Before revision:  Explicit condition number expressions and perturbation bound for the TLS problem can be recovered from our estimates. With TLSE problems solved by the classical QR-SVD or randomized algorithms , numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, the estimates based on mixed and componentwise condition numbers are much tighter.
After revision:  Explicit condition number expressions and perturbation bound for the TLS problem . For TLSE problems with equilibratory input data , numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, the estimates based on mixed and componentwise condition numbers are much tighter.