0.05/0.18 WORST_CASE(?,O(n^2)) 0.05/0.18 0.05/0.18 Preprocessing Cost Relations 0.05/0.18 ===================================== 0.05/0.18 0.05/0.18 #### Computed strongly connected components 0.05/0.18 0. recursive : [eval_ax_bb2_in/4,eval_ax_bb3_in/4] 0.05/0.18 1. recursive : [eval_ax_bb1_in/3,eval_ax_bb2_in_loop_cont/5,eval_ax_bb4_in/4] 0.05/0.18 2. non_recursive : [eval_ax_stop/1] 0.05/0.18 3. non_recursive : [eval_ax_bb5_in/1] 0.05/0.18 4. non_recursive : [eval_ax_bb1_in_loop_cont/2] 0.05/0.18 5. non_recursive : [eval_ax_bb0_in/2] 0.05/0.18 6. non_recursive : [eval_ax_start/4] 0.05/0.18 0.05/0.18 #### Obtained direct recursion through partial evaluation 0.05/0.18 0. SCC is partially evaluated into eval_ax_bb2_in/4 0.05/0.18 1. SCC is partially evaluated into eval_ax_bb1_in/3 0.05/0.18 2. SCC is completely evaluated into other SCCs 0.05/0.18 3. SCC is completely evaluated into other SCCs 0.05/0.18 4. SCC is completely evaluated into other SCCs 0.05/0.18 5. SCC is partially evaluated into eval_ax_bb0_in/2 0.05/0.18 6. SCC is partially evaluated into eval_ax_start/4 0.05/0.18 0.05/0.18 Control-Flow Refinement of Cost Relations 0.05/0.18 ===================================== 0.05/0.18 0.05/0.18 ### Specialization of cost equations eval_ax_bb2_in/4 0.05/0.18 * CE 7 is refined into CE [8] 0.05/0.18 * CE 6 is refined into CE [9] 0.05/0.18 0.05/0.18 0.05/0.18 ### Cost equations --> "Loop" of eval_ax_bb2_in/4 0.05/0.18 * CEs [9] --> Loop 8 0.05/0.18 * CEs [8] --> Loop 9 0.05/0.18 0.05/0.18 ### Ranking functions of CR eval_ax_bb2_in(V_n,V__01,B,C) 0.05/0.18 * RF of phase [8]: [V_n-V__01-1] 0.05/0.18 0.05/0.18 #### Partial ranking functions of CR eval_ax_bb2_in(V_n,V__01,B,C) 0.05/0.18 * Partial RF of phase [8]: 0.05/0.18 - RF of loop [8:1]: 0.05/0.18 V_n-V__01-1 0.05/0.18 0.05/0.18 0.05/0.18 ### Specialization of cost equations eval_ax_bb1_in/3 0.05/0.18 * CE 5 is refined into CE [10] 0.05/0.18 * CE 3 is refined into CE [11,12] 0.05/0.18 * CE 4 is discarded (unfeasible) 0.05/0.18 0.05/0.18 0.05/0.18 ### Cost equations --> "Loop" of eval_ax_bb1_in/3 0.05/0.18 * CEs [11] --> Loop 10 0.05/0.18 * CEs [12] --> Loop 11 0.05/0.18 * CEs [10] --> Loop 12 0.05/0.18 0.05/0.18 ### Ranking functions of CR eval_ax_bb1_in(V_n,V__0,B) 0.05/0.18 * RF of phase [12]: [V_n-V__0-2] 0.05/0.18 0.05/0.18 #### Partial ranking functions of CR eval_ax_bb1_in(V_n,V__0,B) 0.05/0.18 * Partial RF of phase [12]: 0.05/0.18 - RF of loop [12:1]: 0.05/0.18 V_n-V__0-2 0.05/0.18 0.05/0.18 0.05/0.18 ### Specialization of cost equations eval_ax_bb0_in/2 0.05/0.18 * CE 2 is refined into CE [13,14,15] 0.05/0.18 0.05/0.18 0.05/0.18 ### Cost equations --> "Loop" of eval_ax_bb0_in/2 0.05/0.18 * CEs [15] --> Loop 13 0.05/0.18 * CEs [13] --> Loop 14 0.05/0.18 * CEs [14] --> Loop 15 0.05/0.18 0.05/0.18 ### Ranking functions of CR eval_ax_bb0_in(V_n,B) 0.05/0.18 0.05/0.18 #### Partial ranking functions of CR eval_ax_bb0_in(V_n,B) 0.05/0.18 0.05/0.18 0.05/0.18 ### Specialization of cost equations eval_ax_start/4 0.05/0.18 * CE 1 is refined into CE [16,17,18] 0.05/0.18 0.05/0.18 0.05/0.18 ### Cost equations --> "Loop" of eval_ax_start/4 0.05/0.18 * CEs [18] --> Loop 16 0.05/0.18 * CEs [17] --> Loop 17 0.05/0.18 * CEs [16] --> Loop 18 0.05/0.18 0.05/0.18 ### Ranking functions of CR eval_ax_start(V_i,V_j,V_n,B) 0.05/0.18 0.05/0.18 #### Partial ranking functions of CR eval_ax_start(V_i,V_j,V_n,B) 0.05/0.18 0.05/0.18 0.05/0.18 Computing Bounds 0.05/0.18 ===================================== 0.05/0.18 0.05/0.18 #### Cost of chains of eval_ax_bb2_in(V_n,V__01,B,C): 0.05/0.18 * Chain [[8],9]: 1*it(8)+0 0.05/0.18 Such that:it(8) =< -V__01+C 0.05/0.18 0.05/0.18 with precondition: [B=2,V_n=C+1,V__01>=0,V_n>=V__01+2] 0.05/0.18 0.05/0.18 * Chain [9]: 0 0.05/0.18 with precondition: [B=2,V__01=C,V__01>=0,V__01+1>=V_n] 0.05/0.18 0.05/0.18 0.05/0.18 #### Cost of chains of eval_ax_bb1_in(V_n,V__0,B): 0.05/0.18 * Chain [[12],10]: 1*it(12)+1*s(1)+1*s(4)+0 0.05/0.18 Such that:it(12) =< V_n-V__0 0.05/0.18 aux(2) =< V_n 0.05/0.18 s(1) =< aux(2) 0.05/0.18 s(4) =< it(12)*aux(2) 0.05/0.18 0.05/0.18 with precondition: [B=3,V__0>=0,V_n>=V__0+3] 0.05/0.18 0.05/0.18 * Chain [11]: 0 0.05/0.18 with precondition: [V__0=0,B=3,1>=V_n] 0.05/0.18 0.05/0.18 * Chain [10]: 1*s(1)+0 0.05/0.18 Such that:s(1) =< V_n 0.05/0.18 0.05/0.18 with precondition: [B=3,V_n>=2,V__0+2>=V_n] 0.05/0.18 0.05/0.18 0.05/0.18 #### Cost of chains of eval_ax_bb0_in(V_n,B): 0.05/0.18 * Chain [15]: 1*s(5)+0 0.05/0.18 Such that:s(5) =< 2 0.05/0.18 0.05/0.18 with precondition: [V_n=2] 0.05/0.18 0.05/0.18 * Chain [14]: 0 0.05/0.18 with precondition: [1>=V_n] 0.05/0.18 0.05/0.18 * Chain [13]: 2*s(6)+1*s(9)+0 0.05/0.18 Such that:aux(3) =< V_n 0.05/0.18 s(6) =< aux(3) 0.05/0.18 s(9) =< s(6)*aux(3) 0.05/0.18 0.05/0.18 with precondition: [V_n>=3] 0.05/0.18 0.05/0.18 0.05/0.18 #### Cost of chains of eval_ax_start(V_i,V_j,V_n,B): 0.05/0.18 * Chain [18]: 1*s(10)+0 0.05/0.18 Such that:s(10) =< 2 0.05/0.18 0.05/0.18 with precondition: [V_n=2] 0.05/0.18 0.05/0.18 * Chain [17]: 0 0.05/0.18 with precondition: [1>=V_n] 0.05/0.18 0.05/0.18 * Chain [16]: 2*s(12)+1*s(13)+0 0.05/0.18 Such that:s(11) =< V_n 0.05/0.18 s(12) =< s(11) 0.05/0.18 s(13) =< s(12)*s(11) 0.05/0.18 0.05/0.18 with precondition: [V_n>=3] 0.05/0.18 0.05/0.18 0.05/0.18 Closed-form bounds of eval_ax_start(V_i,V_j,V_n,B): 0.05/0.18 ------------------------------------- 0.05/0.18 * Chain [18] with precondition: [V_n=2] 0.05/0.18 - Upper bound: 2 0.05/0.18 - Complexity: constant 0.05/0.18 * Chain [17] with precondition: [1>=V_n] 0.05/0.18 - Upper bound: 0 0.05/0.18 - Complexity: constant 0.05/0.18 * Chain [16] with precondition: [V_n>=3] 0.05/0.18 - Upper bound: 2*V_n+V_n*V_n 0.05/0.18 - Complexity: n^2 0.05/0.18 0.05/0.18 ### Maximum cost of eval_ax_start(V_i,V_j,V_n,B): max([2,nat(V_n)*nat(V_n)+nat(V_n)*2]) 0.05/0.18 Asymptotic class: n^2 0.05/0.18 * Total analysis performed in 107 ms. 0.05/0.18 0.05/0.28 EOF