(* Title: HOL/Real/Float.ML
ID: $Id$
Author: Steven Obua
*)
signature FLOAT_ARITH =
sig
exception Destruct_floatstr of string
val destruct_floatstr: (char -> bool) -> (char -> bool) -> string -> bool * string * string * bool * string
exception Floating_point of string
val approx_dec_by_bin: Intt.int -> Float.float -> Float.float * Float.float
val approx_decstr_by_bin: int -> string -> Float.float * Float.float
val mk_float: Float.float -> term
val dest_float: term -> Float.float
val approx_float: int -> (Float.float * Float.float -> Float.float * Float.float)
-> string -> term * term
end;
structure FloatArith : FLOAT_ARITH =
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
exception Floating_point of string;
val ln2_10 = Math.ln 10.0 / Math.ln 2.0;
val exp5 = Intt.pow (Intt.int 5);
val exp10 = Intt.pow (Intt.int 10);
fun intle a b = not (Intt.cmp (a, b) = GREATER);
fun intless a b = Intt.cmp (a, b) = LESS;
fun find_most_significant q r =
let
fun int2real i =
case (Real.fromString o Intt.string_of_int) i of
SOME r => r
| NONE => raise (Floating_point "int2real")
fun subtract (q, r) (q', r') =
if intle r r' then
(Intt.sub q (Intt.mult q' (exp10 (Intt.sub r' r))), r)
else
(Intt.sub (Intt.mult q (exp10 (Intt.sub r r'))) q', r')
fun bin2dec d =
if intle Intt.zero d then
(Intt.exp d, Intt.zero)
else
(exp5 (Intt.neg d), d)
val L = Intt.int (Real.floor (int2real (Intt.log q) + int2real r * ln2_10))
val L1 = Intt.inc L
val (q1, r1) = subtract (q, r) (bin2dec L1)
in
if intle Intt.zero q1 then
let
val (q2, r2) = subtract (q, r) (bin2dec (Intt.inc L1))
in
if intle Intt.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 Intt.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 (Intt.add acc (Intt.exp (Intt.sub d d'))) d' ds
fun seq2bin [] = (Intt.zero, Intt.zero)
| seq2bin (d::ds) = (Intt.inc (addseq Intt.zero d ds), d)
fun approx d_seq d0 precision (q,r) =
if q = Intt.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 (Intt.sub d0 d) then
let
val d' = Intt.sub d0 precision
val x1 = seq2bin (d_seq)
val x2 = (Intt.inc
(Intt.mult (fst x1)
(Intt.exp (Intt.sub (snd x1) d'))), 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 = Intt.zero then
((Intt.zero, Intt.zero), (Intt.zero, Intt.zero))
else
let
val (d, (q', r')) = find_most_significant q r
in
if intle precision Intt.zero then
let
val x1 = seq2bin [d]
in
if q' = Intt.zero then
(x1, x1)
else
(x1, seq2bin [Intt.inc d])
end
else
approx [d] d precision (q', r')
end
in
if intle Intt.zero q then
approx_start n (q,r)
else
let
val ((a1,b1), (a2, b2)) = approx_start n (Intt.neg q, r)
in
((Intt.neg a2, b2), (Intt.neg a1, b1))
end
end
fun approx_decstr_by_bin n decstr =
let
fun str2int s = the_default Intt.zero (Intt.int_of_string s);
fun signint p x = if p then x else Intt.neg x
val (p, d1, d2, ep, e) = destruct_floatstr Char.isDigit (fn e => e = #"e" orelse e = #"E") decstr
val s = Intt.int (size d2)
val q = signint p (Intt.add (Intt.mult (str2int d1) (exp10 s)) (str2int d2))
val r = Intt.sub (signint ep (str2int e)) s
in
approx_dec_by_bin (Intt.int n) (q,r)
end
fun mk_float (a, b) = @{term "float"} $
HOLogic.mk_prod (pairself (HOLogic.mk_number HOLogic.intT) (a, b));
fun dest_float (Const ("Float.float", _) $ (Const ("Pair", _) $ a $ b)) =
pairself (snd o HOLogic.dest_number) (a, b)
| dest_float t = ((snd o HOLogic.dest_number) t, Intt.zero);
fun approx_float prec f value =
let
val interval = approx_decstr_by_bin prec value
val (flower, fupper) = f interval
in
(mk_float flower, mk_float fupper)
end;
end;