0.03/0.19 WORST_CASE(?,O(n^2)) 0.03/0.19 0.03/0.19 Preprocessing Cost Relations 0.03/0.19 ===================================== 0.03/0.19 0.03/0.19 #### Computed strongly connected components 0.03/0.19 0. recursive : [eval2/5] 0.03/0.19 1. recursive : [eval1/3,eval2_loop_cont/4] 0.03/0.19 2. non_recursive : [exit_location/1] 0.03/0.19 3. non_recursive : [eval1_loop_cont/2] 0.03/0.19 4. non_recursive : [start/3] 0.03/0.19 0.03/0.19 #### Obtained direct recursion through partial evaluation 0.03/0.19 0. SCC is partially evaluated into eval2/5 0.03/0.19 1. SCC is partially evaluated into eval1/3 0.03/0.19 2. SCC is completely evaluated into other SCCs 0.03/0.19 3. SCC is completely evaluated into other SCCs 0.03/0.19 4. SCC is partially evaluated into start/3 0.03/0.19 0.03/0.19 Control-Flow Refinement of Cost Relations 0.03/0.19 ===================================== 0.03/0.19 0.03/0.19 ### Specialization of cost equations eval2/5 0.03/0.19 * CE 7 is refined into CE [8] 0.03/0.19 * CE 6 is refined into CE [9] 0.03/0.19 * CE 5 is refined into CE [10] 0.03/0.19 0.03/0.19 0.03/0.19 ### Cost equations --> "Loop" of eval2/5 0.03/0.19 * CEs [10] --> Loop 8 0.03/0.19 * CEs [8] --> Loop 9 0.03/0.19 * CEs [9] --> Loop 10 0.03/0.19 0.03/0.19 ### Ranking functions of CR eval2(A,B,C,D,E) 0.03/0.19 * RF of phase [8]: [A-B] 0.03/0.19 0.03/0.19 #### Partial ranking functions of CR eval2(A,B,C,D,E) 0.03/0.19 * Partial RF of phase [8]: 0.03/0.19 - RF of loop [8:1]: 0.03/0.19 A-B 0.03/0.19 0.03/0.19 0.03/0.19 ### Specialization of cost equations eval1/3 0.03/0.19 * CE 2 is refined into CE [11,12] 0.03/0.19 * CE 4 is refined into CE [13] 0.03/0.19 * CE 3 is refined into CE [14,15] 0.03/0.19 0.03/0.19 0.03/0.19 ### Cost equations --> "Loop" of eval1/3 0.03/0.19 * CEs [15] --> Loop 11 0.03/0.19 * CEs [14] --> Loop 12 0.03/0.19 * CEs [12] --> Loop 13 0.03/0.19 * CEs [11] --> Loop 14 0.03/0.19 * CEs [13] --> Loop 15 0.03/0.19 0.03/0.19 ### Ranking functions of CR eval1(A,B,C) 0.03/0.19 * RF of phase [11]: [A] 0.03/0.19 0.03/0.19 #### Partial ranking functions of CR eval1(A,B,C) 0.03/0.19 * Partial RF of phase [11]: 0.03/0.19 - RF of loop [11:1]: 0.03/0.19 A 0.03/0.19 0.03/0.19 0.03/0.19 ### Specialization of cost equations start/3 0.03/0.19 * CE 1 is refined into CE [16,17,18,19,20] 0.03/0.19 0.03/0.19 0.03/0.19 ### Cost equations --> "Loop" of start/3 0.03/0.19 * CEs [20] --> Loop 16 0.03/0.19 * CEs [19] --> Loop 17 0.03/0.19 * CEs [18] --> Loop 18 0.03/0.19 * CEs [16] --> Loop 19 0.03/0.19 * CEs [17] --> Loop 20 0.03/0.19 0.03/0.19 ### Ranking functions of CR start(A,B,C) 0.03/0.19 0.03/0.19 #### Partial ranking functions of CR start(A,B,C) 0.03/0.19 0.03/0.19 0.03/0.19 Computing Bounds 0.03/0.19 ===================================== 0.03/0.19 0.03/0.19 #### Cost of chains of eval2(A,B,C,D,E): 0.03/0.19 * Chain [[8],10]: 1*it(8)+0 0.03/0.19 Such that:it(8) =< -B+E 0.03/0.19 0.03/0.19 with precondition: [C=2,A=D+1,A=E,B>=0,A>=B+1] 0.03/0.19 0.03/0.19 * Chain [[8],9]: 1*it(8)+0 0.03/0.19 Such that:it(8) =< A-B 0.03/0.19 0.03/0.19 with precondition: [C=3,B>=0,A>=B+1] 0.03/0.19 0.03/0.19 * Chain [10]: 0 0.03/0.19 with precondition: [C=2,B=A,B=D+1,B=E,B>=0] 0.03/0.19 0.03/0.19 * Chain [9]: 0 0.03/0.19 with precondition: [C=3,B>=0,A>=B] 0.03/0.19 0.03/0.19 0.03/0.19 #### Cost of chains of eval1(A,B,C): 0.03/0.19 * Chain [[11],15]: 1*it(11)+1*s(3)+0 0.03/0.19 Such that:aux(3) =< A 0.03/0.19 it(11) =< aux(3) 0.03/0.19 s(3) =< it(11)*aux(3) 0.03/0.19 0.03/0.19 with precondition: [C=3,A>=1] 0.03/0.19 0.03/0.19 * Chain [[11],14]: 1*it(11)+1*s(3)+0 0.03/0.19 Such that:aux(4) =< A 0.03/0.19 it(11) =< aux(4) 0.03/0.19 s(3) =< it(11)*aux(4) 0.03/0.19 0.03/0.19 with precondition: [C=3,A>=1] 0.03/0.19 0.03/0.19 * Chain [[11],13]: 2*it(11)+1*s(3)+0 0.03/0.19 Such that:aux(5) =< A 0.03/0.19 it(11) =< aux(5) 0.03/0.19 s(3) =< it(11)*aux(5) 0.03/0.19 0.03/0.19 with precondition: [C=3,A>=2] 0.03/0.19 0.03/0.19 * Chain [[11],12,15]: 1*it(11)+1*s(3)+1 0.03/0.19 Such that:aux(6) =< A 0.03/0.19 it(11) =< aux(6) 0.03/0.19 s(3) =< it(11)*aux(6) 0.03/0.19 0.03/0.19 with precondition: [C=3,A>=1] 0.03/0.19 0.03/0.19 * Chain [15]: 0 0.03/0.19 with precondition: [C=3] 0.03/0.19 0.03/0.19 * Chain [14]: 0 0.03/0.19 with precondition: [C=3,A>=0] 0.03/0.19 0.03/0.19 * Chain [13]: 1*s(4)+0 0.03/0.19 Such that:s(4) =< A 0.03/0.19 0.03/0.19 with precondition: [C=3,A>=1] 0.03/0.19 0.03/0.19 * Chain [12,15]: 1 0.03/0.19 with precondition: [A=0,C=3] 0.03/0.19 0.03/0.19 0.03/0.19 #### Cost of chains of start(A,B,C): 0.03/0.19 * Chain [20]: 0 0.03/0.19 with precondition: [] 0.03/0.19 0.03/0.19 * Chain [19]: 1 0.03/0.19 with precondition: [A=0] 0.03/0.19 0.03/0.19 * Chain [18]: 0 0.03/0.19 with precondition: [A>=0] 0.03/0.19 0.03/0.19 * Chain [17]: 4*s(16)+3*s(17)+1 0.03/0.19 Such that:s(15) =< A 0.03/0.19 s(16) =< s(15) 0.03/0.19 s(17) =< s(16)*s(15) 0.03/0.19 0.03/0.19 with precondition: [A>=1] 0.03/0.19 0.03/0.19 * Chain [16]: 2*s(19)+1*s(20)+0 0.03/0.19 Such that:s(18) =< A 0.03/0.19 s(19) =< s(18) 0.03/0.19 s(20) =< s(19)*s(18) 0.03/0.19 0.03/0.19 with precondition: [A>=2] 0.03/0.19 0.03/0.19 0.03/0.19 Closed-form bounds of start(A,B,C): 0.03/0.19 ------------------------------------- 0.03/0.19 * Chain [20] with precondition: [] 0.03/0.19 - Upper bound: 0 0.03/0.19 - Complexity: constant 0.03/0.19 * Chain [19] with precondition: [A=0] 0.03/0.19 - Upper bound: 1 0.03/0.19 - Complexity: constant 0.03/0.19 * Chain [18] with precondition: [A>=0] 0.03/0.19 - Upper bound: 0 0.03/0.19 - Complexity: constant 0.03/0.19 * Chain [17] with precondition: [A>=1] 0.03/0.19 - Upper bound: 4*A+1+3*A*A 0.03/0.19 - Complexity: n^2 0.03/0.19 * Chain [16] with precondition: [A>=2] 0.03/0.19 - Upper bound: 2*A+A*A 0.03/0.19 - Complexity: n^2 0.03/0.19 0.03/0.19 ### Maximum cost of start(A,B,C): max([1,nat(A)*2+1+nat(A)*2*nat(A)+(nat(A)*nat(A)+nat(A)*2)]) 0.03/0.19 Asymptotic class: n^2 0.03/0.19 * Total analysis performed in 134 ms. 0.03/0.19 0.03/0.30 EOF