This paper is devoted to modifications of a two-step algorithm for reducing a matrix to tridiagonal form. At the first stage the matrix is reduced to a symmetric band matrix. A successive band reduction is then used to reduce the symmetric band matrix to tridiagonal form. Both stages apply two-sided orthogonal transformations to the matrix based on Householder reflectors. The underlying idea for our modifications is to use speculative computations of components of the Householder transformations. Performance comparisons between the Intel® Math Kernel Library (Intel® MKL), PLASMA tridiagonal reduction, and our implementation are provided.