(* Title: HOL/Real/Float.ML
ID: $Id$
Author: Steven Obua
*)
structure ExactFloatingPoint :
sig
exception Destruct_floatstr of string
val destruct_floatstr : (char -> bool) -> (char -> bool) -> string -> bool * string * string * bool * string
exception Floating_point of string
type floatrep = IntInf.int * IntInf.int
val approx_dec_by_bin : IntInf.int -> floatrep -> floatrep * floatrep
val approx_decstr_by_bin : int -> string -> floatrep * floatrep
end
=
struct
exception Destruct_floatstr of string;
fun destruct_floatstr isDigit isExp number =
let
val numlist = filter (not o Char.isSpace) (String.explode number)
fun countsigns ((#"+")::cs) = countsigns cs
| countsigns ((#"-")::cs) =
let
val (positive, rest) = countsigns cs
in
(not positive, rest)
end
| countsigns cs = (true, cs)
fun readdigits [] = ([], [])
| readdigits (q as c::cs) =
if (isDigit c) then
let
val (digits, rest) = readdigits cs
in
(c::digits, rest)
end
else
([], q)
fun readfromexp_helper cs =
let
val (positive, rest) = countsigns cs
val (digits, rest') = readdigits rest
in
case rest' of
[] => (positive, digits)
| _ => raise (Destruct_floatstr number)
end
fun readfromexp [] = (true, [])
| readfromexp (c::cs) =
if isExp c then
readfromexp_helper cs
else
raise (Destruct_floatstr number)
fun readfromdot [] = ([], readfromexp [])
| readfromdot ((#".")::cs) =
let
val (digits, rest) = readdigits cs
val exp = readfromexp rest
in
(digits, exp)
end
| readfromdot cs = readfromdot ((#".")::cs)
val (positive, numlist) = countsigns numlist
val (digits1, numlist) = readdigits numlist
val (digits2, exp) = readfromdot numlist
in
(positive, String.implode digits1, String.implode digits2, fst exp, String.implode (snd exp))
end
type floatrep = IntInf.int * IntInf.int
exception Floating_point of string;
val ln2_10 = (Math.ln 10.0)/(Math.ln 2.0)
fun intmul a b = IntInf.* (a,b)
fun intsub a b = IntInf.- (a,b)
fun intadd a b = IntInf.+ (a,b)
fun intpow a b = IntInf.pow (a, IntInf.toInt b);
fun intle a b = IntInf.<= (a, b);
fun intless a b = IntInf.< (a, b);
fun intneg a = IntInf.~ a;
val zero = IntInf.fromInt 0;
val one = IntInf.fromInt 1;
val two = IntInf.fromInt 2;
val ten = IntInf.fromInt 10;
val five = IntInf.fromInt 5;
fun find_most_significant q r =
let
fun int2real i =
case Real.fromString (IntInf.toString i) of
SOME r => r
| NONE => raise (Floating_point "int2real")
fun subtract (q, r) (q', r') =
if intle r r' then
(intsub q (intmul q' (intpow ten (intsub r' r))), r)
else
(intsub (intmul q (intpow ten (intsub r r'))) q', r')
fun bin2dec d =
if intle zero d then
(intpow two d, zero)
else
(intpow five (intneg d), d)
val L = IntInf.fromInt (Real.floor (int2real (IntInf.fromInt (IntInf.log2 q)) + (int2real r) * ln2_10))
val L1 = intadd L one
val (q1, r1) = subtract (q, r) (bin2dec L1)
in
if intle zero q1 then
let
val (q2, r2) = subtract (q, r) (bin2dec (intadd L1 one))
in
if intle zero q2 then
raise (Floating_point "find_most_significant")
else
(L1, (q1, r1))
end
else
let
val (q0, r0) = subtract (q, r) (bin2dec L)
in
if intle zero q0 then
(L, (q0, r0))
else
raise (Floating_point "find_most_significant")
end
end
fun approx_dec_by_bin n (q,r) =
let
fun addseq acc d' [] = acc
| addseq acc d' (d::ds) = addseq (intadd acc (intpow two (intsub d d'))) d' ds
fun seq2bin [] = (zero, zero)
| seq2bin (d::ds) = (intadd (addseq zero d ds) one, d)
fun approx d_seq d0 precision (q,r) =
if q = zero then
let val x = seq2bin d_seq in
(x, x)
end
else
let
val (d, (q', r')) = find_most_significant q r
in
if intless precision (intsub d0 d) then
let
val d' = intsub d0 precision
val x1 = seq2bin (d_seq)
val x2 = (intadd (intmul (fst x1) (intpow two (intsub (snd x1) d'))) one, d') (* = seq2bin (d'::d_seq) *)
in
(x1, x2)
end
else
approx (d::d_seq) d0 precision (q', r')
end
fun approx_start precision (q, r) =
if q = zero then
((zero, zero), (zero, zero))
else
let
val (d, (q', r')) = find_most_significant q r
in
if intle precision zero then
let
val x1 = seq2bin [d]
in
if q' = zero then
(x1, x1)
else
(x1, seq2bin [intadd d one])
end
else
approx [d] d precision (q', r')
end
in
if intle zero q then
approx_start n (q,r)
else
let
val ((a1,b1), (a2, b2)) = approx_start n (intneg q, r)
in
((intneg a2, b2), (intneg a1, b1))
end
end
fun approx_decstr_by_bin n decstr =
let
fun str2int s = case IntInf.fromString s of SOME x => x | NONE => zero
fun signint p x = if p then x else intneg x
val (p, d1, d2, ep, e) = destruct_floatstr Char.isDigit (fn e => e = #"e" orelse e = #"E") decstr
val s = IntInf.fromInt (size d2)
val q = signint p (intadd (intmul (str2int d1) (intpow ten s)) (str2int d2))
val r = intsub (signint ep (str2int e)) s
in
approx_dec_by_bin (IntInf.fromInt n) (q,r)
end
end;
structure FloatArith =
struct
type float = IntInf.int * IntInf.int
val izero = IntInf.fromInt 0
val ione = IntInf.fromInt 1
val imone = IntInf.fromInt ~1
val itwo = IntInf.fromInt 2
fun imul a b = IntInf.* (a,b)
fun isub a b = IntInf.- (a,b)
fun iadd a b = IntInf.+ (a,b)
val floatzero = (izero, izero)
fun positive_part (a,b) =
(if IntInf.< (a,izero) then izero else a, b)
fun negative_part (a,b) =
(if IntInf.< (a,izero) then a else izero, b)
fun is_negative (a,b) =
if IntInf.< (a, izero) then true else false
fun is_positive (a,b) =
if IntInf.< (izero, a) then true else false
fun is_zero (a,b) =
if a = izero then true else false
fun ipow2 a = IntInf.pow ((IntInf.fromInt 2), IntInf.toInt a)
fun add (a1, b1) (a2, b2) =
if IntInf.< (b1, b2) then
(iadd a1 (imul a2 (ipow2 (isub b2 b1))), b1)
else
(iadd (imul a1 (ipow2 (isub b1 b2))) a2, b2)
fun sub (a1, b1) (a2, b2) =
if IntInf.< (b1, b2) then
(isub a1 (imul a2 (ipow2 (isub b2 b1))), b1)
else
(isub (imul a1 (ipow2 (isub b1 b2))) a2, b2)
fun neg (a, b) = (IntInf.~ a, b)
fun is_equal a b = is_zero (sub a b)
fun is_less a b = is_negative (sub a b)
fun max a b = if is_less a b then b else a
fun min a b = if is_less a b then a else b
fun abs a = if is_negative a then neg a else a
fun mul (a1, b1) (a2, b2) = (imul a1 a2, iadd b1 b2)
end;
structure Float:
sig
type float = FloatArith.float
type floatfunc = float * float -> float * float
val mk_intinf : typ -> IntInf.int -> term
val mk_float : float -> term
exception Dest_intinf;
val dest_intinf : term -> IntInf.int
val dest_nat : term -> IntInf.int
exception Dest_float;
val dest_float : term -> float
val float_const : term
val float_add_const : term
val float_diff_const : term
val float_uminus_const : term
val float_pprt_const : term
val float_nprt_const : term
val float_abs_const : term
val float_mult_const : term
val float_le_const : term
val nat_le_const : term
val nat_less_const : term
val nat_eq_const : term
val approx_float : int -> floatfunc -> string -> term * term
val sign_term : term -> cterm
(* exception Float_op_oracle_data of term
exception Nat_op_oracle_data of term
val float_op_oracle : Sign.sg * exn -> term
val nat_op_oracle : Sign.sg * exn -> term
val invoke_float_op : term -> thm
val invoke_nat_op : term -> thm*)
end
=
struct
structure Inttab = TableFun(type key = int val ord = (rev_order o int_ord));
type float = IntInf.int*IntInf.int
type floatfunc = float*float -> float*float
val th = theory "Float"
val sg = sign_of th
val float_const = Const ("Float.float", HOLogic.mk_prodT (HOLogic.intT, HOLogic.intT) --> HOLogic.realT)
val float_add_const = Const ("HOL.plus", HOLogic.realT --> HOLogic.realT --> HOLogic.realT)
val float_diff_const = Const ("HOL.minus", HOLogic.realT --> HOLogic.realT --> HOLogic.realT)
val float_mult_const = Const ("HOL.times", HOLogic.realT --> HOLogic.realT --> HOLogic.realT)
val float_uminus_const = Const ("HOL.uminus", HOLogic.realT --> HOLogic.realT)
val float_abs_const = Const ("HOL.abs", HOLogic.realT --> HOLogic.realT)
val float_le_const = Const ("Orderings.less_eq", HOLogic.realT --> HOLogic.realT --> HOLogic.boolT)
val float_pprt_const = Const ("OrderedGroup.pprt", HOLogic.realT --> HOLogic.realT)
val float_nprt_const = Const ("OrderedGroup.nprt", HOLogic.realT --> HOLogic.realT)
val nat_le_const = Const ("Orderings.less_eq", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT)
val nat_less_const = Const ("Orderings.less", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT)
val nat_eq_const = Const ("op =", HOLogic.natT --> HOLogic.natT --> HOLogic.boolT)
val zero = FloatArith.izero
val minus_one = FloatArith.imone
val two = FloatArith.itwo
exception Dest_intinf;
exception Dest_float;
fun mk_intinf ty n =
let
fun mk_bit n = if n = zero then HOLogic.B0_const else HOLogic.B1_const
fun bin_of n =
if n = zero then HOLogic.pls_const
else if n = minus_one then HOLogic.min_const
else
let
(*val (q,r) = IntInf.divMod (n, two): doesn't work in SML 10.0.7, but in newer versions!!!*)
val q = IntInf.div (n, two)
val r = IntInf.mod (n, two)
in
HOLogic.bit_const $ bin_of q $ mk_bit r
end
in
HOLogic.number_of_const ty $ (bin_of n)
end
fun dest_intinf n =
let
fun dest_bit n =
case n of
Const ("Numeral.bit.B0", _) => FloatArith.izero
| Const ("Numeral.bit.B1", _) => FloatArith.ione
| _ => raise Dest_intinf
fun int_of n =
case n of
Const ("Numeral.Pls", _) => FloatArith.izero
| Const ("Numeral.Min", _) => FloatArith.imone
| Const ("Numeral.Bit", _) $ q $ r => FloatArith.iadd (FloatArith.imul (int_of q) FloatArith.itwo) (dest_bit r)
| _ => raise Dest_intinf
in
case n of
Const ("Numeral.number_of", _) $ n' => int_of n'
| Const ("Numeral0", _) => FloatArith.izero
| Const ("Numeral1", _) => FloatArith.ione
| _ => raise Dest_intinf
end
fun mk_float (a,b) =
float_const $ (HOLogic.mk_prod ((mk_intinf HOLogic.intT a), (mk_intinf HOLogic.intT b)))
fun dest_float f =
case f of
(Const ("Float.float", _) $ (Const ("Pair", _) $ a $ b)) => (dest_intinf a, dest_intinf b)
| Const ("Numeral.number_of",_) $ a => (dest_intinf f, 0)
| Const ("Numeral0", _) => (FloatArith.izero, FloatArith.izero)
| Const ("Numeral1", _) => (FloatArith.ione, FloatArith.izero)
| _ => raise Dest_float
fun dest_nat n =
let
val v = dest_intinf n
in
if IntInf.< (v, FloatArith.izero) then
FloatArith.izero
else
v
end
fun approx_float prec f value =
let
val interval = ExactFloatingPoint.approx_decstr_by_bin prec value
val (flower, fupper) = f interval
in
(mk_float flower, mk_float fupper)
end
fun sign_term t = cterm_of sg t
(*exception Float_op_oracle_data of term;
fun float_op_oracle (sg, exn as Float_op_oracle_data t) =
Logic.mk_equals (t,
case t of
f $ a $ b =>
let
val a' = dest_float a
val b' = dest_float b
in
if f = float_add_const then
mk_float (FloatArith.add a' b')
else if f = float_diff_const then
mk_float (FloatArith.sub a' b')
else if f = float_mult_const then
mk_float (FloatArith.mul a' b')
else if f = float_le_const then
(if FloatArith.is_less b' a' then
HOLogic.false_const
else
HOLogic.true_const)
else raise exn
end
| f $ a =>
let
val a' = dest_float a
in
if f = float_uminus_const then
mk_float (FloatArith.neg a')
else if f = float_abs_const then
mk_float (FloatArith.abs a')
else if f = float_pprt_const then
mk_float (FloatArith.positive_part a')
else if f = float_nprt_const then
mk_float (FloatArith.negative_part a')
else
raise exn
end
| _ => raise exn
)
val th = ref ([]: Theory.theory list)
val sg = ref ([]: Sign.sg list)
fun invoke_float_op c =
let
val th = (if length(!th) = 0 then th := [theory "MatrixLP"] else (); hd (!th))
val sg = (if length(!sg) = 0 then sg := [sign_of th] else (); hd (!sg))
in
invoke_oracle th "float_op" (sg, Float_op_oracle_data c)
end
exception Nat_op_oracle_data of term;
fun nat_op_oracle (sg, exn as Nat_op_oracle_data t) =
Logic.mk_equals (t,
case t of
f $ a $ b =>
let
val a' = dest_nat a
val b' = dest_nat b
in
if f = nat_le_const then
(if IntInf.<= (a', b') then
HOLogic.true_const
else
HOLogic.false_const)
else if f = nat_eq_const then
(if a' = b' then
HOLogic.true_const
else
HOLogic.false_const)
else if f = nat_less_const then
(if IntInf.< (a', b') then
HOLogic.true_const
else
HOLogic.false_const)
else
raise exn
end
| _ => raise exn)
fun invoke_nat_op c =
let
val th = (if length (!th) = 0 then th := [theory "MatrixLP"] else (); hd (!th))
val sg = (if length (!sg) = 0 then sg := [sign_of th] else (); hd (!sg))
in
invoke_oracle th "nat_op" (sg, Nat_op_oracle_data c)
end
*)
end;