculations for the forward and reverse passes of reverse mode autodiff often use custom Eigen methods for matrices, vectors, and arrays. In the above, .val() will return a view of all of the values of the stan::math::var scalars in the matrix while .adj() will return a view for all of the adjoints of the matrices stan::math::var scalars. You can think of .val() and .adj() as returning an Eigen matrix of doubles such as Eigen::Matrix<double, Rows, Cols> where both functions scalars are references to the values or adjoints of the original matrix with stan::math::var scalars.

## (4) Setting up the reverse pass

Once the forward pass is complete, and the data for the reverse pass is in the arena the adjoint calculation for the portion of the reverse pass for this function needs to be written. The reverse pass consists of an adjoint calculation which is placed onto the callback stack. When a user calls stan::math::grad this adjoint calculation will be called so that adjoints are accumulated from the final output to the starting inputs.

The adjoint method for a self dot product is

Once we know the adjoint calculation we can use stan::math::reverse_pass_callback to place the adjoint calculation onto the callback stack for the reverse pass. Calling stan::math::reverse_pass_callback with a lambda will place the lambda on Stan's callback stack so that it can be executed in the reverse pass.

reverse_pass_callback([res, arena_v]() mutable { // (2)
});

Notice that for the lambda we create copies of the objects res and arena_v. The trick here is that anything allocated with Stan's arena using stan::arena_t or stan::math::to_arena is TriviallyCopyable. Note that arithmetic types and Stan Math's stan::math::var type are trivially copyable by definition and so can be passed without having to make an explicit call to stan::math::to_arena . Since the memory is safe and sound within our arena these objects are safe and fast to copy.

An alternative approach to the above function could also have been

template <typename EigVec, require_eigen_vt<is_var, EigVec>* = nullptr>
inline var dot_self(const EigVec& v) {
arena_t<EigVec> arena_v(v); // (1) put v into our arena
return make_callback_var(arena_v.val().dot(arena_v.val()), [arena_v](auto& vi) {
});
}

make_callback_var(x, f) is similar to stan::math::reverse_pass_callback , but it constructs an autodiff type from the argument x. The function f is called on the reverse pass similarly to stan::math::reverse_pass_callback . stan::math::make_callback_var should only be used when returning a var_value<double> or var_value<Matrix>.

# Argument types

The functions in Stan Math are used in the C++ code generated by the Stan compiler. This means that all types in the Stan language map to C++ types. The map between Stan types and types in C++ is a one-to-many relationship because of the different blocks in a Stan program. Objects created in data, transformed data, and generated quantities will hold arithmetic types while objects created in parameters, transformed parameters, and the model block will hold autodiff types. The basic Stan types are listed in the table below as well as arrays of any of these types.

Stan Type Data Model
int int int
int[N] std::vector<int> std::vector<int>
real double var/fvar<T>
real[N] std::vector<double> std::vector<var/fvar<T>>
vector[N] Eigen::Matrix<double, Dynamic, 1>(N) Eigen::Matrix<var/fvar<T> Dynamic, 1>(N)
row_vector[N] Eigen::Matrix<double, 1, Dynamic>(N) Eigen::Matrix<var/fvar<T> 1, Dynamic>(N)
matrix[N,M] Eigen::Matrix<double, Dynamic, Dynamic>(N, M) Eigen::Matrix<var/fvar<T> Dynamic, Dynamic>(N, M)
matrix[N, M] x[K] std::vector<Eigen::Matrix<double, Dynamic, Dynamic>>(K, {M, N}) std::vector<Eigen::Matrix<var/fvar<T> Dynamic, Dynamic>>(K, {M, N})

Any array type (int[], real[], or T[] for T any of the above types) map to std::vector<C> types where C is the C++ equivalent of T.

## Unary functions working over arrays

The Stan function sin() expects the math library has the following signatures available.

real sin(real);
vector sin(vector);
array vector sin(array vector)

where array can be multiple arrays within one another. The above translates to the following C++ signatures

// reals
double sin(double);
var sin(var);
// vector
template <typename T>
Eigen::Vector<T, Dynamic, 1> sin(Eigen::Vector<T, Dynamic, 1>);
// array vector
template <typename T>
std::vector<Eigen::Vector<T, Dynamic, 1>> sin(std::vector<Eigen::Vector<T, Dynamic, 1>>);
// array array vector
template <typename T>
std::vector<std::vector<Eigen::Vector<T, Dynamic, 1>>> sin(std::vector<std::vector<Eigen::Vector<T, Dynamic, 1>>>);

A function that accepts an autodiff type should return an autodiff type. Notice that because we have to support arithmetic and autodiff types the signatures expand out for both arithmetic and autodiff types. It would be tedious to write out this method for all of the array types, so Stan math provides helper classes stan::math::apply_scalar_unary and stan::math::apply_vector_unary to apply a function over arrays.

struct sin_fun {
template <typename T>
static inline T fun(const T& x) {
using std::sin;
return sin(x);
}
};
template <typename T, require_not_container_st<std::is_arithmetic, T>* = nullptr>
inline auto sin(const T& x) {
return apply_scalar_unary<sin_fun, T>::apply(x);
}
template <typename Container, require_container_st<std::is_arithmetic, Container>* = nullptr>
inline auto sin(const Container& x) {
return apply_vector_unary<Container>::apply(
x, [&](const auto& v) { return v.array().sin(); });
}
fvar< T > sin(const fvar< T > &x)
Definition sin.hpp:14

In Stan math, a container type is a std::vector that holds either other std::vectors or Eigen::Matrix types. In the first function we use stan::math::apply_scalar_unary to apply sin() to either Scalars, std::vectors holding scalars, or Eigen::Matrix types. The second function in the above uses stan::math::apply_vector_unary to apply its lambda to a container whose elements are also containers or Eigen matrices.

## Binary functions (and least-upper-bound return type semantics)

Multiple argument functions are more involved. Consider real atan2(real, real) which requires four C++ overloads:

double atan2(double, double);
var atan2(double, var);
var atan2(var, double);
var atan2(var, var);

In this case, the return type is a function of both input types. stan::return_type_t is written to take any number of input types and compute from them a single scalar return type. The return type is the least upper bound of the input types that can be constructed from all of the input types. For reverse mode autodiff, this means that if the scalar type of any input argument is a stan::math::var , then the return type is a var.

A generic signature for real atan2(real, real) is:

template <typename T, typename S>
return_type_t<T, S> atan2(T, S);

stan::return_type_t can be used to construct non-scalar return types as well. For the function vector add(vector, vector), the following generic C++ signature could be used:

template <typename T, typename S>
Eigen::Matrix<return_type_t<T, S>, Eigen::Dynamic, 1> atan2(T, S);

Alternativly, with C++14 one can also delay deducing the return type with auto and the compiler can deduce the return type from the code (which would then internally use return_type_t<T, S>)

## Higher order autodiff

Adding the forward mode autodiff type simply adds more overloads. The forward mode autodiff type itself takes a template argument to represent the types of the value and gradient and allows recursive templating. This technically defines an infinite number of new types, but the ones of interest in Math (and the ones that should be tested are) fvar<double>, fvar<fvar<double>>, fvar<var> and fvar<fvar<var>>. These are useful for computing various high-order derivatives. Going back to the real sin(real) function, Stan Math is now expected to implement the following rather expanded list of functions:

double sin(double);
var sin(var);
fvar<double> sin(fvar<double>);
fvar<fvar<double>> sin(fvar<fvar<double>>);
fvar<var> sin(fvar<var>);
fvar<fvar<var>> sin(fvar<fvar<var>>);

For the sake of performance, it may be desirable to also define var sin(var), or similarly fvar<double> sin(fvar<double>) etc.

The type trait return_type_t is defined similarly as for stan::math::var . Autodiff types such as fvar<double> and var should not be mixed, and so the stan::return_type_t does not need to account for various combinations of stan::math::var , fvar<double>, etc.

## Exceptions

Math functions are expected to validate inputs and throw exceptions. It should be impossible to cause a segfault by calling a user-facing Math function or trigger an exception that is not handled by Stan itself. This means that argument sizes should be checked to be compatible. For instance, a matrix multiply function would need to check that the first matrix can in fact be multiplied by the second, if this is not so it should throw.

If a scalar argument should be positive, an exception should be thrown if it is not. If a matrix argument should be positive definite, the function should check this before operating on the matrix. This can lead to significant overhead in the checks, but the current library policy is to get informative errors back to the user, which means doing these checks.

Error checks should be done with the functions in stan/math/prim/err which throw consistently formatted errors.

## Tests

Tests in Math are written with the Google test framework. At this point, there are enough functions in Math that the way to get started is by copying the tests of a similar Function and editing them to suit the new function. There are two basic sets of functions that every Stan function should have. First, for a function named foo, there should be a file in test/unit/math/prim/fun named foo_test.cpp.

This should include tests that:

1. Check that every exception is throwing in the right way at least once and that all edge cases are handled correctly.
• For instance, if there is a stan::math::check_size_match function, then there should be a test that makes sure that if the sizes don't match, and exception is thrown. It is not necessary to check the error message in the tests, just check that an exception is thrown and the exception type is correct.
2. Check that the function returns the expect values for a couple suitable inputs.
• For some functions, these checks should be more extensive than others, but there should at least be two or three even in simple cases.

The prim tests should only have arithmetic types. No autodiff is tested here. The prim tests check that the function handles edge cases correctly and produces the correct output for different function values. The second set of tests should be in stan/math/mix/fun with the same name foo_test.cpp. These tests will be used to test all forms of the autodiff.

The tests in mix/fun should use the expect_ad() function to test all forms of autodiff. expect_ad() takes as its first argument a function to test and then up to three other arguments, all with arithmetic scalar types. For a given function f, and two arguments a and b, expect_ad() tests that:

1. For every combination of a and b, the returned value of f matches the returned value with non-autodiff types.
2. The finite difference gradients of f match those computed with autodiff.
3. Any time a f throws with arithmetic arguments, it also throws with autodiff arguments.

# Reverse mode

## Values

stan::math::value_of returns the values of the variable x. If x is not an autodiff type, the values are simply the input and the input is forwarded along. For reverse mode autodiff, values are double variables. For higher order autodiff (fvar<var>, etc.) this is not always the case. For instance, value_of(fvar<var>) is a var. value_of_rec() is a function which recursively calls value_of() until a double value is found. So calling value_of_rec(fvar<var>) will return a double. The values of x have the same shape as x.

## Values and adjoint extensions to Eigen

The matrix and vector autodiff types come with an extra .val() and .adj(), member functions called .val_op() and .adj_op(). These *_op() member functions are used as a workaround for a bug in Eigen where transpose expressions will be inaccessible because of an incorrect const reference. See here for the details and other information for when this workaround is needed.

The member functions .val() and .val_op() return expressions that evaluate to the values of the autodiff matrix.

The member functions .adj() and .adj_op() return expressions that evaluate to the adjoints of the autodiff matrix.

The non-_op versions of the functions should be preferred. If the code does not compile (usually an error about const-ness) try the _op versions. This can happen when multiplying the values/adjoints of an autodiff type against other matrices.

It is not safe to use .noalias() with the var<Matrix> value/adjoint expressions. In Eigen, .noalias() is similar to the keyword restrict which assumes that a pointer is unique within a given function. But when performing operations on Eigen matrices of stan::math::var while using .val() and .adj() on the left and right-hand side of an operation, the pointer holding the underlying stan::math::var types will be used on both sides. Since the pointer is used on both sides it will not be unique in the operation and so using .noalias() will lead to undefined behaviour. See the Aliasing and Writing Efficient Matrix Product Expressions Eigen docs for more info.