\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

base_std_math

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Base Type Requirements for Standard Math Functions

Purpose

These definitions are required for the user’s code to use the type AD < Base > :

Unary Standard Math

The type Base must support the following functions unary standard math functions (in the CppAD namespace):

Syntax	Result
y = abs ( x )	absolute value
y = acos ( x )	inverse cosine
y = acosh ( x )	inverse hyperbolic cosine
y = asin ( x )	inverse sine
y = asinh ( x )	inverse hyperbolic sin
y = atan ( x )	inverse tangent
y = atanh ( x )	inverse hyperbolic tangent
y = cos ( x )	cosine
y = cosh ( x )	hyperbolic cosine
y = erf ( x )	error function
y = erfc ( x )	complementary error function
y = exp ( x )	exponential
y = expm1 ( x )	exponential of x minus one
y = fabs ( x )	absolute value
y = log ( x )	natural logarithm
y = log1p ( x )	logarithm of one plus x
y = sin ( x )	sine
y = sinh ( x )	hyperbolic sine
y = sqrt ( x )	square root
y = tan ( x )	tangent

where the arguments and return value have the prototypes

      const Base & x

      Base y

For example, base_alloc ,

CPPAD_STANDARD_MATH_UNARY

The macro invocation, within the CppAD namespace,

CPPAD_STANDARD_MATH_UNARY ( Base , Fun )

defines the syntax

y = CppAD:: Fun ( x )

This macro uses the functions std:: Fun which must be defined and have the same prototype as CppAD:: Fun . For example, float .

sign

The type Base must support the syntax

y = CppAD::sign ( x )

which computes

\[\begin{split}y = \left\{ \begin{array}{ll} +1 & {\rm if} \; x > 0 \\ 0 & {\rm if} \; x = 0 \\ -1 & {\rm if} \; x < 0 \end{array} \right.\end{split}\]

where x and y have the same prototype as above. For example, see base_alloc . Note that, if ordered comparisons are not defined for the type Base , the code sign function should generate an assert if it is used; see complex invalid unary math .

pow

The type Base must support the syntax

z = CppAD::pow ( x , y )

which computes \(z = x^y\). The arguments x and y have prototypes

      const Base & x

      const Base & y

and the return value z has prototype

Base z

For example, see base_alloc .

isnan

If Base defines the isnan function, you may also have to provide a definition in the CppAD namespace (to avoid a function ambiguity). For example, see base_complex .