------------------------------------------------------------------------------
-- --
-- GNAT RUNTIME COMPONENTS --
-- --
-- A D A . T E X T _ I O . F I X E D _ I O --
-- --
-- B o d y --
-- --
-- Copyright (C) 1992-2004 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 2, 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 COPYING. If not, write --
-- to the Free Software Foundation, 59 Temple Place - Suite 330, Boston, --
-- MA 02111-1307, USA. --
-- --
-- As a special exception, if other files instantiate generics from this --
-- unit, or you link this unit with other files to produce an executable, --
-- this unit does not by itself cause the resulting executable to be --
-- covered by the GNU General Public License. This exception does not --
-- however invalidate any other reasons why the executable file might be --
-- covered by the GNU Public License. --
-- --
-- GNAT was originally developed by the GNAT team at New York University. --
-- Extensive contributions were provided by Ada Core Technologies Inc. --
-- --
------------------------------------------------------------------------------
-- Fixed point I/O
-- ---------------
-- The following documents implementation details of the fixed point
-- input/output routines in the GNAT run time. The first part describes
-- general properties of fixed point types as defined by the Ada 95 standard,
-- including the Information Systems Annex.
-- Subsequently these are reduced to implementation constraints and the impact
-- of these constraints on a few possible approaches to I/O are given.
-- Based on this analysis, a specific implementation is selected for use in
-- the GNAT run time. Finally, the chosen algorithm is analyzed numerically in
-- order to provide user-level documentation on limits for range and precision
-- of fixed point types as well as accuracy of input/output conversions.
-- -------------------------------------------
-- - General Properties of Fixed Point Types -
-- -------------------------------------------
-- Operations on fixed point values, other than input and output, are not
-- important for the purposes of this document. Only the set of values that a
-- fixed point type can represent and the input and output operations are
-- significant.
-- Values
-- ------
-- Set set of values of a fixed point type comprise the integral
-- multiples of a number called the small of the type. The small can
-- either be a power of ten, a power of two or (if the implementation
-- allows) an arbitrary strictly positive real value.
-- Implementations need to support fixed-point types with a precision
-- of at least 24 bits, and (in order to comply with the Information
-- Systems Annex) decimal types need to support at least digits 18.
-- For the rest, however, no requirements exist for the minimal small
-- and range that need to be supported.
-- Operations
-- ----------
-- 'Image and 'Wide_Image (see RM 3.5(34))
-- These attributes return a decimal real literal best approximating
-- the value (rounded away from zero if halfway between) with a
-- single leading character that is either a minus sign or a space,
-- one or more digits before the decimal point (with no redundant
-- leading zeros), a decimal point, and N digits after the decimal
-- point. For a subtype S, the value of N is S'Aft, the smallest
-- positive integer such that (10**N)*S'Delta is greater or equal to
-- one, see RM 3.5.10(5).
-- For an arbitrary small, this means large number arithmetic needs
-- to be performed.
-- Put (see RM A.10.9(22-26))
-- The requirements for Put add no extra constraints over the image
-- attributes, although it would be nice to be able to output more
-- than S'Aft digits after the decimal point for values of subtype S.
-- 'Value and 'Wide_Value attribute (RM 3.5(40-55))
-- Since the input can be given in any base in the range 2..16,
-- accurate conversion to a fixed point number may require
-- arbitrary precision arithmetic if there is no limit on the
-- magnitude of the small of the fixed point type.
-- Get (see RM A.10.9(12-21))
-- The requirements for Get are identical to those of the Value
-- attribute.
-- ------------------------------
-- - Implementation Constraints -
-- ------------------------------
-- The requirements listed above for the input/output operations lead to
-- significant complexity, if no constraints are put on supported smalls.
-- Implementation Strategies
-- -------------------------
-- * Float arithmetic
-- * Arbitrary-precision integer arithmetic
-- * Fixed-precision integer arithmetic
-- Although it seems convenient to convert fixed point numbers to floating-
-- point and then print them, this leads to a number of restrictions.
-- The first one is precision. The widest floating-point type generally
-- available has 53 bits of mantissa. This means that Fine_Delta cannot
-- be less than 2.0**(-53).
-- In GNAT, Fine_Delta is 2.0**(-63), and Duration for example is a
-- 64-bit type. It would still be possible to use multi-precision
-- floating-point to perform calculations using longer mantissas,
-- but this is a much harder approach.
-- The base conversions needed for input and output of (non-decimal)
-- fixed point types can be seen as pairs of integer multiplications
-- and divisions.
-- Arbitrary-precision integer arithmetic would be suitable for the job
-- at hand, but has the draw-back that it is very heavy implementation-wise.
-- Especially in embedded systems, where fixed point types are often used,
-- it may not be desirable to require large amounts of storage and time
-- for fixed I/O operations.
-- Fixed-precision integer arithmetic has the advantage of simplicity and
-- speed. For the most common fixed point types this would be a perfect
-- solution. The downside however may be a too limited set of acceptable
-- fixed point types.
-- Extra Precision
-- ---------------
-- Using a scaled divide which truncates and returns a remainder R,
-- another E trailing digits can be calculated by computing the value
-- (R * (10.0**E)) / Z using another scaled divide. This procedure
-- can be repeated to compute an arbitrary number of digits in linear
-- time and storage. The last scaled divide should be rounded, with
-- a possible carry propagating to the more significant digits, to
-- ensure correct rounding of the unit in the last place.
-- An extension of this technique is to limit the value of Q to 9 decimal
-- digits, since 32-bit integers can be much more efficient than 64-bit
-- integers to output.
with Interfaces; use Interfaces;
with System.Arith_64; use System.Arith_64;
with System.Img_Real; use System.Img_Real;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Text_IO.Float_Aux;
with Ada.Text_IO.Generic_Aux;
package body Ada.Text_IO.Fixed_IO is
-- Note: we still use the floating-point I/O routines for input of
-- ordinary fixed-point and output using exponent format. This will
-- result in inaccuracies for fixed point types with a small that is
-- not a power of two, and for types that require more precision than
-- is available in Long_Long_Float.
package Aux renames Ada.Text_IO.Float_Aux;
Extra_Layout_Space : constant Field := 5 + Num'Fore;
-- Extra space that may be needed for output of sign, decimal point,
-- exponent indication and mandatory decimals after and before the
-- decimal point. A string with length
-- Fore + Aft + Exp + Extra_Layout_Space
-- is always long enough for formatting any fixed point number.
-- Implementation of Put routines
-- The following section describes a specific implementation choice for
-- performing base conversions needed for output of values of a fixed
-- point type T with small T'Small. The goal is to be able to output
-- all values of types with a precision of 64 bits and a delta of at
-- least 2.0**(-63), as these are current GNAT limitations already.
-- The chosen algorithm uses fixed precision integer arithmetic for
-- reasons of simplicity and efficiency. It is important to understand
-- in what ways the most simple and accurate approach to fixed point I/O
-- is limiting, before considering more complicated schemes.
-- Without loss of generality assume T has a range (-2.0**63) * T'Small
-- .. (2.0**63 - 1) * T'Small, and is output with Aft digits after the
-- decimal point and T'Fore - 1 before. If T'Small is integer, or
-- 1.0 / T'Small is integer, let S = T'Small and E = 0. For other T'Small,
-- let S and E be integers such that S / 10**E best approximates T'Small
-- and S is in the range 10**17 .. 10**18 - 1. The extra decimal scaling
-- factor 10**E can be trivially handled during final output, by adjusting
-- the decimal point or exponent.
-- Convert a value X * S of type T to a 64-bit integer value Q equal
-- to 10.0**D * (X * S) rounded to the nearest integer.
-- This conversion is a scaled integer divide of the form
-- Q := (X * Y) / Z,
-- where all variables are 64-bit signed integers using 2's complement,
-- and both the multiplication and division are done using full
-- intermediate precision. The final decimal value to be output is
-- Q * 10**(E-D)
-- This value can be written to the output file or to the result string
-- according to the format described in RM A.3.10. The details of this
-- operation are omitted here.
-- A 64-bit value can contain all integers with 18 decimal digits, but
-- not all with 19 decimal digits. If the total number of requested output
-- digits (Fore - 1) + Aft is greater than 18, for purposes of the
-- conversion Aft is adjusted to 18 - (Fore - 1). In that case, or
-- when Fore > 19, trailing zeros can complete the output after writing
-- the first 18 significant digits, or the technique described in the
-- next section can be used.
-- The final expression for D is
-- D := Integer'Max (-18, Integer'Min (Aft, 18 - (Fore - 1)));
-- For Y and Z the following expressions can be derived:
-- Q / (10.0**D) = X * S
-- Q = X * S * (10.0**D) = (X * Y) / Z
-- S * 10.0**D = Y / Z;
-- If S is an integer greater than or equal to one, then Fore must be at
-- least 20 in order to print T'First, which is at most -2.0**63.
-- This means D < 0, so use
-- (1) Y = -S and Z = -10**(-D).
-- If 1.0 / S is an integer greater than one, use
-- (2) Y = -10**D and Z = -(1.0 / S), for D >= 0
-- or
-- (3) Y = 1 and Z = (1.0 / S) * 10**(-D), for D < 0
-- Negative values are used for nominator Y and denominator Z, so that S
-- can have a maximum value of 2.0**63 and a minimum of 2.0**(-63).
-- For Z in -1 .. -9, Fore will still be 20, and D will be negative, as
-- (-2.0**63) / -9 is greater than 10**18. In these cases there is room
-- in the denominator for the extra decimal scaling required, so case (3)
-- will not overflow.
pragma Assert (System.Fine_Delta >= 2.0**(-63));
pragma Assert (Num'Small in 2.0**(-63) .. 2.0**63);
pragma Assert (Num'Fore <= 37);
-- These assertions need to be relaxed to allow for a Small of
-- 2.0**(-64) at least, since there is an ACATS test for this ???
Max_Digits : constant := 18;
-- Maximum number of decimal digits that can be represented in a
-- 64-bit signed number, see above
-- The constants E0 .. E5 implement a binary search for the appropriate
-- power of ten to scale the small so that it has one digit before the
-- decimal point.
subtype Int is Integer;
E0 : constant Int := -20 * Boolean'Pos (Num'Small >= 1.0E1);
E1 : constant Int := E0 + 10 * Boolean'Pos (Num'Small * 10.0**E0 < 1.0E-10);
E2 : constant Int := E1 + 5 * Boolean'Pos (Num'Small * 10.0**E1 < 1.0E-5);
E3 : constant Int := E2 + 3 * Boolean'Pos (Num'Small * 10.0**E2 < 1.0E-3);
E4 : constant Int := E3 + 2 * Boolean'Pos (Num'Small * 10.0**E3 < 1.0E-1);
E5 : constant Int := E4 + 1 * Boolean'Pos (Num'Small * 10.0**E4 < 1.0E-0);
Scale : constant Integer := E5;
pragma Assert (Num'Small * 10.0**Scale >= 1.0
and then Num'Small * 10.0**Scale < 10.0);
Exact : constant Boolean :=
Float'Floor (Num'Small) = Float'Ceiling (Num'Small)
or Float'Floor (1.0 / Num'Small) = Float'Ceiling (1.0 / Num'Small)
or Num'Small >= 10.0**Max_Digits;
-- True iff a numerator and denominator can be calculated such that
-- their ratio exactly represents the small of Num
-- Local Subprograms
procedure Put
(To : out String;
Last : out Natural;
Item : Num;
Fore : Field;
Aft : Field;
Exp : Field);
-- Actual output function, used internally by all other Put routines
---------
-- Get --
---------
procedure Get
(File : in File_Type;
Item : out Num;
Width : in Field := 0)
is
pragma Unsuppress (Range_Check);
begin
Aux.Get (File, Long_Long_Float (Item), Width);
exception
when Constraint_Error => raise Data_Error;
end Get;
procedure Get
(Item : out Num;
Width : in Field := 0)
is
pragma Unsuppress (Range_Check);
begin
Aux.Get (Current_In, Long_Long_Float (Item), Width);
exception
when Constraint_Error => raise Data_Error;
end Get;
procedure Get
(From : in String;
Item : out Num;
Last : out Positive)
is
pragma Unsuppress (Range_Check);
begin
Aux.Gets (From, Long_Long_Float (Item), Last);
exception
when Constraint_Error => raise Data_Error;
end Get;
---------
-- Put --
---------
procedure Put
(File : in File_Type;
Item : in Num;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp)
is
S : String (1 .. Fore + Aft + Exp + Extra_Layout_Space);
Last : Natural;
begin
Put (S, Last, Item, Fore, Aft, Exp);
Generic_Aux.Put_Item (File, S (1 .. Last));
end Put;
procedure Put
(Item : in Num;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp)
is
S : String (1 .. Fore + Aft + Exp + Extra_Layout_Space);
Last : Natural;
begin
Put (S, Last, Item, Fore, Aft, Exp);
Generic_Aux.Put_Item (Text_IO.Current_Out, S (1 .. Last));
end Put;
procedure Put
(To : out String;
Item : in Num;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp)
is
Fore : constant Integer := To'Length
- 1 -- Decimal point
- Field'Max (1, Aft) -- Decimal part
- Boolean'Pos (Exp /= 0) -- Exponent indicator
- Exp; -- Exponent
Last : Natural;
begin
if Fore not in Field'Range then
raise Layout_Error;
end if;
Put (To, Last, Item, Fore, Aft, Exp);
if Last /= To'Last then
raise Layout_Error;
end if;
end Put;
procedure Put
(To : out String;
Last : out Natural;
Item : Num;
Fore : Field;
Aft : Field;
Exp : Field)
is
subtype Digit is Int64 range 0 .. 9;
X : constant Int64 := Int64'Integer_Value (Item);
A : constant Field := Field'Max (Aft, 1);
Neg : constant Boolean := (Item < 0.0);
Pos : Integer; -- Next digit X has value X * 10.0**Pos;
Y, Z : Int64;
E : constant Integer := Boolean'Pos (not Exact)
* (Max_Digits - 1 + Scale);
D : constant Integer := Boolean'Pos (Exact)
* Integer'Min (A, Max_Digits - (Num'Fore - 1))
+ Boolean'Pos (not Exact)
* (Scale - 1);
procedure Put_Character (C : Character);
pragma Inline (Put_Character);
-- Add C to the output string To, updating Last
procedure Put_Digit (X : Digit);
-- Add digit X to the output string (going from left to right),
-- updating Last and Pos, and inserting the sign, leading zeroes
-- or a decimal point when necessary. After outputting the first
-- digit, Pos must not be changed outside Put_Digit anymore
procedure Put_Int64 (X : Int64; Scale : Integer);
-- Output the decimal number X * 10**Scale
procedure Put_Scaled
(X, Y, Z : Int64;
A : Field;
E : Integer);
-- Output the decimal number (X * Y / Z) * 10**E, producing A digits
-- after the decimal point and rounding the final digit. The value
-- X * Y / Z is computed with full precision, but must be in the
-- range of Int64.
-------------------
-- Put_Character --
-------------------
procedure Put_Character (C : Character) is
begin
Last := Last + 1;
To (Last) := C;
end Put_Character;
---------------
-- Put_Digit --
---------------
procedure Put_Digit (X : Digit) is
Digs : constant array (Digit) of Character := "0123456789";
begin
if Last = 0 then
if X /= 0 or Pos <= 0 then
-- Before outputting first digit, include leading space,
-- posible minus sign and, if the first digit is fractional,
-- decimal seperator and leading zeros.
-- The Fore part has Pos + 1 + Boolean'Pos (Neg) characters,
-- if Pos >= 0 and otherwise has a single zero digit plus minus
-- sign if negative. Add leading space if necessary.
for J in Integer'Max (0, Pos) + 2 + Boolean'Pos (Neg) .. Fore
loop
Put_Character (' ');
end loop;
-- Output minus sign, if number is negative
if Neg then
Put_Character ('-');
end if;
-- If starting with fractional digit, output leading zeros
if Pos < 0 then
Put_Character ('0');
Put_Character ('.');
for J in Pos .. -2 loop
Put_Character ('0');
end loop;
end if;
Put_Character (Digs (X));
end if;
else
-- This is not the first digit to be output, so the only
-- special handling is that for the decimal point
if Pos = -1 then
Put_Character ('.');
end if;
Put_Character (Digs (X));
end if;
Pos := Pos - 1;
end Put_Digit;
---------------
-- Put_Int64 --
---------------
procedure Put_Int64 (X : Int64; Scale : Integer) is
begin
if X = 0 then
return;
end if;
Pos := Scale;
if X not in -9 .. 9 then
Put_Int64 (X / 10, Scale + 1);
end if;
Put_Digit (abs (X rem 10));
end Put_Int64;
----------------
-- Put_Scaled --
----------------
procedure Put_Scaled
(X, Y, Z : Int64;
A : Field;
E : Integer)
is
N : constant Natural := (A + Max_Digits - 1) / Max_Digits + 1;
Q : array (1 .. N) of Int64 := (others => 0);
XX : Int64 := X;
YY : Int64 := Y;
AA : Field := A;
begin
for J in Q'Range loop
exit when XX = 0;
Scaled_Divide (XX, YY, Z, Q (J), XX, Round => AA = 0);
-- As the last block of digits is rounded, a carry may have to
-- be propagated to the more significant digits. Since the last
-- block may have less than Max_Digits, the test for this block
-- is specialized.
-- The absolute value of the left-most digit block may equal
-- 10*Max_Digits, as no carry can be propagated from there.
-- The final output routines need to be prepared to handle
-- this specific case.
if (Q (J) = YY or -Q (J) = YY) and then J > Q'First then
if Q (J) < 0 then
Q (J - 1) := Q (J - 1) + 1;
else
Q (J - 1) := Q (J - 1) - 1;
end if;
Q (J) := 0;
Propagate_Carry :
for J in reverse Q'First + 1 .. Q'Last loop
if Q (J) >= 10**Max_Digits then
Q (J - 1) := Q (J - 1) + 1;
Q (J) := Q (J) - 10**Max_Digits;
elsif Q (J) <= -10**Max_Digits then
Q (J - 1) := Q (J - 1) - 1;
Q (J) := Q (J) + 10**Max_Digits;
end if;
end loop Propagate_Carry;
end if;
YY := -10**Integer'Min (Max_Digits, AA);
AA := AA - Integer'Min (Max_Digits, AA);
end loop;
for J in Q'First .. Q'Last - 1 loop
Put_Int64 (Q (J), E - (J - Q'First) * Max_Digits);
end loop;
Put_Int64 (Q (Q'Last), E - A);
end Put_Scaled;
-- Start of processing for Put
begin
Last := To'First - 1;
if Exp /= 0 then
-- With the Exp format, it is not known how many output digits to
-- generate, as leading zeros must be ignored. Computing too many
-- digits and then truncating the output will not give the closest
-- output, it is necessary to round at the correct digit.
-- The general approach is as follows: as long as no digits have
-- been generated, compute the Aft next digits (without rounding).
-- Once a non-zero digit is generated, determine the exact number
-- of digits remaining and compute them with rounding.
-- Since a large number of iterations might be necessary in case
-- of Aft = 1, the following optimization would be desirable.
-- Count the number Z of leading zero bits in the integer
-- representation of X, and start with producing
-- Aft + Z * 1000 / 3322 digits in the first scaled division.
-- However, the floating-point routines are still used now ???
System.Img_Real.Set_Image_Real (Long_Long_Float (Item), To, Last,
Fore, Aft, Exp);
return;
end if;
if Exact then
Y := Int64'Min (Int64 (-Num'Small), -1) * 10**Integer'Max (0, D);
Z := Int64'Min (Int64 (-1.0 / Num'Small), -1)
* 10**Integer'Max (0, -D);
else
Y := Int64 (-Num'Small * 10.0**E);
Z := -10**Max_Digits;
end if;
Put_Scaled (X, Y, Z, A - D, -D);
-- If only zero digits encountered, unit digit has not been output yet
if Last < To'First then
Pos := 0;
end if;
-- Always output digits up to the first one after the decimal point
while Pos >= -A loop
Put_Digit (0);
end loop;
end Put;
end Ada.Text_IO.Fixed_IO;