Tangent and Hyperbolic Tangent Reverse Mode Theory
Keywords: tangent hyperbolic reverse mode theory tan notation eliminating y(t) positive orders z(t) order zero

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Tangent and Hyperbolic Tangent Reverse Mode Theory
Notation
We use the reverse theory standard math function definition for the functions H and G . In addition, we use the forward mode notation in tan_forward for X ( t ) , Y ( t ) and Z ( t ) .
Eliminating Y(t)
For j > 0 , the forward mode coefficients are given by y ( j -1 ) = ∑ k = 0 j -1 z ( k ) z ( j - k -1 ) Fix j > 0 and suppose that H is the same as G except that y ( j -1 ) is replaced as a function of the Taylor coefficients for Z ( t ) . To be specific, for k = 0 , … , j -1 , ∂ H ∂ z ( k ) = ∂ G ∂ z ( k ) + ∂ G ∂ y ( j -1 ) ∂ y ( j -1 ) ∂ z ( k ) = ∂ G ∂ z ( k ) + ∂ G ∂ y ( j -1 ) 2 z ( j - k -1 )
Positive Orders Z(t)
For order j > 0 , suppose that H is the same as G except that z ( j ) is expressed as a function of the coefficients for X ( t ) , and the lower order Taylor coefficients for Y ( t ) , Z ( t ) . z ( j ) = x ( j ) ± 1 j ∑ k = 1 j k x ( k ) y ( j - k ) For k = 1 , … , j , the partial of H with respect to x ( k ) is given by ∂ H ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ x ( k ) = ∂ G ∂ x ( k ) + ∂ G ∂ z ( j ) [δ ( j - k ) ± k j y ( j - k ) ] where δ ( j - k ) is one if j = k and zero otherwise. For k = 1 , … , j The partial of H with respect to y j - k , is given by ∂ H ∂ y ( j - k ) = ∂ G ∂ y ( j - k ) + ∂ G ∂ z ( j ) ∂ z ( j ) ∂ y ( j - k ) = ∂ G ∂ y ( j - k ) ± ∂ G ∂ z ( j ) k j x k
Order Zero Z(t)
The order zero coefficients for the tangent and hyperbolic tangent are z ( 0 ) = { tan ( x ( 0 ) ) tanh ( x ( 0 ) ) Suppose that H is the same as G except that z ( 0 ) is expressed as a function of the Taylor coefficients for X ( t ) . In this case, ∂ H ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) + ∂ G ∂ z ( 0 ) ∂ z ( 0 ) ∂ x ( 0 ) = ∂ G ∂ x ( 0 ) + ∂ G ∂ z ( 0 ) ( 1 ± y ( 0 ) )

Input File: omh/theory/tan_reverse.omh