This brief presents the results of a study of the possibilities of reducing the arithmetic complexity of computing basic operations in octonionic neural networks and also proposes new algorithmic solutions for efficiently performing these operations. Here, we primarily mean the operation of multiplying octonions, the operation of computing the dot product of two octonion-valued vectors, and the operation of multiple multiplications of an octonion by several other octonions. In order to reduce the computational complexity of these operations, it is proposed to use the fast Walsh-Hadamard transform, which is well known in digital signal processing. Using this transform reduces the number of multiplications and additions of real numbers required to perform computations. Thus, the use of the proposed algorithms will speed up computations in octonion-valued neural networks.