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Up-> CppAD AD ADValued std_math_ad

CppAD-> Install Introduction AD ADFun multi_thread library cppad_ipopt_nlp Example preprocessor Appendix

AD-> Default ad_copy Convert ADValued BoolValued VecAD base_require

ADValued-> Arithmetic std_math_ad MathOther CondExp Discrete user_atomic

std_math_ad-> Acos.cpp Asin.cpp Atan.cpp Cos.cpp Cosh.cpp Exp.cpp Log.cpp Log10.cpp Sin.cpp Sinh.cpp Sqrt.cpp Tan.cpp Tanh.cpp

Headings-> Syntax Purpose x y Operation Sequence fun Examples Derivatives ---..acos ---..asin ---..atan ---..cos ---..cosh ---..exp ---..log ---..log10 ---..sin ---..sinh ---..sqrt ---..tan ---..tanh

AD Standard Math Unary Functions
Syntax
y = fun(x)

Purpose
Evaluates the one argument standard math function fun where its argument is an AD of Base object.

x
The argument x has one of the following prototypes
     const AD<Base>               &x
     const VecAD<Base>::reference &x

y
The result y has prototype
     AD<Base> y

Operation Sequence
Most of these functions are AD of Base atomic operations . In all cases, The AD of Base operation sequence used to calculate y is independent of x.

fun
A definition of fun is included for each of the following functions: acos, asin, atan, cos, cosh, exp, log, log10, sin, sinh, sqrt, tan, tanh.

Examples
The following files contain examples and tests of these functions. Each test returns true if it succeeds and false otherwise.

Acos.cpp	The AD acos Function: Example and Test
Asin.cpp	The AD asin Function: Example and Test
Atan.cpp	The AD atan Function: Example and Test
Cos.cpp	The AD cos Function: Example and Test
Cosh.cpp	The AD cosh Function: Example and Test
Exp.cpp	The AD exp Function: Example and Test
Log.cpp	The AD log Function: Example and Test
Log10.cpp	The AD log10 Function: Example and Test
Sin.cpp	The AD sin Function: Example and Test
Sinh.cpp	The AD sinh Function: Example and Test
Sqrt.cpp	The AD sqrt Function: Example and Test
Tan.cpp	The AD tan Function: Example and Test
Tanh.cpp	The AD tanh Function: Example and Test

Derivatives
Each of these functions satisfy a standard math function differential equation. Calculating derivatives using this differential equation is discussed for both forward and reverse mode. The exact form of the differential equation for each of these functions is listed below:

acos $$\begin{array}{ccc}\frac{\partial [\mathrm{acos}(\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill -(1-\mathit{x}*\mathit{x}{)}^{-1/2}\end{array}$$
asin $$\begin{array}{ccc}\frac{\partial [\mathrm{asin}(\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill (1-\mathit{x}*\mathit{x}{)}^{-1/2}\end{array}$$
atan $$\begin{array}{ccc}\frac{\partial [\mathrm{atan}(\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \frac{1}{1+{\mathit{x}}^2}\end{array}$$
cos $$\begin{array}{ccc}\frac{\partial [\cos (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill -\sin (\mathit{x})\\ \frac{\partial [\sin (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \cos (\mathit{x})\end{array}$$
cosh $$\begin{array}{ccc}\frac{\partial [\cosh (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \sinh (\mathit{x})\\ \frac{\partial [\sin (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \cosh (\mathit{x})\end{array}$$
exp $$\begin{array}{ccc}\frac{\partial [\exp (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \exp (\mathit{x})\end{array}$$
log $$\begin{array}{ccc}\frac{\partial [\log (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \frac{1}{\mathit{x}}\end{array}$$
log10
This function is special in that it's derivatives are calculated using the relation $$\begin{array}{ccc}\log 10(\mathit{x})\hfill & \hfill =\hfill & \hfill \log (\mathit{x})/\log (10)\end{array}$$
sin $$\begin{array}{ccc}\frac{\partial [\sin (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \cos (\mathit{x})\\ \frac{\partial [\cos (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill -\sin (\mathit{x})\end{array}$$
sinh $$\begin{array}{ccc}\frac{\partial [\sinh (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \cosh (\mathit{x})\\ \frac{\partial [\cosh (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \sinh (\mathit{x})\end{array}$$
sqrt $$\begin{array}{ccc}\frac{\partial [\mathrm{sqrt}(\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill \frac{1}{2\mathrm{sqrt}(\mathit{x})}\end{array}$$
tan $$\begin{array}{ccc}\frac{\partial [\tan (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill 1+\tan (\mathit{x}{)}^2\end{array}$$
tanh $$\begin{array}{ccc}\frac{\partial [\tanh (\mathit{x})]}{\partial \mathit{x}}\hfill & \hfill =\hfill & \hfill 1-\tanh (\mathit{x}{)}^2\end{array}$$

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Input File: cppad/local/std_math_ad.hpp