src/HOL/Tools/float_arith.ML

+(*  Title:  HOL/Real/float_arith.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: 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;
+fun exp5 x = Integer.pow x 5;
+fun exp10 x = Integer.pow x 10;
+fun exp2 x = Integer.pow x 2;
+
+fun find_most_significant q r =
+  let
+    fun int2real i =
+      case (Real.fromString o string_of_int) i of
+        SOME r => r
+        | NONE => raise (Floating_point "int2real")
+    fun subtract (q, r) (q', r') =
+      if r <= r' then
+        (q - q' * exp10 (r' - r), r)
+      else
+        (q * exp10 (r - r') - q', r')
+    fun bin2dec d =
+      if 0 <= d then
+        (exp2 d, 0)
+      else
+        (exp5 (~ d), d)
+
+    val L = Real.floor (int2real (IntInf.log2 q) + int2real r * ln2_10)
+    val L1 = L + 1
+
+    val (q1, r1) = subtract (q, r) (bin2dec L1) 
+  in
+    if 0 <= q1 then
+      let
+        val (q2, r2) = subtract (q, r) (bin2dec (L1 + 1))
+      in
+        if 0 <= 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 0 <= 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 (acc + exp2 (d - d')) d' ds
+
+    fun seq2bin [] = (0, 0)
+      | seq2bin (d::ds) = (addseq 0 d ds + 1, d)
+
+    fun approx d_seq d0 precision (q,r) =
+      if q = 0 then
+        let val x = seq2bin d_seq in
+          (x, x)
+        end
+      else
+        let
+          val (d, (q', r')) = find_most_significant q r
+        in
+          if precision < d0 - d then
+            let
+              val d' = d0 - precision
+              val x1 = seq2bin (d_seq)
+              val x2 = (fst x1 * exp2 (snd x1 - d') + 1,  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 = 0 then
+        ((0, 0), (0, 0))
+      else
+        let
+          val (d, (q', r')) = find_most_significant q r
+        in
+          if precision <= 0 then
+            let
+              val x1 = seq2bin [d]
+            in
+              if q' = 0 then
+                (x1, x1)
+              else
+                (x1, seq2bin [d + 1])
+            end
+          else
+            approx [d] d precision (q', r')
+        end
+  in
+    if 0 <= q then
+      approx_start n (q,r)
+    else
+      let
+        val ((a1,b1), (a2, b2)) = approx_start n (~ q, r)
+      in
+        ((~ a2, b2), (~ a1, b1))
+      end
+  end
+
+fun approx_decstr_by_bin n decstr =
+  let
+    fun str2int s = the_default 0 (Int.fromString s)
+    fun signint p x = if p then x else ~ x
+
+    val (p, d1, d2, ep, e) = destruct_floatstr Char.isDigit (fn e => e = #"e" orelse e = #"E") decstr
+    val s = size d2
+
+    val q = signint p (str2int d1 * exp10 s + str2int d2)
+    val r = signint ep (str2int e) - s
+  in
+    approx_dec_by_bin 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, 0);
+
+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;