real-valued_basic_functions.Rmd

# Real-Valued Basic Functions

This chapter describes built-in functions that take zero or more real
or integer arguments and return real values.

```{r results='asis', echo=FALSE}
if (knitr::is_html_output()) {
cat(' * <a href="fun-vectorization.html">Vectorization of Real-Valued Functions</a>\n')
cat(' * <a href="built-in-constants.html">Mathematical Constants</a>\n')
cat(' * <a href="special-values.html">Special Values</a>\n')
cat(' * <a href="get-log-prob.html">Log Probability Function</a>\n')
cat(' * <a href="logical-functions.html">Logical Functions</a>\n')
cat(' * <a href="real-valued-arithmetic-operators.html">Real-Valued Arithmetic Operators</a>\n')
cat(' * <a href="step-functions.html">Step-like Functions</a>\n')
cat(' * <a href="power-and-logarithm-functions.html">Power and Logarithm Functions</a>\n')
cat(' * <a href="trigonometric-functions.html">Trigonometric Functions</a>\n')
cat(' * <a href="hyperbolic-trigonometric-functions.html">Hyperbolic Trigonometric Functions</a>\n')
cat(' * <a href="link-functions.html">Link Functions</a>\n')
cat(' * <a href="phi-function.html">Probability-Related Functions</a>\n')
cat(' * <a href="betafun.html">Combinatorial Functions</a>\n')
cat(' * <a href="composed-functions.html">Composed Functions</a>\n')
cat(' * <a href="special-functions.html">Special Functions</a>\n')
}
```

## Vectorization of real-valued functions {#fun-vectorization}

Although listed in this chapter, many of Stan's built-in functions are
vectorized so that they may be applied to any argument type.  The
vectorized form of these functions is not any faster than writing an
explicit loop that iterates over the elements applying the
function---it's just easier to read and write and less error prone.

### Unary function vectorization

Many of Stan's unary functions can be applied to any argument type.
For example, the exponential function, `exp`, can be applied to `real`
arguments or arrays of `real` arguments.  Other than for integer
arguments, the result type is the same as the argument type, including
dimensionality and size.  Integer arguments are first promoted to real
values, but the result will still have the same dimensionality and
size as the argument.

#### Real and real array arguments

When applied to a simple real value, the result is a real value.  When
applied to arrays, vectorized functions like `exp()` are defined
elementwise.  For example,

```stan
 // declare some variables for arguments
 real x0;
 array[5] real x1;
 array[4, 7] real x2;
 // ...
 // declare some variables for results
 real y0;
 array[5] real y1;
 array[4, 7] real y2;
 // ...
 // calculate and assign results
 y0 = exp(x0);
 y1 = exp(x1);
 y2 = exp(x2);
```

When `exp` is applied to an array, it applies elementwise.  For
example, the statement above,

```stan
 y2 = exp(x2);
```

produces the same result for `y2` as the explicit loop

```stan
for (i in 1:4) {
  for (j in 1:7) {
    y2[i, j] = exp(x2[i, j]);
  }
}
```

#### Vector and matrix arguments

Vectorized functions also apply elementwise to vectors and matrices.
For example,

```stan
 vector[5] xv;
 row_vector[7] xrv;
 matrix[10, 20] xm;

 vector[5] yv;
 row_vector[7] yrv;
 matrix[10, 20] ym;

 yv = exp(xv);
 yrv = exp(xrv);
 ym = exp(xm);
```

Arrays of vectors and matrices work the same way.  For example,

```stan
 array[12] matrix[17, 93] u;

 array[12] matrix[17, 93] z;

 z = exp(u);
```

After this has been executed, `z[i, j, k]` will be equal to `exp(u[i,
j, k])`.

#### Integer and integer array arguments

Integer arguments are promoted to real values in vectorized unary
functions.  Thus if `n` is of type `int`, `exp(n)` is of type `real`.
Arrays work the same way, so that if `n2` is a one dimensional array
of integers, then `exp(n2)` will be a one-dimensional array of reals
with the same number of elements as `n2`.  For example,

```stan
 array[23] int n1;
 array[23] real z1;
 z1 = exp(n1);
```

It would be illegal to try to assign `exp(n1)` to an array of
integers; the return type is a real array.

### Binary function vectorization

Like the unary functions, many of Stan's binary functions have been
vectorized, and can be applied elementwise to combinations of both
scalars or container types.

#### Scalar and scalar array arguments

When applied to two scalar values, the result is a scalar value.  When
applied to two arrays, or combination of a scalar value and an array,
vectorized functions like `pow()` are defined elementwise.  For example,

```stan
 // declare some variables for arguments
 real x00;
 real x01;
 array[5] real x10;
 array[5]real x11;
 array[4, 7] real x20;
 array[4, 7] real x21;
 // ...
 // declare some variables for results
 real y0;
 array[5] real y1;
 array[4, 7] real y2;
 // ...
 // calculate and assign results
 y0 = pow(x00, x01);
 y1 = pow(x10, x11);
 y2 = pow(x20, x21);
```

When `pow` is applied to two arrays, it applies elementwise.  For
example, the statement above,

```stan
 y2 = pow(x20, x21);
```

produces the same result for `y2` as the explicit loop

```stan
for (i in 1:4) {
  for (j in 1:7) {
    y2[i, j] = pow(x20[i, j], x21[i, j]);
  }
}
```

Alternatively, if a combination of an array and a scalar are
provided, the scalar value is broadcast to be applied to each
value of the array. For example, the following statement:
```stan
y2 = pow(x20, x00);
```

produces the same result for `y2` as the explicit loop:
```stan
for (i in 1:4) {
  for (j in 1:7) {
    y2[i, j] = pow(x20[i, j], x00);
  }
}
```

#### Vector and matrix arguments

Vectorized binary functions also apply elementwise to vectors and matrices,
and to combinations of these with scalar values.
For example,

```stan
 real x00;
 vector[5] xv00;
 vector[5] xv01;
 row_vector[7] xrv;
 matrix[10, 20] xm;

 vector[5] yv;
 row_vector[7] yrv;
 matrix[10, 20] ym;

 yv = pow(xv00, xv01);
 yrv = pow(xrv, x00);
 ym = pow(x00, xm);
```

Arrays of vectors and matrices work the same way.  For example,

```stan
 array[12] matrix[17, 93] u;

 array[12] matrix[17, 93] z;

 z = pow(u, x00);
```

After this has been executed, `z[i, j, k]` will be equal to `pow(u[i,
j, k], x00)`.

#### Input & return types

Vectorised binary functions require that both inputs, unless one is a real,
be containers of the same type and size. For example, the following statements
are legal:
```stan
 vector[5] xv;
 row_vector[7] xrv;
 matrix[10, 20] xm;

 vector[5] yv = pow(xv, xv)
 row_vector[7] yrv = pow(xrv, xrv)
 matrix[10, 20] = pow(xm, xm)
```

But the following statements are not:
```stan
 vector[5] xv;
 vector[7] xv2;
 row_vector[5] xrv;

 // Cannot mix different types
 vector[5] yv = pow(xv, xrv)

 // Cannot mix different sizes of the same type
 vector[5] yv = pow(xv, xv2)
```

While the vectorized binary functions generally require the same input types,
the only exception to this is for binary functions that require one input to be
an integer and the other to be a real (e.g., `bessel_first_kind`). For these
functions, one argument can be a container of any type while the other can be
an integer array, as long as the dimensions of both are the same. For example,
the following statements are legal:

```stan
 vector[5] xv;
 matrix[5, 5] xm;
 array[5] int xi;
 array[5, 5] int xii;

 vector[5] yv = bessel_first_kind(xi, xv);
 matrix[5, 5] ym = bessel_first_kind(xii, xm);
```

Whereas these are not:
```stan
 vector[5] xv;
 matrix[5, 5] xm;
 array[7] int xi;

 // Dimensions of containers do not match
 vector[5] yv = bessel_first_kind(xi, xv);

 // Function requires first argument be an integer type
 matrix[5, 5] ym = bessel_first_kind(xm, xm);
```

## Mathematical constants {#built-in-constants}

Constants are represented as functions with no arguments and must be
called as such.  For instance, the mathematical constant $\pi$ must be
written in a Stan program as `pi()`.

<!-- real; pi; (); -->
\index{{\tt \bfseries pi }!{\tt (): real}|hyperpage}

`real` **`pi`**`()`<br>\newline
$\pi$, the ratio of a circle's circumference to its diameter
`r since("2.0")`

<!-- real; e; (); -->
\index{{\tt \bfseries e }!{\tt (): real}|hyperpage}

`real` **`e`**`()`<br>\newline
$e$, the base of the natural logarithm
`r since("2.0")`

<!-- real; sqrt2; (); -->
\index{{\tt \bfseries sqrt2 }!{\tt (): real}|hyperpage}

`real` **`sqrt2`**`()`<br>\newline
The square root of 2
`r since("2.0")`

<!-- real; log2; (); -->
\index{{\tt \bfseries log2 }!{\tt (): real}|hyperpage}

`real` **`log2`**`()`<br>\newline
The natural logarithm of 2
`r since("2.0")`

<!-- real; log10; (); -->
\index{{\tt \bfseries log10 }!{\tt (): real}|hyperpage}

`real` **`log10`**`()`<br>\newline
The natural logarithm of 10
`r since("2.0")`

## Special values

<!-- real; not_a_number; (); -->
\index{{\tt \bfseries not\_a\_number }!{\tt (): real}|hyperpage}

`real` **`not_a_number`**`()`<br>\newline
Not-a-number, a special non-finite real value returned to signal an
error
`r since("2.0")`

<!-- real; positive_infinity; (); -->
\index{{\tt \bfseries positive\_infinity }!{\tt (): real}|hyperpage}

`real` **`positive_infinity`**`()`<br>\newline
Positive infinity, a special non-finite real value larger than all
finite numbers
`r since("2.0")`

<!-- real; negative_infinity; (); -->
\index{{\tt \bfseries negative\_infinity }!{\tt (): real}|hyperpage}

`real` **`negative_infinity`**`()`<br>\newline
Negative infinity, a special non-finite real value smaller than all
finite numbers
`r since("2.0")`

<!-- real; machine_precision; (); -->
\index{{\tt \bfseries machine\_precision }!{\tt (): real}|hyperpage}

`real` **`machine_precision`**`()`<br>\newline
The smallest number $x$ such that $(x + 1) \neq 1$ in floating-point
arithmetic on the current hardware platform
`r since("2.0")`

## Log probability function {#get-log-prob}

The basic purpose of a Stan program is to compute a log probability
function and its derivatives.  The log probability function in a Stan
model outputs the log density on the unconstrained scale.  A log
probability accumulator starts at zero and is then incremented in
various ways by a Stan program.  The variables are first transformed
from unconstrained to constrained, and the log Jacobian determinant
added to the log probability accumulator.  Then the model block is
executed on the constrained parameters, with each sampling statement
(`~`) and log probability increment statement (`increment_log_prob`)
adding to the accumulator.  At the end of the model block execution,
the value of the log probability accumulator is the log probability
value returned by the Stan program.

Stan provides a special built-in function `target()` that takes no
arguments and returns the current value of the log probability
accumulator.[^fn_lp]  This function is primarily useful for debugging
purposes, where for instance, it may be used with a print statement to
display the log probability accumulator at various stages of execution
to see where it becomes ill defined.

[^fn_lp]: This function used to be called `get_lp()`, but that   name
has been deprecated; using it will print a warning.  The function
`get_lp()` will be removed in Stan 2.32.0.

<!-- real; target; (); -->
\index{{\tt \bfseries target }!{\tt (): real}|hyperpage}

`real` **`target`**`()`<br>\newline
Return the current value of the log probability accumulator.
`r since("2.10")`

<!-- real; get_lp; (); -->
\index{{\tt \bfseries get\_lp }!{\tt (): real}|hyperpage}

`real` **`get_lp`**`()`<br>\newline
Return the current value of the log probability accumulator;
**deprecated;** - use `target()` instead.
`r since("2.5, scheduled for removal in 2.32.0")`

Both `target` and the deprecated `get_lp` act like other functions
ending in `_lp`, meaning that they may only may only be used in the
model block.

## Logical functions

Like C++, BUGS, and R, Stan uses 0 to encode false, and 1 to encode
true.  Stan supports the usual boolean comparison operations and
boolean operators.  These all have the same syntax and precedence as
in  C++; for the full list of operators and precedences, see the
reference manual.

### Comparison operators

All comparison operators return boolean values, either 0 or 1.  Each
operator has two signatures, one for integer comparisons and one for
floating-point comparisons.  Comparing an integer and real value is
carried out by first promoting the integer value.

<!-- int; operator<; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_less\_than }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator<`**`(int x, int y)`<br>\newline

<!-- int; operator<; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_less\_than }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator<`**`(real x, real y)`<br>\newline
Return 1 if x is less than y and 0 otherwise. \[ \text{operator<}(x,y)
= \begin{cases} 1 & \text{if $x < y$} \\ 0 & \text{otherwise}
\end{cases} \]
`r since("2.0")`

<!-- int; operator<=; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_less\_than\_equal }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator<=`**`(int x, int y)`<br>\newline

<!-- int; operator<=; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_less\_than\_equal }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator<=`**`(real x, real y)`<br>\newline
Return 1 if x is less than or equal y and 0 otherwise. \[
\text{operator<=}(x,y) = \begin{cases} 1 & \text{if $x \leq y$} \\ 0 &
\text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- int; operator>; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_greater\_than }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator>`**`(int x, int y)`<br>\newline

<!-- int; operator>; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_greater\_than }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator>`**`(real x, real y)`<br>\newline
Return 1 if x is greater than y and 0 otherwise. \[ \text{operator>} =
\begin{cases} 1 & \text{if $x > y$} \\ 0 & \text{otherwise}
\end{cases} \]
`r since("2.0")`

<!-- int; operator>=; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_greater\_than\_equal }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator>=`**`(int x, int y)`<br>\newline

<!-- int; operator>=; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_greater\_than\_equal }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator>=`**`(real x, real y)`<br>\newline
Return 1 if x is greater than or equal to y and 0 otherwise. \[
\text{operator>=} = \begin{cases} 1 & \text{if $x \geq y$} \\ 0 &
\text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- int; operator==; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_equal }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator==`**`(int x, int y)`<br>\newline

<!-- int; operator==; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_equal }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator==`**`(real x, real y)`<br>\newline
Return 1 if x is equal to y and 0 otherwise. \[ \text{operator==}(x,y)
= \begin{cases} 1 & \text{if $x = y$} \\ 0 & \text{otherwise}
\end{cases} \]
`r since("2.0")`

<!-- int; operator!=; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_not\_equal }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator!=`**`(int x, int y)`<br>\newline

<!-- int; operator!=; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_not\_equal }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator!=`**`(real x, real y)`<br>\newline
Return 1 if x is not equal to y and 0 otherwise. \[
\text{operator!=}(x,y) = \begin{cases} 1 & \text{if $x \neq y$} \\ 0 &
\text{otherwise} \end{cases} \]
`r since("2.0")`

### Boolean operators

Boolean operators return either 0 for false or 1 for true.  Inputs may
be any real or integer values, with non-zero values being treated as
true and zero values treated as false.  These operators have the usual
precedences, with negation (not) binding the most tightly, conjunction
the next and disjunction the weakest; all of the operators bind more
tightly than the comparisons.  Thus an expression such as `!a && b` is
interpreted as `(!a) && b`, and `a < b || c >= d && e != f` as `(a <
b) || (((c >= d) && (e != f)))`.

<!-- int; operator!; (int x); -->
\index{{\tt \bfseries operator\_negation }!{\tt (int x): int}|hyperpage}

`int` **`operator!`**`(int x)`<br>\newline

<!-- int; operator!; (real x); -->
\index{{\tt \bfseries operator\_negation }!{\tt (real x): int}|hyperpage}

`int` **`operator!`**`(real x)`<br>\newline
Return 1 if x is zero and 0 otherwise. \[ \text{operator!}(x) =
\begin{cases} 0 & \text{if $x \neq 0$} \\ 1 & \text{if $x = 0$}
\end{cases} \]
`r since("2.0")`

<!-- int; operator&&; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_and }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator&&`**`(int x, int y)`<br>\newline

<!-- int; operator&&; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_and }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator&&`**`(real x, real y)`<br>\newline
Return 1 if x is unequal to 0 and y is unequal to 0. \[
\mathrm{operator\&\&}(x,y) = \begin{cases} 1 & \text{if $x \neq 0$}
\text{ and } y \neq 0\\ 0 & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- int; operator||; (int x, int y); -->
\index{{\tt \bfseries operator\_logical\_or }!{\tt (int x, int y): int}|hyperpage}

`int` **`operator||`**`(int x, int y)`<br>\newline

<!-- int; operator||; (real x, real y); -->
\index{{\tt \bfseries operator\_logical\_or }!{\tt (real x, real y): int}|hyperpage}

`int` **`operator||`**`(real x, real y)`<br>\newline
Return 1 if x is unequal to 0 or y is unequal to 0. \[
\text{operator||}(x,y) = \begin{cases} 1 & \text{if $x \neq 0$}
\textrm{ or } y \neq 0\\ 0 & \text{otherwise} \end{cases} \]
`r since("2.0")`

#### Boolean operator short circuiting

Like in  C++, the boolean operators `&&` and `||` and are implemented
to short circuit directly to a return value after evaluating the first
argument if it is sufficient to resolve the result.  In evaluating `a
|| b`, if `a` evaluates to a value other than zero, the expression
returns the value 1 without evaluating the expression `b`.  Similarly,
evaluating `a && b` first evaluates `a`, and if the result is zero,
returns 0 without evaluating `b`.

### Logical functions

The logical functions introduce conditional behavior functionally and
are primarily provided for compatibility with BUGS and JAGS.

<!-- real; step; (real x); -->
\index{{\tt \bfseries step }!{\tt (real x): real}|hyperpage}

`real` **`step`**`(real x)`<br>\newline
Return 1 if x is positive and 0 otherwise. \[ \text{step}(x) =
\begin{cases} 0 & \text{if } x < 0 \\ 1 & \text{otherwise} \end{cases}
\] _**Warning:**_ `int_step(0)` and `int_step(NaN)` return 0 whereas
`step(0)` and `step(NaN)` return 1.

The step function is often used in BUGS to perform conditional
operations.  For instance, `step(a-b)` evaluates to 1 if `a` is
greater than `b` and evaluates to 0 otherwise. `step` is a step-like
functions; see the warning in section [step functions](#step-functions) applied to
expressions dependent on parameters.
`r since("2.0")`

<!-- int; is_inf; (real x); -->
\index{{\tt \bfseries is\_inf }!{\tt (real x): int}|hyperpage}

`int` **`is_inf`**`(real x)`<br>\newline
Return 1 if x is infinite (positive or negative) and 0 otherwise.
`r since("2.5")`

<!-- int; is_nan; (real x); -->
\index{{\tt \bfseries is\_nan }!{\tt (real x): int}|hyperpage}

`int` **`is_nan`**`(real x)`<br>\newline
Return 1 if x is NaN and 0 otherwise.
`r since("2.5")`

Care must be taken because both of these indicator functions are
step-like and thus can cause discontinuities in gradients when applied
to parameters; see section [step-like functions](#step-functions) for details.

## Real-valued arithmetic operators {#real-valued-arithmetic-operators}

The arithmetic operators are presented using  C++ notation.  For
instance `operator+(x,y)` refers to the binary addition operator and
`operator-(x)` to the unary negation operator.  In Stan programs,
these are written using the usual infix and prefix notations as `x +
y` and `-x`, respectively.

### Binary infix operators

<!-- real; operator+; (real x, real y); -->
\index{{\tt \bfseries operator\_add }!{\tt (real x, real y): real}|hyperpage}

`real` **`operator+`**`(real x, real y)`<br>\newline
Return the sum of x and y. \[ (x + y) = \text{operator+}(x,y) = x+y \]
`r since("2.0")`

<!-- real; operator-; (real x, real y); -->
\index{{\tt \bfseries operator\_subtract }!{\tt (real x, real y): real}|hyperpage}

`real` **`operator-`**`(real x, real y)`<br>\newline
Return the difference between x and y. \[ (x - y) =
\text{operator-}(x,y) = x - y \]
`r since("2.0")`

<!-- real; operator*; (real x, real y); -->
\index{{\tt \bfseries operator\_multiply }!{\tt (real x, real y): real}|hyperpage}

`real` **`operator*`**`(real x, real y)`<br>\newline
Return the product of x and y. \[ (x * y) = \text{operator*}(x,y) = xy
\]
`r since("2.0")`

<!-- real; operator/; (real x, real y); -->
\index{{\tt \bfseries operator\_divide }!{\tt (real x, real y): real}|hyperpage}

`real` **`operator/`**`(real x, real y)`<br>\newline
Return the quotient of x and y. \[ (x / y) = \text{operator/}(x,y) =
\frac{x}{y} \]
`r since("2.0")`

<!-- real; operator^; (real x, real y); -->
\index{{\tt \bfseries operator\_pow }!{\tt (real x, real y): real}|hyperpage}

`real` **`operator^`**`(real x, real y)`<br>\newline
Return x raised to the power of y. \[ (x^\mathrm{\wedge}y) =
\text{operator}^\mathrm{\wedge}(x,y) = x^y \]
`r since("2.5")`

### Unary prefix operators

<!-- real; operator-; (real x); -->
\index{{\tt \bfseries operator\_subtract }!{\tt (real x): real}|hyperpage}

`real` **`operator-`**`(real x)`<br>\newline
Return the negation of the subtrahend x. \[ \text{operator-}(x) = (-x)
\]
`r since("2.0")`

<!-- real; operator+; (real x); -->
\index{{\tt \bfseries operator\_add }!{\tt (real x): real}|hyperpage}

`real` **`operator+`**`(real x)`<br>\newline
Return the value of x. \[ \text{operator+}(x) = x \]
`r since("2.0")`

## Step-like functions {#step-functions}

_**Warning:**_  *These functions can seriously hinder sampling and
optimization efficiency for gradient-based methods (e.g., NUTS, HMC,
BFGS) if applied to parameters (including transformed parameters and
local variables in the transformed parameters or model block).  The
problem is that they break gradients due to discontinuities coupled
with   zero gradients elsewhere.  They do not hinder sampling when
used in the   data, transformed data, or generated quantities blocks.*

### Absolute value functions

<!-- R; fabs; (T x); -->
\index{{\tt \bfseries fabs }!{\tt (T x): R}|hyperpage}

`R` **`fabs`**`(T x)`<br>\newline
absolute value of x
`r since("2.0, vectorized in 2.13")`

<!-- real; fdim; (real x, real y); -->
\index{{\tt \bfseries fdim }!{\tt (real x, real y): real}|hyperpage}

`real` **`fdim`**`(real x, real y)`<br>\newline
Return the positive difference between x and y, which is x - y if x is
greater than y and 0 otherwise; see warning above.
\[ \text{fdim}(x,y) = \begin{cases} x-y &
\text{if } x \geq y \\ 0 & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- R; fdim; (T1 x, T2 y); -->
\index{{\tt \bfseries fdim }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`fdim`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `fdim` function
`r since("2.25")`

### Bounds functions

<!-- real; fmin; (real x, real y); -->
\index{{\tt \bfseries fmin }!{\tt (real x, real y): real}|hyperpage}

`real` **`fmin`**`(real x, real y)`<br>\newline
Return the minimum of x and y; see warning above.
\[ \text{fmin}(x,y) = \begin{cases} x &
\text{if } x \leq y \\ y & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- R; fmin; (T1 x, T2 y); -->
\index{{\tt \bfseries fmin }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`fmin`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `fmin` function
`r since("2.25")`

<!-- real; fmax; (real x, real y); -->
\index{{\tt \bfseries fmax }!{\tt (real x, real y): real}|hyperpage}

`real` **`fmax`**`(real x, real y)`<br>\newline
Return the maximum of x and y; see warning above.
\[ \text{fmax}(x,y) = \begin{cases} x &
\text{if } x \geq y \\ y & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- R; fmax; (T1 x, T2 y); -->
\index{{\tt \bfseries fmax }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`fmax`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `fmax` function
`r since("2.25")`

### Arithmetic functions

<!-- real; fmod; (real x, real y); -->
\index{{\tt \bfseries fmod }!{\tt (real x, real y): real}|hyperpage}

`real` **`fmod`**`(real x, real y)`<br>\newline
Return the real value remainder after dividing x by y; see warning above.
\[ \text{fmod}(x,y) = x - \left\lfloor \frac{x}{y} \right\rfloor \, y \]
The operator $\lfloor u \rfloor$ is the floor operation; see below.
`r since("2.0")`

<!-- R; fmod; (T1 x, T2 y); -->
\index{{\tt \bfseries fmod }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`fmod`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `fmod` function
`r since("2.25")`

### Rounding functions

_**Warning:**_  Rounding functions convert real values to integers.
Because the output is an integer, any gradient information resulting
from functions applied to the integer is not passed to the real value
it was derived from.  With MCMC sampling using HMC or NUTS, the MCMC
acceptance procedure will correct for any error due to poor gradient
calculations, but the result is likely to be reduced acceptance
probabilities and less efficient sampling.

The rounding functions cannot be used as indices to arrays because
they return real values.  Stan may introduce integer-valued versions
of these in the future, but as of now, there is no good workaround.

<!-- R; floor; (T x); -->
\index{{\tt \bfseries floor }!{\tt (T x): R}|hyperpage}

`R` **`floor`**`(T x)`<br>\newline
floor of x, which is the largest integer less than or equal to x,
converted to a real value; see warning at start of section
[step-like functions](#step-functions)
`r since("2.0, vectorized in 2.13")`

<!-- R; ceil; (T x); -->
\index{{\tt \bfseries ceil }!{\tt (T x): R}|hyperpage}

`R` **`ceil`**`(T x)`<br>\newline
ceiling of x, which is the smallest integer greater than or equal to
x, converted to a real value; see warning at start of section
[step-like functions](#step-functions)
`r since("2.0, vectorized in 2.13")`

<!-- R; round; (T x); -->
\index{{\tt \bfseries round }!{\tt (T x): R}|hyperpage}

`R` **`round`**`(T x)`<br>\newline
nearest integer to x, converted to a real value; see warning at start
of section [step-like functions](#step-functions)
`r since("2.0, vectorized in 2.13")`

<!-- R; trunc; (T x); -->
\index{{\tt \bfseries trunc }!{\tt (T x): R}|hyperpage}

`R` **`trunc`**`(T x)`<br>\newline
integer nearest to but no larger in magnitude than x, converted to a
double value; see warning at start of section [step-like functions](#step-functions)
`r since("2.0, vectorized in 2.13")`

## Power and logarithm functions

<!-- R; sqrt; (T x); -->
\index{{\tt \bfseries sqrt }!{\tt (T x): R}|hyperpage}

`R` **`sqrt`**`(T x)`<br>\newline
square root of x
`r since("2.0, vectorized in 2.13")`

<!-- R; cbrt; (T x); -->
\index{{\tt \bfseries cbrt }!{\tt (T x): R}|hyperpage}

`R` **`cbrt`**`(T x)`<br>\newline
cube root of x
`r since("2.0, vectorized in 2.13")`

<!-- R; square; (T x); -->
\index{{\tt \bfseries square }!{\tt (T x): R}|hyperpage}

`R` **`square`**`(T x)`<br>\newline
square of x
`r since("2.0, vectorized in 2.13")`

<!-- R; exp; (T x); -->
\index{{\tt \bfseries exp }!{\tt (T x): R}|hyperpage}

`R` **`exp`**`(T x)`<br>\newline
natural exponential of x
`r since("2.0, vectorized in 2.13")`

<!-- R; exp2; (T x); -->
\index{{\tt \bfseries exp2 }!{\tt (T x): R}|hyperpage}

`R` **`exp2`**`(T x)`<br>\newline
base-2 exponential of x
`r since("2.0, vectorized in 2.13")`

<!-- R; log; (T x); -->
\index{{\tt \bfseries log }!{\tt (T x): R}|hyperpage}

`R` **`log`**`(T x)`<br>\newline
natural logarithm of x
`r since("2.0, vectorized in 2.13")`

<!-- R; log2; (T x); -->
\index{{\tt \bfseries log2 }!{\tt (T x): R}|hyperpage}

`R` **`log2`**`(T x)`<br>\newline
base-2 logarithm of x
`r since("2.0, vectorized in 2.13")`

<!-- R; log10; (T x); -->
\index{{\tt \bfseries log10 }!{\tt (T x): R}|hyperpage}

`R` **`log10`**`(T x)`<br>\newline
base-10 logarithm of x
`r since("2.0, vectorized in 2.13")`

<!-- real; pow; (real x, real y); -->
\index{{\tt \bfseries pow }!{\tt (real x, real y): real}|hyperpage}

`real` **`pow`**`(real x, real y)`<br>\newline
Return x raised to the power of y. \[ \text{pow}(x,y) = x^y \]
`r since("2.0")`

<!-- R; pow; (T1 x, T2 y); -->
\index{{\tt \bfseries pow }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`pow`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `pow` function
`r since("2.25")`

<!-- R; inv; (T x); -->
\index{{\tt \bfseries inv }!{\tt (T x): R}|hyperpage}

`R` **`inv`**`(T x)`<br>\newline
inverse of x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_sqrt; (T x); -->
\index{{\tt \bfseries inv\_sqrt }!{\tt (T x): R}|hyperpage}

`R` **`inv_sqrt`**`(T x)`<br>\newline
inverse of the square root of x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_square; (T x); -->
\index{{\tt \bfseries inv\_square }!{\tt (T x): R}|hyperpage}

`R` **`inv_square`**`(T x)`<br>\newline
inverse of the square of x
`r since("2.0, vectorized in 2.13")`

## Trigonometric functions

<!-- real; hypot; (real x, real y); -->
\index{{\tt \bfseries hypot }!{\tt (real x, real y): real}|hyperpage}

`real` **`hypot`**`(real x, real y)`<br>\newline
Return the length of the hypotenuse of a right triangle with sides of
length x and y. \[ \text{hypot}(x,y) = \begin{cases} \sqrt{x^2+y^2} &
\text{if } x,y\geq 0 \\ \textrm{NaN} & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- R; hypot; (T1 x, T2 y); -->
\index{{\tt \bfseries hypot }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`hypot`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `hypot` function
`r since("2.25")`

<!-- R; cos; (T x); -->
\index{{\tt \bfseries cos }!{\tt (T x): R}|hyperpage}

`R` **`cos`**`(T x)`<br>\newline
cosine of the angle x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; sin; (T x); -->
\index{{\tt \bfseries sin }!{\tt (T x): R}|hyperpage}

`R` **`sin`**`(T x)`<br>\newline
sine of the angle x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; tan; (T x); -->
\index{{\tt \bfseries tan }!{\tt (T x): R}|hyperpage}

`R` **`tan`**`(T x)`<br>\newline
tangent of the angle x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; acos; (T x); -->
\index{{\tt \bfseries acos }!{\tt (T x): R}|hyperpage}

`R` **`acos`**`(T x)`<br>\newline
principal arc (inverse) cosine (in radians) of x
`r since("2.0, vectorized in 2.13")`

<!-- R; asin; (T x); -->
\index{{\tt \bfseries asin }!{\tt (T x): R}|hyperpage}

`R` **`asin`**`(T x)`<br>\newline
principal arc (inverse) sine (in radians) of x
`r since("2.0")`

<!-- R; atan; (T x); -->
\index{{\tt \bfseries atan }!{\tt (T x): R}|hyperpage}

`R` **`atan`**`(T x)`<br>\newline
principal arc (inverse) tangent (in radians) of x, with values from
$-\pi/2$ to $\pi/2$
`r since("2.0, vectorized in 2.13")`

<!-- real; atan2; (real y, real x); -->
\index{{\tt \bfseries atan2 }!{\tt (real y, real x): real}|hyperpage}

`real` **`atan2`**`(real y, real x)`<br>\newline
Return the principal arc (inverse) tangent (in radians) of y divided
by x, \[ \text{atan2}(y, x) = \arctan\left(\frac{y}{x}\right) \]
`r since("2.0, vectorized in 2.13")`

## Hyperbolic trigonometric functions

<!-- R; cosh; (T x); -->
\index{{\tt \bfseries cosh }!{\tt (T x): R}|hyperpage}

`R` **`cosh`**`(T x)`<br>\newline
hyperbolic cosine of x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; sinh; (T x); -->
\index{{\tt \bfseries sinh }!{\tt (T x): R}|hyperpage}

`R` **`sinh`**`(T x)`<br>\newline
hyperbolic sine of x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; tanh; (T x); -->
\index{{\tt \bfseries tanh }!{\tt (T x): R}|hyperpage}

`R` **`tanh`**`(T x)`<br>\newline
hyperbolic tangent of x (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; acosh; (T x); -->
\index{{\tt \bfseries acosh }!{\tt (T x): R}|hyperpage}

`R` **`acosh`**`(T x)`<br>\newline
inverse hyperbolic cosine (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; asinh; (T x); -->
\index{{\tt \bfseries asinh }!{\tt (T x): R}|hyperpage}

`R` **`asinh`**`(T x)`<br>\newline
inverse hyperbolic cosine (in radians)
`r since("2.0, vectorized in 2.13")`

<!-- R; atanh; (T x); -->
\index{{\tt \bfseries atanh }!{\tt (T x): R}|hyperpage}

`R` **`atanh`**`(T x)`<br>\newline
inverse hyperbolic tangent (in radians) of x
`r since("2.0, vectorized in 2.13")`

## Link functions {#link-functions}

The following functions are commonly used as link functions in
generalized linear models.  The function $\Phi$ is also commonly used
as a link function (see section [probability-related functions](#Phi-function)).

<!-- R; logit; (T x); -->
\index{{\tt \bfseries logit }!{\tt (T x): R}|hyperpage}

`R` **`logit`**`(T x)`<br>\newline
log odds, or logit, function applied to x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_logit; (T x); -->
\index{{\tt \bfseries inv\_logit }!{\tt (T x): R}|hyperpage}

`R` **`inv_logit`**`(T x)`<br>\newline
logistic sigmoid function applied to x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_cloglog; (T x); -->
\index{{\tt \bfseries inv\_cloglog }!{\tt (T x): R}|hyperpage}

`R` **`inv_cloglog`**`(T x)`<br>\newline
inverse of the complementary log-log function applied to x
`r since("2.0, vectorized in 2.13")`

## Probability-related functions {#Phi-function}

### Normal cumulative distribution functions

The error function `erf` is related to the standard normal cumulative
distribution function $\Phi$ by scaling.  See section
[normal distribution](#normal-distribution) for the general normal cumulative
distribution function (and its complement).

<!-- R; erf; (T x); -->
\index{{\tt \bfseries erf }!{\tt (T x): R}|hyperpage}

`R` **`erf`**`(T x)`<br>\newline
error function, also known as the Gauss error function, of x
`r since("2.0, vectorized in 2.13")`

<!-- R; erfc; (T x); -->
\index{{\tt \bfseries erfc }!{\tt (T x): R}|hyperpage}

`R` **`erfc`**`(T x)`<br>\newline
complementary error function of x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_erfc; (T x); -->
\index{{\tt \bfseries inv\_erfc }!{\tt (T x): R}|hyperpage}

`R` **`inv_erfc`**`(T x)`<br>\newline
inverse of the complementary error function of x
`r since("2.29, vectorized in 2.29")`

<!-- R; Phi; (T x); -->
\index{{\tt \bfseries phi }!{\tt (T x): R}|hyperpage}

`R` **`Phi`**`(T x)`<br>\newline
standard normal cumulative distribution function of x
`r since("2.0, vectorized in 2.13")`

<!-- R; inv_Phi; (T x); -->
\index{{\tt \bfseries inv\_phi }!{\tt (T x): R}|hyperpage}

`R` **`inv_Phi`**`(T x)`<br>\newline
Return the value of the inverse standard normal cdf $\Phi^{-1}$ at the
specified quantile `x`. The details of the algorithm can be found in [@Wichura:1988].
Quantile arguments below 1e-16 are untested; quantiles above 0.999999999 result in increasingly large errors.
`r since("2.0, vectorized in 2.13")`

<!-- R; Phi_approx; (T x); -->
\index{{\tt \bfseries phi\_approx }!{\tt (T x): R}|hyperpage}

`R` **`Phi_approx`**`(T x)`<br>\newline
fast approximation of the unit (may replace `Phi` for probit
regression with maximum absolute error of 0.00014, see
[@BowlingEtAl:2009] for details)
`r since("2.0, vectorized in 2.13")`

### Other probability-related functions

<!-- real; binary_log_loss; (int y, real y_hat); -->
\index{{\tt \bfseries binary\_log\_loss }!{\tt (int y, real y\_hat): real}|hyperpage}

`real` **`binary_log_loss`**`(int y, real y_hat)`<br>\newline
Return the log loss function for for predicting $\hat{y} \in [0,1]$
for boolean outcome $y \in \{0,1\}$. \[
\mathrm{binary\_log\_loss}(y,\hat{y}) = \begin{cases} -\log \hat{y} &
\text{if } y = 0\\ -\log (1 - \hat{y}) & \text{otherwise} \end{cases}
\]
`r since("2.0")`

<!-- R; binary_log_loss; (T1 x, T2 y); -->
\index{{\tt \bfseries binary\_log\_loss }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`binary_log_loss`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `binary_log_loss` function
`r since("2.25")`

<!-- real; owens_t; (real h, real a); -->
\index{{\tt \bfseries owens\_t }!{\tt (real h, real a): real}|hyperpage}

`real` **`owens_t`**`(real h, real a)`<br>\newline
Return the Owen's T function for the probability of the event $X > h$
and $0<Y<aX$ where X and Y are independent standard normal random
variables. \[ \mathrm{owens\_t}(h,a) = \frac{1}{2\pi} \int_0^a
\frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]
`r since("2.25")`

<!-- R; owens_t; (T1 x, T2 y); -->
\index{{\tt \bfseries owens\_t }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`owens_t`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `owens_t` function
`r since("2.25")`

## Combinatorial functions {#betafun}

<!-- real; beta; (real alpha, real beta); -->
\index{{\tt \bfseries beta }!{\tt (real alpha, real beta): real}|hyperpage}

`real` **`beta`**`(real alpha, real beta)`<br>\newline
Return the beta function applied to alpha and beta. The beta function,
$\text{B}(\alpha,\beta)$, computes the normalizing constant for the beta
distribution, and is defined for $\alpha > 0$ and $\beta > 0$. See section
[appendix](#beta-appendix) for definition of $\text{B}(\alpha, \beta)$.
`r since("2.25")`

<!-- R; beta; (T1 x, T2 y); -->
\index{{\tt \bfseries beta }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`beta`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `beta` function
`r since("2.25")`

<!-- real; inc_beta; (real alpha, real beta, real x); -->
\index{{\tt \bfseries inc\_beta }!{\tt (real alpha, real beta, real x): real}|hyperpage}

`real` **`inc_beta`**`(real alpha, real beta, real x)`<br>\newline
Return the regularized incomplete beta function up to x applied to alpha and beta.
See section [appendix](#inc-beta-appendix) for a definition.
`r since("2.10")`

<!-- real; lbeta; (real alpha, real beta); -->
\index{{\tt \bfseries lbeta }!{\tt (real alpha, real beta): real}|hyperpage}

`real` **`lbeta`**`(real alpha, real beta)`<br>\newline
Return the natural logarithm of the beta function applied to alpha and
beta. The beta function, $\text{B}(\alpha,\beta)$, computes the
normalizing constant for the beta distribution, and is defined for
$\alpha > 0$ and $\beta > 0$. \[ \text{lbeta}(\alpha,\beta) = \log
\Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b) \] See section
[appendix](#beta-appendix) for definition of $\text{B}(\alpha, \beta)$.
`r since("2.0")`

<!-- R; lbeta; (T1 x, T2 y); -->
\index{{\tt \bfseries lbeta }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`lbeta`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `lbeta` function
`r since("2.25")`

<!-- R; tgamma; (T x); -->
\index{{\tt \bfseries tgamma }!{\tt (T x): R}|hyperpage}

`R` **`tgamma`**`(T x)`<br>\newline
gamma function applied to x. The gamma function is the generalization
of the factorial function to continuous variables, defined so that
$\Gamma(n+1) = n!$. See for a full definition of $\Gamma(x)$. The
function is defined for positive numbers and non-integral negative
numbers,
`r since("2.0, vectorized in 2.13")`

<!-- R; lgamma; (T x); -->
\index{{\tt \bfseries lgamma }!{\tt (T x): R}|hyperpage}

`R` **`lgamma`**`(T x)`<br>\newline
natural logarithm of the gamma function applied to x,
`r since("2.0, vectorized in 2.15")`

<!-- R; digamma; (T x); -->
\index{{\tt \bfseries digamma }!{\tt (T x): R}|hyperpage}

`R` **`digamma`**`(T x)`<br>\newline
digamma function applied to x. The digamma function is the derivative
of the natural logarithm of the Gamma function. The function is
defined for positive numbers and non-integral negative numbers
`r since("2.0, vectorized in 2.13")`

<!-- R; trigamma; (T x); -->
\index{{\tt \bfseries trigamma }!{\tt (T x): R}|hyperpage}

`R` **`trigamma`**`(T x)`<br>\newline
trigamma function applied to x. The trigamma function is the second
derivative of the natural logarithm of the Gamma function
`r since("2.0, vectorized in 2.13")`

<!-- real; lmgamma; (int n, real x); -->
\index{{\tt \bfseries lmgamma }!{\tt (int n, real x): real}|hyperpage}

`real` **`lmgamma`**`(int n, real x)`<br>\newline
Return the natural logarithm of the multivariate gamma function
$\Gamma_n$ with n dimensions applied to x. \[ \text{lmgamma}(n,x) =
\begin{cases} \frac{n(n-1)}{4} \log \pi + \sum_{j=1}^n \log
\Gamma\left(x + \frac{1 - j}{2}\right) & \text{if } x\not\in
\{\dots,-3,-2,-1,0\}\\ \textrm{error} & \text{otherwise} \end{cases}
\]
`r since("2.0")`

<!-- R; lmgamma; (T1 x, T2 y); -->
\index{{\tt \bfseries lmgamma }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`lmgamma`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `lmgamma` function
`r since("2.25")`

<!-- real; gamma_p; (real a, real z); -->
\index{{\tt \bfseries gamma\_p }!{\tt (real a, real z): real}|hyperpage}

`real` **`gamma_p`**`(real a, real z)`<br>\newline
Return the normalized lower incomplete gamma function of a and z
defined for positive a and nonnegative z. \[ \mathrm{gamma\_p}(a,z) =
\begin{cases} \frac{1}{\Gamma(a)}\int_0^zt^{a-1}e^{-t}dt & \text{if }
a > 0, z \geq 0 \\ \textrm{error} & \text{otherwise} \end{cases} \]
`r since("2.0")`

<!-- R; gamma_p; (T1 x, T2 y); -->
\index{{\tt \bfseries gamma\_p }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`gamma_p`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `gamma_p` function
`r since("2.25")`

<!-- real; gamma_q; (real a, real z); -->
\index{{\tt \bfseries gamma\_q }!{\tt (real a, real z): real}|hyperpage}

`real` **`gamma_q`**`(real a, real z)`<br>\newline
Return the normalized upper incomplete gamma function of a and z
defined for positive a and nonnegative z. \[ \mathrm{gamma\_q}(a,z) =
\begin{cases} \frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t}dt &
\text{if } a > 0, z \geq 0 \\[6pt] \textrm{error} & \text{otherwise}
\end{cases} \]
`r since("2.0")`

<!-- R; gamma_q; (T1 x, T2 y); -->
\index{{\tt \bfseries gamma\_q }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`gamma_q`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `gamma_q` function
`r since("2.25")`

<!-- real; binomial_coefficient_log; (real x, real y); -->
\index{{\tt \bfseries binomial\_coefficient\_log }!{\tt (real x, real y): real}|hyperpage}

`real` **`binomial_coefficient_log`**`(real x, real y)`<br>\newline
_**Warning:**_ This function is deprecated and should be replaced with
`lchoose`. Return the natural logarithm of the binomial coefficient of
x and y. For non-negative integer inputs, the binomial coefficient
function is written as $\binom{x}{y}$ and pronounced "x choose y."
This function generalizes to real numbers using the gamma function.
For $0 \leq y \leq x$, \[ \mathrm{binomial\_coefficient\_log}(x,y) =
\log\Gamma(x+1) - \log\Gamma(y+1) - \log\Gamma(x-y+1). \]
`r since("2.0, deprecated since 2.10, scheduled for removal in 2.32")`

<!-- R; binomial_coefficient_log; (T1 x, T2 y); -->
\index{{\tt \bfseries binomial\_coefficient\_log }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`binomial_coefficient_log`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `binomial_coefficient_log` function
`r since("2.25")`

<!-- int; choose; (int x, int y); -->
\index{{\tt \bfseries choose }!{\tt (int x, int y): int}|hyperpage}

`int` **`choose`**`(int x, int y)`<br>\newline
Return the binomial coefficient of x and y. For non-negative integer
inputs, the binomial coefficient function is written as $\binom{x}{y}$
and pronounced "x choose y." In its the antilog of the `lchoose`
function but returns an integer rather than a real number with no
non-zero decimal places. For $0 \leq y \leq x$, the binomial
coefficient function can be defined via the factorial function \[
\text{choose}(x,y) = \frac{x!}{\left(y!\right)\left(x - y\right)!}. \]
`r since("2.14")`

<!-- R; choose; (T1 x, T2 y); -->
\index{{\tt \bfseries choose }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`choose`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `choose` function
`r since("2.25")`

<!-- real; bessel_first_kind; (int v, real x); -->
\index{{\tt \bfseries bessel\_first\_kind }!{\tt (int v, real x): real}|hyperpage}

`real` **`bessel_first_kind`**`(int v, real x)`<br>\newline
Return the Bessel function of the first kind with order v applied to
x. \[ \mathrm{bessel\_first\_kind}(v,x) = J_v(x), \] where \[
J_v(x)=\left(\frac{1}{2}x\right)^v \sum_{k=0}^\infty
\frac{\left(-\frac{1}{4}x^2\right)^k}{k!\, \Gamma(v+k+1)} \]
`r since("2.5")`

<!-- R; bessel_first_kind; (T1 x, T2 y); -->
\index{{\tt \bfseries bessel\_first\_kind }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`bessel_first_kind`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `bessel_first_kind` function
`r since("2.25")`

<!-- real; bessel_second_kind; (int v, real x); -->
\index{{\tt \bfseries bessel\_second\_kind }!{\tt (int v, real x): real}|hyperpage}

`real` **`bessel_second_kind`**`(int v, real x)`<br>\newline
Return the Bessel function of the second kind with order v applied to
x defined for positive x and v. For $x,v > 0$, \[
\mathrm{bessel\_second\_kind}(v,x) = \begin{cases} Y_v(x) & \text{if }
x > 0 \\ \textrm{error} & \text{otherwise} \end{cases} \] where \[
Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)} \]
`r since("2.5")`

<!-- R; bessel_second_kind; (T1 x, T2 y); -->
\index{{\tt \bfseries bessel\_second\_kind }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`bessel_second_kind`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `bessel_second_kind` function
`r since("2.25")`

<!-- real; modified_bessel_first_kind; (int v, real z); -->
\index{{\tt \bfseries modified\_bessel\_first\_kind }!{\tt (int v, real z): real}|hyperpage}

`real` **`modified_bessel_first_kind`**`(int v, real z)`<br>\newline
Return the modified Bessel function of the first kind with order v
applied to z defined for all z and integer v. \[
\mathrm{modified\_bessel\_first\_kind}(v,z) = I_v(z) \] where \[
{I_v}(z) = \left(\frac{1}{2}z\right)^v\sum_{k=0}^\infty
\frac{\left(\frac{1}{4}z^2\right)^k}{k!\Gamma(v+k+1)} \]
`r since("2.1")`

<!-- R; modified_bessel_first_kind; (T1 x, T2 y); -->
\index{{\tt \bfseries modified\_bessel\_first\_kind }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`modified_bessel_first_kind`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `modified_bessel_first_kind` function
`r since("2.25")`

<!-- real; log_modified_bessel_first_kind; (real v, real z); -->
\index{{\tt \bfseries log\_modified\_bessel\_first\_kind }!{\tt (real v, real z): real}|hyperpage}

`real` **`log_modified_bessel_first_kind`**`(real v, real z)`<br>\newline
Return the log of the modified Bessel function of the first kind. v does
not have to be an integer.
`r since("2.26")`

<!-- R; log_modified_bessel_first_kind; (T1 x, T2 y); -->
\index{{\tt \bfseries log\_modified\_bessel\_first\_kind }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`log_modified_bessel_first_kind`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `log_modified_bessel_first_kind` function
`r since("2.26")`

<!-- real; modified_bessel_second_kind; (int v, real z); -->
\index{{\tt \bfseries modified\_bessel\_second\_kind }!{\tt (int v, real z): real}|hyperpage}

`real` **`modified_bessel_second_kind`**`(int v, real z)`<br>\newline
Return the modified Bessel function of the second kind with order v
applied to z defined for positive z and integer v. \[
\mathrm{modified\_bessel\_second\_kind}(v,z) = \begin{cases} K_v(z) &
\text{if } z > 0 \\ \textrm{error} & \text{if } z \leq 0 \end{cases}
\] where \[ {K_v}(z) = \frac{\pi}{2}\cdot\frac{I_{-v}(z) -
I_{v}(z)}{\sin(v\pi)} \]
`r since("2.1")`

<!-- R; modified_bessel_second_kind; (T1 x, T2 y); -->
\index{{\tt \bfseries modified\_bessel\_second\_kind }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`modified_bessel_second_kind`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `modified_bessel_second_kind` function
`r since("2.25")`

<!-- real; falling_factorial; (real x, real n); -->
\index{{\tt \bfseries falling\_factorial }!{\tt (real x, real n): real}|hyperpage}

`real` **`falling_factorial`**`(real x, real n)`<br>\newline
Return the falling factorial of x with power n defined for positive x
and real n. \[ \mathrm{falling\_factorial}(x,n) = \begin{cases} (x)_n
& \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}
\] where \[ (x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)} \]
`r since("2.0")`

<!-- R; falling_factorial; (T1 x, T2 y); -->
\index{{\tt \bfseries falling\_factorial }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`falling_factorial`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `falling_factorial` function
`r since("2.25")`

<!-- real; lchoose; (real x, real y); -->
\index{{\tt \bfseries lchoose }!{\tt (real x, real y): real}|hyperpage}

`real` **`lchoose`**`(real x, real y)`<br>\newline
Return the natural logarithm of the generalized binomial coefficient
of x and y. For non-negative integer inputs, the binomial coefficient
function is written as $\binom{x}{y}$ and pronounced "x choose y."
This function generalizes to real numbers using the gamma function.
For $0 \leq y \leq x$, \[ \mathrm{binomial\_coefficient\_log}(x,y) =
\log\Gamma(x+1) - \log\Gamma(y+1) - \log\Gamma(x-y+1). \]
`r since("2.10")`

<!-- R; lchoose; (T1 x, T2 y); -->
\index{{\tt \bfseries lchoose }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`lchoose`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `lchoose` function
`r since("2.29")`

<!-- real; log_falling_factorial; (real x, real n); -->
\index{{\tt \bfseries log\_falling\_factorial }!{\tt (real x, real n): real}|hyperpage}

`real` **`log_falling_factorial`**`(real x, real n)`<br>\newline
Return the log of the falling factorial of x with power n defined for
positive x and real n. \[ \mathrm{log\_falling\_factorial}(x,n) =
\begin{cases} \log (x)_n & \text{if } x > 0 \\ \textrm{error} &
\text{if } x \leq 0 \end{cases} \]
`r since("2.0")`

<!-- real; rising_factorial; (real x, int n); -->
\index{{\tt \bfseries rising\_factorial }!{\tt (real x, int n): real}|hyperpage}

`real` **`rising_factorial`**`(real x, int n)`<br>\newline
Return the rising factorial of x with power n defined for positive x
and integer n. \[ \mathrm{rising\_factorial}(x,n) = \begin{cases} x^{(n)}
& \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}
\] where \[ x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)} \]
`r since("2.20")`

<!-- R; rising_factorial; (T1 x, T2 y); -->
\index{{\tt \bfseries rising\_factorial }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`rising_factorial`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `rising_factorial` function
`r since("2.25")`

<!-- real; log_rising_factorial; (real x, real n); -->
\index{{\tt \bfseries log\_rising\_factorial }!{\tt (real x, real n): real}|hyperpage}

`real` **`log_rising_factorial`**`(real x, real n)`<br>\newline
Return the log of the rising factorial of x with power n defined for
positive x and real n. \[ \mathrm{log\_rising\_factorial}(x,n) =
\begin{cases} \log x^{(n)} & \text{if } x > 0 \\ \textrm{error} &
\text{if } x \leq 0 \end{cases} \]
`r since("2.0")`

<!-- R; log_rising_factorial; (T1 x, T2 y); -->
\index{{\tt \bfseries log\_rising\_factorial }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`log_rising_factorial`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `log_rising_factorial` function
`r since("2.25")`

## Composed functions {#composed-functions}

The functions in this section are equivalent in theory to combinations
of other functions.  In practice, they are implemented to be more
efficient and more numerically stable than defining them directly
using more basic Stan functions.

<!-- R; expm1; (T x); -->
\index{{\tt \bfseries expm1 }!{\tt (T x): R}|hyperpage}

`R` **`expm1`**`(T x)`<br>\newline
natural exponential of x minus 1
`r since("2.0, vectorized in 2.13")`

<!-- real; fma; (real x, real y, real z); -->
\index{{\tt \bfseries fma }!{\tt (real x, real y, real z): real}|hyperpage}

`real` **`fma`**`(real x, real y, real z)`<br>\newline
Return z plus the result of x multiplied by y. \[ \text{fma}(x,y,z) =
(x \times y) + z \]
`r since("2.0")`

<!-- real; multiply_log; (real x, real y); -->
\index{{\tt \bfseries multiply\_log }!{\tt (real x, real y): real}|hyperpage}

`real` **`multiply_log`**`(real x, real y)`<br>\newline
_**Warning:**_ This function is deprecated and should be replaced with
`lmultiply`. Return the product of x and the natural logarithm of y.
\[ \mathrm{multiply\_log}(x,y) = \begin{cases} 0 & \text{if } x = y =
0 \\ x \log y & \text{if } x, y \neq 0 \\ \text{NaN} &
\text{otherwise} \end{cases} \]
`r since("2.0, deprecated since 2.10, scheduled for removal in 2.32")`

<!-- R; multiply_log; (T1 x, T2 y); -->
\index{{\tt \bfseries multiply\_log }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`multiply_log`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `multiply_log` function
`r since("2.25")`

<!-- real; ldexp; (real x, int y); -->
\index{{\tt \bfseries ldexp }!{\tt (real x, int y): real}|hyperpage}

`real` **`ldexp`**`(real x, int y)`<br>\newline
Return the product of x and two raised to the y power. \[
\text{ldexp}(x,y) = x 2^y  \]
`r since("2.25")`

<!-- R; ldexp; (T1 x, T2 y); -->
\index{{\tt \bfseries ldexp }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`ldexp`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `ldexp` function
`r since("2.25")`

<!-- real; lmultiply; (real x, real y); -->
\index{{\tt \bfseries lmultiply }!{\tt (real x, real y): real}|hyperpage}

`real` **`lmultiply`**`(real x, real y)`<br>\newline
Return the product of x and the natural logarithm of y. \[
\text{lmultiply}(x,y) = \begin{cases} 0 & \text{if } x = y = 0 \\ x
\log y & \text{if } x, y \neq 0 \\ \text{NaN} & \text{otherwise}
\end{cases} \]
`r since("2.10")`

<!-- R; lmultiply; (T1 x, T2 y); -->
\index{{\tt \bfseries lmultiply }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`lmultiply`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `lmultiply` function
`r since("2.25")`

<!-- R; log1p; (T x); -->
\index{{\tt \bfseries log1p }!{\tt (T x): R}|hyperpage}

`R` **`log1p`**`(T x)`<br>\newline
natural logarithm of 1 plus x
`r since("2.0, vectorized in 2.13")`

<!-- R; log1m; (T x); -->
\index{{\tt \bfseries log1m }!{\tt (T x): R}|hyperpage}

`R` **`log1m`**`(T x)`<br>\newline
natural logarithm of 1 minus x
`r since("2.0, vectorized in 2.13")`

<!-- R; log1p_exp; (T x); -->
\index{{\tt \bfseries log1p\_exp }!{\tt (T x): R}|hyperpage}

`R` **`log1p_exp`**`(T x)`<br>\newline
natural logarithm of one plus the natural exponentiation of x
`r since("2.0, vectorized in 2.13")`

<!-- R; log1m_exp; (T x); -->
\index{{\tt \bfseries log1m\_exp }!{\tt (T x): R}|hyperpage}

`R` **`log1m_exp`**`(T x)`<br>\newline
logarithm of one minus the natural exponentiation of x
`r since("2.0, vectorized in 2.13")`

<!-- real; log_diff_exp; (real x, real y); -->
\index{{\tt \bfseries log\_diff\_exp }!{\tt (real x, real y): real}|hyperpage}

`real` **`log_diff_exp`**`(real x, real y)`<br>\newline
Return the natural logarithm of the difference of the natural
exponentiation of x and the natural exponentiation of y. \[
\mathrm{log\_diff\_exp}(x,y) = \begin{cases} \log(\exp(x)-\exp(y)) &
\text{if } x > y \\[6pt] \textrm{NaN} & \text{otherwise} \end{cases}
\]
`r since("2.0")`

<!-- R; log_diff_exp; (T1 x, T2 y); -->
\index{{\tt \bfseries log\_diff\_exp }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`log_diff_exp`**`(T1 x, T2 y)`<br>\newline
Vectorized implementation of the `log_diff_exp` function
`r since("2.25")`

<!-- real; log_mix; (real theta, real lp1, real lp2); -->
\index{{\tt \bfseries log\_mix }!{\tt (real theta, real lp1, real lp2): real}|hyperpage}

`real` **`log_mix`**`(real theta, real lp1, real lp2)`<br>\newline
Return the log mixture of the log densities lp1 and lp2 with mixing
proportion theta, defined by \begin{eqnarray*}
\mathrm{log\_mix}(\theta, \lambda_1, \lambda_2) & = & \log \!\left(
\theta \exp(\lambda_1) + \left( 1 - \theta \right) \exp(\lambda_2)
\right) \\[3pt] & = & \mathrm{log\_sum\_exp}\!\left(\log(\theta) +
\lambda_1, \ \log(1 - \theta) + \lambda_2\right). \end{eqnarray*}
`r since("2.6")`

<!-- real; log_sum_exp; (real x, real y); -->
\index{{\tt \bfseries log\_sum\_exp }!{\tt (real x, real y): real}|hyperpage}

`real` **`log_sum_exp`**`(real x, real y)`<br>\newline
Return the natural logarithm of the sum of the natural exponentiation
of x and the natural exponentiation of y. \[
\mathrm{log\_sum\_exp}(x,y) = \log(\exp(x)+\exp(y)) \]
`r since("2.0")`

<!-- R; log_inv_logit; (T x); -->
\index{{\tt \bfseries log\_inv\_logit }!{\tt (T x): R}|hyperpage}

`R` **`log_inv_logit`**`(T x)`<br>\newline
natural logarithm of the inverse logit function of x
`r since("2.0, vectorized in 2.13")`

<!-- R; log_inv_logit_diff; (T1 x, T2 y); -->
\index{{\tt \bfseries log\_inv\_logit\_diff }!{\tt (T1 x, T2 y): R}|hyperpage}

`R` **`log_inv_logit_diff`**`(T1 x, T2 y)`<br>\newline
natural logarithm of the difference of the inverse logit function of x and the inverse logit function of y
`r since("2.25")`

<!-- R; log1m_inv_logit; (T x); -->
\index{{\tt \bfseries log1m\_inv\_logit }!{\tt (T x): R}|hyperpage}

`R` **`log1m_inv_logit`**`(T x)`<br>\newline
natural logarithm of 1 minus the inverse logit function of x
`r since("2.0, vectorized in 2.13")`

## Special functions {#special-functions}

<!-- R; lambert_w0; (reals x); -->
\index{{\tt \bfseries lambert\_w0 }!{\tt (T x): R}|hyperpage}

`R` **`lambert_w0`**`(T x)`<br>\newline

Implementation of the $W_0$ branch of the Lambert W function, i.e., solution to the function $W_0(x) \exp^{ W_0(x)} = x$
`r since("2.25")`

<!-- R; lambert_wm1; (T x); -->
\index{{\tt \bfseries lambert\_wm1 }!{\tt (T x): R}|hyperpage}

`R` **`lambert_wm1`**`(T x)`<br>\newline
Implementation of the $W_{-1}$ branch of the Lambert W function, i.e., solution to the function $W_{-1}(x) \exp^{W_{-1}(x)} = x$
`r since("2.25")`