------------------------------------------------------------------------------
-- --
-- GNAT COMPILER COMPONENTS --
-- --
-- S Y S T E M . V A L U E _ F --
-- --
-- B o d y --
-- --
-- Copyright (C) 2020-2023, 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. --
-- --
-- As a special exception under Section 7 of GPL version 3, you are granted --
-- additional permissions described in the GCC Runtime Library Exception, --
-- version 3.1, as published by the Free Software Foundation. --
-- --
-- You should have received a copy of the GNU General Public License and --
-- a copy of the GCC Runtime Library Exception along with this program; --
-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
-- <http://www.gnu.org/licenses/>. --
-- --
-- GNAT was originally developed by the GNAT team at New York University. --
-- Extensive contributions were provided by Ada Core Technologies Inc. --
-- --
------------------------------------------------------------------------------
with System.Unsigned_Types; use System.Unsigned_Types;
with System.Val_Util; use System.Val_Util;
with System.Value_R;
package body System.Value_F is
-- The prerequisite of the implementation is that the computation of the
-- operands of the scaled divide does not unduly overflow when the small
-- is neither an integer nor the reciprocal of an integer, which means
-- that its numerator and denominator must be both not larger than the
-- smallest divide 2**(Int'Size - 1) / Base where Base ranges over the
-- supported values for the base of the literal. Given that the largest
-- supported base is 16, this gives a limit of 2**(Int'Size - 5).
pragma Assert (Int'Size <= Uns'Size);
-- We need an unsigned type large enough to represent the mantissa
package Impl is new Value_R (Uns, 1, 2**(Int'Size - 1), Round => True);
-- We use the Extra digit for ordinary fixed-point types
function Integer_To_Fixed
(Str : String;
Val : Uns;
Base : Unsigned;
ScaleB : Integer;
Extra : Unsigned;
Minus : Boolean;
Num : Int;
Den : Int) return Int;
-- Convert the real value from integer to fixed point representation
-- The goal is to compute Val * (Base ** ScaleB) / (Num / Den) with correct
-- rounding for all decimal values output by Typ'Image, that is to say up
-- to Typ'Aft decimal digits. Unlike for the output, the RM does not say
-- what the rounding must be for the input, but a reasonable exegesis of
-- the intent is that Typ'Value o Typ'Image should be the identity, which
-- is made possible because 'Aft is defined such that 'Image is injective.
-- For a type with a mantissa of M bits including the sign, the number N1
-- of decimal digits required to represent all the numbers is given by:
-- N1 = ceil ((M - 1) * log 2 / log 10) [N1 = 10/19/39 for M = 32/64/128]
-- but this mantissa can represent any set of contiguous numbers with only
-- N2 different decimal digits where:
-- N2 = floor ((M - 1) * log 2 / log 10) [N2 = 9/18/38 for M = 32/64/128]
-- Of course N1 = N2 + 1 holds, which means both that Val may not contain
-- enough significant bits to represent all the values of the type and that
-- 1 extra decimal digit contains the information for the missing bits.
-- Therefore the actual computation to be performed is
-- V = (Val * Base + Extra) * (Base ** (ScaleB - 1)) / (Num / Den)
-- using two steps of scaled divide if Extra is positive and ScaleB too
-- (1) Val * (Den * (Base ** ScaleB)) = Q1 * Num + R1
-- (2) Extra * (Den * (Base ** ScaleB)) = Q2 * -Base + R2
-- which yields after dividing (1) by Num and (2) by Num * Base and summing
-- V = Q1 + (R1 - Q2) / Num + R2 / (Num * Base)
-- but we get rid of the third term by using a rounding divide for (2).
-- This works only if Den * (Base ** ScaleB) does not overflow for inputs
-- corresponding to 'Image. Let S = Num / Den, B = Base and N the scale in
-- base B of S, i.e. the smallest integer such that B**N * S >= 1. Then,
-- for X a positive of the mantissa, i.e. 1 <= X <= 2**(M-1), we have
-- 1/B <= X * S * B**(N-1) < 2**(M-1)
-- which means that the inputs corresponding to the output of 'Image have a
-- ScaleB equal either to 1 - N or (after multiplying the inequality by B)
-- to -N, possibly after renormalizing X, i.e. multiplying it by a suitable
-- power of B. Therefore
-- Den * (Base ** ScaleB) <= Den * (B ** (1 - N)) < Num * B
-- which means that the product does not overflow if Num <= 2**(M-1) / B.
-- On the other hand, if Extra is positive and ScaleB negative, the above
-- two steps are
-- (1b) Val * Den = Q1 * (Num * (Base ** -ScaleB)) + R1
-- (2b) Extra * Den = Q2 * -Base + R2
-- which yields after dividing (1b) by Num * (Base ** -ScaleB) and (2b) by
-- Num * (Base ** (1 - ScaleB)) and summing
-- V = Q1 + (R1 - Q2) / (Num * (Base ** -ScaleB)) + R2 / ...
-- but we get rid of the third term by using a rounding divide for (2b).
-- This works only if Num * (Base ** -ScaleB) does not overflow for inputs
-- corresponding to 'Image. With the determination of ScaleB above, we have
-- Num * (Base ** -ScaleB) <= Num * (B ** N) < Den * B
-- which means that the product does not overflow if Den <= 2**(M-1) / B.
----------------------
-- Integer_To_Fixed --
----------------------
function Integer_To_Fixed
(Str : String;
Val : Uns;
Base : Unsigned;
ScaleB : Integer;
Extra : Unsigned;
Minus : Boolean;
Num : Int;
Den : Int) return Int
is
pragma Assert (Base in 2 .. 16);
pragma Assert (Extra < Base);
-- Accept only one extra digit after those used for Val
pragma Assert (Num < 0 and then Den < 0);
-- Accept only negative numbers to allow -2**(Int'Size - 1)
function Safe_Expont
(Base : Int;
Exp : in out Natural;
Factor : Int) return Int;
-- Return (Base ** Exp) * Factor if the computation does not overflow,
-- or else the number of the form (Base ** K) * Factor with the largest
-- magnitude if the former computation overflows. In both cases, Exp is
-- updated to contain the remaining power in the computation. Note that
-- Factor is expected to be negative in this context.
function Unsigned_To_Signed (Val : Uns) return Int;
-- Convert an integer value from unsigned to signed representation
-----------------
-- Safe_Expont --
-----------------
function Safe_Expont
(Base : Int;
Exp : in out Natural;
Factor : Int) return Int
is
pragma Assert (Base /= 0 and then Factor < 0);
Min : constant Int := Int'First / Base;
Result : Int := Factor;
begin
while Exp > 0 and then Result >= Min loop
Result := Result * Base;
Exp := Exp - 1;
end loop;
return Result;
end Safe_Expont;
------------------------
-- Unsigned_To_Signed --
------------------------
function Unsigned_To_Signed (Val : Uns) return Int is
begin
-- Deal with overflow cases, and also with largest negative number
if Val > Uns (Int'Last) then
if Minus and then Val = Uns (-(Int'First)) then
return Int'First;
else
Bad_Value (Str);
end if;
-- Negative values
elsif Minus then
return -(Int (Val));
-- Positive values
else
return Int (Val);
end if;
end Unsigned_To_Signed;
-- Local variables
B : constant Int := Int (Base);
V : Uns := Val;
E : Uns := Uns (Extra);
Y, Z, Q1, R1, Q2, R2 : Int;
begin
-- We will use a scaled divide operation for which we must control the
-- magnitude of operands so that an overflow exception is not unduly
-- raised during the computation. The only real concern is the exponent.
-- If ScaleB is too negative, then drop trailing digits, but preserve
-- the last dropped digit.
if ScaleB < 0 then
declare
LS : Integer := -ScaleB;
begin
Y := Den;
Z := Safe_Expont (B, LS, Num);
for J in 1 .. LS loop
E := V rem Uns (B);
V := V / Uns (B);
end loop;
end;
-- If ScaleB is too positive, then scale V up, which may then overflow
elsif ScaleB > 0 then
declare
LS : Integer := ScaleB;
begin
Y := Safe_Expont (B, LS, Den);
Z := Num;
for J in 1 .. LS loop
if V <= (Uns'Last - E) / Uns (B) then
V := V * Uns (B) + E;
E := 0;
else
Bad_Value (Str);
end if;
end loop;
end;
-- If ScaleB is zero, then proceed directly
else
Y := Den;
Z := Num;
end if;
-- Perform a scaled divide operation with final rounding to match Image
-- using two steps if there is an extra digit available. The second and
-- third operands are always negative so the sign of the quotient is the
-- sign of the first operand and the sign of the remainder the opposite.
if E > 0 then
Scaled_Divide (Unsigned_To_Signed (V), Y, Z, Q1, R1, Round => False);
Scaled_Divide (Unsigned_To_Signed (E), Y, -B, Q2, R2, Round => True);
-- Avoid an overflow during the subtraction. Note that Q2 is smaller
-- than Y and R1 smaller than Z in magnitude, so it is safe to take
-- their absolute value.
if abs Q2 >= 2 ** (Int'Size - 2)
or else abs R1 >= 2 ** (Int'Size - 2)
then
declare
Bit : constant Int := Q2 rem 2;
begin
Q2 := (Q2 - Bit) / 2;
R1 := (R1 - Bit) / 2;
Y := -2;
end;
else
Y := -1;
end if;
Scaled_Divide (Q2 - R1, Y, Z, Q2, R2, Round => True);
return Q1 + Q2;
else
Scaled_Divide (Unsigned_To_Signed (V), Y, Z, Q1, R1, Round => True);
return Q1;
end if;
exception
when Constraint_Error => Bad_Value (Str);
end Integer_To_Fixed;
----------------
-- Scan_Fixed --
----------------
function Scan_Fixed
(Str : String;
Ptr : not null access Integer;
Max : Integer;
Num : Int;
Den : Int) return Int
is
Base : Unsigned;
Scl : Impl.Scale_Array;
Extra : Unsigned;
Minus : Boolean;
Val : Impl.Value_Array;
begin
Val := Impl.Scan_Raw_Real (Str, Ptr, Max, Base, Scl, Extra, Minus);
return
Integer_To_Fixed (Str, Val (1), Base, Scl (1), Extra, Minus, Num, Den);
end Scan_Fixed;
-----------------
-- Value_Fixed --
-----------------
function Value_Fixed
(Str : String;
Num : Int;
Den : Int) return Int
is
Base : Unsigned;
Scl : Impl.Scale_Array;
Extra : Unsigned;
Minus : Boolean;
Val : Impl.Value_Array;
begin
Val := Impl.Value_Raw_Real (Str, Base, Scl, Extra, Minus);
return
Integer_To_Fixed (Str, Val (1), Base, Scl (1), Extra, Minus, Num, Den);
end Value_Fixed;
end System.Value_F;