0.05/0.17 WORST_CASE(?,O(n^2)) 0.05/0.17 0.05/0.17 Preprocessing Cost Relations 0.05/0.17 ===================================== 0.05/0.17 0.05/0.17 #### Computed strongly connected components 0.05/0.17 0. recursive : [eval_foo_bb2_in/4,eval_foo_bb3_in/4] 0.05/0.17 1. recursive : [eval_foo_bb1_in/2,eval_foo_bb2_in_loop_cont/4,eval_foo_bb4_in/3] 0.05/0.17 2. non_recursive : [eval_foo_stop/1] 0.05/0.17 3. non_recursive : [eval_foo_bb5_in/1] 0.05/0.17 4. non_recursive : [eval_foo_bb1_in_loop_cont/2] 0.05/0.17 5. non_recursive : [eval_foo_bb0_in/2] 0.05/0.17 6. non_recursive : [eval_foo_start/3] 0.05/0.17 0.05/0.17 #### Obtained direct recursion through partial evaluation 0.05/0.17 0. SCC is partially evaluated into eval_foo_bb2_in/4 0.05/0.17 1. SCC is partially evaluated into eval_foo_bb1_in/2 0.05/0.17 2. SCC is completely evaluated into other SCCs 0.05/0.17 3. SCC is completely evaluated into other SCCs 0.05/0.17 4. SCC is completely evaluated into other SCCs 0.05/0.17 5. SCC is partially evaluated into eval_foo_bb0_in/2 0.05/0.17 6. SCC is partially evaluated into eval_foo_start/3 0.05/0.17 0.05/0.17 Control-Flow Refinement of Cost Relations 0.05/0.17 ===================================== 0.05/0.17 0.05/0.17 ### Specialization of cost equations eval_foo_bb2_in/4 0.05/0.17 * CE 6 is refined into CE [7] 0.05/0.17 * CE 5 is refined into CE [8] 0.05/0.17 0.05/0.17 0.05/0.17 ### Cost equations --> "Loop" of eval_foo_bb2_in/4 0.05/0.17 * CEs [8] --> Loop 7 0.05/0.17 * CEs [7] --> Loop 8 0.05/0.17 0.05/0.17 ### Ranking functions of CR eval_foo_bb2_in(V__0,V__01,B,C) 0.05/0.17 * RF of phase [7]: [2*V__0-2*V__01-1] 0.05/0.17 0.05/0.17 #### Partial ranking functions of CR eval_foo_bb2_in(V__0,V__01,B,C) 0.05/0.17 * Partial RF of phase [7]: 0.05/0.17 - RF of loop [7:1]: 0.05/0.17 2*V__0-2*V__01-1 0.05/0.17 0.05/0.17 0.05/0.17 ### Specialization of cost equations eval_foo_bb1_in/2 0.05/0.17 * CE 4 is refined into CE [9] 0.05/0.17 * CE 3 is refined into CE [10,11] 0.05/0.17 0.05/0.17 0.05/0.17 ### Cost equations --> "Loop" of eval_foo_bb1_in/2 0.05/0.17 * CEs [11] --> Loop 9 0.05/0.17 * CEs [10] --> Loop 10 0.05/0.17 * CEs [9] --> Loop 11 0.05/0.17 0.05/0.17 ### Ranking functions of CR eval_foo_bb1_in(V__0,B) 0.05/0.17 * RF of phase [9]: [V__0-1] 0.05/0.17 * RF of phase [10]: [V__0+1] 0.05/0.17 0.05/0.17 #### Partial ranking functions of CR eval_foo_bb1_in(V__0,B) 0.05/0.17 * Partial RF of phase [9]: 0.05/0.17 - RF of loop [9:1]: 0.05/0.17 V__0-1 0.05/0.17 * Partial RF of phase [10]: 0.05/0.17 - RF of loop [10:1]: 0.05/0.17 V__0+1 0.05/0.17 0.05/0.17 0.05/0.17 ### Specialization of cost equations eval_foo_bb0_in/2 0.05/0.17 * CE 2 is refined into CE [12,13,14] 0.05/0.17 0.05/0.17 0.05/0.17 ### Cost equations --> "Loop" of eval_foo_bb0_in/2 0.05/0.17 * CEs [14] --> Loop 12 0.05/0.17 * CEs [13] --> Loop 13 0.05/0.17 * CEs [12] --> Loop 14 0.05/0.17 0.05/0.17 ### Ranking functions of CR eval_foo_bb0_in(V_x,B) 0.05/0.17 0.05/0.17 #### Partial ranking functions of CR eval_foo_bb0_in(V_x,B) 0.05/0.17 0.05/0.17 0.05/0.17 ### Specialization of cost equations eval_foo_start/3 0.05/0.17 * CE 1 is refined into CE [15,16,17] 0.05/0.17 0.05/0.17 0.05/0.17 ### Cost equations --> "Loop" of eval_foo_start/3 0.05/0.17 * CEs [17] --> Loop 15 0.05/0.17 * CEs [16] --> Loop 16 0.05/0.17 * CEs [15] --> Loop 17 0.05/0.17 0.05/0.17 ### Ranking functions of CR eval_foo_start(V_x,V_y,B) 0.05/0.17 0.05/0.17 #### Partial ranking functions of CR eval_foo_start(V_x,V_y,B) 0.05/0.17 0.05/0.17 0.05/0.17 Computing Bounds 0.05/0.17 ===================================== 0.05/0.17 0.05/0.17 #### Cost of chains of eval_foo_bb2_in(V__0,V__01,B,C): 0.05/0.17 * Chain [[7],8]: 1*it(7)+0 0.05/0.17 Such that:it(7) =< 2*V__0-2*V__01 0.05/0.17 0.05/0.17 with precondition: [B=2,V__01>=1,C>=2*V__01,C>=V__0,2*V__0>=C+2] 0.05/0.17 0.05/0.17 * Chain [8]: 0 0.05/0.17 with precondition: [B=2,V__01=C,V__01>=1,V__01>=V__0] 0.05/0.17 0.05/0.17 0.05/0.17 #### Cost of chains of eval_foo_bb1_in(V__0,B): 0.05/0.17 * Chain [[10],11]: 1*it(10)+0 0.05/0.17 Such that:it(10) =< V__0+1 0.05/0.17 0.05/0.17 with precondition: [B=3,1>=V__0,V__0>=0] 0.05/0.17 0.05/0.17 * Chain [[9],[10],11]: 1*it(9)+1*it(10)+1*s(3)+0 0.05/0.17 Such that:it(10) =< 2 0.05/0.17 it(9) =< V__0 0.05/0.17 aux(1) =< 2*V__0 0.05/0.17 s(3) =< it(9)*aux(1) 0.05/0.17 0.05/0.17 with precondition: [B=3,V__0>=2] 0.05/0.17 0.05/0.17 * Chain [11]: 0 0.05/0.17 with precondition: [B=3,0>=V__0+1] 0.05/0.17 0.05/0.17 0.05/0.17 #### Cost of chains of eval_foo_bb0_in(V_x,B): 0.05/0.17 * Chain [14]: 1*s(4)+0 0.05/0.17 Such that:s(4) =< V_x+1 0.05/0.17 0.05/0.17 with precondition: [1>=V_x,V_x>=0] 0.05/0.17 0.05/0.17 * Chain [13]: 0 0.05/0.17 with precondition: [0>=V_x+1] 0.05/0.17 0.05/0.17 * Chain [12]: 1*s(5)+1*s(6)+1*s(8)+0 0.05/0.17 Such that:s(5) =< 2 0.05/0.17 s(6) =< V_x 0.05/0.17 s(7) =< 2*V_x 0.05/0.17 s(8) =< s(6)*s(7) 0.05/0.17 0.05/0.17 with precondition: [V_x>=2] 0.05/0.17 0.05/0.17 0.05/0.17 #### Cost of chains of eval_foo_start(V_x,V_y,B): 0.05/0.17 * Chain [17]: 1*s(9)+0 0.05/0.17 Such that:s(9) =< V_x+1 0.05/0.17 0.05/0.17 with precondition: [1>=V_x,V_x>=0] 0.05/0.17 0.05/0.17 * Chain [16]: 0 0.05/0.17 with precondition: [0>=V_x+1] 0.05/0.17 0.05/0.17 * Chain [15]: 1*s(10)+1*s(11)+1*s(13)+0 0.05/0.17 Such that:s(10) =< 2 0.05/0.17 s(11) =< V_x 0.05/0.17 s(12) =< 2*V_x 0.05/0.17 s(13) =< s(11)*s(12) 0.05/0.17 0.05/0.17 with precondition: [V_x>=2] 0.05/0.17 0.05/0.17 0.05/0.17 Closed-form bounds of eval_foo_start(V_x,V_y,B): 0.05/0.17 ------------------------------------- 0.05/0.17 * Chain [17] with precondition: [1>=V_x,V_x>=0] 0.05/0.17 - Upper bound: V_x+1 0.05/0.17 - Complexity: n 0.05/0.17 * Chain [16] with precondition: [0>=V_x+1] 0.05/0.17 - Upper bound: 0 0.05/0.17 - Complexity: constant 0.05/0.17 * Chain [15] with precondition: [V_x>=2] 0.05/0.17 - Upper bound: V_x+2+2*V_x*V_x 0.05/0.17 - Complexity: n^2 0.05/0.17 0.05/0.17 ### Maximum cost of eval_foo_start(V_x,V_y,B): max([nat(V_x+1),nat(V_x)+2+nat(2*V_x)*nat(V_x)]) 0.05/0.17 Asymptotic class: n^2 0.05/0.17 * Total analysis performed in 99 ms. 0.05/0.17 0.05/0.27 EOF