gcc/ada/urealp.adb

   1 ------------------------------------------------------------------------------
   2 --                                                                          --
   3 --                         GNAT COMPILER COMPONENTS                         --
   4 --                                                                          --
   5 --                               U R E A L P                                --
   6 --                                                                          --
   7 --                                 B o d y                                  --
   8 --                                                                          --
   9 --          Copyright (C) 1992-2003 Free Software Foundation, Inc.          --
  10 --                                                                          --
  11 -- GNAT is free software;  you can  redistribute it  and/or modify it under --
  12 -- terms of the  GNU General Public License as published  by the Free Soft- --
  13 -- ware  Foundation;  either version 2,  or (at your option) any later ver- --
  14 -- sion.  GNAT is distributed in the hope that it will be useful, but WITH- --
  15 -- OUT ANY WARRANTY;  without even the  implied warranty of MERCHANTABILITY --
  16 -- or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License --
  17 -- for  more details.  You should have  received  a copy of the GNU General --
  18 -- Public License  distributed with GNAT;  see file COPYING.  If not, write --
  19 -- to  the Free Software Foundation,  59 Temple Place - Suite 330,  Boston, --
  20 -- MA 02111-1307, USA.                                                      --
  21 --                                                                          --
  22 -- As a special exception,  if other files  instantiate  generics from this --
  23 -- unit, or you link  this unit with other files  to produce an executable, --
  24 -- this  unit  does not  by itself cause  the resulting  executable  to  be --
  25 -- covered  by the  GNU  General  Public  License.  This exception does not --
  26 -- however invalidate  any other reasons why  the executable file  might be --
  27 -- covered by the  GNU Public License.                                      --
  28 --                                                                          --
  29 -- GNAT was originally developed  by the GNAT team at  New York University. --
  30 -- Extensive contributions were provided by Ada Core Technologies Inc.      --
  31 --                                                                          --
  32 ------------------------------------------------------------------------------
  33
  34 with Alloc;
  35 with Output;  use Output;
  36 with Table;
  37 with Tree_IO; use Tree_IO;
  38
  39 package body Urealp is
  40
  41    Ureal_First_Entry : constant Ureal := Ureal'Succ (No_Ureal);
  42    --  First subscript allocated in Ureal table (note that we can't just
  43    --  add 1 to No_Ureal, since "+" means something different for Ureals!
  44
  45    type Ureal_Entry is record
  46       Num  : Uint;
  47       --  Numerator (always non-negative)
  48
  49       Den  : Uint;
  50       --  Denominator (always non-zero, always positive if base is zero)
  51
  52       Rbase : Nat;
  53       --  Base value. If Rbase is zero, then the value is simply Num / Den.
  54       --  If Rbase is non-zero, then the value is Num / (Rbase ** Den)
  55
  56       Negative : Boolean;
  57       --  Flag set if value is negative
  58    end record;
  59
  60    package Ureals is new Table.Table (
  61      Table_Component_Type => Ureal_Entry,
  62      Table_Index_Type     => Ureal,
  63      Table_Low_Bound      => Ureal_First_Entry,
  64      Table_Initial        => Alloc.Ureals_Initial,
  65      Table_Increment      => Alloc.Ureals_Increment,
  66      Table_Name           => "Ureals");
  67
  68    --  The following universal reals are the values returned by the constant
  69    --  functions. They are initialized by the initialization procedure.
  70
  71    UR_0          : Ureal;
  72    UR_M_0        : Ureal;
  73    UR_Tenth      : Ureal;
  74    UR_Half       : Ureal;
  75    UR_1          : Ureal;
  76    UR_2          : Ureal;
  77    UR_10         : Ureal;
  78    UR_10_36      : Ureal;
  79    UR_M_10_36    : Ureal;
  80    UR_100        : Ureal;
  81    UR_2_128      : Ureal;
  82    UR_2_80       : Ureal;
  83    UR_2_M_128    : Ureal;
  84    UR_2_M_80     : Ureal;
  85
  86    Num_Ureal_Constants : constant := 10;
  87    --  This is used for an assertion check in Tree_Read and Tree_Write to
  88    --  help remember to add values to these routines when we add to the list.
  89
  90    Normalized_Real : Ureal := No_Ureal;
  91    --  Used to memoize Norm_Num and Norm_Den, if either of these functions
  92    --  is called, this value is set and Normalized_Entry contains the result
  93    --  of the normalization. On subsequent calls, this is used to avoid the
  94    --  call to Normalize if it has already been made.
  95
  96    Normalized_Entry : Ureal_Entry;
  97    --  Entry built by most recent call to Normalize
  98
  99    -----------------------
 100    -- Local Subprograms --
 101    -----------------------
 102
 103    function Decimal_Exponent_Hi (V : Ureal) return Int;
 104    --  Returns an estimate of the exponent of Val represented as a normalized
 105    --  decimal number (non-zero digit before decimal point), The estimate is
 106    --  either correct, or high, but never low. The accuracy of the estimate
 107    --  affects only the efficiency of the comparison routines.
 108
 109    function Decimal_Exponent_Lo (V : Ureal) return Int;
 110    --  Returns an estimate of the exponent of Val represented as a normalized
 111    --  decimal number (non-zero digit before decimal point), The estimate is
 112    --  either correct, or low, but never high. The accuracy of the estimate
 113    --  affects only the efficiency of the comparison routines.
 114
 115    function Equivalent_Decimal_Exponent (U : Ureal_Entry) return Int;
 116    --  U is a Ureal entry for which the base value is non-zero, the value
 117    --  returned is the equivalent decimal exponent value, i.e. the value of
 118    --  Den, adjusted as though the base were base 10. The value is rounded
 119    --  to the nearest integer, and so can be one off.
 120
 121    function Is_Integer (Num, Den : Uint) return Boolean;
 122    --  Return true if the real quotient of Num / Den is an integer value
 123
 124    function Normalize (Val : Ureal_Entry) return Ureal_Entry;
 125    --  Normalizes the Ureal_Entry by reducing it to lowest terms (with a
 126    --  base value of 0).
 127
 128    function Same (U1, U2 : Ureal) return Boolean;
 129    pragma Inline (Same);
 130    --  Determines if U1 and U2 are the same Ureal. Note that we cannot use
 131    --  the equals operator for this test, since that tests for equality,
 132    --  not identity.
 133
 134    function Store_Ureal (Val : Ureal_Entry) return Ureal;
 135    --  This store a new entry in the universal reals table and return
 136    --  its index in the table.
 137
 138    -------------------------
 139    -- Decimal_Exponent_Hi --
 140    -------------------------
 141
 142    function Decimal_Exponent_Hi (V : Ureal) return Int is
 143       Val : constant Ureal_Entry := Ureals.Table (V);
 144
 145    begin
 146       --  Zero always returns zero
 147
 148       if UR_Is_Zero (V) then
 149          return 0;
 150
 151       --  For numbers in rational form, get the maximum number of digits in the
 152       --  numerator and the minimum number of digits in the denominator, and
 153       --  subtract. For example:
 154
 155       --     1000 / 99 = 1.010E+1
 156       --     9999 / 10 = 9.999E+2
 157
 158       --  This estimate may of course be high, but that is acceptable
 159
 160       elsif Val.Rbase = 0 then
 161          return UI_Decimal_Digits_Hi (Val.Num) -
 162                 UI_Decimal_Digits_Lo (Val.Den);
 163
 164       --  For based numbers, just subtract the decimal exponent from the
 165       --  high estimate of the number of digits in the numerator and add
 166       --  one to accommodate possible round off errors for non-decimal
 167       --  bases. For example:
 168
 169       --     1_500_000 / 10**4 = 1.50E-2
 170
 171       else -- Val.Rbase /= 0
 172          return UI_Decimal_Digits_Hi (Val.Num) -
 173                 Equivalent_Decimal_Exponent (Val) + 1;
 174       end if;
 175    end Decimal_Exponent_Hi;
 176
 177    -------------------------
 178    -- Decimal_Exponent_Lo --
 179    -------------------------
 180
 181    function Decimal_Exponent_Lo (V : Ureal) return Int is
 182       Val : constant Ureal_Entry := Ureals.Table (V);
 183
 184    begin
 185       --  Zero always returns zero
 186
 187       if UR_Is_Zero (V) then
 188          return 0;
 189
 190       --  For numbers in rational form, get min digits in numerator, max digits
 191       --  in denominator, and subtract and subtract one more for possible loss
 192       --  during the division. For example:
 193
 194       --     1000 / 99 = 1.010E+1
 195       --     9999 / 10 = 9.999E+2
 196
 197       --  This estimate may of course be low, but that is acceptable
 198
 199       elsif Val.Rbase = 0 then
 200          return UI_Decimal_Digits_Lo (Val.Num) -
 201                 UI_Decimal_Digits_Hi (Val.Den) - 1;
 202
 203       --  For based numbers, just subtract the decimal exponent from the
 204       --  low estimate of the number of digits in the numerator and subtract
 205       --  one to accommodate possible round off errors for non-decimal
 206       --  bases. For example:
 207
 208       --     1_500_000 / 10**4 = 1.50E-2
 209
 210       else -- Val.Rbase /= 0
 211          return UI_Decimal_Digits_Lo (Val.Num) -
 212                 Equivalent_Decimal_Exponent (Val) - 1;
 213       end if;
 214    end Decimal_Exponent_Lo;
 215
 216    -----------------
 217    -- Denominator --
 218    -----------------
 219
 220    function Denominator (Real : Ureal) return Uint is
 221    begin
 222       return Ureals.Table (Real).Den;
 223    end Denominator;
 224
 225    ---------------------------------
 226    -- Equivalent_Decimal_Exponent --
 227    ---------------------------------
 228
 229    function Equivalent_Decimal_Exponent (U : Ureal_Entry) return Int is
 230
 231       --  The following table is a table of logs to the base 10
 232
 233       Logs : constant array (Nat range 1 .. 16) of Long_Float := (
 234                 1 => 0.000000000000000,
 235                 2 => 0.301029995663981,
 236                 3 => 0.477121254719662,
 237                 4 => 0.602059991327962,
 238                 5 => 0.698970004336019,
 239                 6 => 0.778151250383644,
 240                 7 => 0.845098040014257,
 241                 8 => 0.903089986991944,
 242                 9 => 0.954242509439325,
 243                10 => 1.000000000000000,
 244                11 => 1.041392685158230,
 245                12 => 1.079181246047620,
 246                13 => 1.113943352306840,
 247                14 => 1.146128035678240,
 248                15 => 1.176091259055680,
 249                16 => 1.204119982655920);
 250
 251    begin
 252       pragma Assert (U.Rbase /= 0);
 253       return Int (Long_Float (UI_To_Int (U.Den)) * Logs (U.Rbase));
 254    end Equivalent_Decimal_Exponent;
 255
 256    ----------------
 257    -- Initialize --
 258    ----------------
 259
 260    procedure Initialize is
 261    begin
 262       Ureals.Init;
 263       UR_0       := UR_From_Components (Uint_0, Uint_1,         0, False);
 264       UR_M_0     := UR_From_Components (Uint_0, Uint_1,         0, True);
 265       UR_Half    := UR_From_Components (Uint_1, Uint_1,         2, False);
 266       UR_Tenth   := UR_From_Components (Uint_1, Uint_1,        10, False);
 267       UR_1       := UR_From_Components (Uint_1, Uint_1,         0, False);
 268       UR_2       := UR_From_Components (Uint_1, Uint_Minus_1,   2, False);
 269       UR_10      := UR_From_Components (Uint_1, Uint_Minus_1,  10, False);
 270       UR_10_36   := UR_From_Components (Uint_1, Uint_Minus_36, 10, False);
 271       UR_M_10_36 := UR_From_Components (Uint_1, Uint_Minus_36, 10, True);
 272       UR_100     := UR_From_Components (Uint_1, Uint_Minus_2,  10, False);
 273       UR_2_128   := UR_From_Components (Uint_1, Uint_Minus_128, 2, False);
 274       UR_2_M_128 := UR_From_Components (Uint_1, Uint_128,       2, False);
 275       UR_2_80    := UR_From_Components (Uint_1, Uint_Minus_80,  2, False);
 276       UR_2_M_80  := UR_From_Components (Uint_1, Uint_80,        2, False);
 277    end Initialize;
 278
 279    ----------------
 280    -- Is_Integer --
 281    ----------------
 282
 283    function Is_Integer (Num, Den : Uint) return Boolean is
 284    begin
 285       return (Num / Den) * Den = Num;
 286    end Is_Integer;
 287
 288    ----------
 289    -- Mark --
 290    ----------
 291
 292    function Mark return Save_Mark is
 293    begin
 294       return Save_Mark (Ureals.Last);
 295    end Mark;
 296
 297    --------------
 298    -- Norm_Den --
 299    --------------
 300
 301    function Norm_Den (Real : Ureal) return Uint is
 302    begin
 303       if not Same (Real, Normalized_Real) then
 304          Normalized_Real  := Real;
 305          Normalized_Entry := Normalize (Ureals.Table (Real));
 306       end if;
 307
 308       return Normalized_Entry.Den;
 309    end Norm_Den;
 310
 311    --------------
 312    -- Norm_Num --
 313    --------------
 314
 315    function Norm_Num (Real : Ureal) return Uint is
 316    begin
 317       if not Same (Real, Normalized_Real) then
 318          Normalized_Real  := Real;
 319          Normalized_Entry := Normalize (Ureals.Table (Real));
 320       end if;
 321
 322       return Normalized_Entry.Num;
 323    end Norm_Num;
 324
 325    ---------------
 326    -- Normalize --
 327    ---------------
 328
 329    function Normalize (Val : Ureal_Entry) return Ureal_Entry is
 330       J   : Uint;
 331       K   : Uint;
 332       Tmp : Uint;
 333       Num : Uint;
 334       Den : Uint;
 335       M   : constant Uintp.Save_Mark := Uintp.Mark;
 336
 337    begin
 338       --  Start by setting J to the greatest of the absolute values of the
 339       --  numerator and the denominator (taking into account the base value),
 340       --  and K to the lesser of the two absolute values. The gcd of Num and
 341       --  Den is the gcd of J and K.
 342
 343       if Val.Rbase = 0 then
 344          J := Val.Num;
 345          K := Val.Den;
 346
 347       elsif Val.Den < 0 then
 348          J := Val.Num * Val.Rbase ** (-Val.Den);
 349          K := Uint_1;
 350
 351       else
 352          J := Val.Num;
 353          K := Val.Rbase ** Val.Den;
 354       end if;
 355
 356       Num := J;
 357       Den := K;
 358
 359       if K > J then
 360          Tmp := J;
 361          J := K;
 362          K := Tmp;
 363       end if;
 364
 365       J := UI_GCD (J, K);
 366       Num := Num / J;
 367       Den := Den / J;
 368       Uintp.Release_And_Save (M, Num, Den);
 369
 370       --  Divide numerator and denominator by gcd and return result
 371
 372       return (Num      => Num,
 373               Den      => Den,
 374               Rbase    => 0,
 375               Negative => Val.Negative);
 376    end Normalize;
 377
 378    ---------------
 379    -- Numerator --
 380    ---------------
 381
 382    function Numerator (Real : Ureal) return Uint is
 383    begin
 384       return Ureals.Table (Real).Num;
 385    end Numerator;
 386
 387    --------
 388    -- pr --
 389    --------
 390
 391    procedure pr (Real : Ureal) is
 392    begin
 393       UR_Write (Real);
 394       Write_Eol;
 395    end pr;
 396
 397    -----------
 398    -- Rbase --
 399    -----------
 400
 401    function Rbase (Real : Ureal) return Nat is
 402    begin
 403       return Ureals.Table (Real).Rbase;
 404    end Rbase;
 405
 406    -------------
 407    -- Release --
 408    -------------
 409
 410    procedure Release (M : Save_Mark) is
 411    begin
 412       Ureals.Set_Last (Ureal (M));
 413    end Release;
 414
 415    ----------
 416    -- Same --
 417    ----------
 418
 419    function Same (U1, U2 : Ureal) return Boolean is
 420    begin
 421       return Int (U1) = Int (U2);
 422    end Same;
 423
 424    -----------------
 425    -- Store_Ureal --
 426    -----------------
 427
 428    function Store_Ureal (Val : Ureal_Entry) return Ureal is
 429    begin
 430       Ureals.Increment_Last;
 431       Ureals.Table (Ureals.Last) := Val;
 432
 433       --  Normalize representation of signed values
 434
 435       if Val.Num < 0 then
 436          Ureals.Table (Ureals.Last).Negative := True;
 437          Ureals.Table (Ureals.Last).Num := -Val.Num;
 438       end if;
 439
 440       return Ureals.Last;
 441    end Store_Ureal;
 442
 443    ---------------
 444    -- Tree_Read --
 445    ---------------
 446
 447    procedure Tree_Read is
 448    begin
 449       pragma Assert (Num_Ureal_Constants = 10);
 450
 451       Ureals.Tree_Read;
 452       Tree_Read_Int (Int (UR_0));
 453       Tree_Read_Int (Int (UR_M_0));
 454       Tree_Read_Int (Int (UR_Tenth));
 455       Tree_Read_Int (Int (UR_Half));
 456       Tree_Read_Int (Int (UR_1));
 457       Tree_Read_Int (Int (UR_2));
 458       Tree_Read_Int (Int (UR_10));
 459       Tree_Read_Int (Int (UR_100));
 460       Tree_Read_Int (Int (UR_2_128));
 461       Tree_Read_Int (Int (UR_2_M_128));
 462
 463       --  Clear the normalization cache
 464
 465       Normalized_Real := No_Ureal;
 466    end Tree_Read;
 467
 468    ----------------
 469    -- Tree_Write --
 470    ----------------
 471
 472    procedure Tree_Write is
 473    begin
 474       pragma Assert (Num_Ureal_Constants = 10);
 475
 476       Ureals.Tree_Write;
 477       Tree_Write_Int (Int (UR_0));
 478       Tree_Write_Int (Int (UR_M_0));
 479       Tree_Write_Int (Int (UR_Tenth));
 480       Tree_Write_Int (Int (UR_Half));
 481       Tree_Write_Int (Int (UR_1));
 482       Tree_Write_Int (Int (UR_2));
 483       Tree_Write_Int (Int (UR_10));
 484       Tree_Write_Int (Int (UR_100));
 485       Tree_Write_Int (Int (UR_2_128));
 486       Tree_Write_Int (Int (UR_2_M_128));
 487    end Tree_Write;
 488
 489    ------------
 490    -- UR_Abs --
 491    ------------
 492
 493    function UR_Abs (Real : Ureal) return Ureal is
 494       Val : constant Ureal_Entry := Ureals.Table (Real);
 495
 496    begin
 497       return Store_Ureal (
 498                (Num      => Val.Num,
 499                 Den      => Val.Den,
 500                 Rbase    => Val.Rbase,
 501                 Negative => False));
 502    end UR_Abs;
 503
 504    ------------
 505    -- UR_Add --
 506    ------------
 507
 508    function UR_Add (Left : Uint; Right : Ureal) return Ureal is
 509    begin
 510       return UR_From_Uint (Left) + Right;
 511    end UR_Add;
 512
 513    function UR_Add (Left : Ureal; Right : Uint) return Ureal is
 514    begin
 515       return Left + UR_From_Uint (Right);
 516    end UR_Add;
 517
 518    function UR_Add (Left : Ureal; Right : Ureal) return Ureal is
 519       Lval : Ureal_Entry := Ureals.Table (Left);
 520       Rval : Ureal_Entry := Ureals.Table (Right);
 521
 522       Num  : Uint;
 523
 524    begin
 525       --  Note, in the temporary Ureal_Entry values used in this procedure,
 526       --  we store the sign as the sign of the numerator (i.e. xxx.Num may
 527       --  be negative, even though in stored entries this can never be so)
 528
 529       if Lval.Rbase /= 0 and then Lval.Rbase = Rval.Rbase then
 530
 531          declare
 532             Opd_Min, Opd_Max   : Ureal_Entry;
 533             Exp_Min, Exp_Max   : Uint;
 534
 535          begin
 536             if Lval.Negative then
 537                Lval.Num := (-Lval.Num);
 538             end if;
 539
 540             if Rval.Negative then
 541                Rval.Num := (-Rval.Num);
 542             end if;
 543
 544             if Lval.Den < Rval.Den then
 545                Exp_Min := Lval.Den;
 546                Exp_Max := Rval.Den;
 547                Opd_Min := Lval;
 548                Opd_Max := Rval;
 549             else
 550                Exp_Min := Rval.Den;
 551                Exp_Max := Lval.Den;
 552                Opd_Min := Rval;
 553                Opd_Max := Lval;
 554             end if;
 555
 556             Num :=
 557               Opd_Min.Num * Lval.Rbase ** (Exp_Max - Exp_Min) + Opd_Max.Num;
 558
 559             if Num = 0 then
 560                return Store_Ureal (
 561                         (Num      => Uint_0,
 562                          Den      => Uint_1,
 563                          Rbase    => 0,
 564                          Negative => Lval.Negative));
 565
 566             else
 567                return Store_Ureal (
 568                         (Num      => abs Num,
 569                          Den      => Exp_Max,
 570                          Rbase    => Lval.Rbase,
 571                          Negative => (Num < 0)));
 572             end if;
 573          end;
 574
 575       else
 576          declare
 577             Ln : Ureal_Entry := Normalize (Lval);
 578             Rn : Ureal_Entry := Normalize (Rval);
 579
 580          begin
 581             if Ln.Negative then
 582                Ln.Num := (-Ln.Num);
 583             end if;
 584
 585             if Rn.Negative then
 586                Rn.Num := (-Rn.Num);
 587             end if;
 588
 589             Num := (Ln.Num * Rn.Den) + (Rn.Num * Ln.Den);
 590
 591             if Num = 0 then
 592                return Store_Ureal (
 593                         (Num      => Uint_0,
 594                          Den      => Uint_1,
 595                          Rbase    => 0,
 596                          Negative => Lval.Negative));
 597
 598             else
 599                return Store_Ureal (
 600                         Normalize (
 601                           (Num      => abs Num,
 602                            Den      => Ln.Den * Rn.Den,
 603                            Rbase    => 0,
 604                            Negative => (Num < 0))));
 605             end if;
 606          end;
 607       end if;
 608    end UR_Add;
 609
 610    ----------------
 611    -- UR_Ceiling --
 612    ----------------
 613
 614    function UR_Ceiling (Real : Ureal) return Uint is
 615       Val : constant Ureal_Entry := Normalize (Ureals.Table (Real));
 616
 617    begin
 618       if Val.Negative then
 619          return UI_Negate (Val.Num / Val.Den);
 620       else
 621          return (Val.Num + Val.Den - 1) / Val.Den;
 622       end if;
 623    end UR_Ceiling;
 624
 625    ------------
 626    -- UR_Div --
 627    ------------
 628
 629    function UR_Div (Left : Uint; Right : Ureal) return Ureal is
 630    begin
 631       return UR_From_Uint (Left) / Right;
 632    end UR_Div;
 633
 634    function UR_Div (Left : Ureal; Right : Uint) return Ureal is
 635    begin
 636       return Left / UR_From_Uint (Right);
 637    end UR_Div;
 638
 639    function UR_Div (Left, Right : Ureal) return Ureal is
 640       Lval : constant Ureal_Entry := Ureals.Table (Left);
 641       Rval : constant Ureal_Entry := Ureals.Table (Right);
 642       Rneg : constant Boolean     := Rval.Negative xor Lval.Negative;
 643
 644    begin
 645       pragma Assert (Rval.Num /= Uint_0);
 646
 647       if Lval.Rbase = 0 then
 648
 649          if Rval.Rbase = 0 then
 650             return Store_Ureal (
 651                      Normalize (
 652                        (Num      => Lval.Num * Rval.Den,
 653                         Den      => Lval.Den * Rval.Num,
 654                         Rbase    => 0,
 655                         Negative => Rneg)));
 656
 657          elsif Is_Integer (Lval.Num, Rval.Num * Lval.Den) then
 658             return Store_Ureal (
 659                      (Num      => Lval.Num / (Rval.Num * Lval.Den),
 660                       Den      => (-Rval.Den),
 661                       Rbase    => Rval.Rbase,
 662                       Negative => Rneg));
 663
 664          elsif Rval.Den < 0 then
 665             return Store_Ureal (
 666                      Normalize (
 667                        (Num      => Lval.Num,
 668                         Den      => Rval.Rbase ** (-Rval.Den) *
 669                                     Rval.Num *
 670                                     Lval.Den,
 671                         Rbase    => 0,
 672                         Negative => Rneg)));
 673
 674          else
 675             return Store_Ureal (
 676                      Normalize (
 677                        (Num      => Lval.Num * Rval.Rbase ** Rval.Den,
 678                         Den      => Rval.Num * Lval.Den,
 679                         Rbase    => 0,
 680                         Negative => Rneg)));
 681          end if;
 682
 683       elsif Is_Integer (Lval.Num, Rval.Num) then
 684
 685          if Rval.Rbase = Lval.Rbase then
 686             return Store_Ureal (
 687                      (Num      => Lval.Num / Rval.Num,
 688                       Den      => Lval.Den - Rval.Den,
 689                       Rbase    => Lval.Rbase,
 690                       Negative => Rneg));
 691
 692          elsif Rval.Rbase = 0 then
 693             return Store_Ureal (
 694                      (Num      => (Lval.Num / Rval.Num) * Rval.Den,
 695                       Den      => Lval.Den,
 696                       Rbase    => Lval.Rbase,
 697                       Negative => Rneg));
 698
 699          elsif Rval.Den < 0 then
 700             declare
 701                Num, Den : Uint;
 702
 703             begin
 704                if Lval.Den < 0 then
 705                   Num := (Lval.Num / Rval.Num) * (Lval.Rbase ** (-Lval.Den));
 706                   Den := Rval.Rbase ** (-Rval.Den);
 707                else
 708                   Num := Lval.Num / Rval.Num;
 709                   Den := (Lval.Rbase ** Lval.Den) *
 710                          (Rval.Rbase ** (-Rval.Den));
 711                end if;
 712
 713                return Store_Ureal (
 714                         (Num      => Num,
 715                          Den      => Den,
 716                          Rbase    => 0,
 717                          Negative => Rneg));
 718             end;
 719
 720          else
 721             return Store_Ureal (
 722                      (Num      => (Lval.Num / Rval.Num) *
 723                                   (Rval.Rbase ** Rval.Den),
 724                       Den      => Lval.Den,
 725                       Rbase    => Lval.Rbase,
 726                       Negative => Rneg));
 727          end if;
 728
 729       else
 730          declare
 731             Num, Den : Uint;
 732
 733          begin
 734             if Lval.Den < 0 then
 735                Num := Lval.Num * (Lval.Rbase ** (-Lval.Den));
 736                Den := Rval.Num;
 737
 738             else
 739                Num := Lval.Num;
 740                Den := Rval.Num * (Lval.Rbase ** Lval.Den);
 741             end if;
 742
 743             if Rval.Rbase /= 0 then
 744                if Rval.Den < 0 then
 745                   Den := Den * (Rval.Rbase ** (-Rval.Den));
 746                else
 747                   Num := Num * (Rval.Rbase ** Rval.Den);
 748                end if;
 749
 750             else
 751                Num := Num * Rval.Den;
 752             end if;
 753
 754             return Store_Ureal (
 755                      Normalize (
 756                        (Num      => Num,
 757                         Den      => Den,
 758                         Rbase    => 0,
 759                         Negative => Rneg)));
 760          end;
 761       end if;
 762    end UR_Div;
 763
 764    -----------
 765    -- UR_Eq --
 766    -----------
 767
 768    function UR_Eq (Left, Right : Ureal) return Boolean is
 769    begin
 770       return not UR_Ne (Left, Right);
 771    end UR_Eq;
 772
 773    ---------------------
 774    -- UR_Exponentiate --
 775    ---------------------
 776
 777    function UR_Exponentiate (Real : Ureal; N : Uint) return Ureal is
 778       X    : constant Uint := abs N;
 779       Bas  : Ureal;
 780       Val  : Ureal_Entry;
 781       Neg  : Boolean;
 782       IBas : Uint;
 783
 784    begin
 785       --  If base is negative, then the resulting sign depends on whether
 786       --  the exponent is even or odd (even => positive, odd = negative)
 787
 788       if UR_Is_Negative (Real) then
 789          Neg := (N mod 2) /= 0;
 790          Bas := UR_Negate (Real);
 791       else
 792          Neg := False;
 793          Bas := Real;
 794       end if;
 795
 796       Val := Ureals.Table (Bas);
 797
 798       --  If the base is a small integer, then we can return the result in
 799       --  exponential form, which can save a lot of time for junk exponents.
 800
 801       IBas := UR_Trunc (Bas);
 802
 803       if IBas <= 16
 804         and then UR_From_Uint (IBas) = Bas
 805       then
 806          return Store_Ureal (
 807                  (Num      => Uint_1,
 808                   Den      => -N,
 809                   Rbase    => UI_To_Int (UR_Trunc (Bas)),
 810                   Negative => Neg));
 811
 812       --  If the exponent is negative then we raise the numerator and the
 813       --  denominator (after normalization) to the absolute value of the
 814       --  exponent and we return the reciprocal. An assert error will happen
 815       --  if the numerator is zero.
 816
 817       elsif N < 0 then
 818          pragma Assert (Val.Num /= 0);
 819          Val := Normalize (Val);
 820
 821          return Store_Ureal (
 822                  (Num      => Val.Den ** X,
 823                   Den      => Val.Num ** X,
 824                   Rbase    => 0,
 825                   Negative => Neg));
 826
 827       --  If positive, we distinguish the case when the base is not zero, in
 828       --  which case the new denominator is just the product of the old one
 829       --  with the exponent,
 830
 831       else
 832          if Val.Rbase /= 0 then
 833
 834             return Store_Ureal (
 835                     (Num      => Val.Num ** X,
 836                      Den      => Val.Den * X,
 837                      Rbase    => Val.Rbase,
 838                      Negative => Neg));
 839
 840          --  And when the base is zero, in which case we exponentiate
 841          --  the old denominator.
 842
 843          else
 844             return Store_Ureal (
 845                     (Num      => Val.Num ** X,
 846                      Den      => Val.Den ** X,
 847                      Rbase    => 0,
 848                      Negative => Neg));
 849          end if;
 850       end if;
 851    end UR_Exponentiate;
 852
 853    --------------
 854    -- UR_Floor --
 855    --------------
 856
 857    function UR_Floor (Real : Ureal) return Uint is
 858       Val : constant Ureal_Entry := Normalize (Ureals.Table (Real));
 859
 860    begin
 861       if Val.Negative then
 862          return UI_Negate ((Val.Num + Val.Den - 1) / Val.Den);
 863       else
 864          return Val.Num / Val.Den;
 865       end if;
 866    end UR_Floor;
 867
 868    ------------------------
 869    -- UR_From_Components --
 870    ------------------------
 871
 872    function UR_From_Components
 873      (Num      : Uint;
 874       Den      : Uint;
 875       Rbase    : Nat := 0;
 876       Negative : Boolean := False)
 877       return     Ureal
 878    is
 879    begin
 880       return Store_Ureal (
 881                (Num      => Num,
 882                 Den      => Den,
 883                 Rbase    => Rbase,
 884                 Negative => Negative));
 885    end UR_From_Components;
 886
 887    ------------------
 888    -- UR_From_Uint --
 889    ------------------
 890
 891    function UR_From_Uint (UI : Uint) return Ureal is
 892    begin
 893       return UR_From_Components
 894         (abs UI, Uint_1, Negative => (UI < 0));
 895    end UR_From_Uint;
 896
 897    -----------
 898    -- UR_Ge --
 899    -----------
 900
 901    function UR_Ge (Left, Right : Ureal) return Boolean is
 902    begin
 903       return not (Left < Right);
 904    end UR_Ge;
 905
 906    -----------
 907    -- UR_Gt --
 908    -----------
 909
 910    function UR_Gt (Left, Right : Ureal) return Boolean is
 911    begin
 912       return (Right < Left);
 913    end UR_Gt;
 914
 915    --------------------
 916    -- UR_Is_Negative --
 917    --------------------
 918
 919    function UR_Is_Negative (Real : Ureal) return Boolean is
 920    begin
 921       return Ureals.Table (Real).Negative;
 922    end UR_Is_Negative;
 923
 924    --------------------
 925    -- UR_Is_Positive --
 926    --------------------
 927
 928    function UR_Is_Positive (Real : Ureal) return Boolean is
 929    begin
 930       return not Ureals.Table (Real).Negative
 931         and then Ureals.Table (Real).Num /= 0;
 932    end UR_Is_Positive;
 933
 934    ----------------
 935    -- UR_Is_Zero --
 936    ----------------
 937
 938    function UR_Is_Zero (Real : Ureal) return Boolean is
 939    begin
 940       return Ureals.Table (Real).Num = 0;
 941    end UR_Is_Zero;
 942
 943    -----------
 944    -- UR_Le --
 945    -----------
 946
 947    function UR_Le (Left, Right : Ureal) return Boolean is
 948    begin
 949       return not (Right < Left);
 950    end UR_Le;
 951
 952    -----------
 953    -- UR_Lt --
 954    -----------
 955
 956    function UR_Lt (Left, Right : Ureal) return Boolean is
 957    begin
 958       --  An operand is not less than itself
 959
 960       if Same (Left, Right) then
 961          return False;
 962
 963       --  Deal with zero cases
 964
 965       elsif UR_Is_Zero (Left) then
 966          return UR_Is_Positive (Right);
 967
 968       elsif UR_Is_Zero (Right) then
 969          return Ureals.Table (Left).Negative;
 970
 971       --  Different signs are decisive (note we dealt with zero cases)
 972
 973       elsif Ureals.Table (Left).Negative
 974         and then not Ureals.Table (Right).Negative
 975       then
 976          return True;
 977
 978       elsif not Ureals.Table (Left).Negative
 979         and then Ureals.Table (Right).Negative
 980       then
 981          return False;
 982
 983       --  Signs are same, do rapid check based on worst case estimates of
 984       --  decimal exponent, which will often be decisive. Precise test
 985       --  depends on whether operands are positive or negative.
 986
 987       elsif Decimal_Exponent_Hi (Left) < Decimal_Exponent_Lo (Right) then
 988          return UR_Is_Positive (Left);
 989
 990       elsif Decimal_Exponent_Lo (Left) > Decimal_Exponent_Hi (Right) then
 991          return UR_Is_Negative (Left);
 992
 993       --  If we fall through, full gruesome test is required. This happens
 994       --  if the numbers are close together, or in some weird (/=10) base.
 995
 996       else
 997          declare
 998             Imrk   : constant Uintp.Save_Mark  := Mark;
 999             Rmrk   : constant Urealp.Save_Mark := Mark;
1000             Lval   : Ureal_Entry;
1001             Rval   : Ureal_Entry;
1002             Result : Boolean;
1003
1004          begin
1005             Lval := Ureals.Table (Left);
1006             Rval := Ureals.Table (Right);
1007
1008             --  An optimization. If both numbers are based, then subtract
1009             --  common value of base to avoid unnecessarily giant numbers
1010
1011             if Lval.Rbase = Rval.Rbase and then Lval.Rbase /= 0 then
1012                if Lval.Den < Rval.Den then
1013                   Rval.Den := Rval.Den - Lval.Den;
1014                   Lval.Den := Uint_0;
1015                else
1016                   Lval.Den := Lval.Den - Rval.Den;
1017                   Rval.Den := Uint_0;
1018                end if;
1019             end if;
1020
1021             Lval := Normalize (Lval);
1022             Rval := Normalize (Rval);
1023
1024             if Lval.Negative then
1025                Result := (Lval.Num * Rval.Den) > (Rval.Num * Lval.Den);
1026             else
1027                Result := (Lval.Num * Rval.Den) < (Rval.Num * Lval.Den);
1028             end if;
1029
1030             Release (Imrk);
1031             Release (Rmrk);
1032             return Result;
1033          end;
1034       end if;
1035    end UR_Lt;
1036
1037    ------------
1038    -- UR_Max --
1039    ------------
1040
1041    function UR_Max (Left, Right : Ureal) return Ureal is
1042    begin
1043       if Left >= Right then
1044          return Left;
1045       else
1046          return Right;
1047       end if;
1048    end UR_Max;
1049
1050    ------------
1051    -- UR_Min --
1052    ------------
1053
1054    function UR_Min (Left, Right : Ureal) return Ureal is
1055    begin
1056       if Left <= Right then
1057          return Left;
1058       else
1059          return Right;
1060       end if;
1061    end UR_Min;
1062
1063    ------------
1064    -- UR_Mul --
1065    ------------
1066
1067    function UR_Mul (Left : Uint; Right : Ureal) return Ureal is
1068    begin
1069       return UR_From_Uint (Left) * Right;
1070    end UR_Mul;
1071
1072    function UR_Mul (Left : Ureal; Right : Uint) return Ureal is
1073    begin
1074       return Left * UR_From_Uint (Right);
1075    end UR_Mul;
1076
1077    function UR_Mul (Left, Right : Ureal) return Ureal is
1078       Lval : constant Ureal_Entry := Ureals.Table (Left);
1079       Rval : constant Ureal_Entry := Ureals.Table (Right);
1080       Num  : Uint                 := Lval.Num * Rval.Num;
1081       Den  : Uint;
1082       Rneg : constant Boolean     := Lval.Negative xor Rval.Negative;
1083
1084    begin
1085       if Lval.Rbase = 0 then
1086          if Rval.Rbase = 0 then
1087             return Store_Ureal (
1088                      Normalize (
1089                         (Num      => Num,
1090                          Den      => Lval.Den * Rval.Den,
1091                          Rbase    => 0,
1092                          Negative => Rneg)));
1093
1094          elsif Is_Integer (Num, Lval.Den) then
1095             return Store_Ureal (
1096                      (Num      => Num / Lval.Den,
1097                       Den      => Rval.Den,
1098                       Rbase    => Rval.Rbase,
1099                       Negative => Rneg));
1100
1101          elsif Rval.Den < 0 then
1102             return Store_Ureal (
1103                      Normalize (
1104                        (Num      => Num * (Rval.Rbase ** (-Rval.Den)),
1105                         Den      => Lval.Den,
1106                         Rbase    => 0,
1107                         Negative => Rneg)));
1108
1109          else
1110             return Store_Ureal (
1111                      Normalize (
1112                        (Num      => Num,
1113                         Den      => Lval.Den * (Rval.Rbase ** Rval.Den),
1114                         Rbase    => 0,
1115                         Negative => Rneg)));
1116          end if;
1117
1118       elsif Lval.Rbase = Rval.Rbase then
1119          return Store_Ureal (
1120                   (Num      => Num,
1121                    Den      => Lval.Den + Rval.Den,
1122                    Rbase    => Lval.Rbase,
1123                    Negative => Rneg));
1124
1125       elsif Rval.Rbase = 0 then
1126          if Is_Integer (Num, Rval.Den) then
1127             return Store_Ureal (
1128                      (Num      => Num / Rval.Den,
1129                       Den      => Lval.Den,
1130                       Rbase    => Lval.Rbase,
1131                       Negative => Rneg));
1132
1133          elsif Lval.Den < 0 then
1134             return Store_Ureal (
1135                      Normalize (
1136                        (Num      => Num * (Lval.Rbase ** (-Lval.Den)),
1137                         Den      => Rval.Den,
1138                         Rbase    => 0,
1139                         Negative => Rneg)));
1140
1141          else
1142             return Store_Ureal (
1143                      Normalize (
1144                        (Num      => Num,
1145                         Den      => Rval.Den * (Lval.Rbase ** Lval.Den),
1146                         Rbase    => 0,
1147                         Negative => Rneg)));
1148          end if;
1149
1150       else
1151          Den := Uint_1;
1152
1153          if Lval.Den < 0 then
1154             Num := Num * (Lval.Rbase ** (-Lval.Den));
1155          else
1156             Den := Den * (Lval.Rbase ** Lval.Den);
1157          end if;
1158
1159          if Rval.Den < 0 then
1160             Num := Num * (Rval.Rbase ** (-Rval.Den));
1161          else
1162             Den := Den * (Rval.Rbase ** Rval.Den);
1163          end if;
1164
1165          return Store_Ureal (
1166                   Normalize (
1167                     (Num      => Num,
1168                      Den      => Den,
1169                      Rbase    => 0,
1170                      Negative => Rneg)));
1171       end if;
1172    end UR_Mul;
1173
1174    -----------
1175    -- UR_Ne --
1176    -----------
1177
1178    function UR_Ne (Left, Right : Ureal) return Boolean is
1179    begin
1180       --  Quick processing for case of identical Ureal values (note that
1181       --  this also deals with comparing two No_Ureal values).
1182
1183       if Same (Left, Right) then
1184          return False;
1185
1186       --  Deal with case of one or other operand is No_Ureal, but not both
1187
1188       elsif Same (Left, No_Ureal) or else Same (Right, No_Ureal) then
1189          return True;
1190
1191       --  Do quick check based on number of decimal digits
1192
1193       elsif Decimal_Exponent_Hi (Left) < Decimal_Exponent_Lo (Right) or else
1194             Decimal_Exponent_Lo (Left) > Decimal_Exponent_Hi (Right)
1195       then
1196          return True;
1197
1198       --  Otherwise full comparison is required
1199
1200       else
1201          declare
1202             Imrk   : constant Uintp.Save_Mark  := Mark;
1203             Rmrk   : constant Urealp.Save_Mark := Mark;
1204             Lval   : constant Ureal_Entry := Normalize (Ureals.Table (Left));
1205             Rval   : constant Ureal_Entry := Normalize (Ureals.Table (Right));
1206             Result : Boolean;
1207
1208          begin
1209             if UR_Is_Zero (Left) then
1210                return not UR_Is_Zero (Right);
1211
1212             elsif UR_Is_Zero (Right) then
1213                return not UR_Is_Zero (Left);
1214
1215             --  Both operands are non-zero
1216
1217             else
1218                Result :=
1219                   Rval.Negative /= Lval.Negative
1220                    or else Rval.Num /= Lval.Num
1221                    or else Rval.Den /= Lval.Den;
1222                Release (Imrk);
1223                Release (Rmrk);
1224                return Result;
1225             end if;
1226          end;
1227       end if;
1228    end UR_Ne;
1229
1230    ---------------
1231    -- UR_Negate --
1232    ---------------
1233
1234    function UR_Negate (Real : Ureal) return Ureal is
1235    begin
1236       return Store_Ureal (
1237                (Num      => Ureals.Table (Real).Num,
1238                 Den      => Ureals.Table (Real).Den,
1239                 Rbase    => Ureals.Table (Real).Rbase,
1240                 Negative => not Ureals.Table (Real).Negative));
1241    end UR_Negate;
1242
1243    ------------
1244    -- UR_Sub --
1245    ------------
1246
1247    function UR_Sub (Left : Uint; Right : Ureal) return Ureal is
1248    begin
1249       return UR_From_Uint (Left) + UR_Negate (Right);
1250    end UR_Sub;
1251
1252    function UR_Sub (Left : Ureal; Right : Uint) return Ureal is
1253    begin
1254       return Left + UR_From_Uint (-Right);
1255    end UR_Sub;
1256
1257    function UR_Sub (Left, Right : Ureal) return Ureal is
1258    begin
1259       return Left + UR_Negate (Right);
1260    end UR_Sub;
1261
1262    ----------------
1263    -- UR_To_Uint --
1264    ----------------
1265
1266    function UR_To_Uint (Real : Ureal) return Uint is
1267       Val : constant Ureal_Entry := Normalize (Ureals.Table (Real));
1268       Res : Uint;
1269
1270    begin
1271       Res := (Val.Num + (Val.Den / 2)) / Val.Den;
1272
1273       if Val.Negative then
1274          return UI_Negate (Res);
1275       else
1276          return Res;
1277       end if;
1278    end UR_To_Uint;
1279
1280    --------------
1281    -- UR_Trunc --
1282    --------------
1283
1284    function UR_Trunc (Real : Ureal) return Uint is
1285       Val : constant Ureal_Entry := Normalize (Ureals.Table (Real));
1286
1287    begin
1288       if Val.Negative then
1289          return -(Val.Num / Val.Den);
1290       else
1291          return Val.Num / Val.Den;
1292       end if;
1293    end UR_Trunc;
1294
1295    --------------
1296    -- UR_Write --
1297    --------------
1298
1299    procedure UR_Write (Real : Ureal) is
1300       Val : constant Ureal_Entry := Ureals.Table (Real);
1301
1302    begin
1303       --  If value is negative, we precede the constant by a minus sign
1304       --  and add an extra layer of parentheses on the outside since the
1305       --  minus sign is part of the value, not a negation operator.
1306
1307       if Val.Negative then
1308          Write_Str ("(-");
1309       end if;
1310
1311       --  Constants in base 10 can be written in normal Ada literal style
1312
1313       if Val.Rbase = 10 then
1314          UI_Write (Val.Num / 10);
1315          Write_Char ('.');
1316          UI_Write (Val.Num mod 10);
1317
1318          if Val.Den /= 0 then
1319             Write_Char ('E');
1320             UI_Write (1 - Val.Den);
1321          end if;
1322
1323       --  Constants in a base other than 10 can still be easily written
1324       --  in normal Ada literal style if the numerator is one.
1325
1326       elsif Val.Rbase /= 0 and then Val.Num = 1 then
1327          Write_Int (Val.Rbase);
1328          Write_Str ("#1.0#E");
1329          UI_Write (-Val.Den);
1330
1331       --  Other constants with a base other than 10 are written using one
1332       --  of the following forms, depending on the sign of the number
1333       --  and the sign of the exponent (= minus denominator value)
1334
1335       --    (numerator.0*base**exponent)
1336       --    (numerator.0*base**(-exponent))
1337
1338       elsif Val.Rbase /= 0 then
1339          Write_Char ('(');
1340          UI_Write (Val.Num, Decimal);
1341          Write_Str (".0*");
1342          Write_Int (Val.Rbase);
1343          Write_Str ("**");
1344
1345          if Val.Den <= 0 then
1346             UI_Write (-Val.Den, Decimal);
1347
1348          else
1349             Write_Str ("(-");
1350             UI_Write (Val.Den, Decimal);
1351             Write_Char (')');
1352          end if;
1353
1354          Write_Char (')');
1355
1356       --  Rational constants with a denominator of 1 can be written as
1357       --  a real literal for the numerator integer.
1358
1359       elsif Val.Den = 1 then
1360          UI_Write (Val.Num, Decimal);
1361          Write_Str (".0");
1362
1363       --  Non-based (rational) constants are written in (num/den) style
1364
1365       else
1366          Write_Char ('(');
1367          UI_Write (Val.Num, Decimal);
1368          Write_Str (".0/");
1369          UI_Write (Val.Den, Decimal);
1370          Write_Str (".0)");
1371       end if;
1372
1373       --  Add trailing paren for negative values
1374
1375       if Val.Negative then
1376          Write_Char (')');
1377       end if;
1378    end UR_Write;
1379
1380    -------------
1381    -- Ureal_0 --
1382    -------------
1383
1384    function Ureal_0 return Ureal is
1385    begin
1386       return UR_0;
1387    end Ureal_0;
1388
1389    -------------
1390    -- Ureal_1 --
1391    -------------
1392
1393    function Ureal_1 return Ureal is
1394    begin
1395       return UR_1;
1396    end Ureal_1;
1397
1398    -------------
1399    -- Ureal_2 --
1400    -------------
1401
1402    function Ureal_2 return Ureal is
1403    begin
1404       return UR_2;
1405    end Ureal_2;
1406
1407    --------------
1408    -- Ureal_10 --
1409    --------------
1410
1411    function Ureal_10 return Ureal is
1412    begin
1413       return UR_10;
1414    end Ureal_10;
1415
1416    ---------------
1417    -- Ureal_100 --
1418    ---------------
1419
1420    function Ureal_100 return Ureal is
1421    begin
1422       return UR_100;
1423    end Ureal_100;
1424
1425    -----------------
1426    -- Ureal_10_36 --
1427    -----------------
1428
1429    function Ureal_10_36 return Ureal is
1430    begin
1431       return UR_10_36;
1432    end Ureal_10_36;
1433
1434    -------------------
1435    -- Ureal_M_10_36 --
1436    -------------------
1437
1438    function Ureal_M_10_36 return Ureal is
1439    begin
1440       return UR_M_10_36;
1441    end Ureal_M_10_36;
1442
1443    -----------------
1444    -- Ureal_2_128 --
1445    -----------------
1446
1447    function Ureal_2_128 return Ureal is
1448    begin
1449       return UR_2_128;
1450    end Ureal_2_128;
1451
1452    ----------------
1453    -- Ureal_2_80 --
1454    ----------------
1455
1456    function Ureal_2_80 return Ureal is
1457    begin
1458       return UR_2_80;
1459    end Ureal_2_80;
1460
1461    -------------------
1462    -- Ureal_2_M_128 --
1463    -------------------
1464
1465    function Ureal_2_M_128 return Ureal is
1466    begin
1467       return UR_2_M_128;
1468    end Ureal_2_M_128;
1469
1470    -------------------
1471    -- Ureal_2_M_80 --
1472    -------------------
1473
1474    function Ureal_2_M_80 return Ureal is
1475    begin
1476       return UR_2_M_80;
1477    end Ureal_2_M_80;
1478
1479    ----------------
1480    -- Ureal_Half --
1481    ----------------
1482
1483    function Ureal_Half return Ureal is
1484    begin
1485       return UR_Half;
1486    end Ureal_Half;
1487
1488    ---------------
1489    -- Ureal_M_0 --
1490    ---------------
1491
1492    function Ureal_M_0 return Ureal is
1493    begin
1494       return UR_M_0;
1495    end Ureal_M_0;
1496
1497    -----------------
1498    -- Ureal_Tenth --
1499    -----------------
1500
1501    function Ureal_Tenth return Ureal is
1502    begin
1503       return UR_Tenth;
1504    end Ureal_Tenth;
1505
1506 end Urealp;