0.04/0.28 WORST_CASE(?,O(n^2)) 0.04/0.28 0.04/0.28 Preprocessing Cost Relations 0.04/0.28 ===================================== 0.04/0.28 0.04/0.28 #### Computed strongly connected components 0.04/0.28 0. recursive : [eval_while2_bb2_in/3,eval_while2_bb3_in/3] 0.04/0.28 1. recursive : [eval_while2_8/8,eval_while2_9/8,eval_while2_bb1_in/8,eval_while2_bb2_in_loop_cont/9,eval_while2_bb4_in/8] 0.04/0.28 2. non_recursive : [eval_while2_stop/5] 0.04/0.28 3. non_recursive : [eval_while2_bb5_in/5] 0.04/0.28 4. non_recursive : [exit_location/1] 0.04/0.28 5. non_recursive : [eval_while2_bb1_in_loop_cont/6] 0.04/0.28 6. non_recursive : [eval_while2_4/5] 0.04/0.28 7. non_recursive : [eval_while2_3/5] 0.04/0.28 8. non_recursive : [eval_while2_2/5] 0.04/0.28 9. non_recursive : [eval_while2_1/5] 0.04/0.28 10. non_recursive : [eval_while2_0/5] 0.04/0.28 11. non_recursive : [eval_while2_bb0_in/5] 0.04/0.28 12. non_recursive : [eval_while2_start/5] 0.04/0.28 0.04/0.28 #### Obtained direct recursion through partial evaluation 0.04/0.28 0. SCC is partially evaluated into eval_while2_bb2_in/3 0.04/0.28 1. SCC is partially evaluated into eval_while2_bb1_in/8 0.04/0.28 2. SCC is completely evaluated into other SCCs 0.04/0.28 3. SCC is completely evaluated into other SCCs 0.04/0.28 4. SCC is completely evaluated into other SCCs 0.04/0.28 5. SCC is partially evaluated into eval_while2_bb1_in_loop_cont/6 0.04/0.28 6. SCC is partially evaluated into eval_while2_4/5 0.04/0.28 7. SCC is completely evaluated into other SCCs 0.04/0.28 8. SCC is completely evaluated into other SCCs 0.04/0.28 9. SCC is completely evaluated into other SCCs 0.04/0.28 10. SCC is completely evaluated into other SCCs 0.04/0.28 11. SCC is completely evaluated into other SCCs 0.04/0.28 12. SCC is partially evaluated into eval_while2_start/5 0.04/0.28 0.04/0.28 Control-Flow Refinement of Cost Relations 0.04/0.28 ===================================== 0.04/0.28 0.04/0.28 ### Specialization of cost equations eval_while2_bb2_in/3 0.04/0.28 * CE 11 is refined into CE [12] 0.04/0.28 * CE 10 is refined into CE [13] 0.04/0.28 * CE 9 is refined into CE [14] 0.04/0.28 0.04/0.28 0.04/0.28 ### Cost equations --> "Loop" of eval_while2_bb2_in/3 0.04/0.28 * CEs [14] --> Loop 12 0.04/0.28 * CEs [12] --> Loop 13 0.04/0.28 * CEs [13] --> Loop 14 0.04/0.28 0.04/0.28 ### Ranking functions of CR eval_while2_bb2_in(V_j_0,B,C) 0.04/0.28 * RF of phase [12]: [V_j_0] 0.04/0.28 0.04/0.28 #### Partial ranking functions of CR eval_while2_bb2_in(V_j_0,B,C) 0.04/0.28 * Partial RF of phase [12]: 0.04/0.28 - RF of loop [12:1]: 0.04/0.28 V_j_0 0.04/0.28 0.04/0.28 0.04/0.28 ### Specialization of cost equations eval_while2_bb1_in/8 0.04/0.28 * CE 5 is refined into CE [15] 0.04/0.28 * CE 3 is refined into CE [16,17] 0.04/0.28 * CE 6 is refined into CE [18] 0.04/0.28 * CE 4 is refined into CE [19] 0.04/0.28 0.04/0.28 0.04/0.28 ### Cost equations --> "Loop" of eval_while2_bb1_in/8 0.04/0.28 * CEs [19] --> Loop 15 0.04/0.28 * CEs [15] --> Loop 16 0.04/0.28 * CEs [18] --> Loop 17 0.04/0.28 * CEs [16,17] --> Loop 18 0.04/0.28 0.04/0.28 ### Ranking functions of CR eval_while2_bb1_in(V_3,V_N,V_i_0,V_j_0,B,C,D,E) 0.04/0.28 * RF of phase [15]: [V_i_0] 0.04/0.28 0.04/0.28 #### Partial ranking functions of CR eval_while2_bb1_in(V_3,V_N,V_i_0,V_j_0,B,C,D,E) 0.04/0.28 * Partial RF of phase [15]: 0.04/0.28 - RF of loop [15:1]: 0.04/0.28 V_i_0 0.04/0.28 0.04/0.28 0.04/0.28 ### Specialization of cost equations eval_while2_bb1_in_loop_cont/6 0.04/0.28 * CE 7 is refined into CE [20] 0.04/0.28 * CE 8 is refined into CE [21] 0.04/0.28 0.04/0.28 0.04/0.28 ### Cost equations --> "Loop" of eval_while2_bb1_in_loop_cont/6 0.04/0.28 * CEs [20] --> Loop 19 0.04/0.28 * CEs [21] --> Loop 20 0.04/0.28 0.04/0.28 ### Ranking functions of CR eval_while2_bb1_in_loop_cont(A,B,C,D,E,F) 0.04/0.28 0.04/0.28 #### Partial ranking functions of CR eval_while2_bb1_in_loop_cont(A,B,C,D,E,F) 0.04/0.28 0.04/0.28 0.04/0.28 ### Specialization of cost equations eval_while2_4/5 0.04/0.28 * CE 2 is refined into CE [22,23,24,25,26] 0.04/0.28 0.04/0.28 0.04/0.28 ### Cost equations --> "Loop" of eval_while2_4/5 0.04/0.28 * CEs [23] --> Loop 21 0.04/0.28 * CEs [22,25] --> Loop 22 0.04/0.28 * CEs [26] --> Loop 23 0.04/0.28 * CEs [24] --> Loop 24 0.04/0.28 0.04/0.28 ### Ranking functions of CR eval_while2_4(V_3,V_N,V_i_0,V_j_0,B) 0.04/0.28 0.04/0.28 #### Partial ranking functions of CR eval_while2_4(V_3,V_N,V_i_0,V_j_0,B) 0.04/0.28 0.04/0.28 0.04/0.28 ### Specialization of cost equations eval_while2_start/5 0.04/0.28 * CE 1 is refined into CE [27,28,29,30] 0.04/0.28 0.04/0.28 0.04/0.28 ### Cost equations --> "Loop" of eval_while2_start/5 0.04/0.28 * CEs [30] --> Loop 25 0.04/0.28 * CEs [29] --> Loop 26 0.04/0.28 * CEs [28] --> Loop 27 0.04/0.28 * CEs [27] --> Loop 28 0.04/0.28 0.04/0.28 ### Ranking functions of CR eval_while2_start(V_3,V_N,V_i_0,V_j_0,B) 0.04/0.28 0.04/0.28 #### Partial ranking functions of CR eval_while2_start(V_3,V_N,V_i_0,V_j_0,B) 0.04/0.28 0.04/0.28 0.04/0.28 Computing Bounds 0.04/0.28 ===================================== 0.04/0.28 0.04/0.28 #### Cost of chains of eval_while2_bb2_in(V_j_0,B,C): 0.04/0.28 * Chain [[12],14]: 1*it(12)+0 0.04/0.28 Such that:it(12) =< V_j_0 0.04/0.28 0.04/0.28 with precondition: [B=2,C=0,V_j_0>=1] 0.04/0.28 0.04/0.28 * Chain [[12],13]: 1*it(12)+0 0.04/0.28 Such that:it(12) =< V_j_0 0.04/0.28 0.04/0.28 with precondition: [B=3,V_j_0>=1] 0.04/0.28 0.04/0.28 * Chain [13]: 0 0.04/0.28 with precondition: [B=3,V_j_0>=0] 0.04/0.28 0.04/0.28 0.04/0.28 #### Cost of chains of eval_while2_bb1_in(V_3,V_N,V_i_0,V_j_0,B,C,D,E): 0.04/0.28 * Chain [[15],18]: 1*it(15)+1*s(1)+1*s(4)+0 0.04/0.28 Such that:it(15) =< V_i_0 0.04/0.28 aux(2) =< V_N 0.04/0.28 s(1) =< aux(2) 0.04/0.28 s(4) =< it(15)*aux(2) 0.04/0.28 0.04/0.28 with precondition: [B=3,V_i_0>=2,V_N>=V_i_0] 0.04/0.28 0.04/0.28 * Chain [[15],17]: 1*it(15)+1*s(4)+0 0.04/0.28 Such that:aux(1) =< V_N 0.04/0.28 it(15) =< V_i_0 0.04/0.28 s(4) =< it(15)*aux(1) 0.04/0.28 0.04/0.28 with precondition: [B=3,V_i_0>=1,V_N>=V_i_0] 0.04/0.28 0.04/0.28 * Chain [[15],16]: 1*it(15)+1*s(4)+0 0.04/0.28 Such that:aux(1) =< V_N 0.04/0.28 it(15) =< V_i_0 0.04/0.28 s(4) =< it(15)*aux(1) 0.04/0.28 0.04/0.28 with precondition: [B=4,C=0,D=0,E=0,V_i_0>=1,V_N>=V_i_0] 0.04/0.28 0.04/0.28 * Chain [18]: 1*s(1)+0 0.04/0.28 Such that:s(1) =< V_N 0.04/0.28 0.04/0.28 with precondition: [B=3,V_i_0>=1,V_N>=V_i_0] 0.04/0.28 0.04/0.28 * Chain [17]: 0 0.04/0.28 with precondition: [B=3,V_N>=V_i_0] 0.04/0.28 0.04/0.28 * Chain [16]: 0 0.04/0.28 with precondition: [B=4,C=V_3,E=V_j_0,V_i_0=D,0>=V_i_0,V_N>=V_i_0] 0.04/0.28 0.04/0.28 0.04/0.28 #### Cost of chains of eval_while2_bb1_in_loop_cont(A,B,C,D,E,F): 0.04/0.28 * Chain [20]: 0 0.04/0.28 with precondition: [A=3] 0.04/0.28 0.04/0.28 * Chain [19]: 0 0.04/0.28 with precondition: [A=4] 0.04/0.28 0.04/0.28 0.04/0.28 #### Cost of chains of eval_while2_4(V_3,V_N,V_i_0,V_j_0,B): 0.04/0.28 * Chain [24]: 0 0.04/0.28 with precondition: [] 0.04/0.28 0.04/0.28 * Chain [23]: 0 0.04/0.28 with precondition: [0>=V_N] 0.04/0.28 0.04/0.28 * Chain [22]: 3*s(9)+2*s(12)+0 0.04/0.28 Such that:aux(6) =< V_N 0.04/0.28 s(9) =< aux(6) 0.04/0.28 s(12) =< s(9)*aux(6) 0.04/0.28 0.04/0.28 with precondition: [V_N>=1] 0.04/0.28 0.04/0.28 * Chain [21]: 2*s(16)+1*s(19)+0 0.04/0.28 Such that:aux(7) =< V_N 0.04/0.28 s(16) =< aux(7) 0.04/0.28 s(19) =< s(16)*aux(7) 0.04/0.28 0.04/0.28 with precondition: [V_N>=2] 0.04/0.28 0.04/0.28 0.04/0.28 #### Cost of chains of eval_while2_start(V_3,V_N,V_i_0,V_j_0,B): 0.04/0.28 * Chain [28]: 0 0.04/0.28 with precondition: [] 0.04/0.28 0.04/0.28 * Chain [27]: 0 0.04/0.28 with precondition: [0>=V_N] 0.04/0.28 0.04/0.28 * Chain [26]: 3*s(21)+2*s(22)+0 0.04/0.28 Such that:s(20) =< V_N 0.04/0.28 s(21) =< s(20) 0.04/0.28 s(22) =< s(21)*s(20) 0.04/0.28 0.04/0.28 with precondition: [V_N>=1] 0.04/0.28 0.04/0.28 * Chain [25]: 2*s(24)+1*s(25)+0 0.04/0.28 Such that:s(23) =< V_N 0.04/0.28 s(24) =< s(23) 0.04/0.28 s(25) =< s(24)*s(23) 0.04/0.28 0.04/0.28 with precondition: [V_N>=2] 0.04/0.28 0.04/0.28 0.04/0.28 Closed-form bounds of eval_while2_start(V_3,V_N,V_i_0,V_j_0,B): 0.04/0.28 ------------------------------------- 0.04/0.28 * Chain [28] with precondition: [] 0.04/0.28 - Upper bound: 0 0.04/0.28 - Complexity: constant 0.04/0.28 * Chain [27] with precondition: [0>=V_N] 0.04/0.28 - Upper bound: 0 0.04/0.28 - Complexity: constant 0.04/0.28 * Chain [26] with precondition: [V_N>=1] 0.04/0.28 - Upper bound: 2*V_N*V_N+3*V_N 0.04/0.28 - Complexity: n^2 0.04/0.28 * Chain [25] with precondition: [V_N>=2] 0.04/0.28 - Upper bound: 2*V_N+V_N*V_N 0.04/0.28 - Complexity: n^2 0.04/0.28 0.04/0.28 ### Maximum cost of eval_while2_start(V_3,V_N,V_i_0,V_j_0,B): nat(V_N)*nat(V_N)+nat(V_N)*2+(nat(V_N)*nat(V_N)+nat(V_N)) 0.04/0.28 Asymptotic class: n^2 0.04/0.28 * Total analysis performed in 203 ms. 0.04/0.28 0.04/0.38 EOF