The anomalies of supercooled water in thermodynamic response functions at atmospheric pressure, the phase transition between low and high density amorphous ices (LDA and HDA), and a predicted fragile-strong transition are accounted for in a unified manner by reconciling an idea due to Stanley and co-workers introducing a second critical point separating LDA and HDA ices with a conjecture proposed by Speedy that LDA is a different phase from a normal water, called water II. The reconciliation is made on the basis of results from extensive molecular dynamics simulations at constant pressure and temperature. It is found that there exist large gaps around temperature 213 K in thermodynamic, structural, and dynamic propel-ties at atmospheric pressure, suggesting liquid-liquid phase transition. This transition is identified with an extension of the experimentally observed LDA-HDA transition in high pressure to atmospheric pressure. Thus, we propose a new phase diagram where the locus oi. the second critical point is moved into negative pressure region. With this simple modification, it becomes possible to account for the divergence of the thermodynamic response functions at atmospheric pressure in terms of the critical point and the spinodal-like instability of HDA. The unstable HDA undergoes a transition to LDA phase in lower temperature. The transition is also observed in high pressure region such as 200 MPa while it disappears at negative pressure, -200 MPa. This reinforces our proposed phase diagram in which there is no continuous path from a supercooled state to LDA at atmospheric pressure. It is argued that the HDA-LDA transition is accompanied by a fragile-strong transition A possible mechanism of avoiding crystallization of aqueous solutions is also discussed in terms of a difference in hydrogen bond number distribution between LDA and HDA.