Presburger Arithmetic and Its Use in Verification

Appendix A
Examples of multicore parallelism in F#
A.1 Source code of π calculation
module PI
open System
open System.Linq
open System.Threading.Tasks
open System.Collections.Concurrent
let NUM_STEPS =100000000
let steps =1.0/(float NUM_STEPS)
let compute1() =
let rec computeUtil(i, acc) =
if i =0then acc ∗ steps
else
let x =(float i +0.5)∗steps
computeUtil (i−1, acc +4.0/(1.0+x ∗ x))
computeUtil(NUM_STEPS, 0.0)
let compute2() =
let sum = ref 0.0
for i in 1..NUM_STEPS do
let x =(float i +0.5)∗steps
sum := !sum +4.0/(1.0+x ∗ x)
!sum ∗ steps
module Parallel =
let compute1() =
let sum = ref 0.0
let monitor = new Object()
Parallel.For(
0, NUM_STEPS, new ParallelOptions(),
(fun () −> 0.0),
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