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 --          Copyright (C) 1992-2009, 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 3,  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.                                     --
  17 --                                                                          --
  18 -- As a special exception under Section 7 of GPL version 3, you are granted --
  19 -- additional permissions described in the GCC Runtime Library Exception,   --
  20 -- version 3.1, as published by the Free Software Foundation.               --
  21 --                                                                          --
  22 -- You should have received a copy of the GNU General Public License and    --
  23 -- a copy of the GCC Runtime Library Exception along with this program;     --
  24 -- see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see    --
  25 -- <http://www.gnu.org/licenses/>.                                          --
  26 --                                                                          --
  27 -- GNAT was originally developed  by the GNAT team at  New York University. --
  28 -- Extensive contributions were provided by Ada Core Technologies Inc.      --
  29 --                                                                          --
  30 ------------------------------------------------------------------------------
  31
  32 --  The implementation here is portable to any IEEE implementation. It does
  33 --  not handle non-binary radix, and also assumes that model numbers and
  34 --  machine numbers are basically identical, which is not true of all possible
  35 --  floating-point implementations. On a non-IEEE machine, this body must be
  36 --  specialized appropriately, or better still, its generic instantiations
  37 --  should be replaced by efficient machine-specific code.
  38
  39 with Ada.Unchecked_Conversion;
  40 with System;
  41 package body System.Fat_Gen is
  42
  43    Float_Radix        : constant T := T (T'Machine_Radix);
  44    Radix_To_M_Minus_1 : constant T := Float_Radix ** (T'Machine_Mantissa - 1);
  45
  46    pragma Assert (T'Machine_Radix = 2);
  47    --  This version does not handle radix 16
  48
  49    --  Constants for Decompose and Scaling
  50
  51    Rad    : constant T := T (T'Machine_Radix);
  52    Invrad : constant T := 1.0 / Rad;
  53
  54    subtype Expbits is Integer range 0 .. 6;
  55    --  2 ** (2 ** 7) might overflow.  How big can radix-16 exponents get?
  56
  57    Log_Power : constant array (Expbits) of Integer := (1, 2, 4, 8, 16, 32, 64);
  58
  59    R_Power : constant array (Expbits) of T :=
  60      (Rad **  1,
  61       Rad **  2,
  62       Rad **  4,
  63       Rad **  8,
  64       Rad ** 16,
  65       Rad ** 32,
  66       Rad ** 64);
  67
  68    R_Neg_Power : constant array (Expbits) of T :=
  69      (Invrad **  1,
  70       Invrad **  2,
  71       Invrad **  4,
  72       Invrad **  8,
  73       Invrad ** 16,
  74       Invrad ** 32,
  75       Invrad ** 64);
  76
  77    -----------------------
  78    -- Local Subprograms --
  79    -----------------------
  80
  81    procedure Decompose (XX : T; Frac : out T; Expo : out UI);
  82    --  Decomposes a floating-point number into fraction and exponent parts.
  83    --  Both results are signed, with Frac having the sign of XX, and UI has
  84    --  the sign of the exponent. The absolute value of Frac is in the range
  85    --  0.0 <= Frac < 1.0. If Frac = 0.0 or -0.0, then Expo is always zero.
  86
  87    function Gradual_Scaling  (Adjustment : UI) return T;
  88    --  Like Scaling with a first argument of 1.0, but returns the smallest
  89    --  denormal rather than zero when the adjustment is smaller than
  90    --  Machine_Emin. Used for Succ and Pred.
  91
  92    --------------
  93    -- Adjacent --
  94    --------------
  95
  96    function Adjacent (X, Towards : T) return T is
  97    begin
  98       if Towards = X then
  99          return X;
 100       elsif Towards > X then
 101          return Succ (X);
 102       else
 103          return Pred (X);
 104       end if;
 105    end Adjacent;
 106
 107    -------------
 108    -- Ceiling --
 109    -------------
 110
 111    function Ceiling (X : T) return T is
 112       XT : constant T := Truncation (X);
 113    begin
 114       if X <= 0.0 then
 115          return XT;
 116       elsif X = XT then
 117          return X;
 118       else
 119          return XT + 1.0;
 120       end if;
 121    end Ceiling;
 122
 123    -------------
 124    -- Compose --
 125    -------------
 126
 127    function Compose (Fraction : T; Exponent : UI) return T is
 128       Arg_Frac : T;
 129       Arg_Exp  : UI;
 130       pragma Unreferenced (Arg_Exp);
 131    begin
 132       Decompose (Fraction, Arg_Frac, Arg_Exp);
 133       return Scaling (Arg_Frac, Exponent);
 134    end Compose;
 135
 136    ---------------
 137    -- Copy_Sign --
 138    ---------------
 139
 140    function Copy_Sign (Value, Sign : T) return T is
 141       Result : T;
 142
 143       function Is_Negative (V : T) return Boolean;
 144       pragma Import (Intrinsic, Is_Negative);
 145
 146    begin
 147       Result := abs Value;
 148
 149       if Is_Negative (Sign) then
 150          return -Result;
 151       else
 152          return Result;
 153       end if;
 154    end Copy_Sign;
 155
 156    ---------------
 157    -- Decompose --
 158    ---------------
 159
 160    procedure Decompose (XX : T; Frac : out T; Expo : out UI) is
 161       X : constant T := T'Machine (XX);
 162
 163    begin
 164       if X = 0.0 then
 165          Frac := X;
 166          Expo := 0;
 167
 168          --  More useful would be defining Expo to be T'Machine_Emin - 1 or
 169          --  T'Machine_Emin - T'Machine_Mantissa, which would preserve
 170          --  monotonicity of the exponent function ???
 171
 172       --  Check for infinities, transfinites, whatnot
 173
 174       elsif X > T'Safe_Last then
 175          Frac := Invrad;
 176          Expo := T'Machine_Emax + 1;
 177
 178       elsif X < T'Safe_First then
 179          Frac := -Invrad;
 180          Expo := T'Machine_Emax + 2;    -- how many extra negative values?
 181
 182       else
 183          --  Case of nonzero finite x. Essentially, we just multiply
 184          --  by Rad ** (+-2**N) to reduce the range.
 185
 186          declare
 187             Ax : T  := abs X;
 188             Ex : UI := 0;
 189
 190          --  Ax * Rad ** Ex is invariant
 191
 192          begin
 193             if Ax >= 1.0 then
 194                while Ax >= R_Power (Expbits'Last) loop
 195                   Ax := Ax * R_Neg_Power (Expbits'Last);
 196                   Ex := Ex + Log_Power (Expbits'Last);
 197                end loop;
 198
 199                --  Ax < Rad ** 64
 200
 201                for N in reverse Expbits'First .. Expbits'Last - 1 loop
 202                   if Ax >= R_Power (N) then
 203                      Ax := Ax * R_Neg_Power (N);
 204                      Ex := Ex + Log_Power (N);
 205                   end if;
 206
 207                   --  Ax < R_Power (N)
 208                end loop;
 209
 210                --  1 <= Ax < Rad
 211
 212                Ax := Ax * Invrad;
 213                Ex := Ex + 1;
 214
 215             else
 216                --  0 < ax < 1
 217
 218                while Ax < R_Neg_Power (Expbits'Last) loop
 219                   Ax := Ax * R_Power (Expbits'Last);
 220                   Ex := Ex - Log_Power (Expbits'Last);
 221                end loop;
 222
 223                --  Rad ** -64 <= Ax < 1
 224
 225                for N in reverse Expbits'First .. Expbits'Last - 1 loop
 226                   if Ax < R_Neg_Power (N) then
 227                      Ax := Ax * R_Power (N);
 228                      Ex := Ex - Log_Power (N);
 229                   end if;
 230
 231                   --  R_Neg_Power (N) <= Ax < 1
 232                end loop;
 233             end if;
 234
 235             Frac := (if X > 0.0 then Ax else -Ax);
 236             Expo := Ex;
 237          end;
 238       end if;
 239    end Decompose;
 240
 241    --------------
 242    -- Exponent --
 243    --------------
 244
 245    function Exponent (X : T) return UI is
 246       X_Frac : T;
 247       X_Exp  : UI;
 248       pragma Unreferenced (X_Frac);
 249    begin
 250       Decompose (X, X_Frac, X_Exp);
 251       return X_Exp;
 252    end Exponent;
 253
 254    -----------
 255    -- Floor --
 256    -----------
 257
 258    function Floor (X : T) return T is
 259       XT : constant T := Truncation (X);
 260    begin
 261       if X >= 0.0 then
 262          return XT;
 263       elsif XT = X then
 264          return X;
 265       else
 266          return XT - 1.0;
 267       end if;
 268    end Floor;
 269
 270    --------------
 271    -- Fraction --
 272    --------------
 273
 274    function Fraction (X : T) return T is
 275       X_Frac : T;
 276       X_Exp  : UI;
 277       pragma Unreferenced (X_Exp);
 278    begin
 279       Decompose (X, X_Frac, X_Exp);
 280       return X_Frac;
 281    end Fraction;
 282
 283    ---------------------
 284    -- Gradual_Scaling --
 285    ---------------------
 286
 287    function Gradual_Scaling  (Adjustment : UI) return T is
 288       Y  : T;
 289       Y1 : T;
 290       Ex : UI := Adjustment;
 291
 292    begin
 293       if Adjustment < T'Machine_Emin - 1 then
 294          Y  := 2.0 ** T'Machine_Emin;
 295          Y1 := Y;
 296          Ex := Ex - T'Machine_Emin;
 297          while Ex < 0 loop
 298             Y := T'Machine (Y / 2.0);
 299
 300             if Y = 0.0 then
 301                return Y1;
 302             end if;
 303
 304             Ex := Ex + 1;
 305             Y1 := Y;
 306          end loop;
 307
 308          return Y1;
 309
 310       else
 311          return Scaling (1.0, Adjustment);
 312       end if;
 313    end Gradual_Scaling;
 314
 315    ------------------
 316    -- Leading_Part --
 317    ------------------
 318
 319    function Leading_Part (X : T; Radix_Digits : UI) return T is
 320       L    : UI;
 321       Y, Z : T;
 322
 323    begin
 324       if Radix_Digits >= T'Machine_Mantissa then
 325          return X;
 326
 327       elsif Radix_Digits <= 0 then
 328          raise Constraint_Error;
 329
 330       else
 331          L := Exponent (X) - Radix_Digits;
 332          Y := Truncation (Scaling (X, -L));
 333          Z := Scaling (Y, L);
 334          return Z;
 335       end if;
 336    end Leading_Part;
 337
 338    -------------
 339    -- Machine --
 340    -------------
 341
 342    --  The trick with Machine is to force the compiler to store the result
 343    --  in memory so that we do not have extra precision used. The compiler
 344    --  is clever, so we have to outwit its possible optimizations! We do
 345    --  this by using an intermediate pragma Volatile location.
 346
 347    function Machine (X : T) return T is
 348       Temp : T;
 349       pragma Volatile (Temp);
 350    begin
 351       Temp := X;
 352       return Temp;
 353    end Machine;
 354
 355    ----------------------
 356    -- Machine_Rounding --
 357    ----------------------
 358
 359    --  For now, the implementation is identical to that of Rounding, which is
 360    --  a permissible behavior, but is not the most efficient possible approach.
 361
 362    function Machine_Rounding (X : T) return T is
 363       Result : T;
 364       Tail   : T;
 365
 366    begin
 367       Result := Truncation (abs X);
 368       Tail   := abs X - Result;
 369
 370       if Tail >= 0.5  then
 371          Result := Result + 1.0;
 372       end if;
 373
 374       if X > 0.0 then
 375          return Result;
 376
 377       elsif X < 0.0 then
 378          return -Result;
 379
 380       --  For zero case, make sure sign of zero is preserved
 381
 382       else
 383          return X;
 384       end if;
 385    end Machine_Rounding;
 386
 387    -----------
 388    -- Model --
 389    -----------
 390
 391    --  We treat Model as identical to Machine. This is true of IEEE and other
 392    --  nice floating-point systems, but not necessarily true of all systems.
 393
 394    function Model (X : T) return T is
 395    begin
 396       return Machine (X);
 397    end Model;
 398
 399    ----------
 400    -- Pred --
 401    ----------
 402
 403    --  Subtract from the given number a number equivalent to the value of its
 404    --  least significant bit. Given that the most significant bit represents
 405    --  a value of 1.0 * radix ** (exp - 1), the value we want is obtained by
 406    --  shifting this by (mantissa-1) bits to the right, i.e. decreasing the
 407    --  exponent by that amount.
 408
 409    --  Zero has to be treated specially, since its exponent is zero
 410
 411    function Pred (X : T) return T is
 412       X_Frac : T;
 413       X_Exp  : UI;
 414
 415    begin
 416       if X = 0.0 then
 417          return -Succ (X);
 418
 419       else
 420          Decompose (X, X_Frac, X_Exp);
 421
 422          --  A special case, if the number we had was a positive power of
 423          --  two, then we want to subtract half of what we would otherwise
 424          --  subtract, since the exponent is going to be reduced.
 425
 426          --  Note that X_Frac has the same sign as X, so if X_Frac is 0.5,
 427          --  then we know that we have a positive number (and hence a
 428          --  positive power of 2).
 429
 430          if X_Frac = 0.5 then
 431             return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
 432
 433          --  Otherwise the exponent is unchanged
 434
 435          else
 436             return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa);
 437          end if;
 438       end if;
 439    end Pred;
 440
 441    ---------------
 442    -- Remainder --
 443    ---------------
 444
 445    function Remainder (X, Y : T) return T is
 446       A        : T;
 447       B        : T;
 448       Arg      : T;
 449       P        : T;
 450       P_Frac   : T;
 451       Sign_X   : T;
 452       IEEE_Rem : T;
 453       Arg_Exp  : UI;
 454       P_Exp    : UI;
 455       K        : UI;
 456       P_Even   : Boolean;
 457
 458       Arg_Frac : T;
 459       pragma Unreferenced (Arg_Frac);
 460
 461    begin
 462       if Y = 0.0 then
 463          raise Constraint_Error;
 464       end if;
 465
 466       if X > 0.0 then
 467          Sign_X :=  1.0;
 468          Arg := X;
 469       else
 470          Sign_X := -1.0;
 471          Arg := -X;
 472       end if;
 473
 474       P := abs Y;
 475
 476       if Arg < P then
 477          P_Even := True;
 478          IEEE_Rem := Arg;
 479          P_Exp := Exponent (P);
 480
 481       else
 482          Decompose (Arg, Arg_Frac, Arg_Exp);
 483          Decompose (P,   P_Frac,   P_Exp);
 484
 485          P := Compose (P_Frac, Arg_Exp);
 486          K := Arg_Exp - P_Exp;
 487          P_Even := True;
 488          IEEE_Rem := Arg;
 489
 490          for Cnt in reverse 0 .. K loop
 491             if IEEE_Rem >= P then
 492                P_Even := False;
 493                IEEE_Rem := IEEE_Rem - P;
 494             else
 495                P_Even := True;
 496             end if;
 497
 498             P := P * 0.5;
 499          end loop;
 500       end if;
 501
 502       --  That completes the calculation of modulus remainder. The final
 503       --  step is get the IEEE remainder. Here we need to compare Rem with
 504       --  (abs Y) / 2. We must be careful of unrepresentable Y/2 value
 505       --  caused by subnormal numbers
 506
 507       if P_Exp >= 0 then
 508          A := IEEE_Rem;
 509          B := abs Y * 0.5;
 510
 511       else
 512          A := IEEE_Rem * 2.0;
 513          B := abs Y;
 514       end if;
 515
 516       if A > B or else (A = B and then not P_Even) then
 517          IEEE_Rem := IEEE_Rem - abs Y;
 518       end if;
 519
 520       return Sign_X * IEEE_Rem;
 521    end Remainder;
 522
 523    --------------
 524    -- Rounding --
 525    --------------
 526
 527    function Rounding (X : T) return T is
 528       Result : T;
 529       Tail   : T;
 530
 531    begin
 532       Result := Truncation (abs X);
 533       Tail   := abs X - Result;
 534
 535       if Tail >= 0.5  then
 536          Result := Result + 1.0;
 537       end if;
 538
 539       if X > 0.0 then
 540          return Result;
 541
 542       elsif X < 0.0 then
 543          return -Result;
 544
 545       --  For zero case, make sure sign of zero is preserved
 546
 547       else
 548          return X;
 549       end if;
 550    end Rounding;
 551
 552    -------------
 553    -- Scaling --
 554    -------------
 555
 556    --  Return x * rad ** adjustment quickly,
 557    --  or quietly underflow to zero, or overflow naturally.
 558
 559    function Scaling (X : T; Adjustment : UI) return T is
 560    begin
 561       if X = 0.0 or else Adjustment = 0 then
 562          return X;
 563       end if;
 564
 565       --  Nonzero x essentially, just multiply repeatedly by Rad ** (+-2**n)
 566
 567       declare
 568          Y  : T  := X;
 569          Ex : UI := Adjustment;
 570
 571       --  Y * Rad ** Ex is invariant
 572
 573       begin
 574          if Ex < 0 then
 575             while Ex <= -Log_Power (Expbits'Last) loop
 576                Y := Y * R_Neg_Power (Expbits'Last);
 577                Ex := Ex + Log_Power (Expbits'Last);
 578             end loop;
 579
 580             --  -64 < Ex <= 0
 581
 582             for N in reverse Expbits'First .. Expbits'Last - 1 loop
 583                if Ex <= -Log_Power (N) then
 584                   Y := Y * R_Neg_Power (N);
 585                   Ex := Ex + Log_Power (N);
 586                end if;
 587
 588                --  -Log_Power (N) < Ex <= 0
 589             end loop;
 590
 591             --  Ex = 0
 592
 593          else
 594             --  Ex >= 0
 595
 596             while Ex >= Log_Power (Expbits'Last) loop
 597                Y := Y * R_Power (Expbits'Last);
 598                Ex := Ex - Log_Power (Expbits'Last);
 599             end loop;
 600
 601             --  0 <= Ex < 64
 602
 603             for N in reverse Expbits'First .. Expbits'Last - 1 loop
 604                if Ex >= Log_Power (N) then
 605                   Y := Y * R_Power (N);
 606                   Ex := Ex - Log_Power (N);
 607                end if;
 608
 609                --  0 <= Ex < Log_Power (N)
 610
 611             end loop;
 612
 613             --  Ex = 0
 614          end if;
 615
 616          return Y;
 617       end;
 618    end Scaling;
 619
 620    ----------
 621    -- Succ --
 622    ----------
 623
 624    --  Similar computation to that of Pred: find value of least significant
 625    --  bit of given number, and add. Zero has to be treated specially since
 626    --  the exponent can be zero, and also we want the smallest denormal if
 627    --  denormals are supported.
 628
 629    function Succ (X : T) return T is
 630       X_Frac : T;
 631       X_Exp  : UI;
 632       X1, X2 : T;
 633
 634    begin
 635       if X = 0.0 then
 636          X1 := 2.0 ** T'Machine_Emin;
 637
 638          --  Following loop generates smallest denormal
 639
 640          loop
 641             X2 := T'Machine (X1 / 2.0);
 642             exit when X2 = 0.0;
 643             X1 := X2;
 644          end loop;
 645
 646          return X1;
 647
 648       else
 649          Decompose (X, X_Frac, X_Exp);
 650
 651          --  A special case, if the number we had was a negative power of
 652          --  two, then we want to add half of what we would otherwise add,
 653          --  since the exponent is going to be reduced.
 654
 655          --  Note that X_Frac has the same sign as X, so if X_Frac is -0.5,
 656          --  then we know that we have a negative number (and hence a
 657          --  negative power of 2).
 658
 659          if X_Frac = -0.5 then
 660             return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
 661
 662          --  Otherwise the exponent is unchanged
 663
 664          else
 665             return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa);
 666          end if;
 667       end if;
 668    end Succ;
 669
 670    ----------------
 671    -- Truncation --
 672    ----------------
 673
 674    --  The basic approach is to compute
 675
 676    --    T'Machine (RM1 + N) - RM1
 677
 678    --  where N >= 0.0 and RM1 = radix ** (mantissa - 1)
 679
 680    --  This works provided that the intermediate result (RM1 + N) does not
 681    --  have extra precision (which is why we call Machine). When we compute
 682    --  RM1 + N, the exponent of N will be normalized and the mantissa shifted
 683    --  shifted appropriately so the lower order bits, which cannot contribute
 684    --  to the integer part of N, fall off on the right. When we subtract RM1
 685    --  again, the significant bits of N are shifted to the left, and what we
 686    --  have is an integer, because only the first e bits are different from
 687    --  zero (assuming binary radix here).
 688
 689    function Truncation (X : T) return T is
 690       Result : T;
 691
 692    begin
 693       Result := abs X;
 694
 695       if Result >= Radix_To_M_Minus_1 then
 696          return Machine (X);
 697
 698       else
 699          Result := Machine (Radix_To_M_Minus_1 + Result) - Radix_To_M_Minus_1;
 700
 701          if Result > abs X  then
 702             Result := Result - 1.0;
 703          end if;
 704
 705          if X > 0.0 then
 706             return  Result;
 707
 708          elsif X < 0.0 then
 709             return -Result;
 710
 711          --  For zero case, make sure sign of zero is preserved
 712
 713          else
 714             return X;
 715          end if;
 716       end if;
 717    end Truncation;
 718
 719    -----------------------
 720    -- Unbiased_Rounding --
 721    -----------------------
 722
 723    function Unbiased_Rounding (X : T) return T is
 724       Abs_X  : constant T := abs X;
 725       Result : T;
 726       Tail   : T;
 727
 728    begin
 729       Result := Truncation (Abs_X);
 730       Tail   := Abs_X - Result;
 731
 732       if Tail > 0.5  then
 733          Result := Result + 1.0;
 734
 735       elsif Tail = 0.5 then
 736          Result := 2.0 * Truncation ((Result / 2.0) + 0.5);
 737       end if;
 738
 739       if X > 0.0 then
 740          return Result;
 741
 742       elsif X < 0.0 then
 743          return -Result;
 744
 745       --  For zero case, make sure sign of zero is preserved
 746
 747       else
 748          return X;
 749       end if;
 750    end Unbiased_Rounding;
 751
 752    -----------
 753    -- Valid --
 754    -----------
 755
 756    --  Note: this routine does not work for VAX float. We compensate for this
 757    --  in Exp_Attr by using the Valid functions in Vax_Float_Operations rather
 758    --  than the corresponding instantiation of this function.
 759
 760    function Valid (X : not null access T) return Boolean is
 761
 762       IEEE_Emin : constant Integer := T'Machine_Emin - 1;
 763       IEEE_Emax : constant Integer := T'Machine_Emax - 1;
 764
 765       IEEE_Bias : constant Integer := -(IEEE_Emin - 1);
 766
 767       subtype IEEE_Exponent_Range is
 768         Integer range IEEE_Emin - 1 .. IEEE_Emax + 1;
 769
 770       --  The implementation of this floating point attribute uses a
 771       --  representation type Float_Rep that allows direct access to the
 772       --  exponent and mantissa parts of a floating point number.
 773
 774       --  The Float_Rep type is an array of Float_Word elements. This
 775       --  representation is chosen to make it possible to size the type based
 776       --  on a generic parameter. Since the array size is known at compile
 777       --  time, efficient code can still be generated. The size of Float_Word
 778       --  elements should be large enough to allow accessing the exponent in
 779       --  one read, but small enough so that all floating point object sizes
 780       --  are a multiple of the Float_Word'Size.
 781
 782       --  The following conditions must be met for all possible
 783       --  instantiations of the attributes package:
 784
 785       --    - T'Size is an integral multiple of Float_Word'Size
 786
 787       --    - The exponent and sign are completely contained in a single
 788       --      component of Float_Rep, named Most_Significant_Word (MSW).
 789
 790       --    - The sign occupies the most significant bit of the MSW and the
 791       --      exponent is in the following bits. Unused bits (if any) are in
 792       --      the least significant part.
 793
 794       type Float_Word is mod 2**Positive'Min (System.Word_Size, 32);
 795       type Rep_Index is range 0 .. 7;
 796
 797       Rep_Words : constant Positive :=
 798          (T'Size + Float_Word'Size - 1) / Float_Word'Size;
 799       Rep_Last  : constant Rep_Index := Rep_Index'Min
 800         (Rep_Index (Rep_Words - 1), (T'Mantissa + 16) / Float_Word'Size);
 801       --  Determine the number of Float_Words needed for representing the
 802       --  entire floating-point value. Do not take into account excessive
 803       --  padding, as occurs on IA-64 where 80 bits floats get padded to 128
 804       --  bits. In general, the exponent field cannot be larger than 15 bits,
 805       --  even for 128-bit floating-point types, so the final format size
 806       --  won't be larger than T'Mantissa + 16.
 807
 808       type Float_Rep is
 809          array (Rep_Index range 0 .. Rep_Index (Rep_Words - 1)) of Float_Word;
 810
 811       pragma Suppress_Initialization (Float_Rep);
 812       --  This pragma suppresses the generation of an initialization procedure
 813       --  for type Float_Rep when operating in Initialize/Normalize_Scalars
 814       --  mode. This is not just a matter of efficiency, but of functionality,
 815       --  since Valid has a pragma Inline_Always, which is not permitted if
 816       --  there are nested subprograms present.
 817
 818       Most_Significant_Word : constant Rep_Index :=
 819                                 Rep_Last * Standard'Default_Bit_Order;
 820       --  Finding the location of the Exponent_Word is a bit tricky. In general
 821       --  we assume Word_Order = Bit_Order. This expression needs to be refined
 822       --  for VMS.
 823
 824       Exponent_Factor : constant Float_Word :=
 825                           2**(Float_Word'Size - 1) /
 826                             Float_Word (IEEE_Emax - IEEE_Emin + 3) *
 827                               Boolean'Pos (Most_Significant_Word /= 2) +
 828                                 Boolean'Pos (Most_Significant_Word = 2);
 829       --  Factor that the extracted exponent needs to be divided by to be in
 830       --  range 0 .. IEEE_Emax - IEEE_Emin + 2. Special kludge: Exponent_Factor
 831       --  is 1 for x86/IA64 double extended as GCC adds unused bits to the
 832       --  type.
 833
 834       Exponent_Mask : constant Float_Word :=
 835                         Float_Word (IEEE_Emax - IEEE_Emin + 2) *
 836                           Exponent_Factor;
 837       --  Value needed to mask out the exponent field. This assumes that the
 838       --  range IEEE_Emin - 1 .. IEEE_Emax + contains 2**N values, for some N
 839       --  in Natural.
 840
 841       function To_Float is new Ada.Unchecked_Conversion (Float_Rep, T);
 842
 843       type Float_Access is access all T;
 844       function To_Address is
 845          new Ada.Unchecked_Conversion (Float_Access, System.Address);
 846
 847       XA : constant System.Address := To_Address (Float_Access (X));
 848
 849       R : Float_Rep;
 850       pragma Import (Ada, R);
 851       for R'Address use XA;
 852       --  R is a view of the input floating-point parameter. Note that we
 853       --  must avoid copying the actual bits of this parameter in float
 854       --  form (since it may be a signalling NaN.
 855
 856       E  : constant IEEE_Exponent_Range :=
 857              Integer ((R (Most_Significant_Word) and Exponent_Mask) /
 858                                                         Exponent_Factor)
 859                - IEEE_Bias;
 860       --  Mask/Shift T to only get bits from the exponent. Then convert biased
 861       --  value to integer value.
 862
 863       SR : Float_Rep;
 864       --  Float_Rep representation of significant of X.all
 865
 866    begin
 867       if T'Denorm then
 868
 869          --  All denormalized numbers are valid, so the only invalid numbers
 870          --  are overflows and NaNs, both with exponent = Emax + 1.
 871
 872          return E /= IEEE_Emax + 1;
 873
 874       end if;
 875
 876       --  All denormalized numbers except 0.0 are invalid
 877
 878       --  Set exponent of X to zero, so we end up with the significand, which
 879       --  definitely is a valid number and can be converted back to a float.
 880
 881       SR := R;
 882       SR (Most_Significant_Word) :=
 883            (SR (Most_Significant_Word)
 884              and not Exponent_Mask) + Float_Word (IEEE_Bias) * Exponent_Factor;
 885
 886       return (E in IEEE_Emin .. IEEE_Emax) or else
 887          ((E = IEEE_Emin - 1) and then abs To_Float (SR) = 1.0);
 888    end Valid;
 889
 890    ---------------------
 891    -- Unaligned_Valid --
 892    ---------------------
 893
 894    function Unaligned_Valid (A : System.Address) return Boolean is
 895       subtype FS is String (1 .. T'Size / Character'Size);
 896       type FSP is access FS;
 897
 898       function To_FSP is new Ada.Unchecked_Conversion (Address, FSP);
 899
 900       Local_T : aliased T;
 901
 902    begin
 903       --  Note that we have to be sure that we do not load the value into a
 904       --  floating-point register, since a signalling NaN may cause a trap.
 905       --  The following assignment is what does the actual alignment, since
 906       --  we know that the target Local_T is aligned.
 907
 908       To_FSP (Local_T'Address).all := To_FSP (A).all;
 909
 910       --  Now that we have an aligned value, we can use the normal aligned
 911       --  version of Valid to obtain the required result.
 912
 913       return Valid (Local_T'Access);
 914    end Unaligned_Valid;
 915
 916 end System.Fat_Gen;