  The tempered evolution equation describes the trapped dynamics, widely
appearing in nature, e.g., the motion of living particles in viscous liquid.
This paper proposes the fast predictor-corrector approach for the tempered
fractional ordinary differential equations by digging out the potential 'very'
short memory principle. The algorithms basing on the idea of equidistributing
are detailedly described; their effectiveness and low computation cost, being
linearly increasing with time $t$, are numerically demonstrated.
