0.04/0.42 WORST_CASE(?,O(n^2)) 0.04/0.42 0.04/0.42 Preprocessing Cost Relations 0.04/0.42 ===================================== 0.04/0.42 0.04/0.42 #### Computed strongly connected components 0.04/0.42 0. recursive : [eval_abc_bb2_in/4,eval_abc_bb3_in/4] 0.04/0.42 1. recursive : [eval_abc_10/9,eval_abc_9/9,eval_abc_bb1_in/9,eval_abc_bb2_in_loop_cont/10,eval_abc_bb4_in/9] 0.04/0.42 2. non_recursive : [eval_abc_stop/6] 0.04/0.42 3. non_recursive : [eval_abc_bb5_in/6] 0.04/0.42 4. non_recursive : [exit_location/1] 0.04/0.42 5. non_recursive : [eval_abc_bb1_in_loop_cont/7] 0.04/0.42 6. non_recursive : [eval_abc_5/6] 0.04/0.42 7. non_recursive : [eval_abc_4/6] 0.04/0.42 8. non_recursive : [eval_abc_3/6] 0.04/0.42 9. non_recursive : [eval_abc_2/6] 0.04/0.42 10. non_recursive : [eval_abc_1/6] 0.04/0.42 11. non_recursive : [eval_abc_0/6] 0.04/0.42 12. non_recursive : [eval_abc_bb0_in/6] 0.04/0.42 13. non_recursive : [eval_abc_start/6] 0.04/0.42 0.04/0.42 #### Obtained direct recursion through partial evaluation 0.04/0.42 0. SCC is partially evaluated into eval_abc_bb2_in/4 0.04/0.42 1. SCC is partially evaluated into eval_abc_bb1_in/9 0.04/0.42 2. SCC is completely evaluated into other SCCs 0.04/0.42 3. SCC is completely evaluated into other SCCs 0.04/0.42 4. SCC is completely evaluated into other SCCs 0.04/0.42 5. SCC is partially evaluated into eval_abc_bb1_in_loop_cont/7 0.04/0.42 6. SCC is partially evaluated into eval_abc_5/6 0.04/0.42 7. SCC is completely evaluated into other SCCs 0.04/0.42 8. SCC is completely evaluated into other SCCs 0.04/0.42 9. SCC is completely evaluated into other SCCs 0.04/0.42 10. SCC is completely evaluated into other SCCs 0.04/0.42 11. SCC is completely evaluated into other SCCs 0.04/0.42 12. SCC is completely evaluated into other SCCs 0.04/0.42 13. SCC is partially evaluated into eval_abc_start/6 0.04/0.42 0.04/0.42 Control-Flow Refinement of Cost Relations 0.04/0.42 ===================================== 0.04/0.42 0.04/0.42 ### Specialization of cost equations eval_abc_bb2_in/4 0.04/0.42 * CE 11 is refined into CE [12] 0.04/0.42 * CE 10 is refined into CE [13] 0.04/0.42 * CE 9 is refined into CE [14] 0.04/0.42 0.04/0.42 0.04/0.42 ### Cost equations --> "Loop" of eval_abc_bb2_in/4 0.04/0.42 * CEs [14] --> Loop 12 0.04/0.42 * CEs [12] --> Loop 13 0.04/0.42 * CEs [13] --> Loop 14 0.04/0.42 0.04/0.42 ### Ranking functions of CR eval_abc_bb2_in(V_i_0,V_m,B,C) 0.04/0.42 * RF of phase [12]: [-V_i_0+V_m+1] 0.04/0.42 0.04/0.42 #### Partial ranking functions of CR eval_abc_bb2_in(V_i_0,V_m,B,C) 0.04/0.42 * Partial RF of phase [12]: 0.04/0.42 - RF of loop [12:1]: 0.04/0.42 -V_i_0+V_m+1 0.04/0.42 0.04/0.42 0.04/0.42 ### Specialization of cost equations eval_abc_bb1_in/9 0.04/0.42 * CE 5 is refined into CE [15] 0.04/0.42 * CE 3 is refined into CE [16,17] 0.04/0.42 * CE 6 is refined into CE [18] 0.04/0.42 * CE 4 is refined into CE [19,20] 0.04/0.42 0.04/0.42 0.04/0.42 ### Cost equations --> "Loop" of eval_abc_bb1_in/9 0.04/0.42 * CEs [20] --> Loop 15 0.04/0.42 * CEs [19] --> Loop 16 0.04/0.42 * CEs [15] --> Loop 17 0.04/0.42 * CEs [16] --> Loop 18 0.04/0.42 * CEs [17] --> Loop 19 0.04/0.42 * CEs [18] --> Loop 20 0.04/0.42 0.04/0.42 ### Ranking functions of CR eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_m,V_n,B,C,D,E) 0.04/0.42 * RF of phase [15]: [-V_j_0+V_n+1] 0.04/0.42 * RF of phase [16]: [-V_j_0+V_n+1] 0.04/0.42 0.04/0.42 #### Partial ranking functions of CR eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_m,V_n,B,C,D,E) 0.04/0.42 * Partial RF of phase [15]: 0.04/0.42 - RF of loop [15:1]: 0.04/0.42 -V_j_0+V_n+1 0.04/0.42 * Partial RF of phase [16]: 0.04/0.42 - RF of loop [16:1]: 0.04/0.42 -V_j_0+V_n+1 0.04/0.42 0.04/0.42 0.04/0.42 ### Specialization of cost equations eval_abc_bb1_in_loop_cont/7 0.04/0.42 * CE 7 is refined into CE [21] 0.04/0.42 * CE 8 is refined into CE [22] 0.04/0.42 0.04/0.42 0.04/0.42 ### Cost equations --> "Loop" of eval_abc_bb1_in_loop_cont/7 0.04/0.42 * CEs [21] --> Loop 21 0.04/0.42 * CEs [22] --> Loop 22 0.04/0.42 0.04/0.42 ### Ranking functions of CR eval_abc_bb1_in_loop_cont(A,B,C,D,E,F,G) 0.04/0.42 0.04/0.42 #### Partial ranking functions of CR eval_abc_bb1_in_loop_cont(A,B,C,D,E,F,G) 0.04/0.42 0.04/0.42 0.04/0.42 ### Specialization of cost equations eval_abc_5/6 0.04/0.42 * CE 2 is refined into CE [23,24,25,26,27,28,29,30,31] 0.04/0.42 0.04/0.42 0.04/0.42 ### Cost equations --> "Loop" of eval_abc_5/6 0.04/0.42 * CEs [28] --> Loop 23 0.04/0.42 * CEs [27] --> Loop 24 0.04/0.42 * CEs [26,31] --> Loop 25 0.04/0.42 * CEs [30] --> Loop 26 0.04/0.42 * CEs [24] --> Loop 27 0.04/0.42 * CEs [23,29] --> Loop 28 0.04/0.42 * CEs [25] --> Loop 29 0.04/0.42 0.04/0.42 ### Ranking functions of CR eval_abc_5(V_3,V_i_0,V_j_0,V_m,V_n,B) 0.04/0.42 0.04/0.42 #### Partial ranking functions of CR eval_abc_5(V_3,V_i_0,V_j_0,V_m,V_n,B) 0.04/0.42 0.04/0.42 0.04/0.42 ### Specialization of cost equations eval_abc_start/6 0.04/0.42 * CE 1 is refined into CE [32,33,34,35,36,37,38] 0.04/0.42 0.04/0.42 0.04/0.42 ### Cost equations --> "Loop" of eval_abc_start/6 0.04/0.42 * CEs [38] --> Loop 30 0.04/0.42 * CEs [37] --> Loop 31 0.04/0.42 * CEs [36] --> Loop 32 0.04/0.42 * CEs [35] --> Loop 33 0.04/0.42 * CEs [34] --> Loop 34 0.04/0.42 * CEs [33] --> Loop 35 0.04/0.42 * CEs [32] --> Loop 36 0.04/0.42 0.04/0.42 ### Ranking functions of CR eval_abc_start(V_3,V_i_0,V_j_0,V_m,V_n,B) 0.04/0.42 0.04/0.42 #### Partial ranking functions of CR eval_abc_start(V_3,V_i_0,V_j_0,V_m,V_n,B) 0.04/0.42 0.04/0.42 0.04/0.42 Computing Bounds 0.04/0.42 ===================================== 0.04/0.42 0.04/0.42 #### Cost of chains of eval_abc_bb2_in(V_i_0,V_m,B,C): 0.04/0.42 * Chain [[12],14]: 1*it(12)+0 0.04/0.42 Such that:it(12) =< -V_i_0+C 0.04/0.42 0.04/0.42 with precondition: [B=2,V_m+1=C,V_i_0>=1,V_m>=V_i_0] 0.04/0.42 0.04/0.42 * Chain [[12],13]: 1*it(12)+0 0.04/0.42 Such that:it(12) =< -V_i_0+V_m+1 0.04/0.42 0.04/0.42 with precondition: [B=3,V_i_0>=1,V_m>=V_i_0] 0.04/0.42 0.04/0.42 * Chain [14]: 0 0.04/0.42 with precondition: [B=2,V_i_0=C,V_i_0>=1,V_i_0>=V_m+1] 0.04/0.42 0.04/0.42 * Chain [13]: 0 0.04/0.42 with precondition: [B=3,V_i_0>=1] 0.04/0.42 0.04/0.42 0.04/0.42 #### Cost of chains of eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_m,V_n,B,C,D,E): 0.04/0.42 * Chain [[16],20]: 1*it(16)+0 0.04/0.42 Such that:it(16) =< -V_j_0+V_n+1 0.04/0.42 0.04/0.42 with precondition: [B=3,0>=V_m,V_j_0>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [[16],18]: 1*it(16)+0 0.04/0.42 Such that:it(16) =< -V_j_0+V_n 0.04/0.42 0.04/0.42 with precondition: [B=3,0>=V_m,V_j_0>=1,V_n>=V_j_0+1] 0.04/0.42 0.04/0.42 * Chain [[16],17]: 1*it(16)+0 0.04/0.42 Such that:it(16) =< -V_j_0+C 0.04/0.42 0.04/0.42 with precondition: [B=4,D=1,V_n+1=C,V_n+1=E,0>=V_m,V_j_0>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [[15],20]: 1*it(15)+1*s(3)+0 0.04/0.42 Such that:it(15) =< -V_j_0+V_n+1 0.04/0.42 aux(1) =< V_m 0.04/0.42 s(3) =< it(15)*aux(1) 0.04/0.42 0.04/0.42 with precondition: [B=3,V_j_0>=1,V_m>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [[15],19]: 1*it(15)+1*s(3)+1*s(4)+0 0.04/0.42 Such that:it(15) =< -V_j_0+V_n 0.04/0.42 aux(2) =< V_m 0.04/0.42 s(4) =< aux(2) 0.04/0.42 s(3) =< it(15)*aux(2) 0.04/0.42 0.04/0.42 with precondition: [B=3,V_j_0>=1,V_m>=1,V_n>=V_j_0+1] 0.04/0.42 0.04/0.42 * Chain [[15],18]: 1*it(15)+1*s(3)+0 0.04/0.42 Such that:it(15) =< -V_j_0+V_n 0.04/0.42 aux(1) =< V_m 0.04/0.42 s(3) =< it(15)*aux(1) 0.04/0.42 0.04/0.42 with precondition: [B=3,V_j_0>=1,V_m>=1,V_n>=V_j_0+1] 0.04/0.42 0.04/0.42 * Chain [[15],17]: 1*it(15)+1*s(3)+0 0.04/0.42 Such that:it(15) =< -V_j_0+C 0.04/0.42 aux(1) =< D 0.04/0.42 s(3) =< it(15)*aux(1) 0.04/0.42 0.04/0.42 with precondition: [B=4,V_n+1=C,V_m+1=D,V_n+1=E,V_j_0>=1,V_m>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [20]: 0 0.04/0.42 with precondition: [B=3,V_j_0>=1] 0.04/0.42 0.04/0.42 * Chain [19]: 1*s(4)+0 0.04/0.42 Such that:s(4) =< V_m 0.04/0.42 0.04/0.42 with precondition: [B=3,V_j_0>=1,V_m>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [18]: 0 0.04/0.42 with precondition: [B=3,V_j_0>=1,V_n>=V_j_0] 0.04/0.42 0.04/0.42 * Chain [17]: 0 0.04/0.42 with precondition: [B=4,C=V_3,D=V_i_0,V_j_0=E,V_j_0>=1,V_j_0>=V_n+1] 0.04/0.42 0.04/0.42 0.04/0.42 #### Cost of chains of eval_abc_bb1_in_loop_cont(A,B,C,D,E,F,G): 0.04/0.42 * Chain [22]: 0 0.04/0.42 with precondition: [A=3] 0.04/0.42 0.04/0.42 * Chain [21]: 0 0.04/0.42 with precondition: [A=4] 0.04/0.42 0.04/0.42 0.04/0.42 #### Cost of chains of eval_abc_5(V_3,V_i_0,V_j_0,V_m,V_n,B): 0.04/0.42 * Chain [29]: 0 0.04/0.42 with precondition: [] 0.04/0.42 0.04/0.42 * Chain [28]: 2*s(16)+0 0.04/0.42 Such that:aux(6) =< V_n 0.04/0.42 s(16) =< aux(6) 0.04/0.42 0.04/0.42 with precondition: [0>=V_m,V_n>=1] 0.04/0.42 0.04/0.42 * Chain [27]: 1*s(18)+0 0.04/0.42 Such that:s(18) =< V_n 0.04/0.42 0.04/0.42 with precondition: [0>=V_m,V_n>=2] 0.04/0.42 0.04/0.42 * Chain [26]: 0 0.04/0.42 with precondition: [0>=V_n] 0.04/0.42 0.04/0.42 * Chain [25]: 2*s(19)+1*s(21)+1*s(22)+1*s(25)+0 0.04/0.42 Such that:s(20) =< V_m 0.04/0.42 s(24) =< V_m+1 0.04/0.42 aux(7) =< V_n 0.04/0.42 s(19) =< aux(7) 0.04/0.42 s(21) =< s(20) 0.04/0.42 s(22) =< s(19)*s(20) 0.04/0.42 s(25) =< s(19)*s(24) 0.04/0.42 0.04/0.42 with precondition: [V_m>=1,V_n>=1] 0.04/0.42 0.04/0.42 * Chain [24]: 2*s(28)+2*s(29)+1*s(30)+0 0.04/0.42 Such that:s(27) =< V_m 0.04/0.42 s(26) =< V_n 0.04/0.42 s(28) =< s(26) 0.04/0.42 s(29) =< s(28)*s(27) 0.04/0.42 s(30) =< s(27) 0.04/0.42 0.04/0.42 with precondition: [V_m>=1,V_n>=2] 0.04/0.42 0.04/0.42 * Chain [23]: 0 0.04/0.42 with precondition: [V_n>=1] 0.04/0.42 0.04/0.42 0.04/0.42 #### Cost of chains of eval_abc_start(V_3,V_i_0,V_j_0,V_m,V_n,B): 0.04/0.42 * Chain [36]: 0 0.04/0.42 with precondition: [] 0.04/0.42 0.04/0.42 * Chain [35]: 2*s(32)+0 0.04/0.42 Such that:s(31) =< V_n 0.04/0.42 s(32) =< s(31) 0.04/0.42 0.04/0.42 with precondition: [0>=V_m,V_n>=1] 0.04/0.42 0.04/0.42 * Chain [34]: 1*s(33)+0 0.04/0.42 Such that:s(33) =< V_n 0.04/0.42 0.04/0.42 with precondition: [0>=V_m,V_n>=2] 0.04/0.42 0.04/0.42 * Chain [33]: 0 0.04/0.42 with precondition: [0>=V_n] 0.04/0.42 0.04/0.42 * Chain [32]: 2*s(37)+1*s(38)+1*s(39)+1*s(40)+0 0.04/0.42 Such that:s(34) =< V_m 0.04/0.42 s(35) =< V_m+1 0.04/0.42 s(36) =< V_n 0.04/0.42 s(37) =< s(36) 0.04/0.42 s(38) =< s(34) 0.04/0.42 s(39) =< s(37)*s(34) 0.04/0.42 s(40) =< s(37)*s(35) 0.04/0.42 0.04/0.42 with precondition: [V_m>=1,V_n>=1] 0.04/0.42 0.04/0.42 * Chain [31]: 2*s(43)+2*s(44)+1*s(45)+0 0.04/0.42 Such that:s(41) =< V_m 0.04/0.42 s(42) =< V_n 0.04/0.42 s(43) =< s(42) 0.04/0.42 s(44) =< s(43)*s(41) 0.04/0.42 s(45) =< s(41) 0.04/0.42 0.04/0.42 with precondition: [V_m>=1,V_n>=2] 0.04/0.42 0.04/0.42 * Chain [30]: 0 0.04/0.42 with precondition: [V_n>=1] 0.04/0.42 0.04/0.42 0.04/0.42 Closed-form bounds of eval_abc_start(V_3,V_i_0,V_j_0,V_m,V_n,B): 0.04/0.42 ------------------------------------- 0.04/0.42 * Chain [36] with precondition: [] 0.04/0.42 - Upper bound: 0 0.04/0.42 - Complexity: constant 0.04/0.42 * Chain [35] with precondition: [0>=V_m,V_n>=1] 0.04/0.42 - Upper bound: 2*V_n 0.04/0.42 - Complexity: n 0.04/0.42 * Chain [34] with precondition: [0>=V_m,V_n>=2] 0.04/0.42 - Upper bound: V_n 0.04/0.42 - Complexity: n 0.04/0.42 * Chain [33] with precondition: [0>=V_n] 0.04/0.42 - Upper bound: 0 0.04/0.42 - Complexity: constant 0.04/0.42 * Chain [32] with precondition: [V_m>=1,V_n>=1] 0.04/0.42 - Upper bound: V_n*V_m+V_m+2*V_n+(V_m+1)*V_n 0.04/0.42 - Complexity: n^2 0.04/0.42 * Chain [31] with precondition: [V_m>=1,V_n>=2] 0.04/0.42 - Upper bound: 2*V_m*V_n+V_m+2*V_n 0.04/0.42 - Complexity: n^2 0.04/0.42 * Chain [30] with precondition: [V_n>=1] 0.04/0.42 - Upper bound: 0 0.04/0.42 - Complexity: constant 0.04/0.42 0.04/0.42 ### Maximum cost of eval_abc_start(V_3,V_i_0,V_j_0,V_m,V_n,B): nat(V_n)*nat(V_m)+nat(V_m)+max([nat(V_n)*nat(V_m),nat(V_m+1)*nat(V_n)])+nat(V_n)+nat(V_n) 0.04/0.42 Asymptotic class: n^2 0.04/0.42 * Total analysis performed in 342 ms. 0.04/0.42 0.42/0.52 EOF