1 (* =========================================================== *)
2 (* OCaml interval arithmetic *)
3 (* Author: Thomas C. Hales *)
5 (* =========================================================== *)
7 (* port of interval.cc,
9 This file gives a simple implementation of interval arithmetic,
10 together with the basic arithmetic operations on intervals.
12 It has been incompletely implemented.
14 For now, I am not implementing directed roundings.
15 However, McLaughlin implemented directed rounding several years ago:
16 See http://perso.ens-lyon.fr/nathalie.revol/mpfi.html
17 ~/Library/McLaughlinOCAML/ocaml/src/extensions/ocaml-mpfi/
21 needs "verifier/interval_m/types.ml";;
23 module Interval = struct
27 let mk_interval (a,b) = { lo = a; hi = b; };;
29 let string_of_interval x = Printf.sprintf "[%f;%f]" x.lo x.hi;;
31 (* let izero = mk_interval(0.0,0.0);; *)
32 let zero = mk_interval(0.0,0.0);;
33 let one = mk_interval(1.0,1.0);;
34 let two = mk_interval(2.0,2.0);;
35 let four = mk_interval(4.0,4.0);;
37 let is_zero x =(x.lo=0.0)&&(x.hi=0.0);;
39 let pos x = if (x.lo >= 0.0) then x else
40 mk_interval(0.0, if (x.hi < 0.0) then 0.0 else x.hi );;
42 let imax (x,y) = let t=max x.hi y.hi in mk_interval(t,t);;
44 let imin (x,y) = let t = min x.lo y.lo in mk_interval(t,t);;
46 let imin3(x,y,z) = imin(x,imin(y,z));;
48 let imax3(x,y,z) = imax(x,imax(y,z));;
50 let imax4(w,x,y,z) = imax(imax(w,x),imax(y,z));;
56 let iabs x = max x.hi (~-. (x.lo));;
58 let ilt x y = (x.hi < y.lo);;
60 let igt x y = (x.lo > y.hi);;
62 let ieq x y = (x.lo = y.lo && x.hi = y.hi);;
64 (* need rounding modes -- BUG *)
67 (* start of bug section *)
69 let up() = ( (* bug *) );;
70 let down() = ( (* bug *) );;
71 let nearest() = ( (* bug *) );;
72 let upadd x y = ( x +. y);; (* bug *)
73 let upmul x y = (x *. y);;
74 let updiv x y = (x /. y);;
75 let upsub x y = (x -. y);;
76 let downadd x y = (x +. y);;
77 let downmul x y = (x *. y);;
78 let downdiv x y = (x /. y);;
79 let downsub x y = (x -. y);;
81 (* end of bug section *)
83 let interval_of_string =
84 let dbl_min =1.0e-300 in
86 let ( - ) = (down(); downsub) in
87 let lo = float_of_string s1 - dbl_min in
88 let ( + ) = (up(); upadd) in
89 let hi = float_of_string s2 + dbl_min in
92 let interval_of_single s = interval_of_string (s,s);;
94 let ineg x = mk_interval(~-. (x.hi), ~-. (x.lo));;
96 let iadd x y = mk_interval((down(); downadd x.lo y.lo), (up(); upadd x.hi y.hi));;
100 (if (y.lo >= 0.0) then (x.lo,y.lo,x.hi,y.hi)
101 else if (y.hi <= 0.0) then (x.hi,y.lo,x.lo,y.hi ) else (x.hi,y.lo,x.hi,y.hi))
102 else if (x.hi <= 0.0) then
103 (if (y.hi <= 0.0) then (x.hi,y.hi,x.lo,y.lo)
104 else if (y.lo >= 0.0) then (x.lo,y.hi,x.hi,y.lo) else (x.lo,y.hi,x.lo,y.lo))
106 (if (y.lo >=0.0) then (x.lo,y.hi,x.hi,y.hi)
107 else if (y.hi <=0.0) then (x.hi,y.lo,x.lo,y.lo)
108 else (let lo = (down(); min (downmul x.hi y.lo) (downmul x.lo y.hi)) in
109 let hi = (up(); max (upmul x.hi y.hi) (upmul x.lo y.lo)) in (lo,1.0,hi,1.0)));;
112 let (xlo,ylo,xhi,yhi) = slowcases x y in
113 mk_interval((down(); downmul xlo ylo),(up(); upmul xhi yhi));;
116 let test_slowmul x y =
117 let all = [x.lo *. y.lo; x.hi *. y.lo; x.lo *. y.hi; x.hi *. y.hi] in
118 let m = end_itlist min all in
119 let M = end_itlist max all in
120 ( mk_interval(m,M) = slowmul x y) in
121 let xs = map mk_interval [(~-. 7.0, ~-. 5.0);(~-. 3.0,9.0);(11.0,13.0)] in
122 let ys = map mk_interval [(~-. 16.0, ~-. 14.0);(~-. 10.0,12.0); (18.0,22.0)] in
123 let test i j = test_slowmul (List.nth xs i) (List.nth ys j) or
124 failwith (Printf.sprintf "%d %d" i j) in
127 let _ = test i j in ();
130 let imul x y = if (x.lo > 0.0 && y.lo > 0.0) then
131 mk_interval((down(); downmul x.lo y.lo, (up(); upmul x.hi y.hi))) else slowmul x y;;
133 let isub x y = mk_interval((down();downsub x.lo y.hi),(up(); upsub x.hi y.lo));;
136 let sqrt = Pervasives.sqrt in
137 fun x -> mk_interval(
138 (if (x.lo <= 0.0) then 0.0 else (down(); sqrt(x.lo))),
139 (if (x.hi <= 0.0) then 0.0 else (up(); sqrt(x.hi))));;
143 mk_interval((down(); atan x.lo),(up(); atan x.hi));;
147 mk_interval((down(); acos x.hi),(up(); acos x.lo));;
149 let combine x y = mk_interval(inf(imin(x,y)),sup(imax(x,y)));;
152 let random_int_seed = 81757 in
153 let _ = Random.init(random_int_seed) in
154 fun _ -> Random.float(1.0);;
156 let bounded_from_zero =
157 let dbl_epsilon = 1.0e-8 in
158 fun x-> (x.hi < ~-. dbl_epsilon or x.lo > dbl_epsilon);;
160 let idiv x y = if (bounded_from_zero y) then
161 imul x (mk_interval((down(); downdiv 1.0 y.hi),(up(); updiv 1.0 y.lo)))
162 else raise Unstable;;
164 (* overload arithmetic ops *)