e-support vector regression (e-SVR) can be converted into an unconstrained convex and non-smooth quadratic programming problem. It is not solved by the typical algorithm. In order to solve this non-smooth problem, a class of piecewise smooth functions is introduced to approximate the e-insensitive loss function of e-SVR, which generates a e-piecewise smooth support vector regression (e-dPWSSVR) model. The fast Newton-Armijo algorithm is used to solve the e-dPWSSVR. The piecewise functions can get higher and higher approximation accuracy as required with increase of parameter d. The reduced kernel technique is applied to avoid the computational difficulties in nonlinear e-dPWSSVR for massive datasets. Experimental results show that the proposed e-dPWSSVR has the better regression performance and the learning efficiency than other competitive baselines.