src/HOL/Matrix_LP/float_arith.ML

update documentation: prefer control-symbol-cartouche form of "cite" antiquotations;

+
−(*  Title:      HOL/Matrix_LP/float_arith.ML
+
−    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 (Integer.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>\<open>float\<close> $
+
−  HOLogic.mk_prod (apply2 (HOLogic.mk_number HOLogic.intT) (a, b));
+
−
+
−fun dest_float (Const (\<^const_name>\<open>float\<close>, _) $ (Const (\<^const_name>\<open>Pair\<close>, _) $ a $ b)) =
+
−      apply2 (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;