Inverse Tangent and Hyperbolic Tangent Forward Mode Theory
Keywords: inverse tangent hyperbolic forward mode theory atan atanh derivatives taylor coefficients recursion

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Inverse Tangent and Hyperbolic Tangent Forward Mode Theory
Derivatives atan ( 1 ) ( x ) = 1 / ( 1 + x * x ) atanh ( 1 ) ( x ) = 1 / ( 1 - x * x ) If F ( x ) is atan ( x ) or atanh ( x ) , the corresponding derivative satisfies the equation ( 1 ± x * x ) * F ( 1 ) ( x ) - 0 * F ( x ) = 1 and in the standard math function differential equation , A ( x ) = 0 , B ( x ) = 1 ± x * x , and D ( x ) = 1 . We use a , b , d and z to denote the Taylor coefficients for A [ X ( t ) ] , B [ X ( t ) ] , D [ X ( t ) ] , and F [ X ( t ) ] respectively.
Taylor Coefficients Recursion
For j = 0 , 1 , … , z ( 0 ) = F ( x ( 0 ) ) b ( j ) = { 1 ± x ( 0 ) * x ( 0 ) if j = 0 ± ∑ k = 0 j x ( k ) x ( j - k ) otherwise e ( j ) = d ( j ) + ∑ k = 0 j a ( j - k ) * z ( k ) = { 1 if j = 0 0 otherwise z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (∑ k = 0 j e ( k ) ( j + 1 - k ) x ( j + 1 - k ) - ∑ k = 1 j b ( k ) ( j + 1 - k ) z ( j + 1 - k ) ) z ( j + 1 ) = 1 j + 1 1 b ( 0 ) (( j + 1 ) x ( j + 1 ) - ∑ k = 1 j k z ( k ) b ( j + 1 - k ) )

Input File: omh/theory/atan_forward.omh