------------------------------------------------------------------------------ | |

-- -- | |

-- GNAT COMPILER COMPONENTS -- | |

-- -- | |

-- E V A L _ F A T -- | |

-- -- | |

-- B o d y -- | |

-- -- | |

-- Copyright (C) 1992-2021, Free Software Foundation, Inc. -- | |

-- -- | |

-- GNAT is free software; you can redistribute it and/or modify it under -- | |

-- terms of the GNU General Public License as published by the Free Soft- -- | |

-- ware Foundation; either version 3, or (at your option) any later ver- -- | |

-- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- | |

-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- | |

-- or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License -- | |

-- for more details. You should have received a copy of the GNU General -- | |

-- Public License distributed with GNAT; see file COPYING3. If not, go to -- | |

-- http://www.gnu.org/licenses for a complete copy of the license. -- | |

-- -- | |

-- GNAT was originally developed by the GNAT team at New York University. -- | |

-- Extensive contributions were provided by Ada Core Technologies Inc. -- | |

-- -- | |

------------------------------------------------------------------------------ | |

with Einfo; use Einfo; | |

with Einfo.Utils; use Einfo.Utils; | |

with Errout; use Errout; | |

with Opt; use Opt; | |

with Sem_Util; use Sem_Util; | |

package body Eval_Fat is | |

Radix : constant Int := 2; | |

-- This code is currently only correct for the radix 2 case. We use the | |

-- symbolic value Radix where possible to help in the unlikely case of | |

-- anyone ever having to adjust this code for another value, and for | |

-- documentation purposes. | |

-- Another assumption is that the range of the floating-point type is | |

-- symmetric around zero. | |

type Radix_Power_Table is array (Int range 1 .. 4) of Int; | |

Radix_Powers : constant Radix_Power_Table := | |

(Radix ** 1, Radix ** 2, Radix ** 3, Radix ** 4); | |

----------------------- | |

-- Local Subprograms -- | |

----------------------- | |

procedure Decompose | |

(RT : R; | |

X : T; | |

Fraction : out T; | |

Exponent : out UI; | |

Mode : Rounding_Mode := Round); | |

-- Decomposes a non-zero floating-point number into fraction and exponent | |

-- parts. The fraction is in the interval 1.0 / Radix .. T'Pred (1.0) and | |

-- uses Rbase = Radix. The result is rounded to a nearest machine number. | |

-------------- | |

-- Adjacent -- | |

-------------- | |

function Adjacent (RT : R; X, Towards : T) return T is | |

begin | |

if Towards = X then | |

return X; | |

elsif Towards > X then | |

return Succ (RT, X); | |

else | |

return Pred (RT, X); | |

end if; | |

end Adjacent; | |

------------- | |

-- Ceiling -- | |

------------- | |

function Ceiling (RT : R; X : T) return T is | |

XT : constant T := Truncation (RT, X); | |

begin | |

if UR_Is_Negative (X) then | |

return XT; | |

elsif X = XT then | |

return X; | |

else | |

return XT + Ureal_1; | |

end if; | |

end Ceiling; | |

------------- | |

-- Compose -- | |

------------- | |

function Compose (RT : R; Fraction : T; Exponent : UI) return T is | |

Arg_Frac : T; | |

Arg_Exp : UI; | |

pragma Warnings (Off, Arg_Exp); | |

begin | |

Decompose (RT, Fraction, Arg_Frac, Arg_Exp); | |

return Scaling (RT, Arg_Frac, Exponent); | |

end Compose; | |

--------------- | |

-- Copy_Sign -- | |

--------------- | |

function Copy_Sign (RT : R; Value, Sign : T) return T is | |

pragma Warnings (Off, RT); | |

Result : T; | |

begin | |

Result := abs Value; | |

if UR_Is_Negative (Sign) then | |

return -Result; | |

else | |

return Result; | |

end if; | |

end Copy_Sign; | |

--------------- | |

-- Decompose -- | |

--------------- | |

procedure Decompose | |

(RT : R; | |

X : T; | |

Fraction : out T; | |

Exponent : out UI; | |

Mode : Rounding_Mode := Round) | |

is | |

Int_F : UI; | |

begin | |

Decompose_Int (RT, abs X, Int_F, Exponent, Mode); | |

Fraction := UR_From_Components | |

(Num => Int_F, | |

Den => Machine_Mantissa_Value (RT), | |

Rbase => Radix, | |

Negative => False); | |

if UR_Is_Negative (X) then | |

Fraction := -Fraction; | |

end if; | |

return; | |

end Decompose; | |

------------------- | |

-- Decompose_Int -- | |

------------------- | |

-- This procedure should be modified with care, as there are many non- | |

-- obvious details that may cause problems that are hard to detect. For | |

-- zero arguments, Fraction and Exponent are set to zero. Note that sign | |

-- of zero cannot be preserved. | |

procedure Decompose_Int | |

(RT : R; | |

X : T; | |

Fraction : out UI; | |

Exponent : out UI; | |

Mode : Rounding_Mode) | |

is | |

Base : Int := Rbase (X); | |

N : UI := abs Numerator (X); | |

D : UI := Denominator (X); | |

N_Times_Radix : UI; | |

Even : Boolean; | |

-- True iff Fraction is even | |

Most_Significant_Digit : constant UI := | |

Radix ** (Machine_Mantissa_Value (RT) - 1); | |

Uintp_Mark : Uintp.Save_Mark; | |

-- The code is divided into blocks that systematically release | |

-- intermediate values (this routine generates lots of junk). | |

begin | |

if N = Uint_0 then | |

Fraction := Uint_0; | |

Exponent := Uint_0; | |

return; | |

end if; | |

Calculate_D_And_Exponent_1 : begin | |

Uintp_Mark := Mark; | |

Exponent := Uint_0; | |

-- In cases where Base > 1, the actual denominator is Base**D. For | |

-- cases where Base is a power of Radix, use the value 1 for the | |

-- Denominator and adjust the exponent. | |

-- Note: Exponent has different sign from D, because D is a divisor | |

for Power in 1 .. Radix_Powers'Last loop | |

if Base = Radix_Powers (Power) then | |

Exponent := -D * Power; | |

Base := 0; | |

D := Uint_1; | |

exit; | |

end if; | |

end loop; | |

Release_And_Save (Uintp_Mark, D, Exponent); | |

end Calculate_D_And_Exponent_1; | |

if Base > 0 then | |

Calculate_Exponent : begin | |

Uintp_Mark := Mark; | |

-- For bases that are a multiple of the Radix, divide the base by | |

-- Radix and adjust the Exponent. This will help because D will be | |

-- much smaller and faster to process. | |

-- This occurs for decimal bases on machines with binary floating- | |

-- point for example. When calculating 1E40, with Radix = 2, N | |

-- will be 93 bits instead of 133. | |

-- N E | |

-- ------ * Radix | |

-- D | |

-- Base | |

-- N E | |

-- = -------------------------- * Radix | |

-- D D | |

-- (Base/Radix) * Radix | |

-- N E-D | |

-- = --------------- * Radix | |

-- D | |

-- (Base/Radix) | |

-- This code is commented out, because it causes numerous | |

-- failures in the regression suite. To be studied ??? | |

while False and then Base > 0 and then Base mod Radix = 0 loop | |

Base := Base / Radix; | |

Exponent := Exponent + D; | |

end loop; | |

Release_And_Save (Uintp_Mark, Exponent); | |

end Calculate_Exponent; | |

-- For remaining bases we must actually compute the exponentiation | |

-- Because the exponentiation can be negative, and D must be integer, | |

-- the numerator is corrected instead. | |

Calculate_N_And_D : begin | |

Uintp_Mark := Mark; | |

if D < 0 then | |

N := N * Base ** (-D); | |

D := Uint_1; | |

else | |

D := Base ** D; | |

end if; | |

Release_And_Save (Uintp_Mark, N, D); | |

end Calculate_N_And_D; | |

Base := 0; | |

end if; | |

-- Now scale N and D so that N / D is a value in the interval [1.0 / | |

-- Radix, 1.0) and adjust Exponent accordingly, so the value N / D * | |

-- Radix ** Exponent remains unchanged. | |

-- Step 1 - Adjust N so N / D >= 1 / Radix, or N = 0 | |

-- N and D are positive, so N / D >= 1 / Radix implies N * Radix >= D. | |

-- As this scaling is not possible for N is Uint_0, zero is handled | |

-- explicitly at the start of this subprogram. | |

Calculate_N_And_Exponent : begin | |

Uintp_Mark := Mark; | |

N_Times_Radix := N * Radix; | |

while not (N_Times_Radix >= D) loop | |

N := N_Times_Radix; | |

Exponent := Exponent - 1; | |

N_Times_Radix := N * Radix; | |

end loop; | |

Release_And_Save (Uintp_Mark, N, Exponent); | |

end Calculate_N_And_Exponent; | |

-- Step 2 - Adjust D so N / D < 1 | |

-- Scale up D so N / D < 1, so N < D | |

Calculate_D_And_Exponent_2 : begin | |

Uintp_Mark := Mark; | |

while not (N < D) loop | |

-- As N / D >= 1, N / (D * Radix) will be at least 1 / Radix, so | |

-- the result of Step 1 stays valid | |

D := D * Radix; | |

Exponent := Exponent + 1; | |

end loop; | |

Release_And_Save (Uintp_Mark, D, Exponent); | |

end Calculate_D_And_Exponent_2; | |

-- Here the value N / D is in the range [1.0 / Radix .. 1.0) | |

-- Now find the fraction by doing a very simple-minded division until | |

-- enough digits have been computed. | |

-- This division works for all radices, but is only efficient for a | |

-- binary radix. It is just like a manual division algorithm, but | |

-- instead of moving the denominator one digit right, we move the | |

-- numerator one digit left so the numerator and denominator remain | |

-- integral. | |

Fraction := Uint_0; | |

Even := True; | |

Calculate_Fraction_And_N : begin | |

Uintp_Mark := Mark; | |

loop | |

while N >= D loop | |

N := N - D; | |

Fraction := Fraction + 1; | |

Even := not Even; | |

end loop; | |

-- Stop when the result is in [1.0 / Radix, 1.0) | |

exit when Fraction >= Most_Significant_Digit; | |

N := N * Radix; | |

Fraction := Fraction * Radix; | |

Even := True; | |

end loop; | |

Release_And_Save (Uintp_Mark, Fraction, N); | |

end Calculate_Fraction_And_N; | |

Calculate_Fraction_And_Exponent : begin | |

Uintp_Mark := Mark; | |

-- Determine correct rounding based on the remainder which is in | |

-- N and the divisor D. The rounding is performed on the absolute | |

-- value of X, so Ceiling and Floor need to check for the sign of | |

-- X explicitly. | |

case Mode is | |

when Round_Even => | |

-- This rounding mode corresponds to the unbiased rounding | |

-- method that is used at run time. When the real value is | |

-- exactly between two machine numbers, choose the machine | |

-- number with its least significant bit equal to zero. | |

-- The recommendation advice in RM 4.9(38) is that static | |

-- expressions are rounded to machine numbers in the same | |

-- way as the target machine does. | |

if (Even and then N * 2 > D) | |

or else | |

(not Even and then N * 2 >= D) | |

then | |

Fraction := Fraction + 1; | |

end if; | |

when Round => | |

-- Do not round to even as is done with IEEE arithmetic, but | |

-- instead round away from zero when the result is exactly | |

-- between two machine numbers. This biased rounding method | |

-- should not be used to convert static expressions to | |

-- machine numbers, see AI95-268. | |

if N * 2 >= D then | |

Fraction := Fraction + 1; | |

end if; | |

when Ceiling => | |

if N > Uint_0 and then not UR_Is_Negative (X) then | |

Fraction := Fraction + 1; | |

end if; | |

when Floor => | |

if N > Uint_0 and then UR_Is_Negative (X) then | |

Fraction := Fraction + 1; | |

end if; | |

end case; | |

-- The result must be normalized to [1.0/Radix, 1.0), so adjust if | |

-- the result is 1.0 because of rounding. | |

if Fraction = Most_Significant_Digit * Radix then | |

Fraction := Most_Significant_Digit; | |

Exponent := Exponent + 1; | |

end if; | |

-- Put back sign after applying the rounding | |

if UR_Is_Negative (X) then | |

Fraction := -Fraction; | |

end if; | |

Release_And_Save (Uintp_Mark, Fraction, Exponent); | |

end Calculate_Fraction_And_Exponent; | |

end Decompose_Int; | |

-------------- | |

-- Exponent -- | |

-------------- | |

function Exponent (RT : R; X : T) return UI is | |

X_Frac : UI; | |

X_Exp : UI; | |

pragma Warnings (Off, X_Frac); | |

begin | |

Decompose_Int (RT, X, X_Frac, X_Exp, Round_Even); | |

return X_Exp; | |

end Exponent; | |

----------- | |

-- Floor -- | |

----------- | |

function Floor (RT : R; X : T) return T is | |

XT : constant T := Truncation (RT, X); | |

begin | |

if UR_Is_Positive (X) then | |

return XT; | |

elsif XT = X then | |

return X; | |

else | |

return XT - Ureal_1; | |

end if; | |

end Floor; | |

-------------- | |

-- Fraction -- | |

-------------- | |

function Fraction (RT : R; X : T) return T is | |

X_Frac : T; | |

X_Exp : UI; | |

pragma Warnings (Off, X_Exp); | |

begin | |

Decompose (RT, X, X_Frac, X_Exp); | |

return X_Frac; | |

end Fraction; | |

------------------ | |

-- Leading_Part -- | |

------------------ | |

function Leading_Part (RT : R; X : T; Radix_Digits : UI) return T is | |

RD : constant UI := UI_Min (Radix_Digits, Machine_Mantissa_Value (RT)); | |

L : UI; | |

Y : T; | |

begin | |

L := Exponent (RT, X) - RD; | |

Y := UR_From_Uint (UR_Trunc (Scaling (RT, X, -L))); | |

return Scaling (RT, Y, L); | |

end Leading_Part; | |

------------- | |

-- Machine -- | |

------------- | |

function Machine | |

(RT : R; | |

X : T; | |

Mode : Rounding_Mode; | |

Enode : Node_Id) return T | |

is | |

X_Frac : T; | |

X_Exp : UI; | |

Emin : constant UI := Machine_Emin_Value (RT); | |

begin | |

Decompose (RT, X, X_Frac, X_Exp, Mode); | |

-- Case of denormalized number or (gradual) underflow | |

-- A denormalized number is one with the minimum exponent Emin, but that | |

-- breaks the assumption that the first digit of the mantissa is a one. | |

-- This allows the first non-zero digit to be in any of the remaining | |

-- Mant - 1 spots. The gap between subsequent denormalized numbers is | |

-- the same as for the smallest normalized numbers. However, the number | |

-- of significant digits left decreases as a result of the mantissa now | |

-- having leading seros. | |

if X_Exp < Emin then | |

declare | |

Emin_Den : constant UI := Machine_Emin_Value (RT) - | |

Machine_Mantissa_Value (RT) + Uint_1; | |

begin | |

-- Do not issue warnings about underflows in GNATprove mode, | |

-- as calling Machine as part of interval checking may lead | |

-- to spurious warnings. | |

if X_Exp < Emin_Den or not Has_Denormals (RT) then | |

if Has_Signed_Zeros (RT) and then UR_Is_Negative (X) then | |

if not GNATprove_Mode then | |

Error_Msg_N | |

("floating-point value underflows to -0.0??", Enode); | |

end if; | |

return Ureal_M_0; | |

else | |

if not GNATprove_Mode then | |

Error_Msg_N | |

("floating-point value underflows to 0.0??", Enode); | |

end if; | |

return Ureal_0; | |

end if; | |

elsif Has_Denormals (RT) then | |

-- Emin - Mant <= X_Exp < Emin, so result is denormal. Handle | |

-- gradual underflow by first computing the number of | |

-- significant bits still available for the mantissa and | |

-- then truncating the fraction to this number of bits. | |

-- If this value is different from the original fraction, | |

-- precision is lost due to gradual underflow. | |

-- We probably should round here and prevent double rounding as | |

-- a result of first rounding to a model number and then to a | |

-- machine number. However, this is an extremely rare case that | |

-- is not worth the extra complexity. In any case, a warning is | |

-- issued in cases where gradual underflow occurs. | |

declare | |

Denorm_Sig_Bits : constant UI := X_Exp - Emin_Den + 1; | |

X_Frac_Denorm : constant T := UR_From_Components | |

(UR_Trunc (Scaling (RT, abs X_Frac, Denorm_Sig_Bits)), | |

Denorm_Sig_Bits, | |

Radix, | |

UR_Is_Negative (X)); | |

begin | |

-- Do not issue warnings about loss of precision in | |

-- GNATprove mode, as calling Machine as part of interval | |

-- checking may lead to spurious warnings. | |

if X_Frac_Denorm /= X_Frac then | |

if not GNATprove_Mode then | |

Error_Msg_N | |

("gradual underflow causes loss of precision??", | |

Enode); | |

end if; | |

X_Frac := X_Frac_Denorm; | |

end if; | |

end; | |

end if; | |

end; | |

end if; | |

return Scaling (RT, X_Frac, X_Exp); | |

end Machine; | |

----------- | |

-- Model -- | |

----------- | |

function Model (RT : R; X : T) return T is | |

X_Frac : T; | |

X_Exp : UI; | |

begin | |

Decompose (RT, X, X_Frac, X_Exp); | |

return Compose (RT, X_Frac, X_Exp); | |

end Model; | |

---------- | |

-- Pred -- | |

---------- | |

function Pred (RT : R; X : T) return T is | |

begin | |

return -Succ (RT, -X); | |

end Pred; | |

--------------- | |

-- Remainder -- | |

--------------- | |

function Remainder (RT : R; X, Y : T) return T is | |

A : T; | |

B : T; | |

Arg : T; | |

P : T; | |

Arg_Frac : T; | |

P_Frac : T; | |

Sign_X : T; | |

IEEE_Rem : T; | |

Arg_Exp : UI; | |

P_Exp : UI; | |

K : UI; | |

P_Even : Boolean; | |

pragma Warnings (Off, Arg_Frac); | |

begin | |

if UR_Is_Positive (X) then | |

Sign_X := Ureal_1; | |

else | |

Sign_X := -Ureal_1; | |

end if; | |

Arg := abs X; | |

P := abs Y; | |

if Arg < P then | |

P_Even := True; | |

IEEE_Rem := Arg; | |

P_Exp := Exponent (RT, P); | |

else | |

-- ??? what about zero cases? | |

Decompose (RT, Arg, Arg_Frac, Arg_Exp); | |

Decompose (RT, P, P_Frac, P_Exp); | |

P := Compose (RT, P_Frac, Arg_Exp); | |

K := Arg_Exp - P_Exp; | |

P_Even := True; | |

IEEE_Rem := Arg; | |

for Cnt in reverse 0 .. UI_To_Int (K) loop | |

if IEEE_Rem >= P then | |

P_Even := False; | |

IEEE_Rem := IEEE_Rem - P; | |

else | |

P_Even := True; | |

end if; | |

P := P * Ureal_Half; | |

end loop; | |

end if; | |

-- That completes the calculation of modulus remainder. The final step | |

-- is get the IEEE remainder. Here we compare Rem with (abs Y) / 2. | |

if P_Exp >= 0 then | |

A := IEEE_Rem; | |

B := abs Y * Ureal_Half; | |

else | |

A := IEEE_Rem * Ureal_2; | |

B := abs Y; | |

end if; | |

if A > B or else (A = B and then not P_Even) then | |

IEEE_Rem := IEEE_Rem - abs Y; | |

end if; | |

return Sign_X * IEEE_Rem; | |

end Remainder; | |

-------------- | |

-- Rounding -- | |

-------------- | |

function Rounding (RT : R; X : T) return T is | |

Result : T; | |

Tail : T; | |

begin | |

Result := Truncation (RT, abs X); | |

Tail := abs X - Result; | |

if Tail >= Ureal_Half then | |

Result := Result + Ureal_1; | |

end if; | |

if UR_Is_Negative (X) then | |

return -Result; | |

else | |

return Result; | |

end if; | |

end Rounding; | |

------------- | |

-- Scaling -- | |

------------- | |

function Scaling (RT : R; X : T; Adjustment : UI) return T is | |

pragma Warnings (Off, RT); | |

begin | |

if Rbase (X) = Radix then | |

return UR_From_Components | |

(Num => Numerator (X), | |

Den => Denominator (X) - Adjustment, | |

Rbase => Radix, | |

Negative => UR_Is_Negative (X)); | |

elsif Adjustment >= 0 then | |

return X * Radix ** Adjustment; | |

else | |

return X / Radix ** (-Adjustment); | |

end if; | |

end Scaling; | |

---------- | |

-- Succ -- | |

---------- | |

function Succ (RT : R; X : T) return T is | |

Emin : constant UI := Machine_Emin_Value (RT); | |

Mantissa : constant UI := Machine_Mantissa_Value (RT); | |

Exp : UI := UI_Max (Emin, Exponent (RT, X)); | |

Frac : T; | |

New_Frac : T; | |

begin | |

-- Treat zero as a regular denormalized number if they are supported, | |

-- otherwise return the smallest normalized number. | |

if UR_Is_Zero (X) then | |

if Has_Denormals (RT) then | |

Exp := Emin; | |

else | |

return Scaling (RT, Ureal_Half, Emin); | |

end if; | |

end if; | |

-- Multiply the number by 2.0**(Mantissa-Exp) so that the radix point | |

-- will be directly following the mantissa after scaling. | |

Exp := Exp - Mantissa; | |

Frac := Scaling (RT, X, -Exp); | |

-- Round to the neareast integer towards +Inf | |

New_Frac := Ceiling (RT, Frac); | |

-- If the rounding was a NOP, add one, except for -2.0**(Mantissa-1) | |

-- because the exponent is going to be reduced. | |

if New_Frac = Frac then | |

if New_Frac = Scaling (RT, -Ureal_1, Mantissa - 1) then | |

New_Frac := New_Frac + Ureal_Half; | |

else | |

New_Frac := New_Frac + Ureal_1; | |

end if; | |

end if; | |

-- Divide back by 2.0**(Mantissa-Exp) to get the final result | |

return Scaling (RT, New_Frac, Exp); | |

end Succ; | |

---------------- | |

-- Truncation -- | |

---------------- | |

function Truncation (RT : R; X : T) return T is | |

pragma Warnings (Off, RT); | |

begin | |

return UR_From_Uint (UR_Trunc (X)); | |

end Truncation; | |

----------------------- | |

-- Unbiased_Rounding -- | |

----------------------- | |

function Unbiased_Rounding (RT : R; X : T) return T is | |

Abs_X : constant T := abs X; | |

Result : T; | |

Tail : T; | |

begin | |

Result := Truncation (RT, Abs_X); | |

Tail := Abs_X - Result; | |

if Tail > Ureal_Half then | |

Result := Result + Ureal_1; | |

elsif Tail = Ureal_Half then | |

Result := Ureal_2 * | |

Truncation (RT, (Result / Ureal_2) + Ureal_Half); | |

end if; | |

if UR_Is_Negative (X) then | |

return -Result; | |

elsif UR_Is_Positive (X) then | |

return Result; | |

-- For zero case, make sure sign of zero is preserved | |

else | |

return X; | |

end if; | |

end Unbiased_Rounding; | |

end Eval_Fat; |