1 ------------------------------------------------------------------------------
3 -- GNAT COMPILER COMPONENTS --
5 -- S Y S T E M . F A T _ G E N --
9 -- Copyright (C) 1992-2010, Free Software Foundation, Inc. --
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. --
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. --
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/>. --
27 -- GNAT was originally developed by the GNAT team at New York University. --
28 -- Extensive contributions were provided by Ada Core Technologies Inc. --
30 ------------------------------------------------------------------------------
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.
39 with Ada.Unchecked_Conversion;
41 package body System.Fat_Gen is
43 Float_Radix : constant T := T (T'Machine_Radix);
44 Radix_To_M_Minus_1 : constant T := Float_Radix ** (T'Machine_Mantissa - 1);
46 pragma Assert (T'Machine_Radix = 2);
47 -- This version does not handle radix 16
49 -- Constants for Decompose and Scaling
51 Rad : constant T := T (T'Machine_Radix);
52 Invrad : constant T := 1.0 / Rad;
54 subtype Expbits is Integer range 0 .. 6;
55 -- 2 ** (2 ** 7) might overflow. How big can radix-16 exponents get?
57 Log_Power : constant array (Expbits) of Integer := (1, 2, 4, 8, 16, 32, 64);
59 R_Power : constant array (Expbits) of T :=
68 R_Neg_Power : constant array (Expbits) of T :=
77 -----------------------
78 -- Local Subprograms --
79 -----------------------
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.
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.
96 function Adjacent (X, Towards : T) return T is
100 elsif Towards > X then
111 function Ceiling (X : T) return T is
112 XT : constant T := Truncation (X);
127 function Compose (Fraction : T; Exponent : UI) return T is
130 pragma Unreferenced (Arg_Exp);
132 Decompose (Fraction, Arg_Frac, Arg_Exp);
133 return Scaling (Arg_Frac, Exponent);
140 function Copy_Sign (Value, Sign : T) return T is
143 function Is_Negative (V : T) return Boolean;
144 pragma Import (Intrinsic, Is_Negative);
149 if Is_Negative (Sign) then
160 procedure Decompose (XX : T; Frac : out T; Expo : out UI) is
161 X : constant T := T'Machine (XX);
166 -- The normalized exponent of zero is zero, see RM A.5.2(15)
171 -- Check for infinities, transfinites, whatnot
173 elsif X > T'Safe_Last then
175 Expo := T'Machine_Emax + 1;
177 elsif X < T'Safe_First then
179 Expo := T'Machine_Emax + 2; -- how many extra negative values?
182 -- Case of nonzero finite x. Essentially, we just multiply
183 -- by Rad ** (+-2**N) to reduce the range.
189 -- Ax * Rad ** Ex is invariant
193 while Ax >= R_Power (Expbits'Last) loop
194 Ax := Ax * R_Neg_Power (Expbits'Last);
195 Ex := Ex + Log_Power (Expbits'Last);
200 for N in reverse Expbits'First .. Expbits'Last - 1 loop
201 if Ax >= R_Power (N) then
202 Ax := Ax * R_Neg_Power (N);
203 Ex := Ex + Log_Power (N);
218 while Ax < R_Neg_Power (Expbits'Last) loop
219 Ax := Ax * R_Power (Expbits'Last);
220 Ex := Ex - Log_Power (Expbits'Last);
223 -- Rad ** -64 <= Ax < 1
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);
231 -- R_Neg_Power (N) <= Ax < 1
236 Frac := (if X > 0.0 then Ax else -Ax);
246 function Exponent (X : T) return UI is
249 pragma Unreferenced (X_Frac);
251 Decompose (X, X_Frac, X_Exp);
259 function Floor (X : T) return T is
260 XT : constant T := Truncation (X);
275 function Fraction (X : T) return T is
278 pragma Unreferenced (X_Exp);
280 Decompose (X, X_Frac, X_Exp);
284 ---------------------
285 -- Gradual_Scaling --
286 ---------------------
288 function Gradual_Scaling (Adjustment : UI) return T is
291 Ex : UI := Adjustment;
294 if Adjustment < T'Machine_Emin - 1 then
295 Y := 2.0 ** T'Machine_Emin;
297 Ex := Ex - T'Machine_Emin;
299 Y := T'Machine (Y / 2.0);
312 return Scaling (1.0, Adjustment);
320 function Leading_Part (X : T; Radix_Digits : UI) return T is
325 if Radix_Digits >= T'Machine_Mantissa then
328 elsif Radix_Digits <= 0 then
329 raise Constraint_Error;
332 L := Exponent (X) - Radix_Digits;
333 Y := Truncation (Scaling (X, -L));
343 -- The trick with Machine is to force the compiler to store the result
344 -- in memory so that we do not have extra precision used. The compiler
345 -- is clever, so we have to outwit its possible optimizations! We do
346 -- this by using an intermediate pragma Volatile location.
348 function Machine (X : T) return T is
350 pragma Volatile (Temp);
356 ----------------------
357 -- Machine_Rounding --
358 ----------------------
360 -- For now, the implementation is identical to that of Rounding, which is
361 -- a permissible behavior, but is not the most efficient possible approach.
363 function Machine_Rounding (X : T) return T is
368 Result := Truncation (abs X);
369 Tail := abs X - Result;
372 Result := Result + 1.0;
381 -- For zero case, make sure sign of zero is preserved
386 end Machine_Rounding;
392 -- We treat Model as identical to Machine. This is true of IEEE and other
393 -- nice floating-point systems, but not necessarily true of all systems.
395 function Model (X : T) return T is
404 -- Subtract from the given number a number equivalent to the value of its
405 -- least significant bit. Given that the most significant bit represents
406 -- a value of 1.0 * radix ** (exp - 1), the value we want is obtained by
407 -- shifting this by (mantissa-1) bits to the right, i.e. decreasing the
408 -- exponent by that amount.
410 -- Zero has to be treated specially, since its exponent is zero
412 function Pred (X : T) return T is
421 Decompose (X, X_Frac, X_Exp);
423 -- A special case, if the number we had was a positive power of
424 -- two, then we want to subtract half of what we would otherwise
425 -- subtract, since the exponent is going to be reduced.
427 -- Note that X_Frac has the same sign as X, so if X_Frac is 0.5,
428 -- then we know that we have a positive number (and hence a
429 -- positive power of 2).
432 return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
434 -- Otherwise the exponent is unchanged
437 return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa);
446 function Remainder (X, Y : T) return T is
460 pragma Unreferenced (Arg_Frac);
464 raise Constraint_Error;
480 P_Exp := Exponent (P);
483 Decompose (Arg, Arg_Frac, Arg_Exp);
484 Decompose (P, P_Frac, P_Exp);
486 P := Compose (P_Frac, Arg_Exp);
487 K := Arg_Exp - P_Exp;
491 for Cnt in reverse 0 .. K loop
492 if IEEE_Rem >= P then
494 IEEE_Rem := IEEE_Rem - P;
503 -- That completes the calculation of modulus remainder. The final
504 -- step is get the IEEE remainder. Here we need to compare Rem with
505 -- (abs Y) / 2. We must be careful of unrepresentable Y/2 value
506 -- caused by subnormal numbers
517 if A > B or else (A = B and then not P_Even) then
518 IEEE_Rem := IEEE_Rem - abs Y;
521 return Sign_X * IEEE_Rem;
528 function Rounding (X : T) return T is
533 Result := Truncation (abs X);
534 Tail := abs X - Result;
537 Result := Result + 1.0;
546 -- For zero case, make sure sign of zero is preserved
557 -- Return x * rad ** adjustment quickly, or quietly underflow to zero,
558 -- or overflow naturally.
560 function Scaling (X : T; Adjustment : UI) return T is
562 if X = 0.0 or else Adjustment = 0 then
566 -- Nonzero x essentially, just multiply repeatedly by Rad ** (+-2**n)
570 Ex : UI := Adjustment;
572 -- Y * Rad ** Ex is invariant
576 while Ex <= -Log_Power (Expbits'Last) loop
577 Y := Y * R_Neg_Power (Expbits'Last);
578 Ex := Ex + Log_Power (Expbits'Last);
583 for N in reverse Expbits'First .. Expbits'Last - 1 loop
584 if Ex <= -Log_Power (N) then
585 Y := Y * R_Neg_Power (N);
586 Ex := Ex + Log_Power (N);
589 -- -Log_Power (N) < Ex <= 0
598 while Ex >= Log_Power (Expbits'Last) loop
599 Y := Y * R_Power (Expbits'Last);
600 Ex := Ex - Log_Power (Expbits'Last);
605 for N in reverse Expbits'First .. Expbits'Last - 1 loop
606 if Ex >= Log_Power (N) then
607 Y := Y * R_Power (N);
608 Ex := Ex - Log_Power (N);
611 -- 0 <= Ex < Log_Power (N)
627 -- Similar computation to that of Pred: find value of least significant
628 -- bit of given number, and add. Zero has to be treated specially since
629 -- the exponent can be zero, and also we want the smallest denormal if
630 -- denormals are supported.
632 function Succ (X : T) return T is
639 X1 := 2.0 ** T'Machine_Emin;
641 -- Following loop generates smallest denormal
644 X2 := T'Machine (X1 / 2.0);
652 Decompose (X, X_Frac, X_Exp);
654 -- A special case, if the number we had was a negative power of two,
655 -- then we want to add half of what we would otherwise add, since the
656 -- exponent is going to be reduced.
658 -- Note that X_Frac has the same sign as X, so if X_Frac is -0.5,
659 -- then we know that we have a negative number (and hence a negative
662 if X_Frac = -0.5 then
663 return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
665 -- Otherwise the exponent is unchanged
668 return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa);
677 -- The basic approach is to compute
679 -- T'Machine (RM1 + N) - RM1
681 -- where N >= 0.0 and RM1 = radix ** (mantissa - 1)
683 -- This works provided that the intermediate result (RM1 + N) does not
684 -- have extra precision (which is why we call Machine). When we compute
685 -- RM1 + N, the exponent of N will be normalized and the mantissa shifted
686 -- shifted appropriately so the lower order bits, which cannot contribute
687 -- to the integer part of N, fall off on the right. When we subtract RM1
688 -- again, the significant bits of N are shifted to the left, and what we
689 -- have is an integer, because only the first e bits are different from
690 -- zero (assuming binary radix here).
692 function Truncation (X : T) return T is
698 if Result >= Radix_To_M_Minus_1 then
702 Result := Machine (Radix_To_M_Minus_1 + Result) - Radix_To_M_Minus_1;
704 if Result > abs X then
705 Result := Result - 1.0;
714 -- For zero case, make sure sign of zero is preserved
722 -----------------------
723 -- Unbiased_Rounding --
724 -----------------------
726 function Unbiased_Rounding (X : T) return T is
727 Abs_X : constant T := abs X;
732 Result := Truncation (Abs_X);
733 Tail := Abs_X - Result;
736 Result := Result + 1.0;
738 elsif Tail = 0.5 then
739 Result := 2.0 * Truncation ((Result / 2.0) + 0.5);
748 -- For zero case, make sure sign of zero is preserved
753 end Unbiased_Rounding;
759 -- Note: this routine does not work for VAX float. We compensate for this
760 -- in Exp_Attr by using the Valid functions in Vax_Float_Operations rather
761 -- than the corresponding instantiation of this function.
763 function Valid (X : not null access T) return Boolean is
765 IEEE_Emin : constant Integer := T'Machine_Emin - 1;
766 IEEE_Emax : constant Integer := T'Machine_Emax - 1;
768 IEEE_Bias : constant Integer := -(IEEE_Emin - 1);
770 subtype IEEE_Exponent_Range is
771 Integer range IEEE_Emin - 1 .. IEEE_Emax + 1;
773 -- The implementation of this floating point attribute uses a
774 -- representation type Float_Rep that allows direct access to the
775 -- exponent and mantissa parts of a floating point number.
777 -- The Float_Rep type is an array of Float_Word elements. This
778 -- representation is chosen to make it possible to size the type based
779 -- on a generic parameter. Since the array size is known at compile
780 -- time, efficient code can still be generated. The size of Float_Word
781 -- elements should be large enough to allow accessing the exponent in
782 -- one read, but small enough so that all floating point object sizes
783 -- are a multiple of the Float_Word'Size.
785 -- The following conditions must be met for all possible instantiations
786 -- of the attributes package:
788 -- - T'Size is an integral multiple of Float_Word'Size
790 -- - The exponent and sign are completely contained in a single
791 -- component of Float_Rep, named Most_Significant_Word (MSW).
793 -- - The sign occupies the most significant bit of the MSW and the
794 -- exponent is in the following bits. Unused bits (if any) are in
795 -- the least significant part.
797 type Float_Word is mod 2**Positive'Min (System.Word_Size, 32);
798 type Rep_Index is range 0 .. 7;
800 Rep_Words : constant Positive :=
801 (T'Size + Float_Word'Size - 1) / Float_Word'Size;
802 Rep_Last : constant Rep_Index :=
804 (Rep_Index (Rep_Words - 1),
805 (T'Mantissa + 16) / Float_Word'Size);
806 -- Determine the number of Float_Words needed for representing the
807 -- entire floating-point value. Do not take into account excessive
808 -- padding, as occurs on IA-64 where 80 bits floats get padded to 128
809 -- bits. In general, the exponent field cannot be larger than 15 bits,
810 -- even for 128-bit floating-point types, so the final format size
811 -- won't be larger than T'Mantissa + 16.
814 array (Rep_Index range 0 .. Rep_Index (Rep_Words - 1)) of Float_Word;
816 pragma Suppress_Initialization (Float_Rep);
817 -- This pragma suppresses the generation of an initialization procedure
818 -- for type Float_Rep when operating in Initialize/Normalize_Scalars
819 -- mode. This is not just a matter of efficiency, but of functionality,
820 -- since Valid has a pragma Inline_Always, which is not permitted if
821 -- there are nested subprograms present.
823 Most_Significant_Word : constant Rep_Index :=
824 Rep_Last * Standard'Default_Bit_Order;
825 -- Finding the location of the Exponent_Word is a bit tricky. In general
826 -- we assume Word_Order = Bit_Order. This expression needs to be refined
829 Exponent_Factor : constant Float_Word :=
830 2**(Float_Word'Size - 1) /
831 Float_Word (IEEE_Emax - IEEE_Emin + 3) *
832 Boolean'Pos (Most_Significant_Word /= 2) +
833 Boolean'Pos (Most_Significant_Word = 2);
834 -- Factor that the extracted exponent needs to be divided by to be in
835 -- range 0 .. IEEE_Emax - IEEE_Emin + 2. Special kludge: Exponent_Factor
836 -- is 1 for x86/IA64 double extended as GCC adds unused bits to the
839 Exponent_Mask : constant Float_Word :=
840 Float_Word (IEEE_Emax - IEEE_Emin + 2) *
842 -- Value needed to mask out the exponent field. This assumes that the
843 -- range IEEE_Emin - 1 .. IEEE_Emax + contains 2**N values, for some N
846 function To_Float is new Ada.Unchecked_Conversion (Float_Rep, T);
848 type Float_Access is access all T;
849 function To_Address is
850 new Ada.Unchecked_Conversion (Float_Access, System.Address);
852 XA : constant System.Address := To_Address (Float_Access (X));
855 pragma Import (Ada, R);
856 for R'Address use XA;
857 -- R is a view of the input floating-point parameter. Note that we
858 -- must avoid copying the actual bits of this parameter in float
859 -- form (since it may be a signalling NaN.
861 E : constant IEEE_Exponent_Range :=
862 Integer ((R (Most_Significant_Word) and Exponent_Mask) /
865 -- Mask/Shift T to only get bits from the exponent. Then convert biased
866 -- value to integer value.
869 -- Float_Rep representation of significant of X.all
874 -- All denormalized numbers are valid, so the only invalid numbers
875 -- are overflows and NaNs, both with exponent = Emax + 1.
877 return E /= IEEE_Emax + 1;
881 -- All denormalized numbers except 0.0 are invalid
883 -- Set exponent of X to zero, so we end up with the significand, which
884 -- definitely is a valid number and can be converted back to a float.
887 SR (Most_Significant_Word) :=
888 (SR (Most_Significant_Word)
889 and not Exponent_Mask) + Float_Word (IEEE_Bias) * Exponent_Factor;
891 return (E in IEEE_Emin .. IEEE_Emax) or else
892 ((E = IEEE_Emin - 1) and then abs To_Float (SR) = 1.0);
895 ---------------------
896 -- Unaligned_Valid --
897 ---------------------
899 function Unaligned_Valid (A : System.Address) return Boolean is
900 subtype FS is String (1 .. T'Size / Character'Size);
901 type FSP is access FS;
903 function To_FSP is new Ada.Unchecked_Conversion (Address, FSP);
908 -- Note that we have to be sure that we do not load the value into a
909 -- floating-point register, since a signalling NaN may cause a trap.
910 -- The following assignment is what does the actual alignment, since
911 -- we know that the target Local_T is aligned.
913 To_FSP (Local_T'Address).all := To_FSP (A).all;
915 -- Now that we have an aligned value, we can use the normal aligned
916 -- version of Valid to obtain the required result.
918 return Valid (Local_T'Access);