Source file floatItvNan.ml
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(**
FloatItvNan - Floating-point intervals with special IEEE numbers.
Adds special IEEE number managenement (NaN, infinities) to FloatItv.
*)
open Bot
module F = Float
module FI = FloatItv
module B = IntBound
module II = IntItv
(** {2 Types} *)
type t =
{ itv: FI.t with_bot;
(** Interval of non-special values.
Bounds cannot be NaN.
Bounds can be infinities to represent non-infinity floats outside the range of doubles.
*)
nan: bool; (** Whether to include NaN. *)
pinf: bool; (** Whether to include +∞. *)
minf: bool; (** Whether to include -∞. *)
}
(** A set of IEEE floating-point values.
Represented as a set of non-special values, and boolean flags to
indicate the presence of each special IEEE value.
The value 0 in the interval represents both IEEE +0 and -0.
Note: the type can naturally represent the empty set.
*)
let is_valid (a:t) : bool =
match a.itv with
| BOT -> true
| Nb i ->
not (F.is_nan i.FI.lo || i.FI.lo = infinity) &&
not (F.is_nan i.FI.up || i.FI.up = neg_infinity) &&
i.FI.lo <= i.FI.up
(** All elements of type t whould satisfy this predicate. *)
type prec =
[ `SINGLE (** 32-bit single precision *)
| `DOUBLE (** 64-bit double precision *)
| `EXTRA (** anything larger than 64-bit double precision floats *)
| `REAL (** real arithmetic *)
]
(** Precision.
All bounds are represented as double, whatever the precision.
*)
type round =
[ `NEAR (** To nearest *)
| `UP (** Upwards *)
| `DOWN (** Downwards *)
| `ZERO (** Towards 0 *)
| `ANY (** Any rounding mode *)
]
(** Rounding direction. *)
(** {2 Constructors} *)
let bot = { itv = BOT; nan = false; pinf = false; minf = false; }
(** Empty float set. *)
let pinf : t = { bot with pinf = true; }
let minf : t = { bot with minf = true; }
let nan : t = { bot with nan = true; }
let infinities : t = { itv = BOT; nan = false; pinf = true; minf = true; }
let specials : t = { itv = BOT; nan = true; pinf = true; minf = true; }
(** Special values. *)
let of_float (lo:float) (up:float) : t =
if lo > up then bot
else if F.is_nan lo || lo = infinity || F.is_nan up || up = neg_infinity
then invalid_arg (Printf.sprintf "FloatItvNan.of_float: invalid bound [%g,%g]" lo up)
else { bot with itv = Nb { FI.lo; FI.up; }; }
(** Float set reduced to an interval of non-special values.
lo should not be +oo nor NaN.
up should not be -oo nor NaN.
We can have lo = -oo and up = +oo to represent sets of non-infinity
floats larger than the range of double.
*)
let of_interval (a:FI.t) : t =
of_float a.FI.lo a.FI.up
let of_interval_bot (a:FI.t_with_bot) : t =
match a with BOT -> bot | Nb aa -> of_interval aa
let hull (a:float) (b:float) : t =
of_float (min a b) (max a b)
(** Constructs the smallest interval containing a and b. *)
let cst (x:float) : t =
match classify_float x with
| FP_nan -> nan
| FP_infinite -> if x > 0. then pinf else minf
| _ -> { bot with itv = Nb { FI.lo = x; FI.up = x; }; }
(** Singleton (possibly infinity or NaN). *)
let zero : t = cst 0.
let one : t = cst 1.
let two : t = cst 2.
let mone : t = cst (-1.)
let zero_one : t = of_float 0. 1.
let mone_zero : t = of_float (-1.) 0.
let mone_one : t = of_float (-1.) 1.
let mhalf_half : t = of_float (-0.5) 0.5
(** Useful intervals. *)
let add_special (x:t) : t =
{ x with nan = true; pinf = true; minf = true; }
(** Adds NaN and infinities to a set. *)
let remove_special (x:t) : t =
{ x with nan = false; pinf = false; minf = false; }
(** Removes NaN and infinities from a set. *)
let single : t =
of_float (-. F.Single.max_normal) F.Single.max_normal
(** Non-special single precision floats. *)
let double : t =
of_float (-. F.Double.max_normal) F.Double.max_normal
(** Non-special double precision floats. *)
let : t =
of_float neg_infinity infinity
(** Non-special extra (> double) precision. *)
let real : t =
of_float neg_infinity infinity
(** Reals. *)
let single_special : t =
add_special single
(** Single precision floats with specials. *)
let double_special : t =
add_special double
(** Double precision floats with specials. *)
let : t =
add_special extra
(** Extra (> double) precision floats with specials. *)
(** {2 Set-theoretic} *)
let equal (a:t) (b:t) : bool =
a = b
(** Set equality. = also works. *)
let included (a:t) (b:t) : bool =
(match a.itv, b.itv with
| BOT, _ -> true
| _, BOT -> false
| Nb aa, Nb bb -> aa.FI.lo >= bb.FI.lo && aa.FI.up <= bb.FI.up
) &&
(b.nan || not a.nan) &&
(b.minf || not a.minf) &&
(b.pinf || not a.pinf)
(** Set inclusion. *)
let intersect_finite (a:t) (b:t) : bool =
(match a.itv, b.itv with
| BOT, _ | _, BOT -> false
| Nb aa, Nb bb -> aa.FI.lo <= bb.FI.up && aa.FI.up >= bb.FI.lo
)
(** Whether the finite parts of the sets have a non-empty intersection. *)
let intersect (a:t) (b:t) : bool =
(intersect_finite a b) ||
(a.nan && b.nan) ||
(a.minf && b.minf) ||
(a.pinf && b.pinf)
(** Whether the sets have an non-empty intersection. *)
let contains (x:float) (a:t) =
match classify_float x with
| FP_nan -> a.nan
| FP_infinite -> (x > 0. && a.pinf) || (x < 0. && a.minf)
| _ ->
match a.itv with
| BOT -> false
| Nb aa -> aa.lo <= x && aa.up >= x
(** Whether the set contains a certain (finite, infinite or NaN) value. *)
let compare (a:t) (b:t) : int =
Compare.compose
[(fun () -> FI.compare_bot a.itv b.itv);
(fun () -> Stdlib.compare a.nan b.nan);
(fun () -> Stdlib.compare a.minf b.minf);
(fun () -> Stdlib.compare a.pinf b.pinf);
]
(** A total ordering returning -1, 0, or 1. *)
let is_bot (a:t) : bool =
(a.itv = BOT) && (not a.nan) && (not a.pinf) && (not a.minf)
(** Whether the argument is the empty set. *)
let is_finite (a:t) : bool =
(not a.nan) && (not a.pinf) && (not a.minf)
(** Whether the argument contains only finite values (or is empty). *)
let is_infinity (a:t) : bool =
(not a.nan) && (a.itv = BOT)
(** Whether the argument contains only infinities (or is empty). *)
let is_special (a:t) : bool =
a.itv = BOT
(** Whether the argument contains only special values (or is empty). *)
let is_zero (a:t) : bool =
a = zero
(** Whether the argument is the singleton 0. *)
let is_null (a:t) : bool =
a = zero || a = bot
(** Whether the argument contains only 0 (or is empty). *)
let is_positive (a:t) : bool =
(not a.nan) && (not a.minf) &&
(match a.itv with BOT -> true | Nb x -> x.FI.lo >= 0.)
(** Whether the argument contains only (possibly infinite) positive non-NaN values (or is empty). *)
let is_negative (a:t) : bool =
(not a.nan) && (not a.pinf) &&
(match a.itv with BOT -> true | Nb x -> x.FI.up <= 0.)
(** Whether the argument contains only (possibly infinite) negative non-NaN values (or is empty). *)
let is_positive_strict (a:t) : bool =
(not a.nan) && (not a.minf) &&
(match a.itv with BOT -> true | Nb x -> x.FI.lo > 0.)
(** Whether the argument contains only (possibly infinite) strictly positive non-NaN values (or is empty). *)
let is_negative_strict (a:t) : bool =
(not a.nan) && (not a.pinf) &&
(match a.itv with BOT -> true | Nb x -> x.FI.up < 0.)
(** Whether the argument contains only (possibly infinite) strictly negative non-NaN values (or is empty). *)
let is_nonzero (a:t) : bool =
(match a.itv with BOT -> true | Nb x -> x.FI.lo > 0. || x.FI.up < 0.)
(** Whether the argument contains only (possibly infinite or NaN) non-zero values (or is empty). *)
let approx_size (a:t) : int =
(if a.nan then 1 else 0)
+ (if a.pinf then 1 else 0)
+ (if a.minf then 1 else 0)
+ (match a.itv with
| BOT -> 0
| Nb x -> if x.FI.lo = x.FI.up then 1 else 2
)
let is_singleton (a:t) : bool =
approx_size a == 1
(** Whether the argument contains only a single element. *)
let contains_finite (a:t) : bool =
a.itv <> BOT
(** Whether the argument contains at least one finite value. *)
let contains_infinity (a:t) : bool =
a.pinf || a.minf
(** Whether the argument contains an infinity. *)
let contains_special (a:t) : bool =
a.nan || a.pinf || a.minf
(** Whether the argument contains an infinity or NaN. *)
let contains_zero (a:t) : bool =
match a.itv with Nb x -> x.FI.lo <= 0. && x.FI.up >= 0. | BOT -> false
(** Whether the argument contains 0. *)
let contains_nonzero (a:t) : bool =
a.nan || a.pinf || a.minf ||
(match a.itv with BOT -> false | Nb x -> x.FI.lo <> 0. || x.FI.up <> 0.)
(** Whether the argument contains a (possibly NaN or infinite) non-0 value. *)
let contains_positive (a:t) : bool =
a.pinf || (match a.itv with BOT -> false | Nb x -> x.FI.up >= 0.)
(** Whether the argument contains a (possibly infinite) positive value. *)
let contains_negative (a:t) : bool =
a.minf || (match a.itv with BOT -> false | Nb x -> x.FI.lo <= 0.)
(** Whether the argument contains a (possibly infinite) negative value. *)
let contains_positive_strict (a:t) : bool =
a.pinf || (match a.itv with BOT -> false | Nb x -> x.FI.up > 0.)
(** Whether the argument contains a (possibly infinite) strictly positive value. *)
let contains_negative_strict (a:t) : bool =
a.minf || (match a.itv with BOT -> false | Nb x -> x.FI.lo < 0.)
(** Whether the argument contains a (possibly infinite) strictly negative value. *)
let contains_non_nan (a:t) : bool =
a.minf || a.pinf || a.itv <> BOT
(** Whether the argument contains a non-NaN value. *)
let is_in_range (a:t) (lo:float) (up:float) =
(not a.nan) && (not a.pinf) && (not a.minf) &&
(match a.itv with BOT -> true | Nb x -> x.FI.lo >= lo || x.FI.up <= up)
(** Whether the argument contains only finite values, and they are included in the range [lo,up]. *)
let join (a:t) (b:t) =
{ itv = bot_neutral2 FI.join a.itv b.itv;
nan = a.nan || b.nan;
minf = a.minf || b.minf;
pinf = a.pinf || b.pinf;
}
let join_list : t list -> t =
List.fold_left join bot
let meet (a:t) (b:t) =
{ itv = bot_absorb2 FI.meet a.itv b.itv;
nan = a.nan && b.nan;
minf = a.minf && b.minf;
pinf = a.pinf && b.pinf;
}
let widen (a:t) (b:t) =
{ itv = bot_neutral2 FI.widen a.itv b.itv;
nan = a.nan || b.nan;
minf = a.minf || b.minf;
pinf = a.pinf || b.pinf;
}
let positive (x:t) : t =
{ itv = bot_absorb1 FI.positive x.itv;
nan = false;
pinf = x.pinf;
minf = false;
}
(** Positive part of the argument, excluding NaN. *)
let negative (x:t) : t =
{ itv = bot_absorb1 FI.negative x.itv;
nan = false;
minf = x.minf;
pinf = false;
}
(** Negative part of the argument, excluding NaN. *)
let meet_zero (a:t) : t =
meet a zero
(** Intersects with {0} (excluding infinities and NaN). *)
(** {2 Printing} *)
type print_format = FI.print_format
let dfl_fmt = FI.dfl_fmt
let to_string (fmt:print_format) (x:t) : string =
let app x y = if x = "" then y else x ^ "∨" ^ y in
let r = (match x.itv with Nb i -> FI.to_string fmt i | BOT -> "") in
let r = if x.pinf then app r "+∞" else r in
let r = if x.minf then app r "-∞" else r in
let r = if x.nan then app r "NaN" else r in
if r = "" then bot_string else r
let print fmt ch (x:t) = output_string ch (to_string fmt x)
let fprint fmt ch (x:t) = Format.pp_print_string ch (to_string fmt x)
let bprint fmt ch (x:t) = Buffer.add_string ch (to_string fmt x)
(** {2 C predicates} *)
let is_log_eq (a:t) (b:t) : bool =
(intersect_finite a b) || (a.pinf && b.pinf) || (a.minf && b.minf)
let is_log_leq (a:t) (b:t) : bool =
(match a.itv, b.itv with Nb x, Nb y -> x.FI.lo <= y.FI.up | _ -> false) ||
(a.minf && contains_non_nan b) ||
(b.pinf && contains_non_nan a)
let is_log_lt (a:t) (b:t) : bool =
(match a.itv, b.itv with Nb x, Nb y -> x.FI.lo < y.FI.up | _ -> false) ||
(a.minf && (b.pinf || b.itv <> BOT)) ||
(b.pinf && (a.minf || a.itv <> BOT))
let is_log_geq (a:t) (b:t) : bool =
is_log_leq b a
let is_log_gt (a:t) (b:t) : bool =
is_log_lt b a
let is_log_neq (a:t) (b:t) : bool =
match approx_size a, approx_size b with
| 0,_ | _,0 -> false
| 1,1 -> a.nan || b.nan || not (equal a b)
| _ -> true
(** C comparison tests.
Returns true if the test may succeed, false if it cannot.
Note that NaN always compares different to all values (including NaN).
*)
let is_log_leq_false (a:t) (b:t) : bool =
is_log_gt a b || (a.nan && not (is_bot b)) || (b.nan && not (is_bot a))
let is_log_lt_false (a:t) (b:t) : bool =
is_log_geq a b || (a.nan && not (is_bot b)) || (b.nan && not (is_bot a))
let is_log_geq_false (a:t) (b:t) : bool =
is_log_leq_false b a
let is_log_gt_false (a:t) (b:t) : bool =
is_log_lt_false b a
let is_log_eq_false = is_log_neq
let is_log_neq_false = is_log_eq
(** Returns true if the test may fail, false if it cannot.
Due to NaN, which compare always different, <= (resp. >) do not
return the boolean negation of > (resp. <).
However, == is the negation of != even for NaN.
*)
(** {2 Forward arithmetic} *)
let fix_itv (prec:prec) (x:t) : t =
match prec with
| (`SINGLE | `DOUBLE) as prec ->
let m = F.max_normal prec in
(match x.itv with
| BOT -> x
| Nb i ->
let lo,minf,nan1 =
if F.is_nan i.FI.lo then -. m, false, true
else if i.FI.lo < -. m then -. m, true, false
else i.FI.lo, false, false
and up,pinf,nan2 =
if F.is_nan i.FI.up then m, false, true
else if i.FI.up > m then m, true, false
else i.FI.up, false, false
in
{ itv = FI.of_float_bot lo up;
nan = x.nan || nan1 || nan2;
minf = x.minf || minf;
pinf = x.pinf || pinf;
})
| `EXTRA ->
(match x.itv with
| BOT -> x
| Nb i ->
let lo,minf,nan1 =
if F.is_nan i.FI.lo then neg_infinity, true, true
else if i.FI.lo = neg_infinity then neg_infinity, true, false
else i.FI.lo, false, false
and up,pinf,nan2 =
if F.is_nan i.FI.up then infinity, true, true
else if i.FI.up = infinity then infinity, true, false
else i.FI.up, false, false
in
{ itv = FI.of_float_bot lo up;
nan = x.nan || nan1 || nan2;
minf = x.minf || minf;
pinf = x.pinf || pinf;
})
| `REAL ->
{ bot with itv = x.itv; }
let neg (x:t) : t =
{ itv = bot_lift1 FI.neg x.itv;
nan = x.nan;
pinf = x.minf;
minf = x.pinf;
}
(** Negation. *)
let abs (x:t) : t =
{ itv = bot_lift1 FI.abs x.itv;
nan = x.nan;
pinf = x.pinf || x.minf;
minf = false;
}
(** Absolute value. *)
let fix_prec (prec:prec) : FI.prec = match prec with
| `SINGLE -> `SINGLE
| `DOUBLE -> `DOUBLE
| `REAL | `EXTRA -> `REAL
let add (prec:prec) (round:round) (x:t) (y:t) =
fix_itv
prec
{ itv = bot_lift2 (FI.add (fix_prec prec) round) x.itv y.itv;
nan = x.nan || y.nan || (x.pinf && y.minf) || (x.minf && y.pinf);
pinf = (x.pinf && y.pinf) || (x.pinf && contains_finite y) || (y.pinf && contains_finite x);
minf = (x.minf && y.minf) || (x.minf && contains_finite y) || (y.minf && contains_finite x);
}
(** Addition. *)
let sub (prec:prec) (round:round) (x:t) (y:t) =
fix_itv
prec
{ itv = bot_lift2 (FI.sub (fix_prec prec) round) x.itv y.itv;
nan = x.nan || y.nan || (x.pinf && y.pinf) || (x.minf && y.minf);
pinf = (x.pinf && y.minf) || (x.pinf && contains_finite y) || (y.minf && contains_finite x);
minf = (x.minf && y.pinf) || (x.minf && contains_finite y) || (y.pinf && contains_finite x);
}
(** Subtraction. *)
let mul (prec:prec) (round:round) (x:t) (y:t) =
let xm, xz, xp, xi = contains_negative_strict x, contains_zero x,
contains_positive_strict x, contains_infinity x
and ym, yz, yp, yi = contains_negative_strict y, contains_zero y,
contains_positive_strict y, contains_infinity y
in
fix_itv
prec
{ itv = bot_lift2 (FI.mul (fix_prec prec) round) x.itv y.itv;
nan = x.nan || y.nan || (xz && yi) || (xi && yz);
pinf = (x.pinf && yp) || (x.minf && ym) || (y.pinf && xp) || (y.minf && xm);
minf = (x.pinf && ym) || (x.minf && yp) || (y.pinf && xm) || (y.minf && xp);
}
(** Multiplication. *)
let div (prec:prec) (round:round) (x:t) (y:t) =
let xm, xz, xp, xi = contains_negative_strict x, contains_zero x,
contains_positive_strict x, contains_infinity x
and ym, yz, yp, yi = contains_negative y, contains_zero y,
contains_positive y, contains_infinity y
in
let r =
fix_itv
prec
{ itv = bot_absorb2 (FI.div (fix_prec prec) round) x.itv y.itv;
nan = x.nan || y.nan || (xi && yi) || (xz && yz);
pinf = yz || (x.pinf && yp) || (x.minf && ym);
minf = yz || (x.pinf && ym) || (x.minf && yp);
}
in
if yi then join zero r else r
(** Division. *)
let fmod (prec:prec) (round:round) (x:t) (y:t) : t =
fix_itv
prec
{ itv = bot_absorb2 FI.fmod x.itv y.itv;
nan = (contains_special x) || y.nan || (contains_zero y);
minf = false;
pinf = false;
}
(** Remainder (modulo). *)
let square (prec:prec) (round:round) (x:t) : t =
fix_itv
prec
{ itv = bot_lift1 (FI.square (fix_prec prec) round) x.itv;
nan = x.nan;
pinf = x.pinf || x.minf;
minf = false;
}
(** Square. *)
let sqrt (prec:prec) (round:round) (x:t) : t =
fix_itv
prec
{ itv = bot_absorb1 (FI.sqrt (fix_prec prec) round) x.itv;
nan = x.nan || (contains_negative_strict x);
pinf = x.pinf;
minf = false;
}
(** Square root. *)
let round_int (prec:prec) (round:round) (x:t) : t =
fix_itv prec { x with itv = bot_lift1 (FI.round_int (fix_prec prec) round) x.itv; }
(** Round to integer. *)
let round (prec:prec) (round:round) (x:t) : t =
fix_itv prec { x with itv = bot_lift1 (FI.round (fix_prec prec) round) x.itv; }
(** Round to float. *)
let of_int (prec:prec) (round:round) (x:int) (y:int) : t =
fix_itv prec { bot with itv = Nb (FI.of_int (fix_prec prec) round x y); }
(** Conversion from integer range. *)
let of_int64 (prec:prec) (round:round) (x:int64) (y:int64) : t =
fix_itv prec { bot with itv = Nb (FI.of_int64 (fix_prec prec) round x y); }
(** Conversion from int64 range. *)
let of_z (prec:prec) (round:round) (x:Z.t) (y:Z.t) : t =
fix_itv prec { bot with itv = Nb (FI.of_z (fix_prec prec) round x y); }
(** Conversion from integer range. *)
let to_z (x:t) : (Z.t * Z.t) with_bot =
bot_lift1 FI.to_z x.itv
(** Conversion to integer range with truncation. NaN and infinities are discarded. *)
let of_float_prec (prec:prec) (round:round) (lo:float) (up:float) : t =
let r = FI.of_float_bot lo up in
fix_itv prec { bot with itv = bot_lift1 (FI.round (fix_prec prec) round) r; }
(** From bounds, with rounding, precision and handling of specials. *)
let of_int_itv (prec:prec) (round:round) ((lo,up):II.t) : t =
let prec,round = match prec with
| `SINGLE -> `SINGLE, round
| `DOUBLE -> `DOUBLE, round
| `REAL | `EXTRA -> `DOUBLE, `ANY
in
let lo = match lo, round with
| B.Finite l, `NEAR -> F.of_z prec `NEAR l
| B.Finite l, (`DOWN | `ANY) -> F.of_z prec `DOWN l
| B.Finite l, `UP -> F.of_z prec `UP l
| B.Finite l, `ZERO -> F.of_z prec `ZERO l
| _ -> neg_infinity
and up = match up, round with
| B.Finite l, `NEAR -> F.of_z prec `NEAR l
| B.Finite l, `DOWN -> F.of_z prec `DOWN l
| B.Finite l, (`ANY | `UP) -> F.of_z prec `UP l
| B.Finite l, `ZERO -> F.of_z prec `ZERO l
| _ -> infinity
in
if lo > up then bot
else fix_itv prec { bot with itv = Nb { lo; up; }; }
(** Conversion from integer intervals (handling overflows to infinities). *)
let of_int_itv_bot (prec:prec) (round:round) (i:II.t with_bot) : t =
match i with
| BOT -> bot
| Nb (lo,up) -> of_int_itv prec round (lo,up)
(** Conversion from integer intervals (handling overflows to infinities). *)
let to_int_itv (r:t) : II.t with_bot =
match r.itv with
| BOT -> if is_bot r then BOT else Nb II.minf_inf
| Nb i ->
II.of_bound_bot
(if r.minf || Float.is_infinite i.lo then B.MINF else B.Finite (Z.of_float i.lo))
(if r.pinf || Float.is_infinite i.up then B.PINF else B.Finite (Z.of_float i.up))
(** Conversion to integer interval with truncation. Handles infinities. *)
(** {2 Filters} *)
let filter_eq (prec:prec) (x:t) (y:t) : t * t =
let r = meet x y in
let r = { r with nan = false; } in
r, r
let lift_filter_itv f x y =
match bot_absorb2 f x.itv y.itv with
| BOT -> BOT, BOT
| Nb (xx,yy) -> Nb xx, Nb yy
let filter_leq (prec:prec) (x:t) (y:t) : t * t =
let ix, iy = lift_filter_itv (FI.filter_leq (fix_prec prec)) x y in
let ix = if y.pinf then x.itv else ix
and iy = if x.minf then y.itv else iy
in
{ itv = ix; nan = false; minf = x.minf; pinf = x.pinf && y.pinf; },
{ itv = iy; nan = false; pinf = y.pinf; minf = x.minf && y.minf; }
let filter_lt (prec:prec) (x:t) (y:t) : t * t =
let ix, iy = lift_filter_itv (FI.filter_lt (fix_prec prec)) x y in
let ix = if y.pinf then x.itv else ix
and iy = if x.minf then y.itv else iy
in
{ itv = ix; nan = false; minf = x.minf; pinf = false; },
{ itv = iy; nan = false; pinf = y.pinf; minf = false; }
(** C comparison filters.
Keep the parts of the arguments that can satisfy the condition.
NaN is assumed to be different from any value (including NaN).
*)
let filter_geq (prec:prec) (x:t) (y:t) : t * t =
let yy, xx = filter_leq prec y x in xx, yy
let filter_gt (prec:prec) (x:t) (y:t) : t * t =
let yy, xx = filter_lt prec y x in xx, yy
let rec filter_neq (prec:prec) (x:t) (y:t) : t * t =
if x.nan || y.nan then x, y
else if is_singleton x then
if x.pinf then x, { y with pinf = false; }
else if x.minf then x, { y with minf = false; }
else
let ix,iy = match bot_absorb2 (FI.filter_neq (fix_prec prec)) x.itv y.itv with
| BOT -> BOT, BOT
| Nb (xx,yy) -> Nb xx, Nb yy
in
{ x with itv = ix; }, { y with itv = iy; }
else if is_singleton y then
let yy,xx = filter_neq prec y x in xx, yy
else x, y
let filter_nonzero (prec:prec) (x:t) : t =
match bot_absorb2 (FI.filter_neq (fix_prec prec)) x.itv (Nb FI.zero) with
| BOT -> { x with itv = BOT; }
| Nb (xx,_) -> {x with itv = Nb xx; }
let filter_zero (prec:prec) (x:t) : t =
let r = meet x zero in
{ r with nan = false; }
(** Refine both arguments assuming that the test is true. *)
let filter_leq_false (prec:prec) (x:t) (y:t) : t * t =
if x.nan || y.nan then x, y else filter_gt prec x y
let filter_lt_false (prec:prec) (x:t) (y:t) : t * t =
if x.nan || y.nan then x, y else filter_geq prec x y
let filter_geq_false (prec:prec) (x:t) (y:t) : t * t =
let yy, xx = filter_leq_false prec y x in xx, yy
let filter_gt_false (prec:prec) (x:t) (y:t) : t * t =
let yy, xx = filter_lt_false prec y x in xx, yy
let filter_eq_false = filter_neq
let filter_neq_false = filter_eq
let filter_zero_false = filter_nonzero
let filter_nonzero_false = filter_zero
(** Refine both arguments assuming that the test is false. *)
(** {2 Backward arithmetic} *)
let bwd_neg (a:t) (r:t) : t =
meet a (neg r)
(** Backward negation. *)
let bwd_abs (a:t) (r:t) : t =
join (meet a r) (meet a (neg r))
(** Backward absolute value. *)
let bwd_generic2 (prec:prec) (round:round) f (x:t) (y:t) (r:t) : t * t =
if contains_special r then
x, y
else
match x.itv, y.itv, r.itv with
| _, _, BOT -> bot, bot
| BOT,_,_ | _,BOT,_ -> x, y
| Nb xx, Nb yy, Nb rr ->
match f (fix_prec prec) round xx yy rr with
| BOT -> x, y
| Nb (ix,iy) ->
meet x (fix_itv prec { bot with itv = Nb ix; }),
meet y (fix_itv prec { bot with itv = Nb iy; })
let bwd_add (prec:prec) (round:round) (x:t) (y:t) (r:t) : t * t =
bwd_generic2 prec round FI.bwd_add x y r
(** Backward addition. *)
let bwd_sub (prec:prec) (round:round) (x:t) (y:t) (r:t) : t * t =
bwd_generic2 prec round FI.bwd_sub x y r
(** Backward subtraction. *)
let bwd_mul (prec:prec) (round:round) (x:t) (y:t) (r:t) : t * t =
bwd_generic2 prec round FI.bwd_mul x y r
(** Backward multiplication. *)
let bwd_div (prec:prec) (round:round) (x:t) (y:t) (r:t) : t * t =
let xx, yy = bwd_generic2 prec round FI.bwd_div x y r in
let yy =
if contains_zero r then { yy with pinf = y.pinf; minf = y.minf; }
else yy
in
xx, yy
(** Backward division. *)
let bwd_fmod (prec:prec) (round:round) (x:t) (y:t) (r:t) : t * t =
bwd_generic2 prec round (fun _ _ -> FI.bwd_fmod) x y r
(** Backward modulo. *)
let bwd_generic1 (prec:prec) (round:round) f (x:t) (r:t) : t =
if contains_special r then
x
else
match x.itv, r.itv with
| _, BOT -> bot
| BOT, _ -> x
| Nb ix, Nb ir ->
let itv = f (fix_prec prec) round ix ir in
meet x (fix_itv prec { bot with itv; })
let bwd_round_int (prec:prec) (round:round) (x:t) (r:t) : t =
bwd_generic1 prec round FI.bwd_round_int x r
(** Backward rounding to int. *)
let bwd_round (prec:prec) (round:round) (x:t) (r:t) : t =
match x.itv, r.itv with
| _, BOT -> bot
| BOT, _ -> x
| Nb ix, Nb ir ->
let m = match prec with
| (`SINGLE | `DOUBLE) as prec -> F.max_normal prec
| `EXTRA -> F.max_normal `DOUBLE
| `REAL -> infinity
in
let pinf = r.pinf && (x.pinf || ix.FI.up > m)
and minf = r.minf && (x.minf || ix.FI.lo < -.m)
and nan = r.nan && x.nan
in
let itv = FI.bwd_round (fix_prec prec) round ix ir in
meet x (fix_itv prec { itv; pinf; minf; nan; })
(** Backward rounding to float. *)
let bwd_square (prec:prec) (round:round) (x:t) (r:t) : t =
bwd_generic1 prec round FI.bwd_square x r
(** Backward square. *)
let bwd_sqrt (prec:prec) (round:round) (x:t) (r:t) : t =
bwd_generic1 prec round FI.bwd_sqrt x r
(** Backward square root. *)
let bwd_of_int_itv (prec:prec) (round:round) ((lo,up):II.t) (r:t)
: II.t_with_bot =
match r.itv with
| Nb i ->
let i = FI.unround_int (fix_prec prec) round i in
let l =
if F.is_finite i.lo && not r.minf
then B.Finite (Z.of_float i.lo)
else B.MINF
and u =
if F.is_finite i.up && not r.pinf
then B.Finite (Z.of_float i.up)
else B.PINF
in
II.meet (lo,up) (l,u)
| BOT ->
if is_bot r then BOT else Nb II.minf_inf
(** Backward conversion from integer interval. *)
let bwd_to_int_itv (a:t) ((lo,up):II.t) : t =
let l = match lo with
| B.Finite x -> F.of_z `DOUBLE `DOWN x
| _ -> neg_infinity
and u = match up with
| B.Finite x -> F.of_z `DOUBLE `UP x
| _ -> infinity
in
let itv =
bot_absorb1
(fun i -> FI.bwd_round_int `DOUBLE `ZERO i (FI.mk l u)) a.itv
in
{ itv;
nan = a.nan;
minf = a.pinf && (l == neg_infinity);
pinf = a.minf && (u == infinity);
}
(** Backward conversion to integer interval (with truncation). *)