Inverse Tangent and Hyperbolic Tangent Reverse Mode Theory
Keywords: inverse tangent hyperbolic reverse mode theory

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Inverse Tangent and Hyperbolic Tangent Reverse Mode Theory
We use the reverse theory standard math function definition for the functions H and G . In addition, we use the forward mode notation in atan_forward for B ( t ) = 1 ± X ( t ) * X ( t ) We use b for the `p`-th order Taylor coefficient row vectors corresponding to B ( t ) and replace z ( j ) by ( z ( j ) , b ( j ) ) in the definition for G and H . The zero order forward mode formulas for the atan function are z ( 0 ) = F ( x ( 0 ) ) b ( 0 ) = 1 ± x ( 0 ) x ( 0 ) where F ( x ) = atan ( x ) for + and F ( x ) = atanh ( x ) for - . For orders j greater than zero we have b ( j ) = ± ∑ k = 0 j x ( k ) x ( j - k ) z ( j ) = 1 j 1 b ( 0 ) (j x ( j ) - ∑ k = 1 j -1 k z ( k ) b ( j - k ) ) If j = 0 , we note that F ( 1 ) ( x ( 0 ) ) = 1 / b ( 0 ) and hence ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( 0 ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ x ( 0 ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) 1 b ( 0 ) ± ∂ G ∂ b ( j ) 2 x ( 0 ) If j > 0 , then for k = 1 , … , j -1 ∂ H ∂ b ( 0 ) = ∂ G ∂ b ( 0 ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ b ( 0 ) = ∂ G ∂ b ( 0 ) - ∂ G ∂ z ( j ) z ( j ) b ( 0 ) ∂ H ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( j ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ x ( j ) = ∂ G ∂ x ( j ) + ∂ G ∂ z ( j ) 1 b ( 0 ) ± ∂ G ∂ b ( j ) 2 x ( 0 ) ∂ H ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( 0 ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) ± ∂ G ∂ b ( j ) 2 x ( j ) ∂ H ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( k ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ x ( k ) = ∂ G ∂ x ( k ) ± ∂ G ∂ b ( j ) 2 x ( j - k ) ∂ H ∂ z ( k ) = ∂ G ∂ z ( k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ z ( k ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ z ( k ) = ∂ G ∂ z ( k ) - ∂ G ∂ z ( j ) k b ( j - k ) j b ( 0 ) ∂ H ∂ b ( j - k ) = ∂ G ∂ b ( j - k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ b ( j - k ) + ∂ G ∂ b ( j ) ∂ b ( j ) ∂ b ( j - k ) = ∂ G ∂ b ( j - k ) - ∂ G ∂ z ( j ) k z ( k ) j b ( 0 )

Input File: omh/theory/atan_reverse.omh