2. Floating-point arithmetic

3. FPA semantics

4. FPA representation sign

5. SMT floating-point arithmetic

8. double Kp=1.0, Ki=0.25; double set_point=20.0, integral=0.0; double error, in, out; for (int i=0; i < N; i++) { in = read_input(); error = set_point – in; integral = integral + error; out = Kp*error + Ki*integral; set_output(out); } An example

9. Example strategy

10. (declare-fun Kp () (_ FloatingPoint 5 11)) (declare-fun … (define-fun rm () RoundingMode RNE) (assert (= c18 (fp #b0 #b10011 #b0010000000))) ;; = 18.0 (assert (= c22 (fp #b0 #b10011 #b0110000000))) ;; = 22.0 (assert (= cp3 (fp #b0 #b10000 #b1000000000))) ;; = +3.0 (assert (= cn3 (fp #b1 #b10000 #b1000000000))) ;; = -3.0 (assert (= Kp (fp #b0 #b01111 #b0000000000))) ;; = 1.0 (assert (= Ki (fp #b0 #b01101 #b0000000000))) ;; = 0.25 (assert (= set_point (fp #b0 #b10011 #b0100000000))) ;; = 20.0 (assert (= integral (_ +zero 5 11))) ;; = +0.0 An example in SMT2

11. (assert (not (=> (and (fp.leq c18 in) (fp.leq in c22) (fp.eq error_post (fp.sub rm set_point in)) (fp.eq integral_post (fp.add rm integral error_post)) (fp.eq out (fp.add rm (fp.mul rm Kp error_post) (fp.mul rm Ki integral_post)))) (and (fp.leq cn3 out) (fp.leq out cp3)) ))) > z3 pi_controller_1.smt2 unsat An example in SMT2

12. Checking properties

13. Related techniques

14. 1. Translate to QF_BV (build circuits) 2. Translate to SAT (e.g., Tseitin translation, bit-blasting) 3. Run SAT Solver 4. Translate models/proofs (back to QF_BV, then QF_FP) Solving QF_FP via SAT Circuit for and method of providing a floating-point adder US 8463835 B1 (Xilinx)

15. Conversion QF_FP -> QF_BV -> SAT Example performance Bad

16. 1. Mixed abstractions (CBMC) 2. Non-conservative approximations (Z3) 3. Abstraction into interval arithmetic (MathSAT) 4. Translation into non-linear reals (Realizer) Advanced QF_FP solving

17. Abstraction refinement Model sat Proof unsat no refinement possible failed

18. Mixed abstractions Brillout, Kroening, Wahl: Mixed Abstractions for Floating-Point Arithmetic. FMCAD 2009

19. Mixed abstractions Brillout, Kroening, Wahl: Mixed Abstractions for Floating-Point Arithmetic. FMCAD 2009 sign ORM URM RM

20. Small-float approximation sign RM Zeljic, Wintersteiger, Rümmer: Approximations for Model Construction. IJCAR 2014

21. For the theory of floating-point numbers: Small-floats inclusion Zeljic, Wintersteiger, Rümmer: Approximations for Model Construction. IJCAR 2014

22. Approximation framework lift reconstruct Zeljic, Wintersteiger, Rümmer: Approximations for Model Construction. IJCAR 2014

23. Lifting

24. Example precision terms

25. Refinement scheme Model sat Proof unsat no refinement possible failed

26. Approximate model construction

27. Model-guided refinement Model sat Proof unsat no refinement possible failed

28. Precise model reconstruction

29. Model-guided refinement

30. Proof-guided refinement Model sat Proof unsat no refinement possible failed

31. Proof-guided refinement

32. Algorithm properties

33. Fröhlich, Kovásznai, Biere: More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding. CSR 2013 Kovásznai, Fröhlich, Biere: On the Complexity of Fixed-Size Bit-Vector Logics with Binary Encoded Bit-Width. SMT 2012 A note on complexity

34. Non-conservative approximation Fröhlich, Biere, Wintersteiger, Hamadi: Stochastic Local Search for Satisfiability Modulo Theories. AAAI 2015

35. Abstract CDCL Brain, D’Silva, Griggio, Haller, Kroening: Interpolation-Based Verification of Floating-Point Programs with Abstract CDCL. SAS 2013 Brain, D’Silva, Griggio, Haller, Kroening: Deciding floating-point logic with abstract conflict driven clause learning. FMSD 45(2) 2014

36. FP-ACDCL

37. Realizer Leeser, Mukherjee, Ramachandran, Wahl: Make it real: Effective floating-point reasoning via exact arithmetic. DATE 2014

38. Mixed abstraction performance Brillout, Kroening, Wahl: Mixed Abstractions for Floating-Point Arithmetic. FMCAD 2009

39. FP-ACDCL performance Brain, D’Silva, Griggio, Haller, Kroening: Interpolation-Based Verification of Floating-Point Programs with Abstract CDCL. SAS 2013 Brain, D’Silva, Griggio, Haller, Kroening: Deciding floating-point logic with abstract conflict driven clause learning. FMSD 45(2) 2014

40. Experimental evaluation SATUNSAT Z37656 MathSAT76 Ours8646

41. Data (SAT)

42. Data (UNSAT)

43. Complexity of operations 5.3 Homogeneous general-computational operations 5.3.0 5.3.1 General operations 5.3.1.0 Implementations shall provide the following homogeneous general- computational operations for all supported arithmetic formats; … ― sourceFormat remainder(source, source) When y ≠ 0, the remainder r = remainder(x, y) is defined for finite x and y regardless of the … … FPREM1 … can reduce the exponent of ST(0) by no more than 63 in one execution of the instruction … Software can re-execute the instruction (using the partial remainder in ST(0) as the dividend) until … IEEE 754-2008, Sec 5.3. Intel® 64 and IA-32 Architectures Software Developer’s Manual. May 2012

44. Conclusion