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Up-> CppAD AD ADValued Arithmetic ad_binary

AD-> Default ad_copy Convert ADValued BoolValued VecAD base_require

ADValued-> Arithmetic std_math_ad MathOther CondExp Discrete user_atomic

Arithmetic-> UnaryPlus UnaryMinus ad_binary compute_assign

ad_binary-> Add.cpp Sub.cpp Mul.cpp Div.cpp

Headings-> Syntax Purpose Op Base x y z Operation Sequence Example Derivative ---..Addition ---..Subtraction ---..Multiplication ---..Division

AD Binary Arithmetic Operators
Syntax
z = x Op y

Purpose
Performs arithmetic operations where either x or y has type AD<Base> or VecAD<Base>::reference .

Op
The operator Op is one of the following

Op	Meaning
+	z is x plus y
-	z is x minus y
*	z is x times y
/	z is x divided by y

Base
The type Base is determined by the operand that has type AD<Base> or VecAD<Base>::reference.

x
The operand x has the following prototype
     const Type &x
where Type is VecAD<Base>::reference, AD<Base>, Base, or double.

y
The operand y has the following prototype
     const Type &y
where Type is VecAD<Base>::reference, AD<Base>, Base, or double.

z
The result z has the following prototype
     Type z
where Type is AD<Base>.

Operation Sequence
This is an atomic AD of Base operation and hence it is part of the current AD of Base operation sequence .

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

Add.cpp	AD Binary Addition: Example and Test
Sub.cpp	AD Binary Subtraction: Example and Test
Mul.cpp	AD Binary Multiplication: Example and Test
Div.cpp	AD Binary Division: Example and Test

Derivative
If $\mathit{f}$ and $\mathit{g}$ are Base functions

Addition $$\frac{\partial [\mathit{f}(\mathit{x})+\mathit{g}(\mathit{x})]}{\partial \mathit{x}}=\frac{\partial \mathit{f}(\mathit{x})}{\partial \mathit{x}}+\frac{\partial \mathit{g}(\mathit{x})}{\partial \mathit{x}}$$
Subtraction $$\frac{\partial [\mathit{f}(\mathit{x})-\mathit{g}(\mathit{x})]}{\partial \mathit{x}}=\frac{\partial \mathit{f}(\mathit{x})}{\partial \mathit{x}}-\frac{\partial \mathit{g}(\mathit{x})}{\partial \mathit{x}}$$
Multiplication $$\frac{\partial [\mathit{f}(\mathit{x})*\mathit{g}(\mathit{x})]}{\partial \mathit{x}}=\mathit{g}(\mathit{x})*\frac{\partial \mathit{f}(\mathit{x})}{\partial \mathit{x}}+\mathit{f}(\mathit{x})*\frac{\partial \mathit{g}(\mathit{x})}{\partial \mathit{x}}$$
Division $$\frac{\partial [\mathit{f}(\mathit{x})/\mathit{g}(\mathit{x})]}{\partial \mathit{x}}=[1/\mathit{g}(\mathit{x})]*\frac{\partial \mathit{f}(\mathit{x})}{\partial \mathit{x}}-[\mathit{f}(\mathit{x})/\mathit{g}(\mathit{x}{)}^2]*\frac{\partial \mathit{g}(\mathit{x})}{\partial \mathit{x}}$$

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Input File: cppad/local/ad_binary.hpp