1 ------------------------------------------------------------------------------
3 -- GNAT COMPILER COMPONENTS --
5 -- S Y S T E M . F A T _ G E N --
10 -- Copyright (C) 1992-2001 Free Software Foundation, Inc. --
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. --
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. --
30 -- GNAT was originally developed by the GNAT team at New York University. --
31 -- Extensive contributions were provided by Ada Core Technologies Inc. --
33 ------------------------------------------------------------------------------
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.
42 with Ada.Unchecked_Conversion;
44 package body System.Fat_Gen is
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);
50 pragma Assert (T'Machine_Radix = 2);
51 -- This version does not handle radix 16
53 -- Constants for Decompose and Scaling
55 Rad : constant T := T (T'Machine_Radix);
56 Invrad : constant T := 1.0 / Rad;
58 subtype Expbits is Integer range 0 .. 6;
59 -- 2 ** (2 ** 7) might overflow. how big can radix-16 exponents get?
61 Log_Power : constant array (Expbits) of Integer := (1, 2, 4, 8, 16, 32, 64);
63 R_Power : constant array (Expbits) of T :=
72 R_Neg_Power : constant array (Expbits) of T :=
81 -----------------------
82 -- Local Subprograms --
83 -----------------------
85 procedure Decompose (XX : T; Frac : out T; Expo : out UI);
86 -- Decomposes a floating-point number into fraction and exponent parts
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.
97 function Adjacent (X, Towards : T) return T is
102 elsif Towards > X then
114 function Ceiling (X : T) return T is
115 XT : constant T := Truncation (X);
133 function Compose (Fraction : T; Exponent : UI) return T is
138 Decompose (Fraction, Arg_Frac, Arg_Exp);
139 return Scaling (Arg_Frac, Exponent);
146 function Copy_Sign (Value, Sign : T) return T is
149 function Is_Negative (V : T) return Boolean;
150 pragma Import (Intrinsic, Is_Negative);
155 if Is_Negative (Sign) then
166 procedure Decompose (XX : T; Frac : out T; Expo : out UI) is
167 X : T := T'Machine (XX);
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 ???
178 -- Check for infinities, transfinites, whatnot.
180 elsif X > T'Safe_Last then
182 Expo := T'Machine_Emax + 1;
184 elsif X < T'Safe_First then
186 Expo := T'Machine_Emax + 2; -- how many extra negative values?
189 -- Case of nonzero finite x. Essentially, we just multiply
190 -- by Rad ** (+-2**N) to reduce the range.
196 -- Ax * Rad ** Ex is invariant.
200 while Ax >= R_Power (Expbits'Last) loop
201 Ax := Ax * R_Neg_Power (Expbits'Last);
202 Ex := Ex + Log_Power (Expbits'Last);
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);
224 while Ax < R_Neg_Power (Expbits'Last) loop
225 Ax := Ax * R_Power (Expbits'Last);
226 Ex := Ex - Log_Power (Expbits'Last);
229 -- Rad ** -64 <= Ax < 1
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);
237 -- R_Neg_Power (N) <= Ax < 1
256 function Exponent (X : T) return UI is
261 Decompose (X, X_Frac, X_Exp);
269 function Floor (X : T) return T is
270 XT : constant T := Truncation (X);
288 function Fraction (X : T) return T is
293 Decompose (X, X_Frac, X_Exp);
297 ---------------------
298 -- Gradual_Scaling --
299 ---------------------
301 function Gradual_Scaling (Adjustment : UI) return T is
304 Ex : UI := Adjustment;
307 if Adjustment < T'Machine_Emin then
308 Y := 2.0 ** T'Machine_Emin;
310 Ex := Ex - T'Machine_Emin;
313 Y := T'Machine (Y / 2.0);
326 return Scaling (1.0, Adjustment);
334 function Leading_Part (X : T; Radix_Digits : UI) return T is
339 if Radix_Digits >= T'Machine_Mantissa then
343 L := Exponent (X) - Radix_Digits;
344 Y := Truncation (Scaling (X, -L));
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.
360 function Machine (X : T) return T is
362 pragma Volatile (Temp);
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.
376 function Model (X : T) return T is
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.
391 -- Zero has to be treated specially, since its exponent is zero
393 function Pred (X : T) return T is
402 Decompose (X, X_Frac, X_Exp);
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.
408 if X_Frac = 0.5 and then X > 0.0 then
409 return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
411 -- Otherwise the exponent stays the same
414 return X - Gradual_Scaling (X_Exp - T'Machine_Mantissa);
423 function Remainder (X, Y : T) return T is
451 P_Exp := Exponent (P);
454 Decompose (Arg, Arg_Frac, Arg_Exp);
455 Decompose (P, P_Frac, P_Exp);
457 P := Compose (P_Frac, Arg_Exp);
458 K := Arg_Exp - P_Exp;
462 for Cnt in reverse 0 .. K loop
463 if IEEE_Rem >= P then
465 IEEE_Rem := IEEE_Rem - P;
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
488 if A > B or else (A = B and then not P_Even) then
489 IEEE_Rem := IEEE_Rem - abs Y;
492 return Sign_X * IEEE_Rem;
500 function Rounding (X : T) return T is
505 Result := Truncation (abs X);
506 Tail := abs X - Result;
509 Result := Result + 1.0;
518 -- For zero case, make sure sign of zero is preserved
530 -- Return x * rad ** adjustment quickly,
531 -- or quietly underflow to zero, or overflow naturally.
533 function Scaling (X : T; Adjustment : UI) return T is
535 if X = 0.0 or else Adjustment = 0 then
539 -- Nonzero x. essentially, just multiply repeatedly by Rad ** (+-2**n).
543 Ex : UI := Adjustment;
545 -- Y * Rad ** Ex is invariant
549 while Ex <= -Log_Power (Expbits'Last) loop
550 Y := Y * R_Neg_Power (Expbits'Last);
551 Ex := Ex + Log_Power (Expbits'Last);
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);
562 -- -Log_Power (N) < Ex <= 0
570 while Ex >= Log_Power (Expbits'Last) loop
571 Y := Y * R_Power (Expbits'Last);
572 Ex := Ex - Log_Power (Expbits'Last);
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);
583 -- 0 <= Ex < Log_Power (N)
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.
601 function Succ (X : T) return T is
608 X1 := 2.0 ** T'Machine_Emin;
610 -- Following loop generates smallest denormal
613 X2 := T'Machine (X1 / 2.0);
621 Decompose (X, X_Frac, X_Exp);
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.
627 if X_Frac = 0.5 and then X < 0.0 then
628 return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa - 1);
630 -- Otherwise the exponent stays the same
633 return X + Gradual_Scaling (X_Exp - T'Machine_Mantissa);
642 -- The basic approach is to compute
644 -- T'Machine (RM1 + N) - RM1.
646 -- where N >= 0.0 and RM1 = radix ** (mantissa - 1)
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).
657 function Truncation (X : T) return T is
663 if Result >= Radix_To_M_Minus_1 then
667 Result := Machine (Radix_To_M_Minus_1 + Result) - Radix_To_M_Minus_1;
669 if Result > abs X then
670 Result := Result - 1.0;
679 -- For zero case, make sure sign of zero is preserved
688 -----------------------
689 -- Unbiased_Rounding --
690 -----------------------
692 function Unbiased_Rounding (X : T) return T is
693 Abs_X : constant T := abs X;
698 Result := Truncation (Abs_X);
699 Tail := Abs_X - Result;
702 Result := Result + 1.0;
704 elsif Tail = 0.5 then
705 Result := 2.0 * Truncation ((Result / 2.0) + 0.5);
714 -- For zero case, make sure sign of zero is preserved
720 end Unbiased_Rounding;
726 function Valid (X : access T) return Boolean is
728 IEEE_Emin : constant Integer := T'Machine_Emin - 1;
729 IEEE_Emax : constant Integer := T'Machine_Emax - 1;
731 IEEE_Bias : constant Integer := -(IEEE_Emin - 1);
733 subtype IEEE_Exponent_Range is
734 Integer range IEEE_Emin - 1 .. IEEE_Emax + 1;
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.
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.
744 -- The following conditions must be met for all possible
745 -- instantiations of the attributes package:
747 -- - T'Size is an integral multiple of Float_Word'Size
749 -- - The exponent and sign are completely contained in a single
750 -- component of Float_Rep, named Most_Significant_Word (MSW).
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.
756 type Float_Word is mod 2**32;
757 type Rep_Index is range 0 .. 7;
759 Rep_Last : constant Rep_Index := (T'Size - 1) / Float_Word'Size;
761 type Float_Rep is array (Rep_Index range 0 .. Rep_Last) of Float_Word;
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.
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.
779 Exponent_Mask : constant Float_Word :=
780 Float_Word (IEEE_Emax - IEEE_Emin + 2) *
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.
786 function To_Float is new Ada.Unchecked_Conversion (Float_Rep, T);
788 type Float_Access is access all T;
789 function To_Address is
790 new Ada.Unchecked_Conversion (Float_Access, System.Address);
792 XA : constant System.Address := To_Address (Float_Access (X));
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.
801 E : constant IEEE_Exponent_Range :=
802 Integer ((R (Most_Significant_Word) and Exponent_Mask) /
805 -- Mask/Shift T to only get bits from the exponent
806 -- Then convert biased value to integer value.
809 -- Float_Rep representation of significant of X.all
814 -- All denormalized numbers are valid, so only invalid numbers
815 -- are overflows and NaN's, both with exponent = Emax + 1.
817 return E /= IEEE_Emax + 1;
821 -- All denormalized numbers except 0.0 are invalid
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.
827 SR (Most_Significant_Word) :=
828 (SR (Most_Significant_Word)
829 and not Exponent_Mask) + Float_Word (IEEE_Bias) * Exponent_Factor;
831 return (E in IEEE_Emin .. IEEE_Emax) or else
832 ((E = IEEE_Emin - 1) and then abs To_Float (SR) = 1.0);