gcc/ada/s-fatgen.adb

   1 ------------------------------------------------------------------------------
   2 --                                                                          --
   3 --                         GNAT COMPILER COMPONENTS                         --
   4 --                                                                          --
   5 --                       S Y S T E M . F A T _ G E N                        --
   6 --                                                                          --
   7 --                                 B o d y                                  --
   8 --                                                                          --
   9 --                                                                          --
  10 --          Copyright (C) 1992-2001 Free Software Foundation, Inc.          --
  11 --                                                                          --
  12 -- GNAT is free software;  you can  redistribute it  and/or modify it under --
  13 -- terms of the  GNU General Public License as published  by the Free Soft- --
  14 -- ware  Foundation;  either version 2,  or (at your option) any later ver- --
  15 -- sion.  GNAT is distributed in the hope that it will be useful, but WITH- --
  16 -- OUT ANY WARRANTY;  without even the  implied warranty of MERCHANTABILITY --
  17 -- or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License --
  18 -- for  more details.  You should have  received  a copy of the GNU General --
  19 -- Public License  distributed with GNAT;  see file COPYING.  If not, write --
  20 -- to  the Free Software Foundation,  59 Temple Place - Suite 330,  Boston, --
  21 -- MA 02111-1307, USA.                                                      --
  22 --                                                                          --
  23 -- As a special exception,  if other files  instantiate  generics from this --
  24 -- unit, or you link  this unit with other files  to produce an executable, --
  25 -- this  unit  does not  by itself cause  the resulting  executable  to  be --
  26 -- covered  by the  GNU  General  Public  License.  This exception does not --
  27 -- however invalidate  any other reasons why  the executable file  might be --
  28 -- covered by the  GNU Public License.                                      --
  29 --                                                                          --
  30 -- GNAT was originally developed  by the GNAT team at  New York University. --
  31 -- Extensive contributions were provided by Ada Core Technologies Inc.      --
  32 --                                                                          --
  33 ------------------------------------------------------------------------------
  34
  35 --  The implementation here is portable to any IEEE implementation. It does
  36 --  not handle non-binary radix, and also assumes that model numbers and
  37 --  machine numbers are basically identical, which is not true of all possible
  38 --  floating-point implementations. On a non-IEEE machine, this body must be
  39 --  specialized appropriately, or better still, its generic instantiations
  40 --  should be replaced by efficient machine-specific code.
  41
  42 with Ada.Unchecked_Conversion;
  43 with System;
  44 package body System.Fat_Gen is
  45
  46    Float_Radix        : constant T := T (T'Machine_Radix);
  47    Float_Radix_Inv    : constant T := 1.0 / Float_Radix;
  48    Radix_To_M_Minus_1 : constant T := Float_Radix ** (T'Machine_Mantissa - 1);
  49
  50    pragma Assert (T'Machine_Radix = 2);
  51    --  This version does not handle radix 16
  52
  53    --  Constants for Decompose and Scaling
  54
  55    Rad    : constant T := T (T'Machine_Radix);
  56    Invrad : constant T := 1.0 / Rad;
  57
  58    subtype Expbits is Integer range 0 .. 6;
  59    --  2 ** (2 ** 7) might overflow.  how big can radix-16 exponents get?
  60
  61    Log_Power : constant array (Expbits) of Integer := (1, 2, 4, 8, 16, 32, 64);
  62
  63    R_Power : constant array (Expbits) of T :=
  64      (Rad **  1,
  65       Rad **  2,
  66       Rad **  4,
  67       Rad **  8,
  68       Rad ** 16,
  69       Rad ** 32,
  70       Rad ** 64);
  71
  72    R_Neg_Power : constant array (Expbits) of T :=
  73      (Invrad **  1,
  74       Invrad **  2,
  75       Invrad **  4,
  76       Invrad **  8,
  77       Invrad ** 16,
  78       Invrad ** 32,
  79       Invrad ** 64);
  80
  81    -----------------------
  82    -- Local Subprograms --
  83    -----------------------
  84
  85    procedure Decompose (XX : T; Frac : out T; Expo : out UI);
  86    --  Decomposes a floating-point number into fraction and exponent parts
  87
  88    function Gradual_Scaling  (Adjustment : UI) return T;
  89    --  Like Scaling with a first argument of 1.0, but returns the smallest
  90    --  denormal rather than zero when the adjustment is smaller than
  91    --  Machine_Emin. Used for Succ and Pred.
  92
  93    --------------
  94    -- Adjacent --
  95    --------------
  96
  97    function Adjacent (X, Towards : T) return T is
  98    begin
  99       if Towards = X then
 100          return X;
 101
 102       elsif Towards > X then
 103          return Succ (X);
 104
 105       else
 106          return Pred (X);
 107       end if;
 108    end Adjacent;
 109
 110    -------------
 111    -- Ceiling --
 112    -------------
 113
 114    function Ceiling (X : T) return T is
 115       XT : constant T := Truncation (X);
 116
 117    begin
 118       if X <= 0.0 then
 119          return XT;
 120
 121       elsif X = XT then
 122          return X;
 123
 124       else
 125          return XT + 1.0;
 126       end if;
 127    end Ceiling;
 128
 129    -------------
 130    -- Compose --
 131    -------------
 132
 133    function Compose (Fraction : T; Exponent : UI) return T is
 134       Arg_Frac : T;
 135       Arg_Exp  : UI;
 136
 137    begin
 138       Decompose (Fraction, Arg_Frac, Arg_Exp);
 139       return Scaling (Arg_Frac, Exponent);
 140    end Compose;
 141
 142    ---------------
 143    -- Copy_Sign --
 144    ---------------
 145
 146    function Copy_Sign (Value, Sign : T) return T is
 147       Result : T;
 148
 149       function Is_Negative (V : T) return Boolean;
 150       pragma Import (Intrinsic, Is_Negative);
 151
 152    begin
 153       Result := abs Value;
 154
 155       if Is_Negative (Sign) then
 156          return -Result;
 157       else
 158          return Result;
 159       end if;
 160    end Copy_Sign;
 161
 162    ---------------
 163    -- Decompose --
 164    ---------------
 165
 166    procedure Decompose (XX : T; Frac : out T; Expo : out UI) is
 167       X : T := T'Machine (XX);
 168
 169    begin
 170       if X = 0.0 then
 171          Frac := X;
 172          Expo := 0;
 173
 174          --  More useful would be defining Expo to be T'Machine_Emin - 1 or
 175          --  T'Machine_Emin - T'Machine_Mantissa, which would preserve
 176          --  monotonicity of the exponent function ???
 177
 178       --  Check for infinities, transfinites, whatnot.
 179
 180       elsif X > T'Safe_Last then
 181          Frac := Invrad;
 182          Expo := T'Machine_Emax + 1;
 183
 184       elsif X < T'Safe_First then
 185          Frac := -Invrad;
 186          Expo := T'Machine_Emax + 2;    -- how many extra negative values?
 187
 188       else
 189          --  Case of nonzero finite x. Essentially, we just multiply
 190          --  by Rad ** (+-2**N) to reduce the range.
 191
 192          declare
 193             Ax : T  := abs X;
 194             Ex : UI := 0;
 195
 196          --  Ax * Rad ** Ex is invariant.
 197
 198          begin
 199             if Ax >= 1.0 then
 200                while Ax >= R_Power (Expbits'Last) loop
 201                   Ax := Ax * R_Neg_Power (Expbits'Last);
 202                   Ex := Ex + Log_Power (Expbits'Last);
 203                end loop;
 204
 205                --  Ax < Rad ** 64
 206
 207                for N in reverse Expbits'First .. Expbits'Last - 1 loop
 208                   if Ax >= R_Power (N) then
 209                      Ax := Ax * R_Neg_Power (N);
 210                      Ex := Ex + Log_Power (N);
 211                   end if;
 212
 213                   --  Ax < R_Power (N)
 214                end loop;
 215
 216                --  1 <= Ax < Rad
 217
 218                Ax := Ax * Invrad;
 219                Ex := Ex + 1;
 220
 221             else
 222                --  0 < ax < 1
 223
 224                while Ax < R_Neg_Power (Expbits'Last) loop
 225                   Ax := Ax * R_Power (Expbits'Last);
 226                   Ex := Ex - Log_Power (Expbits'Last);
 227                end loop;
 228
 229                --  Rad ** -64 <= Ax < 1
 230
 231                for N in reverse Expbits'First .. Expbits'Last - 1 loop
 232                   if Ax < R_Neg_Power (N) then
 233                      Ax := Ax * R_Power (N);
 234                      Ex := Ex - Log_Power (N);
 235                   end if;
 236
 237                   --  R_Neg_Power (N) <= Ax < 1
 238                end loop;
 239             end if;
 240
 241             if X > 0.0 then
 242                Frac := Ax;
 243             else
 244                Frac := -Ax;
 245             end if;
 246
 247             Expo := Ex;
 248          end;
 249       end if;
 250    end Decompose;
 251
 252    --------------
 253    -- Exponent --
 254    --------------
 255
 256    function Exponent (X : T) return UI is
 257       X_Frac : T;
 258       X_Exp  : UI;
 259
 260    begin
 261       Decompose (X, X_Frac, X_Exp);
 262       return X_Exp;
 263    end Exponent;
 264
 265    -----------
 266    -- Floor --
 267    -----------
 268
 269    function Floor (X : T) return T is
 270       XT : constant T := Truncation (X);
 271
 272    begin
 273       if X >= 0.0 then
 274          return XT;
 275
 276       elsif XT = X then
 277          return X;
 278
 279       else
 280          return XT - 1.0;
 281       end if;
 282    end Floor;
 283
 284    --------------
 285    -- Fraction --
 286    --------------
 287
 288    function Fraction (X : T) return T is
 289       X_Frac : T;
 290       X_Exp  : UI;
 291
 292    begin
 293       Decompose (X, X_Frac, X_Exp);
 294       return X_Frac;
 295    end Fraction;
 296
 297    ---------------------
 298    -- Gradual_Scaling --
 299    ---------------------
 300
 301    function Gradual_Scaling  (Adjustment : UI) return T is
 302       Y  : T;
 303       Y1 : T;
 304       Ex : UI := Adjustment;
 305
 306    begin
 307       if Adjustment < T'Machine_Emin then
 308          Y  := 2.0 ** T'Machine_Emin;
 309          Y1 := Y;
 310          Ex := Ex - T'Machine_Emin;
 311
 312          while Ex <= 0 loop
 313             Y := T'Machine (Y / 2.0);
 314
 315             if Y = 0.0 then
 316                return Y1;
 317             end if;
 318
 319             Ex := Ex + 1;
 320             Y1 := Y;
 321          end loop;
 322
 323          return Y1;
 324
 325       else
 326          return Scaling (1.0, Adjustment);
 327       end if;
 328    end Gradual_Scaling;
 329
 330    ------------------
 331    -- Leading_Part --
 332    ------------------
 333
 334    function Leading_Part (X : T; Radix_Digits : UI) return T is
 335       L    : UI;
 336       Y, Z : T;
 337
 338    begin
 339       if Radix_Digits >= T'Machine_Mantissa then
 340          return X;
 341
 342       else
 343          L := Exponent (X) - Radix_Digits;
 344          Y := Truncation (Scaling (X, -L));
 345          Z := Scaling (Y, L);
 346          return Z;
 347       end if;
 348
 349    end Leading_Part;
 350
 351    -------------
 352    -- Machine --
 353    -------------
 354
 355    --  The trick with Machine is to force the compiler to store the result
 356    --  in memory so that we do not have extra precision used. The compiler
 357    --  is clever, so we have to outwit its possible optimizations! We do
 358    --  this by using an intermediate pragma Volatile location.
 359
 360    function Machine (X : T) return T is
 361       Temp : T;
 362       pragma Volatile (Temp);
 363
 364    begin
 365       Temp := X;
 366       return Temp;
 367    end Machine;
 368
 369    -----------
 370    -- Model --
 371    -----------
 372
 373    --  We treat Model as identical to Machine. This is true of IEEE and other
 374    --  nice floating-point systems, but not necessarily true of all systems.
 375
 376    function Model (X : T) return T is
 377    begin
 378       return Machine (X);
 379    end Model;
 380
 381    ----------
 382    -- Pred --
 383    ----------
 384
 385    --  Subtract from the given number a number equivalent to the value of its
 386    --  least significant bit. Given that the most significant bit represents
 387    --  a value of 1.0 * radix ** (exp - 1), the value we want is obtained by
 388    --  shifting this by (mantissa-1) bits to the right, i.e. decreasing the
 389    --  exponent by that amount.
 390
 391    --  Zero has to be treated specially, since its exponent is zero
 392
 393    function Pred (X : T) return T is
 394       X_Frac : T;
 395       X_Exp  : UI;
 396
 397    begin
 398       if X = 0.0 then
 399          return -Succ (X);
 400
 401       else
 402          Decompose (X, X_Frac, X_Exp);
 403
 404          --  A special case, if the number we had was a positive power of
 405          --  two, then we want to subtract half of what we would otherwise
 406          --  subtract, since the exponent is going to be reduced.
 407
 408          if X_Frac = 0.5 and then X > 0.0 then
 409             return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
 410
 411          --  Otherwise the exponent stays the same
 412
 413          else
 414             return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa);
 415          end if;
 416       end if;
 417    end Pred;
 418
 419    ---------------
 420    -- Remainder --
 421    ---------------
 422
 423    function Remainder (X, Y : T) return T is
 424       A        : T;
 425       B        : T;
 426       Arg      : T;
 427       P        : T;
 428       Arg_Frac : T;
 429       P_Frac   : T;
 430       Sign_X   : T;
 431       IEEE_Rem : T;
 432       Arg_Exp  : UI;
 433       P_Exp    : UI;
 434       K        : UI;
 435       P_Even   : Boolean;
 436
 437    begin
 438       if X > 0.0 then
 439          Sign_X :=  1.0;
 440          Arg := X;
 441       else
 442          Sign_X := -1.0;
 443          Arg := -X;
 444       end if;
 445
 446       P := abs Y;
 447
 448       if Arg < P then
 449          P_Even := True;
 450          IEEE_Rem := Arg;
 451          P_Exp := Exponent (P);
 452
 453       else
 454          Decompose (Arg, Arg_Frac, Arg_Exp);
 455          Decompose (P,   P_Frac,   P_Exp);
 456
 457          P := Compose (P_Frac, Arg_Exp);
 458          K := Arg_Exp - P_Exp;
 459          P_Even := True;
 460          IEEE_Rem := Arg;
 461
 462          for Cnt in reverse 0 .. K loop
 463             if IEEE_Rem >= P then
 464                P_Even := False;
 465                IEEE_Rem := IEEE_Rem - P;
 466             else
 467                P_Even := True;
 468             end if;
 469
 470             P := P * 0.5;
 471          end loop;
 472       end if;
 473
 474       --  That completes the calculation of modulus remainder. The final
 475       --  step is get the IEEE remainder. Here we need to compare Rem with
 476       --  (abs Y) / 2. We must be careful of unrepresentable Y/2 value
 477       --  caused by subnormal numbers
 478
 479       if P_Exp >= 0 then
 480          A := IEEE_Rem;
 481          B := abs Y * 0.5;
 482
 483       else
 484          A := IEEE_Rem * 2.0;
 485          B := abs Y;
 486       end if;
 487
 488       if A > B or else (A = B and then not P_Even) then
 489          IEEE_Rem := IEEE_Rem - abs Y;
 490       end if;
 491
 492       return Sign_X * IEEE_Rem;
 493
 494    end Remainder;
 495
 496    --------------
 497    -- Rounding --
 498    --------------
 499
 500    function Rounding (X : T) return T is
 501       Result : T;
 502       Tail   : T;
 503
 504    begin
 505       Result := Truncation (abs X);
 506       Tail   := abs X - Result;
 507
 508       if Tail >= 0.5  then
 509          Result := Result + 1.0;
 510       end if;
 511
 512       if X > 0.0 then
 513          return Result;
 514
 515       elsif X < 0.0 then
 516          return -Result;
 517
 518       --  For zero case, make sure sign of zero is preserved
 519
 520       else
 521          return X;
 522       end if;
 523
 524    end Rounding;
 525
 526    -------------
 527    -- Scaling --
 528    -------------
 529
 530    --  Return x * rad ** adjustment quickly,
 531    --  or quietly underflow to zero, or overflow naturally.
 532
 533    function Scaling (X : T; Adjustment : UI) return T is
 534    begin
 535       if X = 0.0 or else Adjustment = 0 then
 536          return X;
 537       end if;
 538
 539       --  Nonzero x. essentially, just multiply repeatedly by Rad ** (+-2**n).
 540
 541       declare
 542          Y  : T  := X;
 543          Ex : UI := Adjustment;
 544
 545       --  Y * Rad ** Ex is invariant
 546
 547       begin
 548          if Ex < 0 then
 549             while Ex <= -Log_Power (Expbits'Last) loop
 550                Y := Y * R_Neg_Power (Expbits'Last);
 551                Ex := Ex + Log_Power (Expbits'Last);
 552             end loop;
 553
 554             --  -64 < Ex <= 0
 555
 556             for N in reverse Expbits'First .. Expbits'Last - 1 loop
 557                if Ex <= -Log_Power (N) then
 558                   Y := Y * R_Neg_Power (N);
 559                   Ex := Ex + Log_Power (N);
 560                end if;
 561
 562                --  -Log_Power (N) < Ex <= 0
 563             end loop;
 564
 565             --  Ex = 0
 566
 567          else
 568             --  Ex >= 0
 569
 570             while Ex >= Log_Power (Expbits'Last) loop
 571                Y := Y * R_Power (Expbits'Last);
 572                Ex := Ex - Log_Power (Expbits'Last);
 573             end loop;
 574
 575             --  0 <= Ex < 64
 576
 577             for N in reverse Expbits'First .. Expbits'Last - 1 loop
 578                if Ex >= Log_Power (N) then
 579                   Y := Y * R_Power (N);
 580                   Ex := Ex - Log_Power (N);
 581                end if;
 582
 583                --  0 <= Ex < Log_Power (N)
 584             end loop;
 585
 586             --  Ex = 0
 587          end if;
 588          return Y;
 589       end;
 590    end Scaling;
 591
 592    ----------
 593    -- Succ --
 594    ----------
 595
 596    --  Similar computation to that of Pred: find value of least significant
 597    --  bit of given number, and add. Zero has to be treated specially since
 598    --  the exponent can be zero, and also we want the smallest denormal if
 599    --  denormals are supported.
 600
 601    function Succ (X : T) return T is
 602       X_Frac : T;
 603       X_Exp  : UI;
 604       X1, X2 : T;
 605
 606    begin
 607       if X = 0.0 then
 608          X1 := 2.0 ** T'Machine_Emin;
 609
 610          --  Following loop generates smallest denormal
 611
 612          loop
 613             X2 := T'Machine (X1 / 2.0);
 614             exit when X2 = 0.0;
 615             X1 := X2;
 616          end loop;
 617
 618          return X1;
 619
 620       else
 621          Decompose (X, X_Frac, X_Exp);
 622
 623          --  A special case, if the number we had was a negative power of
 624          --  two, then we want to add half of what we would otherwise add,
 625          --  since the exponent is going to be reduced.
 626
 627          if X_Frac = 0.5 and then X < 0.0 then
 628             return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
 629
 630          --  Otherwise the exponent stays the same
 631
 632          else
 633             return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa);
 634          end if;
 635       end if;
 636    end Succ;
 637
 638    ----------------
 639    -- Truncation --
 640    ----------------
 641
 642    --  The basic approach is to compute
 643
 644    --    T'Machine (RM1 + N) - RM1.
 645
 646    --  where N >= 0.0 and RM1 = radix ** (mantissa - 1)
 647
 648    --  This works provided that the intermediate result (RM1 + N) does not
 649    --  have extra precision (which is why we call Machine). When we compute
 650    --  RM1 + N, the exponent of N will be normalized and the mantissa shifted
 651    --  shifted appropriately so the lower order bits, which cannot contribute
 652    --  to the integer part of N, fall off on the right. When we subtract RM1
 653    --  again, the significant bits of N are shifted to the left, and what we
 654    --  have is an integer, because only the first e bits are different from
 655    --  zero (assuming binary radix here).
 656
 657    function Truncation (X : T) return T is
 658       Result : T;
 659
 660    begin
 661       Result := abs X;
 662
 663       if Result >= Radix_To_M_Minus_1 then
 664          return Machine (X);
 665
 666       else
 667          Result := Machine (Radix_To_M_Minus_1 + Result) - Radix_To_M_Minus_1;
 668
 669          if Result > abs X  then
 670             Result := Result - 1.0;
 671          end if;
 672
 673          if X > 0.0 then
 674             return  Result;
 675
 676          elsif X < 0.0 then
 677             return -Result;
 678
 679          --  For zero case, make sure sign of zero is preserved
 680
 681          else
 682             return X;
 683          end if;
 684       end if;
 685
 686    end Truncation;
 687
 688    -----------------------
 689    -- Unbiased_Rounding --
 690    -----------------------
 691
 692    function Unbiased_Rounding (X : T) return T is
 693       Abs_X  : constant T := abs X;
 694       Result : T;
 695       Tail   : T;
 696
 697    begin
 698       Result := Truncation (Abs_X);
 699       Tail   := Abs_X - Result;
 700
 701       if Tail > 0.5  then
 702          Result := Result + 1.0;
 703
 704       elsif Tail = 0.5 then
 705          Result := 2.0 * Truncation ((Result / 2.0) + 0.5);
 706       end if;
 707
 708       if X > 0.0 then
 709          return Result;
 710
 711       elsif X < 0.0 then
 712          return -Result;
 713
 714       --  For zero case, make sure sign of zero is preserved
 715
 716       else
 717          return X;
 718       end if;
 719
 720    end Unbiased_Rounding;
 721
 722    -----------
 723    -- Valid --
 724    -----------
 725
 726    function Valid (X : access T) return Boolean is
 727
 728       IEEE_Emin : constant Integer := T'Machine_Emin - 1;
 729       IEEE_Emax : constant Integer := T'Machine_Emax - 1;
 730
 731       IEEE_Bias : constant Integer := -(IEEE_Emin - 1);
 732
 733       subtype IEEE_Exponent_Range is
 734         Integer range IEEE_Emin - 1 .. IEEE_Emax + 1;
 735
 736       --  The implementation of this floating point attribute uses
 737       --  a representation type Float_Rep that allows direct access to
 738       --  the exponent and mantissa parts of a floating point number.
 739
 740       --  The Float_Rep type is an array of Float_Word elements. This
 741       --  representation is chosen to make it possible to size the
 742       --  type based on a generic parameter.
 743
 744       --  The following conditions must be met for all possible
 745       --  instantiations of the attributes package:
 746
 747       --    - T'Size is an integral multiple of Float_Word'Size
 748
 749       --    - The exponent and sign are completely contained in a single
 750       --      component of Float_Rep, named Most_Significant_Word (MSW).
 751
 752       --    - The sign occupies the most significant bit of the MSW
 753       --      and the exponent is in the following bits.
 754       --      Unused bits (if any) are in the least significant part.
 755
 756       type Float_Word is mod 2**32;
 757       type Rep_Index is range 0 .. 7;
 758
 759       Rep_Last : constant Rep_Index := (T'Size - 1) / Float_Word'Size;
 760
 761       type Float_Rep is array (Rep_Index range 0 .. Rep_Last) of Float_Word;
 762
 763       Most_Significant_Word : constant Rep_Index :=
 764                                 Rep_Last * Standard'Default_Bit_Order;
 765       --  Finding the location of the Exponent_Word is a bit tricky.
 766       --  In general we assume Word_Order = Bit_Order.
 767       --  This expression needs to be refined for VMS.
 768
 769       Exponent_Factor : constant Float_Word :=
 770                           2**(Float_Word'Size - 1) /
 771                             Float_Word (IEEE_Emax - IEEE_Emin + 3) *
 772                               Boolean'Pos (T'Size /= 96) +
 773                                 Boolean'Pos (T'Size = 96);
 774       --  Factor that the extracted exponent needs to be divided by
 775       --  to be in range 0 .. IEEE_Emax - IEEE_Emin + 2.
 776       --  Special kludge: Exponent_Factor is 0 for x86 double extended
 777       --  as GCC adds 16 unused bits to the type.
 778
 779       Exponent_Mask : constant Float_Word :=
 780                         Float_Word (IEEE_Emax - IEEE_Emin + 2) *
 781                           Exponent_Factor;
 782       --  Value needed to mask out the exponent field.
 783       --  This assumes that the range IEEE_Emin - 1 .. IEEE_Emax + 1
 784       --  contains 2**N values, for some N in Natural.
 785
 786       function To_Float is new Ada.Unchecked_Conversion (Float_Rep, T);
 787
 788       type Float_Access is access all T;
 789       function To_Address is
 790          new Ada.Unchecked_Conversion (Float_Access, System.Address);
 791
 792       XA : constant System.Address := To_Address (Float_Access (X));
 793
 794       R : Float_Rep;
 795       pragma Import (Ada, R);
 796       for R'Address use XA;
 797       --  R is a view of the input floating-point parameter. Note that we
 798       --  must avoid copying the actual bits of this parameter in float
 799       --  form (since it may be a signalling NaN.
 800
 801       E  : constant IEEE_Exponent_Range :=
 802              Integer ((R (Most_Significant_Word) and Exponent_Mask) /
 803                                                         Exponent_Factor)
 804                - IEEE_Bias;
 805       --  Mask/Shift T to only get bits from the exponent
 806       --  Then convert biased value to integer value.
 807
 808       SR : Float_Rep;
 809       --  Float_Rep representation of significant of X.all
 810
 811    begin
 812       if T'Denorm then
 813
 814          --  All denormalized numbers are valid, so only invalid numbers
 815          --  are overflows and NaN's, both with exponent = Emax + 1.
 816
 817          return E /= IEEE_Emax + 1;
 818
 819       end if;
 820
 821       --  All denormalized numbers except 0.0 are invalid
 822
 823       --  Set exponent of X to zero, so we end up with the significand, which
 824       --  definitely is a valid number and can be converted back to a float.
 825
 826       SR := R;
 827       SR (Most_Significant_Word) :=
 828            (SR (Most_Significant_Word)
 829              and not Exponent_Mask) + Float_Word (IEEE_Bias) * Exponent_Factor;
 830
 831       return (E in IEEE_Emin .. IEEE_Emax) or else
 832          ((E = IEEE_Emin - 1) and then abs To_Float (SR) = 1.0);
 833    end Valid;
 834
 835 end System.Fat_Gen;