Numerics

Modular Types

In the Introduction to Ada course, we've seen that Ada has two kinds of integer type: signed and modular types. For example:

    
    
    
        
package Num_Types is

   type Signed_Integer is range 1 .. 1_000_000;
   type Modular is mod 2**32;

end Num_Types;

In this section, we discuss two attributes of modular types: Modulus and Mod. We also discuss operations on modular types.

In the Ada Reference Manual

Modulus Attribute

The Modulus attribute returns the modulus of the modular type as a universal integer value. Let's get the modulus of the 32-bit Modular type that we've declared in the Num_Types package of the previous example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

with Num_Types;   use Num_Types;

procedure Show_Modular is
   Modulus_Value : constant := Modular'Modulus;
begin
   Put_Line (Modulus_Value'Image);
end Show_Modular;

When we run this example, we get 4294967296, which is equal to 2**32.

Mod Attribute

Note

This section was originally written by Robert A. Duff and published as Gem #26: The Mod Attribute.

Operations on signed integers can overflow: if the result is outside the base range, Constraint_Error will be raised. In our previous example, we declared the Signed_Integer type:

type Signed_Integer is range 1 .. 1_000_000;

The base range of Signed_Integer is the range of Signed_Integer'Base, which is chosen by the compiler, but is likely to be something like -2**31 .. 2**31 - 1. (Note: we discussed the Base attribute in this section.)

Operations on modular integers use modular (wraparound) arithmetic. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

with Num_Types;   use Num_Types;

procedure Show_Modular is
   X : Modular;
begin
   X := 1;
   Put_Line (X'Image);

   X := -X;
   Put_Line (X'Image);
end Show_Modular;

Negating X gives -1, which wraps around to 2**32 - 1, i.e. all-one-bits.

But what about a type conversion from signed to modular? Is that a signed operation (so it should overflow) or is it a modular operation (so it should wrap around)? The answer in Ada is the former — that is, if you try to convert, say, Integer'(-1) to Modular, you will get Constraint_Error:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

with Num_Types;   use Num_Types;

procedure Show_Modular is
   I : Integer := -1;
   X : Modular := 1;
begin
   X := Modular (I);  --  raises Constraint_Error
   Put_Line (X'Image);
end Show_Modular;

To solve this problem, we can use the Mod attribute:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

with Num_Types;   use Num_Types;

procedure Show_Modular is
   I : constant Integer := -1;
   X : Modular := 1;
begin
   X := Modular'Mod (I);
   Put_Line (X'Image);
end Show_Modular;

The Mod attribute will correctly convert from any integer type to a given modular type, using wraparound semantics.

Historically

In older versions of Ada — such as Ada 95 —, the only way to do this conversion is to use Unchecked_Conversion, which is somewhat uncomfortable. Furthermore, if you're trying to convert to a generic formal modular type, how do you know what size of signed integer type to use? Note that Unchecked_Conversion might malfunction if the source and target types are of different sizes.

The Mod attribute was added to Ada 2005 to solve this problem. Also, we can now safely use this attribute in generics. For example:

    
    
    
        
generic
   type Formal_Modular is mod <>;
package Mod_Attribute is
   function F return Formal_Modular;
end Mod_Attribute;

    
        
package body Mod_Attribute is

   A_Signed_Integer : Integer := -1;

   function F return Formal_Modular is
   begin
      return Formal_Modular'Mod
               (A_Signed_Integer);
   end F;

end Mod_Attribute;

In this example, F will return the all-ones bit pattern, for whatever modular type is passed to Formal_Modular.

Operations on modular types

Modular types are particularly useful for bit manipulation. For example, we can use the and, or, xor and not operators for modular types.

Also, we can perform bit-shifting by multiplying or dividing a modular object with a power of two. For example, if M is a variable of modular type, then M := M * 2 ** 3; shifts the bits to the left by three bits. Likewise, M := M / 2 ** 3 shifts the bits to the right. Note that the compiler selects the appropriate shifting operator when translating these operations to machine code — no actual multiplication or division will be performed.

Let's see a simple implementation of the CRC-CCITT (0x1D0F) algorithm:

    
    
    
        
package Crc_Defs is

    type Byte is mod 2 ** 8;
    type Crc  is mod 2 ** 16;

    type Byte_Array is
      array (Positive range <>) of Byte;

    function Crc_CCITT (A : Byte_Array)
                        return Crc;

    procedure Display (Crc_A : Crc);

    procedure Display (A : Byte_Array);

end Crc_Defs;

    
        
with Ada.Text_IO; use Ada.Text_IO;

package body Crc_Defs is

    package Byte_IO is new Modular_IO (Byte);
    package Crc_IO  is new Modular_IO (Crc);

    function Crc_CCITT (A : Byte_Array)
                        return Crc
    is
       X     : Byte;
       Crc_A : Crc := 16#1d0f#;
    begin
       for I in A'Range loop
          X := Byte (Crc_A / 2 ** 8) xor A (I);
          X := X xor (X / 2 ** 4);
          declare
             Crc_X : constant Crc := Crc (X);
          begin
             Crc_A := Crc_A * 2 ** 8  xor
                      Crc_X * 2 ** 12 xor
                      Crc_X * 2 ** 5  xor
                      Crc_X;
          end;
       end loop;

       return Crc_A;
    end Crc_CCITT;

    procedure Display (Crc_A : Crc) is
    begin
       Crc_IO.Put (Crc_A);
       New_Line;
    end Display;

    procedure Display (A : Byte_Array) is
    begin
       for E of A loop
          Byte_IO.Put (E);
          Put (", ");
       end loop;
       New_Line;
    end Display;

begin
   Byte_IO.Default_Width := 1;
   Byte_IO.Default_Base  := 16;
   Crc_IO.Default_Width  := 1;
   Crc_IO.Default_Base   := 16;
end Crc_Defs;

    
        
with Ada.Text_IO; use Ada.Text_IO;
with Crc_Defs;    use Crc_Defs;

procedure Show_Crc is
   AA    : constant Byte_Array :=
             (16#0#, 16#20#, 16#30#);
   Crc_A : Crc;
begin
   Crc_A := Crc_CCITT (AA);

   Put ("Input array: ");
   Display (AA);

   Put ("CRC-CCITT: ");
   Display (Crc_A);
end Show_Crc;

In this example, the core of the algorithm is implemented in the Crc_CCITT function. There, we use bit shifting — for instance, * 2 ** 8 and / 2 ** 8, which shift left and right, respectively, by eight bits. We also use the xor operator.

Numeric Literals

Classification

We've already discussed basic characteristics of numeric literals in the Introduction to Ada course — although we haven't used this terminology there. There are two kinds of numeric literals in Ada: integer literals and real literals. They are distinguished by the absence or presence of a radix point. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Real_Integer_Literals is
   Integer_Literal : constant := 365;
   Real_Literal    : constant := 365.2564;
begin
   Put_Line ("Integer Literal: "
             & Integer_Literal'Image);
   Put_Line ("Real Literal:    "
             & Real_Literal'Image);
end Real_Integer_Literals;

Another classification takes the use of a base indicator into account. (Remember that, when writing a literal such as 2#1011#, the base is the element before the first # sign.) So here we distinguish between decimal literals and based literals. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Decimal_Based_Literals is

   package F_IO is new
     Ada.Text_IO.Float_IO (Float);

   --
   --  DECIMAL LITERALS
   --

   Dec_Integer  : constant := 365;

   Dec_Real     : constant := 365.2564;
   Dec_Real_Exp : constant := 0.365_256_4e3;

   --
   --  BASED LITERALS
   --

   Based_Integer     : constant := 16#16D#;
   Based_Integer_Exp : constant := 5#243#e1;

   Based_Real        : constant :=
     2#1_0110_1101.0100_0001_1010_0011_0111#;
   Based_Real_Exp    : constant :=
     7#1.031_153_643#e3;
begin
   F_IO.Default_Fore := 3;
   F_IO.Default_Aft  := 4;
   F_IO.Default_Exp  := 0;

   Put_Line ("Dec_Integer:       "
             & Dec_Integer'Image);

   Put ("Dec_Real:           ");
   F_IO.Put (Item => Dec_Real);
   New_Line;

   Put ("Dec_Real_Exp:       ");
   F_IO.Put (Item => Dec_Real_Exp);
   New_Line;

   Put_Line ("Based_Integer:     "
             & Based_Integer'Image);
   Put_Line ("Based_Integer_Exp: "
             & Based_Integer_Exp'Image);

   Put ("Based_Real:         ");
   F_IO.Put (Item => Based_Real);
   New_Line;

   Put ("Based_Real_Exp:     ");
   F_IO.Put (Item => Based_Real_Exp);
   New_Line;
end Decimal_Based_Literals;

Based literals use the base#number# format. Also, they aren't limited to simple integer literals such as 16#16D#. In fact, we can use a radix point or an exponent in based literals, as well as underscores. In addition, we can use any base from 2 up to 16. We discuss these aspects further in the next section.

Features and Flexibility

Note

This section was originally written by Franco Gasperoni and published as Gem #7: The Beauty of Numeric Literals in Ada.

Ada provides a simple and elegant way of expressing numeric literals. One of those simple, yet powerful aspects is the ability to use underscores to separate groups of digits. For example, 3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510 is more readable and less error prone to type than 3.14159265358979323846264338327950288419716939937510. Here's the complete code:

    
    
    
        
with Ada.Text_IO;

 procedure Ada_Numeric_Literals is
    Pi   : constant :=
      3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510;

    Pi2  : constant :=
      3.14159265358979323846264338327950288419716939937510;

    Z    : constant := Pi - Pi2;
    pragma Assert (Z = 0.0);

    use Ada.Text_IO;
 begin
    Put_Line ("Z = " & Float'Image (Z));
 end Ada_Numeric_Literals;

Also, when using based literals, Ada allows any base from 2 to 16. Thus, we can write the decimal number 136 in any one of the following notations:

    
    
    
        
with Ada.Text_IO;

procedure Ada_Numeric_Literals is
   Bin_136 : constant := 2#1000_1000#;
   Oct_136 : constant := 8#210#;
   Dec_136 : constant := 10#136#;
   Hex_136 : constant := 16#88#;
   pragma Assert (Bin_136 = 136);
   pragma Assert (Oct_136 = 136);
   pragma Assert (Dec_136 = 136);
   pragma Assert (Hex_136 = 136);

   use Ada.Text_IO;

begin
   Put_Line ("Bin_136 = "
             & Integer'Image (Bin_136));
   Put_Line ("Oct_136 = "
             & Integer'Image (Oct_136));
   Put_Line ("Dec_136 = "
             & Integer'Image (Dec_136));
   Put_Line ("Hex_136 = "
             & Integer'Image (Hex_136));
end Ada_Numeric_Literals;

In other languages

The rationale behind the method to specify based literals in the C programming language is strange and unintuitive. Here, you have only three possible bases: 8, 10, and 16 (why no base 2?). Furthermore, requiring that numbers in base 8 be preceded by a zero feels like a bad joke on us programmers. For example, what values do 0210 and 210 represent in C?

When dealing with microcontrollers, we might encounter I/O devices that are memory mapped. Here, we have the ability to write:

Lights_On  : constant := 2#1000_1000#;
Lights_Off : constant := 2#0111_0111#;

and have the ability to turn on/off the lights as follows:

Output_Devices := Output_Devices or  Lights_On;
Output_Devices := Output_Devices and Lights_Off;

Here's the complete example:

    
    
    
        
with Ada.Text_IO;

procedure Ada_Numeric_Literals is
   Lights_On  : constant := 2#1000_1000#;
   Lights_Off : constant := 2#0111_0111#;

   type Byte is mod 256;
   Output_Devices : Byte := 0;

   --  for Output_Devices'Address
   --    use 16#DEAD_BEEF#;
   --  ^^^^^^^^^^^^^^^^^^^^^^^^^^
   --  Memory mapped Output

   use Ada.Text_IO;
begin
   Output_Devices := Output_Devices or
                       Lights_On;

   Put_Line ("Output_Devices (lights on ) = "
             & Byte'Image (Output_Devices));

   Output_Devices := Output_Devices and
                       Lights_Off;

   Put_Line ("Output_Devices (lights off) = "
             & Byte'Image (Output_Devices));
end Ada_Numeric_Literals;

Of course, we can also use records with representation clauses to do the above, which is even more elegant.

The notion of base in Ada allows for exponents, which is particularly pleasant. For instance, we can write:

    
    
    
        
package Literal_Binaries is
   Kilobyte  : constant := 2#1#e+10;
   Megabyte  : constant := 2#1#e+20;
   Gigabyte  : constant := 2#1#e+30;
   Terabyte  : constant := 2#1#e+40;
   Petabyte  : constant := 2#1#e+50;
   Exabyte   : constant := 2#1#e+60;
   Zettabyte : constant := 2#1#e+70;
   Yottabyte : constant := 2#1#e+80;
end Literal_Binaries;

In based literals, the exponent — like the base — uses the regular decimal notation and specifies the power of the base that the based literal should be multiplied with to obtain the final value. For instance 2#1#e+10 = 1 x 210 = 1_024 (in base 10), whereas 16#F#e+2 = 15 x 162 = 15 x 256 = 3_840 (in base 10).

Based numbers apply equally well to real literals. We can, for instance, write:

One_Third : constant := 3#0.1#;
--                      ^^^^^^
--                  same as 1.0/3

Whether we write 3#0.1# or 1.0 / 3, or even 3#1.0#e-1, Ada allows us to specify exactly rational numbers for which decimal literals cannot be written.

The last nice feature is that Ada has an open-ended set of integer and real types. As a result, numeric literals in Ada do not carry with them their type as, for example, in C. The actual type of the literal is determined from the context. This is particularly helpful in avoiding overflows, underflows, and loss of precision.

In other languages

In C, a source of confusion can be the distinction between 32l and 321. Although both look similar, they're actually very different from each other.

And this is not all: all constant computations done at compile time are done in infinite precision, be they integer or real. This allows us to write constants with whatever size and precision without having to worry about overflow or underflow. We can for instance write:

Zero : constant := 1.0 - 3.0 * One_Third;

and be guaranteed that constant Zero has indeed value zero. This is very different from writing:

One_Third_Approx : constant :=
  0.33333333333333333333333333333;
Zero_Approx      : constant :=
  1.0 - 3.0 * One_Third_Approx;

where Zero_Approx is really 1.0e-29 — and that will show up in your numerical computations. The above is quite handy when we want to write fractions without any loss of precision. Here's the complete code:

    
    
    
        
with Ada.Text_IO;

procedure Ada_Numeric_Literals is
   One_Third : constant := 3#1.0#e-1;
   --  same as 1.0/3.0

   Zero      : constant := 1.0 - 3.0 * One_Third;
   pragma Assert (Zero = 0.0);

   One_Third_Approx : constant :=
     0.33333333333333333333333333333;
   Zero_Approx      : constant :=
     1.0 - 3.0 * One_Third_Approx;

   use Ada.Text_IO;

begin
   Put_Line ("Zero        = "
             & Float'Image (Zero));
   Put_Line ("Zero_Approx = "
             & Float'Image (Zero_Approx));
end Ada_Numeric_Literals;

Along these same lines, we can write:

    
    
    
        
with Ada.Text_IO;

with Literal_Binaries; use Literal_Binaries;

procedure Ada_Numeric_Literals is

   Big_Sum : constant := 1         +
                         Kilobyte  +
                         Megabyte  +
                         Gigabyte  +
                         Terabyte  +
                         Petabyte  +
                         Exabyte   +
                         Zettabyte;

   Result : constant := (Yottabyte - 1) /
                        (Kilobyte - 1);

   Nil    : constant := Result - Big_Sum;
   pragma Assert (Nil = 0);

   use Ada.Text_IO;

begin
   Put_Line ("Nil         = "
             & Integer'Image (Nil));
end Ada_Numeric_Literals;

and be guaranteed that Nil is equal to zero.

Floating-Point Types

In this section, we discuss various attributes related to floating-point types.

In the Ada Reference Manual

Representation-oriented attributes

In this section, we discuss attributes related to the representation of floating-point types.

Attribute: Machine_Radix

Machine_Radix is an attribute that returns the radix of the hardware representation of a type. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Machine_Radix is
begin
   Put_Line
     ("Float'Machine_Radix:           "
      & Float'Machine_Radix'Image);
   Put_Line
     ("Long_Float'Machine_Radix:      "
      & Long_Float'Machine_Radix'Image);
   Put_Line
     ("Long_Long_Float'Machine_Radix: "
      & Long_Long_Float'Machine_Radix'Image);
end Show_Machine_Radix;

Usually, this value is two, as the radix is based on a binary system.

Attributes: Machine_Mantissa

Machine_Mantissa is an attribute that returns the number of bits reserved for the mantissa of the floating-point type. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Machine_Mantissa is
begin
   Put_Line
     ("Float'Machine_Mantissa:           "
      & Float'Machine_Mantissa'Image);
   Put_Line
     ("Long_Float'Machine_Mantissa:      "
      & Long_Float'Machine_Mantissa'Image);
   Put_Line
     ("Long_Long_Float'Machine_Mantissa: "
      & Long_Long_Float'Machine_Mantissa'Image);
end Show_Machine_Mantissa;

On a typical desktop PC, as indicated by Machine_Mantissa, we have 24 bits for the floating-point mantissa of the Float type.

Machine_Emin and Machine_Emax

The Machine_Emin and Machine_Emax attributes return the minimum and maximum value, respectively, of the machine exponent the floating-point type. Note that, in all cases, the returned value is a universal integer. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Machine_Emin_Emax is
begin
   Put_Line
     ("Float'Machine_Emin:               "
      & Float'Machine_Emin'Image);
   Put_Line
     ("Float'Machine_Emax:               "
      & Float'Machine_Emax'Image);
   Put_Line
     ("Long_Float'Machine_Emin:          "
      & Long_Float'Machine_Emin'Image);
   Put_Line
     ("Long_Float'Machine_Emax:          "
      & Long_Float'Machine_Emax'Image);
   Put_Line
     ("Long_Long_Float'Machine_Emin:     "
      & Long_Long_Float'Machine_Emin'Image);
   Put_Line
     ("Long_Long_Float'Machine_Emax:     "
      & Long_Long_Float'Machine_Emax'Image);
end Show_Machine_Emin_Emax;

On a typical desktop PC, the value of Float'Machine_Emin and Float'Machine_Emax is -125 and 128, respectively.

To get the actual minimum and maximum value of the exponent for a specific type, we need to use the Machine_Radix attribute that we've seen previously. Let's calculate the minimum and maximum value of the exponent for the Float type on a typical PC:

  • Value of minimum exponent: Float'Machine_Radix ** Float'Machine_Emin.

    • In our target platform, this is 2-125 = 2.35098870164457501594 x 10-38.

  • Value of maximum exponent: Float'Machine_Radix ** Float'Machine_Emax.

    • In our target platform, this is 2128 = 3.40282366920938463463 x 1038.

Attribute: Digits

Digits is an attribute that returns the requested decimal precision of a floating-point subtype. Let's see an example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Digits is
begin
   Put_Line ("Float'Digits:           "
             & Float'Digits'Image);
   Put_Line ("Long_Float'Digits:      "
             & Long_Float'Digits'Image);
   Put_Line ("Long_Long_Float'Digits: "
             & Long_Long_Float'Digits'Image);
end Show_Digits;

Here, the requested decimal precision of the Float type is six digits.

Note that we said that Digits is the requested level of precision, which is specified as part of declaring a floating point type. We can retrieve the actual decimal precision with Base'Digits. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Base_Digits is
   type Float_D3 is new Float digits 3;
begin
   Put_Line ("Float_D3'Digits:           "
             & Float_D3'Digits'Image);
   Put_Line ("Float_D3'Base'Digits:      "
             & Float_D3'Base'Digits'Image);
end Show_Base_Digits;

The requested decimal precision of the Float_D3 type is three digits, while the actual decimal precision is six digits (on a typical desktop PC).

Attributes: Denorm, Signed_Zeros, Machine_Rounds, Machine_Overflows

In this section, we discuss attributes that return Boolean values indicating whether a feature is available or not in the target architecture:

  • Denorm is an attribute that indicates whether the target architecture uses denormalized numbers.

  • Signed_Zeros is an attribute that indicates whether the type uses a sign for zero values, so it can represent both -0.0 and 0.0.

  • Machine_Rounds is an attribute that indicates whether rounding-to-nearest is used, rather than some other choice (such as rounding-toward-zero).

  • Machine_Overflows is an attribute that indicates whether a Constraint_Error exception is (or is not) guaranteed to be raised when an operation with that type produces an overflow or divide-by-zero.

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Boolean_Attributes is
begin
   Put_Line
     ("Float'Denorm:           "
      & Float'Denorm'Image);
   Put_Line
     ("Long_Float'Denorm:      "
      & Long_Float'Denorm'Image);
   Put_Line
     ("Long_Long_Float'Denorm: "
      & Long_Long_Float'Denorm'Image);
   Put_Line
     ("Float'Signed_Zeros:           "
      & Float'Signed_Zeros'Image);
   Put_Line
     ("Long_Float'Signed_Zeros:      "
      & Long_Float'Signed_Zeros'Image);
   Put_Line
     ("Long_Long_Float'Signed_Zeros: "
      & Long_Long_Float'Signed_Zeros'Image);
   Put_Line
     ("Float'Machine_Rounds:           "
      & Float'Machine_Rounds'Image);
   Put_Line
     ("Long_Float'Machine_Rounds:      "
      & Long_Float'Machine_Rounds'Image);
   Put_Line
     ("Long_Long_Float'Machine_Rounds: "
      & Long_Long_Float'Machine_Rounds'Image);
   Put_Line
     ("Float'Machine_Overflows:           "
      & Float'Machine_Overflows'Image);
   Put_Line
     ("Long_Float'Machine_Overflows:      "
      & Long_Float'Machine_Overflows'Image);
   Put_Line
     ("Long_Long_Float'Machine_Overflows: "
      & Long_Long_Float'Machine_Overflows'Image);
end Show_Boolean_Attributes;

On a typical PC, we have the following information:

  • Denorm is true (i.e. the architecture uses denormalized numbers);

  • Signed_Zeros is true (i.e. the standard floating-point types use a sign for zero values);

  • Machine_Rounds is true (i.e. rounding-to-nearest is used for floating-point types);

  • Machine_Overflows is false (i.e. there's no guarantee that a Constraint_Error exception is raised when an operation with a floating-point type produces an overflow or divide-by-zero).

Primitive function attributes

In this section, we discuss attributes that we can use to manipulate floating-point values.

Attributes: Fraction, Exponent and Compose

The Exponent and Fraction attributes return "parts" of a floating-point value:

  • Exponent returns the machine exponent, and

  • Fraction returns the mantissa part.

Compose is used to return a floating-point value based on a fraction (the mantissa part) and the machine exponent.

Let's see some examples:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Exponent_Fraction_Compose is
begin
   Put_Line
     ("Float'Fraction (1.0):     "
      & Float'Fraction (1.0)'Image);
   Put_Line
     ("Float'Fraction (0.25):    "
      & Float'Fraction (0.25)'Image);
   Put_Line
     ("Float'Fraction (1.0e-25): "
      & Float'Fraction (1.0e-25)'Image);
   Put_Line
     ("Float'Exponent (1.0):     "
      & Float'Exponent (1.0)'Image);
   Put_Line
     ("Float'Exponent (0.25):    "
      & Float'Exponent (0.25)'Image);
   Put_Line
     ("Float'Exponent (1.0e-25): "
      & Float'Exponent (1.0e-25)'Image);
   Put_Line
     ("Float'Compose (5.00000e-01, 1):   "
      & Float'Compose (5.00000e-01, 1)'Image);
   Put_Line
     ("Float'Compose (5.00000e-01, -1):  "
      & Float'Compose (5.00000e-01, -1)'Image);
   Put_Line
     ("Float'Compose (9.67141E-01, -83): "
      & Float'Compose (9.67141E-01, -83)'Image);
end Show_Exponent_Fraction_Compose;

To understand this code example, we have to take this formula into account:

Value = Fraction x Machine_RadixExponent

Considering that the value of Float'Machine_Radix on a typical PC is two, we see that the value 1.0 is composed by a fraction of 0.5 and a machine exponent of one. In other words:

0.5 x 21 = 1.0

For the value 0.25, we get a fraction of 0.5 and a machine exponent of -1, which is the result of 0.5 x 2-1 = 0.25. We can use the Compose attribute to perform this calculation. For example, Float'Compose (0.5, -1) = 0.25.

Note that Fraction is always between 0.5 and 0.999999 (i.e < 1.0), except for denormalized numbers, where it can be < 0.5.

Attribute: Scaling

Scaling is an attribute that scales a floating-point value based on the machine radix and a machine exponent passed to the function. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Scaling is
begin
   Put_Line ("Float'Scaling (0.25, 1): "
             & Float'Scaling (0.25, 1)'Image);
   Put_Line ("Float'Scaling (0.25, 2): "
             & Float'Scaling (0.25, 2)'Image);
   Put_Line ("Float'Scaling (0.25, 3): "
             & Float'Scaling (0.25, 3)'Image);
end Show_Scaling;

The scaling is calculated with this formula:

scaling = value x Machine_Radixmachine exponent

For example, on a typical PC with a machine radix of two, Float'Scaling (0.25, 3) = 2.0 corresponds to

0.25 x 23 = 2.0

Round-up and round-down attributes

Floor and Ceiling are attributes that returned the rounded-down or rounded-up value, respectively, of a floating-point value. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Floor_Ceiling is
begin
   Put_Line ("Float'Floor (0.25):   "
             & Float'Floor (0.25)'Image);
   Put_Line ("Float'Ceiling (0.25): "
             & Float'Ceiling (0.25)'Image);
end Show_Floor_Ceiling;

As we can see in this example, the rounded-down value (floor) of 0.25 is 0.0, while the rounded-up value (ceiling) of 0.25 is 1.0.

Round-to-nearest attributes

In this section, we discuss three attributes used for rounding: Rounding, Unbiased_Rounding, Machine_Rounding In all cases, the rounding attributes return the nearest integer value (as a floating-point value). For example, the rounded value for 4.8 is 5.0 because 5 is the closest integer value.

Let's see a code example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Roundings is
begin
   Put_Line
     ("Float'Rounding (0.5):  "
      & Float'Rounding (0.5)'Image);
   Put_Line
     ("Float'Rounding (1.5):  "
      & Float'Rounding (1.5)'Image);
   Put_Line
     ("Float'Rounding (4.5):  "
      & Float'Rounding (4.5)'Image);
   Put_Line
     ("Float'Rounding (-4.5): "
      & Float'Rounding (-4.5)'Image);
   Put_Line
     ("Float'Unbiased_Rounding (0.5): "
      & Float'Unbiased_Rounding (0.5)'Image);
   Put_Line
     ("Float'Unbiased_Rounding (1.5): "
      & Float'Unbiased_Rounding (1.5)'Image);
   Put_Line
     ("Float'Machine_Rounding (0.5): "
      & Float'Machine_Rounding (0.5)'Image);
   Put_Line
     ("Float'Machine_Rounding (1.5): "
      & Float'Machine_Rounding (1.5)'Image);
end Show_Roundings;

The difference between these attributes is the way they handle the case when a value is exactly in between two integer values. For example, 4.5 could be rounded up to 5.0 or rounded down to 4.0. This is the way each rounding attribute works in this case:

  • Rounding rounds away from zero. Positive floating-point values are rounded up, while negative floating-point values are rounded down when the value is between two integer values. For example:

    • 4.5 is rounded-up to 5.0, i.e. Float'Rounding (4.5) = Float'Ceiling (4.5) = 5.0.

    • -4.5 is rounded-down to -5.0, i.e. Float'Rounding (-4.5) = Float'Floor (-4.5) = -5.0.

  • Unbiased_Rounding rounds toward the even integer. For example,

    • Float'Unbiased_Rounding (0.5) = 0.0 because zero is the closest even integer, while

    • Float'Unbiased_Rounding (1.5) = 2.0 because two is the closest even integer.

  • Machine_Rounding uses the most appropriate rounding instruction available on the target platform. While this rounding attribute can potentially have the best performance, its result may be non-portable. For example, whether the rounding of 4.5 becomes 4.0 or 5.0 depends on the target platform.

    • If an algorithm depends on a specific rounding behavior, it's best to avoid the Machine_Rounding attribute. On the other hand, if the rounding behavior won't have a significant impact on the results, we can safely use this attribute.

Attributes: Truncation, Remainder, Adjacent

The Truncation attribute returns the truncated value of a floating-point value, i.e. the value corresponding to the integer part of a number rounded toward zero. This corresponds to the number before the radix point. For example, the truncation of 1.55 is 1.0 because the integer part of 1.55 is 1.

The Remainder attribute returns the remainder part of a division. For example, Float'Remainder (1.25, 0.5) = 0.25. Let's briefly discuss the details of this operations. The result of the division 1.25 / 0.5 is 2.5. Here, 1.25 is the dividend and 0.5 is the divisor. The quotient and remainder of this division are 2 and 0.25, respectively. (Here, the quotient is an integer number, and the remainder is the floating-point part that remains.)

Note that the relation between quotient and remainder is defined in such a way that we get the original dividend back when we use the formula: "quotient x divisor + remainder = dividend". For the previous example, this means 2 x 0.5 + 0.25 = 1.25.

The Adjacent attribute is the next machine value towards another value. For example, on a typical PC, the adjacent value of a small value — say, 1.0 x 10-83 — towards zero is +0.0, while the adjacent value of this small value towards 1.0 is another small, but greater value — in fact, it's 1.40130 x 10-45. Note that the first parameter of the Adjacent attribute is the value we want to analyze and the second parameter is the Towards value.

Let's see a code example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Truncation_Remainder_Adjacent is
begin
   Put_Line
     ("Float'Truncation (1.55):  "
      & Float'Truncation (1.55)'Image);
   Put_Line
     ("Float'Truncation (-1.55): "
      & Float'Truncation (-1.55)'Image);
   Put_Line
     ("Float'Remainder (1.25, 0.25): "
      & Float'Remainder (1.25, 0.25)'Image);
   Put_Line
     ("Float'Remainder (1.25, 0.5):  "
      & Float'Remainder (1.25, 0.5)'Image);
   Put_Line
     ("Float'Remainder (1.25, 1.0):  "
      & Float'Remainder (1.25, 1.0)'Image);
   Put_Line
     ("Float'Remainder (1.25, 2.0):  "
      & Float'Remainder (1.25, 2.0)'Image);
   Put_Line
     ("Float'Adjacent (1.0e-83, 0.0): "
      & Float'Adjacent (1.0e-83, 0.0)'Image);
   Put_Line
     ("Float'Adjacent (1.0e-83, 1.0): "
      & Float'Adjacent (1.0e-83, 1.0)'Image);
end Show_Truncation_Remainder_Adjacent;

Attributes: Copy_Sign and Leading_Part

Copy_Sign is an attribute that returns a value where the sign of the second floating-point argument is multiplied by the magnitude of the first floating-point argument. For example, Float'Copy_Sign (1.0, -10.0) is -1.0. Here, the sign of the second argument (-10.0) is multiplied by the magnitude of the first argument (1.0), so the result is -1.0.

Leading_Part is an attribute that returns the approximated version of the mantissa of a value based on the specified number of leading bits for the mantissa. Let's see some examples:

  • Float'Leading_Part (3.1416, 1) is 2.0 because that's the value we can represent with one leading bit.

    • Note that Float'Fraction (2.0) = 0.5 — which can be represented with one leading bit in the mantissa — and Float'Exponent (2.0) = 2.)

  • If we increase the number of leading bits of the mantissa to two — by writing Float'Leading_Part (3.1416, 2) —, we get 3.0 because that's the value we can represent with two leading bits.

  • If we increase again the number of leading bits to five — Float'Leading_Part (3.1416, 5) —, we get 3.125.

    • Note that, in this case Float'Fraction (3.125) = 0.78125 and Float'Exponent (3.125) = 2.

    • The binary mantissa is actually 2#110_0100_0000_0000_0000_0000#, which can be represented with five leading bits as expected: 2#110_01#.

      • We can get the binary mantissa by calculating Float'Fraction (3.125) * Float (Float'Machine_Radix) ** (Float'Machine_Mantissa - 1) and converting the result to binary format. The -1 value in the formula corresponds to the sign bit.

Attention

In this explanation about the Leading_Part attribute, we're talking about leading bits. Strictly speaking, however, this is actually a simplification, and it's only correct if Machine_Radix is equal to two — which is the case for most machines. Therefore, in most cases, the explanation above is perfectly acceptable.

However, if Machine_Radix is not equal to two, we cannot use the term "bits" anymore, but rather digits of the Machine_Radix.

Let's see some examples:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Copy_Sign_Leading_Part_Machine is
begin
   Put_Line
     ("Float'Copy_Sign (1.0, -10.0): "
      & Float'Copy_Sign (1.0, -10.0)'Image);
   Put_Line
     ("Float'Copy_Sign (-1.0, -10.0): "
      & Float'Copy_Sign (-1.0, -10.0)'Image);
   Put_Line
     ("Float'Copy_Sign (1.0,  10.0): "
      & Float'Copy_Sign (1.0,  10.0)'Image);
   Put_Line
     ("Float'Copy_Sign (1.0, -0.0):  "
      & Float'Copy_Sign (1.0, -0.0)'Image);
   Put_Line
     ("Float'Copy_Sign (1.0,  0.0):  "
      & Float'Copy_Sign (1.0,  0.0)'Image);
   Put_Line
     ("Float'Leading_Part (1.75, 1): "
      & Float'Leading_Part (1.75, 1)'Image);
   Put_Line
     ("Float'Leading_Part (1.75, 2): "
      & Float'Leading_Part (1.75, 2)'Image);
   Put_Line
     ("Float'Leading_Part (1.75, 3): "
      & Float'Leading_Part (1.75, 3)'Image);
end Show_Copy_Sign_Leading_Part_Machine;

Attribute: Machine

Not every real number is directly representable as a floating-point value on a specific machine. For example, let's take a value such as 1.0 x 1015 (or 1,000,000,000,000,000):

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Float_Value is
   package F_IO is new
     Ada.Text_IO.Float_IO (Float);

   V : Float;
begin
   F_IO.Default_Fore := 3;
   F_IO.Default_Aft  := 1;
   F_IO.Default_Exp  := 0;

   V := 1.0E+15;
   Put ("1.0E+15 = ");
   F_IO.Put (Item => V);
   New_Line;

end Show_Float_Value;

If we run this example on a typical PC, we see that the expected value 1_000_000_000_000_000.0 was displayed as 999_999_986_991_000.0. This is because 1.0 x 1015 isn't directly representable on this machine, so it has to be modified to a value that is actually representable (on the machine).

This automatic modification we've just described is actually hidden, so to say, in the assignment. However, we can make it more visible by using the Machine (X) attribute, which returns a version of X that is representable on the target machine. The Machine (X) attribute rounds (or truncates) X to either one of the adjacent machine numbers for the specific floating-point type of X. (Of course, if the real value of X is directly representable on the target machine, no modification is performed.)

In fact, we could rewrite the V := 1.0E+15 assignment of the code example as V := Float'Machine (1.0E+15), as we're never assigning a real value directly to a floating-pointing variable — instead, we're first converting it to a version of the real value that is representable on the target machine. In this case, 999999986991000.0 is a representable version of the real value 1.0 x 1015. Of course, writing V := 1.0E+15 or V := Float'Machine (1.0E+15) doesn't make any difference to the actual value that is assigned to V (in the case of this specific target architecture), as the conversion to a representable value happens automatically during the assignment to V.

There are, however, instances where using the Machine attribute does make a difference in the result. For example, let's say we want to calculate the difference between the original real value in our example (1.0 x 1015) and the actual value that is assigned to V. We can do this by using the Machine attribute in the calculation:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Machine_Attribute is
   package F_IO is new
     Ada.Text_IO.Float_IO (Float);

   V : Float;
begin
   F_IO.Default_Fore := 3;
   F_IO.Default_Aft  := 1;
   F_IO.Default_Exp  := 0;

   Put_Line
     ("Original value: 1_000_000_000_000_000.0");

   V := 1.0E+15;
   Put ("Machine value:  ");
   F_IO.Put (Item => V);
   New_Line;

   V := 1.0E+15 - Float'Machine (1.0E+15);
   Put ("Difference:     ");
   F_IO.Put (Item => V);
   New_Line;

end Show_Machine_Attribute;

When we run this example on a typical PC, we see that the difference is roughly 1.3009 x 107. (Actually, the value that we might see is 1.3008896 x 107, which is a version of 1.3009 x 107 that is representable on the target machine.)

When we write 1.0E+15 - Float'Machine (1.0E+15):

  • the first value in the operation is the universal real value 1.0 x 1015, while

  • the second value in the operation is a version of the universal real value 1.0 x 1015 that is representable on the target machine.

This also means that, in the assignment to V, we're actually writing V := Float'Machine (1.0E+15 - Float'Machine (1.0E+15)), so that:

  1. we first get the intermediate real value that represents the difference between these values; and then

  2. we get a version of this intermediate real value that is representable on the target machine.

This is the reason why we see 1.3008896 x 107 instead of 1.3009 x 107 when we run this application.

Fixed-Point Types

In this section, we discuss various attributes and operations related to fixed-point types.

In the Ada Reference Manual

Attributes of fixed-point types

Attribute: Machine_Radix

Machine_Radix is an attribute that returns the radix of the hardware representation of a type. For example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Fixed_Machine_Radix is
   type T3_D3 is delta 10.0 ** (-3) digits 3;

   D : constant := 2.0 ** (-31);
   type TQ31 is delta D range -1.0 .. 1.0 - D;
begin
   Put_Line ("T3_D3'Machine_Radix: "
             & T3_D3'Machine_Radix'Image);
   Put_Line ("TQ31'Machine_Radix:  "
             & TQ31'Machine_Radix'Image);
end Show_Fixed_Machine_Radix;

Usually, this value is two, as the radix is based on a binary system.

Attribute: Machine_Rounds and Machine_Overflows

In this section, we discuss attributes that return Boolean values indicating whether a feature is available or not in the target architecture:

  • Machine_Rounds is an attribute that indicates what happens when the result of a fixed-point operation is inexact:

    • T'Machine_Rounds = True: inexact result is rounded;

    • T'Machine_Rounds = False: inexact result is truncated.

  • Machine_Overflows is an attribute that indicates whether a Constraint_Error is guaranteed to be raised when a fixed-point operation with that type produces an overflow or divide-by-zero.

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Boolean_Attributes is
   type T3_D3 is delta 10.0 ** (-3) digits 3;

   D : constant := 2.0 ** (-31);
   type TQ31 is delta D range -1.0 .. 1.0 - D;
begin
   Put_Line ("T3_D3'Machine_Rounds:    "
             & T3_D3'Machine_Rounds'Image);
   Put_Line ("TQ31'Machine_Rounds:     "
             & TQ31'Machine_Rounds'Image);
   Put_Line ("T3_D3'Machine_Overflows: "
             & T3_D3'Machine_Overflows'Image);
   Put_Line ("TQ31'Machine_Overflows:  "
             & TQ31'Machine_Overflows'Image);
end Show_Boolean_Attributes;

Attribute: Small and Delta

The Small and Delta attributes return numbers that indicate the numeric precision of a fixed-point type. In many cases, the Small of a type T is equal to the Delta of that type — i.e. T'Small = T'Delta. Let's discuss each attribute and how they distinguish from each other.

The Delta attribute returns the value of the delta that was used in the type definition. For example, if we declare type T3_D3 is delta 10.0 ** (-3) digits D, then the value of T3_D3'Delta is the 10.0 ** (-3) that we used in the type definition.

The Small attribute returns the "small" of a type, i.e. the smallest value used in the machine representation of the type. The small must be at least equal to or smaller than the delta — in other words, it must conform to the T'Small <= T'Delta rule.

For further reading...

The Small and the Delta need not actually be small numbers. They can be arbitrarily large. For instance, they could be 1.0, or 1000.0. Consider the following example:

    
    
    
        
package Fixed_Point_Defs is
   S     : constant := 32;
   Exp   : constant := 128;
   D     : constant := 2.0 ** (-S + Exp + 1);

   type Fixed is delta D
     range -1.0 * 2.0 ** Exp ..
            1.0 * 2.0 ** Exp - D;

   pragma Assert (Fixed'Size = S);
end Fixed_Point_Defs;

    
        
with Fixed_Point_Defs; use Fixed_Point_Defs;
with Ada.Text_IO;      use Ada.Text_IO;

procedure Show_Fixed_Type_Info is
begin
   Put_Line ("Size : "
             & Fixed'Size'Image);
   Put_Line ("Small : "
             & Fixed'Small'Image);
   Put_Line ("Delta : "
             & Fixed'Delta'Image);
   Put_Line ("First : "
             & Fixed'First'Image);
   Put_Line ("Last : "
             & Fixed'Last'Image);
end Show_Fixed_Type_Info;

In this example, the small of the Fixed type is actually quite large: 1.5845632502852867529. (Also, the first and the last values are large: -340,282,366,920,938,463,463,374,607,431,768,211,456.0 and 340,282,366,762,482,138,434,845,932,244,680,310,784.0, or approximately -3.402838 and 3.402838.)

In this case, if we assign 1 or 1,000 to a variable F of this type, the actual value stored in F is zero. Feel free to try this out!

When we declare an ordinary fixed-point data type, we must specify the delta. Specifying the small, however, is optional:

  • If the small isn't specified, it is automatically selected by the compiler. In this case, the actual value of the small is an implementation-defined power of two — always following the rule that says: T'Small <= T'Delta.

  • If we want, however, to specify the small, we can do that by using the Small aspect. In this case, it doesn't need to be a power of two.

For decimal fixed-point types, we cannot specify the small. In this case, it's automatically selected by the compiler, and it's always equal to the delta.

Let's see an example:

    
    
    
        
package Fixed_Small_Delta is
   D3 : constant := 10.0 ** (-3);

   type T3_D3 is delta D3 digits 3;

   type TD3   is delta D3 range -1.0 .. 1.0 - D3;

   D31 : constant := 2.0 ** (-31);
   D15 : constant := 2.0 ** (-15);

   type TQ31 is delta D31 range -1.0 .. 1.0 - D31;

   type TQ15 is delta D15 range -1.0 .. 1.0 - D15
     with Small => D31;
end Fixed_Small_Delta;

    
        
with Ada.Text_IO;       use Ada.Text_IO;

with Fixed_Small_Delta; use Fixed_Small_Delta;

procedure Show_Fixed_Small_Delta is
begin
   Put_Line ("T3_D3'Small: "
             & T3_D3'Small'Image);
   Put_Line ("T3_D3'Delta: "
             & T3_D3'Delta'Image);
   Put_Line ("T3_D3'Size: "
             & T3_D3'Size'Image);
   Put_Line ("--------------------");

   Put_Line ("TD3'Small: "
             & TD3'Small'Image);
   Put_Line ("TD3'Delta: "
             & TD3'Delta'Image);
   Put_Line ("TD3'Size: "
             & TD3'Size'Image);
   Put_Line ("--------------------");

   Put_Line ("TQ31'Small: "
             & TQ31'Small'Image);
   Put_Line ("TQ31'Delta: "
             & TQ31'Delta'Image);
   Put_Line ("TQ32'Size: "
             & TQ31'Size'Image);
   Put_Line ("--------------------");

   Put_Line ("TQ15'Small: "
             & TQ15'Small'Image);
   Put_Line ("TQ15'Delta: "
             & TQ15'Delta'Image);
   Put_Line ("TQ15'Size: "
             & TQ15'Size'Image);
end Show_Fixed_Small_Delta;

As we can see in the output of the code example, the Delta attribute returns the value we used for delta in the type definition of the T3_D3, TD3, TQ31 and TQ15 types.

The TD3 type is an ordinary fixed-point type with the the same delta as the decimal T3_D3 type. In this case, however, TD3'Small is not the same as the TD3'Delta. On a typical desktop PC, TD3'Small is 2-10, while the delta is 10-3. (Remember that, for ordinary fixed-point types, if we don't specify the small, it's automatically selected by the compiler as a power of two smaller than or equal to the delta.)

In the case of the TQ15 type, we're specifying the small by using the Small aspect. In this case, the underlying size of the TQ15 type is 32 bits, while the precision we get when operating with this type is 16 bits. Let's see a specific example for this type:

    
    
    
        
with Ada.Text_IO;       use Ada.Text_IO;

with Fixed_Small_Delta; use Fixed_Small_Delta;

procedure Show_Fixed_Small_Delta is
   V : TQ15;
begin
   Put_Line ("V'Size: " & V'Size'Image);

   V := TQ15'Small;
   Put_Line ("V: " & V'Image);

   V := TQ15'Delta;
   Put_Line ("V: " & V'Image);
end Show_Fixed_Small_Delta;

In the first assignment, we assign TQ15'Small (2-31) to V. This value is smaller than the type's delta (2-15). Even though V'Size is 32 bits, V'Delta indicates 16-bit precision, and TQ15'Small requires 32-bit precision to be represented correctly. As a result, V has a value of zero after this assignment.

In contrast, after the second assignment — where we assign TQ15'Delta (2-15) to V — we see, as expected, that V has the same value as the delta.

Attributes: Fore and Aft

The Fore and Aft attributes indicate the number of characters or digits needed for displaying a value in decimal representation. To be more precise:

  • The Fore attribute refers to the digits before the decimal point and it returns the number of digits plus one for the sign indicator (which is either - or space), and it's always at least two.

  • The Aft attribute returns the number of decimal digits that is needed to represent the delta after the decimal point.

Let's see an example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Fixed_Fore_Aft is
   type T3_D3 is delta 10.0 ** (-3) digits 3;

   D : constant := 2.0 ** (-31);
   type TQ31 is delta D range -1.0 .. 1.0 - D;

   Dec : constant T3_D3 := -0.123;
   Fix : constant TQ31  := -TQ31'Delta;
begin
   Put_Line ("T3_D3'Fore: "
             & T3_D3'Fore'Image);
   Put_Line ("T3_D3'Aft:  "
             & T3_D3'Aft'Image);

   Put_Line ("TQ31'Fore: "
             & TQ31'Fore'Image);
   Put_Line ("TQ31'Aft:  "
             & TQ31'Aft'Image);
   Put_Line ("----");
   Put_Line ("Dec: "
             & Dec'Image);
   Put_Line ("Fix: "
             & Fix'Image);
end Show_Fixed_Fore_Aft;

As we can see in the output of the Dec and Fix variables at the bottom, the value of Fore is two for both T3_D3 and TQ31. This value corresponds to the length of the string "-0" displayed in the output for these variables (the first two characters of "-0.123" and "-0.0000000005").

The value of Dec'Aft is three, which matches the number of digits after the decimal point in "-0.123". Similarly, the value of Fix'Aft is 10, which matches the number of digits after the decimal point in "-0.0000000005".

Attributes of decimal fixed-point types

The attributes presented in this subsection are only available for decimal fixed-point types.

Attribute: Digits

Digits is an attribute that returns the number of significant decimal digits of a decimal fixed-point subtype. This corresponds to the value that we use for the digits in the definition of a decimal fixed-point type.

Let's see an example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Decimal_Digits is
   type T3_D6 is delta 10.0 ** (-3) digits 6;
   subtype T3_D2 is T3_D6 digits 2;
begin
   Put_Line ("T3_D6'Digits: "
             & T3_D6'Digits'Image);
   Put_Line ("T3_D2'Digits: "
             & T3_D2'Digits'Image);
end Show_Decimal_Digits;

In this example, T3_D6'Digits is six, which matches the value that we used for digits in the type definition of T3_D6. The same logic applies for subtypes, as we can see in the value of T3_D2'Digits. Here, the value is two, which was used in the declaration of the T3_D2 subtype.

Attribute: Scale

According to the Ada Reference Manual, the Scale attribute "indicates the position of the point relative to the rightmost significant digits of values" of a decimal type. For example:

  • If the value of Scale is two, then there are two decimal digits after the decimal point.

  • If the value of Scale is negative, that implies that the Delta is a power of 10 greater than 1, and it would be the number of zero digits that every value would end in.

The Scale corresponds to the N used in the delta 10.0 ** (-N) expression of the type declaration. For example, if we write delta 10.0 ** (-3) in the declaration of a type T, then the value of T'Scale is three.

Let's look at this complete example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Decimal_Scale is
   type TM3_D6 is delta 10.0 **   3  digits 6;
   type T3_D6  is delta 10.0 ** (-3) digits 6;
   type T9_D12 is delta 10.0 ** (-9) digits 12;
begin
   Put_Line ("TM3_D6'Scale: "
             & TM3_D6'Scale'Image);
   Put_Line ("T3_D6'Scale: "
             & T3_D6'Scale'Image);
   Put_Line ("T9_D12'Scale: "
             & T9_D12'Scale'Image);
end Show_Decimal_Scale;

In this example, we get the following values for the scales:

  • TM3_D6'Scale = -3,

  • T3_D6'Scale = 3,

  • T9_D12 = 9.

As you can see, the value of Scale is directly related to the delta of the corresponding type declaration.

Attribute: Round

The Round attribute rounds a value of any real type to the nearest value that is a multiple of the delta of the decimal fixed-point type, rounding away from zero if exactly between two such multiples.

For example, if we have a type T with three digits, and we use a value with 10 digits after the decimal point in a call to T'Round, the resulting value will have three digits after the decimal point.

Note that the X input of an S'Round (X) call is a universal real value, while the returned value is of S'Base type.

Let's look at this example:

    
    
    
        
with Ada.Text_IO; use Ada.Text_IO;

procedure Show_Decimal_Round is
   type T3_D3 is delta 10.0 ** (-3) digits 3;
begin
   Put_Line ("T3_D3'Round (0.2774): "
             & T3_D3'Round (0.2774)'Image);
   Put_Line ("T3_D3'Round (0.2777): "
             & T3_D3'Round (0.2777)'Image);
end Show_Decimal_Round;

Here, the T3_D3 has a precision of three digits. Therefore, to fit this precision, 0.2774 is rounded to 0.277, and 0.2777 is rounded to 0.278.

Big Numbers

As we've seen before, we can define numeric types in Ada with a high degree of precision. However, these normal numeric types in Ada are limited to what the underlying hardware actually supports. For example, any signed integer type — whether defined by the language or the user — cannot have a range greater than that of System.Min_Int .. System.Max_Int because those constants reflect the actual hardware's signed integer types. In certain applications, that precision might not be enough, so we have to rely on arbitrary-precision arithmetic. These so-called "big numbers" are limited conceptually only by available memory, in contrast to the underlying hardware-defined numeric types.

Ada supports two categories of big numbers: big integers and big reals — both are specified in child packages of the Ada.Numerics.Big_Numbers package:

Category

Package

Big Integers

Ada.Numerics.Big_Numbers.Big_Integers

Big Reals

Ada.Numerics.Big_Numbers.Big_Real

In the Ada Reference Manual

Overview

Let's start with a simple declaration of big numbers:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Simple_Big_Numbers is
   BI : Big_Integer;
   BR : Big_Real;
begin
   BI := 12345678901234567890;
   BR := 2.0 ** 1234;

   Put_Line ("BI: " & BI'Image);
   Put_Line ("BR: " & BR'Image);

   BI := BI + 1;
   BR := BR + 1.0;

   Put_Line ("BI: " & BI'Image);
   Put_Line ("BR: " & BR'Image);
end Show_Simple_Big_Numbers;

In this example, we're declaring the big integer BI and the big real BR, and we're incrementing them by one.

Naturally, we're not limited to using the + operator (such as in this example). We can use the same operators on big numbers that we can use with normal numeric types. In fact, the common unary operators (+, -, abs) and binary operators (+, -, *, /, **, Min and Max) are available to us. For example:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Simple_Big_Numbers_Operators is
   BI : Big_Integer;
begin
   BI := 12345678901234567890;

   Put_Line ("BI: " & BI'Image);

   BI := -BI + BI / 2;
   BI :=  BI - BI * 2;

   Put_Line ("BI: " & BI'Image);
end Show_Simple_Big_Numbers_Operators;

In this example, we're applying the four basic operators (+, -, *, /) on big integers.

Factorial

A typical example is the factorial: a sequence of the factorial of consecutive small numbers can quickly lead to big numbers. Let's take this implementation as an example:

    
    
    
        
function Factorial (N : Integer)
                    return Long_Long_Integer;

    
        
function Factorial (N : Integer)
                    return Long_Long_Integer is
   Fact : Long_Long_Integer := 1;
begin
   for I in 2 .. N loop
      Fact := Fact * Long_Long_Integer (I);
   end loop;

   return Fact;
end Factorial;

    
        
with Ada.Text_IO; use Ada.Text_IO;

with Factorial;

procedure Show_Factorial is
begin
   for I in 1 .. 50 loop
      Put_Line (I'Image & "! = "
                & Factorial (I)'Image);
   end loop;
end Show_Factorial;

Here, we're using Long_Long_Integer for the computation and return type of the Factorial function. (We're using Long_Long_Integer because its range is probably the biggest possible on the machine, although that is not necessarily so.) The last number we're able to calculate before getting an exception is 20!, which basically shows the limitation of standard integers for this kind of algorithm. If we use big integers instead, we can easily display all numbers up to 50! (and more!):

    
    
    
        
pragma Ada_2022;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

function Factorial (N : Integer)
                    return Big_Integer;

    
        
function Factorial (N : Integer)
                    return Big_Integer is
   Fact : Big_Integer := 1;
begin
   for I in 2 .. N loop
      Fact := Fact * To_Big_Integer (I);
   end loop;

   return Fact;
end Factorial;

    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Factorial;

procedure Show_Big_Number_Factorial is
begin
   for I in 1 .. 50 loop
      Put_Line (I'Image & "! = "
                & Factorial (I)'Image);
   end loop;
end Show_Big_Number_Factorial;

As we can see in this example, replacing the Long_Long_Integer type by the Big_Integer type fixes the problem (the runtime exception) that we had in the previous version. (Note that we're using the To_Big_Integer function to convert from Integer to Big_Integer: we discuss these conversions next.)

Note that there is a limit to the upper bounds for big integers. However, this limit isn't dependent on the hardware types — as it's the case for normal numeric types —, but rather compiler specific. In other words, the compiler can decide how much memory it wants to use to represent big integers.

Conversions

Most probably, we want to mix big numbers and standard numbers (i.e. integer and real numbers) in our application. In this section, we talk about the conversion between big numbers and standard types.

Validity

The package specifications of big numbers include subtypes that ensure that a actual value of a big number is valid:

Type

Subtype for valid values

Big Integers

Valid_Big_Integer

Big Reals

Valid_Big_Real

These subtypes include a contract for this check. For example, this is the definition of the Valid_Big_Integer subtype:

subtype Valid_Big_Integer is Big_Integer
  with Dynamic_Predicate =>
           Is_Valid (Valid_Big_Integer),
       Predicate_Failure =>
           (raise Program_Error);

Any operation on big numbers is actually performing this validity check (via a call to the Is_Valid function). For example, this is the addition operator for big integers:

function "+" (L, R : Valid_Big_Integer)
              return Valid_Big_Integer;

As we can see, both the input values to the operator as well as the return value are expected to be valid — the Valid_Big_Integer subtype triggers this check, so to say. This approach ensures that an algorithm operating on big numbers won't be using invalid values.

Conversion functions

These are the most important functions to convert between big number and standard types:

Category

To big number

From big number

Big Integers

  • To_Big_Integer

  • To_Integer (Integer)

  • From_Big_Integer (other integer types)

Big Reals

  • To_Big_Real (floating-point types or fixed-point types)

  • From_Big_Real

  • To_Big_Real (Valid_Big_Integer)

  • To_Real (Integer)

  • Numerator, Denominator (Integer)

In the following sections, we discuss these functions in more detail.

Big integer to integer

We use the To_Big_Integer and To_Integer functions to convert back and forth between Big_Integer and Integer types:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Simple_Big_Integer_Conversion is
   BI : Big_Integer;
   I  : Integer := 10000;
begin
   BI := To_Big_Integer (I);
   Put_Line ("BI: " & BI'Image);

   I := To_Integer (BI + 1);
   Put_Line ("I:  " & I'Image);
end Show_Simple_Big_Integer_Conversion;

In addition, we can use the generic Signed_Conversions and Unsigned_Conversions packages to convert between Big_Integer and any signed or unsigned integer types:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Arbitrary_Big_Integer_Conversion is

   type Mod_32_Bit is mod 2 ** 32;

   package Long_Long_Integer_Conversions is new
     Signed_Conversions (Long_Long_Integer);
   use Long_Long_Integer_Conversions;

   package Mod_32_Bit_Conversions is new
     Unsigned_Conversions (Mod_32_Bit);
   use Mod_32_Bit_Conversions;

   BI   : Big_Integer;
   LLI  : Long_Long_Integer := 10000;
   U_32 : Mod_32_Bit        := 2 ** 32 + 1;

begin
   BI := To_Big_Integer (LLI);
   Put_Line ("BI:   " & BI'Image);

   LLI := From_Big_Integer (BI + 1);
   Put_Line ("LLI:  " & LLI'Image);

   BI := To_Big_Integer (U_32);
   Put_Line ("BI:   " & BI'Image);

   U_32 := From_Big_Integer (BI + 1);
   Put_Line ("U_32: " & U_32'Image);

end Show_Arbitrary_Big_Integer_Conversion;

In this examples, we declare the Long_Long_Integer_Conversions and the Mod_32_Bit_Conversions to be able to convert between big integers and the Long_Long_Integer and the Mod_32_Bit types, respectively.

Note that, when converting from big integer to integer, we used the To_Integer function, while, when using the instances of the generic packages, the function is named From_Big_Integer.

Big real to floating-point types

When converting between big real and floating-point types, we have to instantiate the generic Float_Conversions package:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Real_Floating_Point_Conversion
is
   type D10 is digits 10;

   package D10_Conversions is new
     Float_Conversions (D10);
   use D10_Conversions;

   package Long_Float_Conversions is new
     Float_Conversions (Long_Float);
   use Long_Float_Conversions;

   BR  : Big_Real;
   LF  : Long_Float := 2.0;
   F10 : D10        := 1.999;

begin
   BR := To_Big_Real (LF);
   Put_Line ("BR:   " & BR'Image);

   LF := From_Big_Real (BR + 1.0);
   Put_Line ("LF:   " & LF'Image);

   BR := To_Big_Real (F10);
   Put_Line ("BR:   " & BR'Image);

   F10 := From_Big_Real (BR + 0.1);
   Put_Line ("F10:  " & F10'Image);

end Show_Big_Real_Floating_Point_Conversion;

In this example, we declare the D10_Conversions and the Long_Float_Conversions to be able to convert between big reals and the custom floating-point type D10 and the Long_Float type, respectively. To do that, we use the To_Big_Real and the From_Big_Real functions.

Big real to fixed-point types

When converting between big real and ordinary fixed-point types, we have to instantiate the generic Fixed_Conversions package:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Real_Fixed_Point_Conversion
is
   D : constant := 2.0 ** (-31);
   type TQ31 is delta D range -1.0 .. 1.0 - D;

   package TQ31_Conversions is new
     Fixed_Conversions (TQ31);
   use TQ31_Conversions;

   BR   : Big_Real;
   FQ31 : TQ31 := 0.25;

begin
   BR := To_Big_Real (FQ31);
   Put_Line ("BR:   " & BR'Image);

   FQ31 := From_Big_Real (BR * 2.0);
   Put_Line ("FQ31: " & FQ31'Image);

end Show_Big_Real_Fixed_Point_Conversion;

In this example, we declare the TQ31_Conversions to be able to convert between big reals and the custom fixed-point type TQ31 type. Again, we use the To_Big_Real and the From_Big_Real functions for the conversions.

Note that there's no direct way to convert between decimal fixed-point types and big real types. (Of course, you could perform this conversion indirectly by using a floating-point or an ordinary fixed-point type in between.)

Big reals to (big) integers

We can also convert between big reals and big integers (or standard integers):

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Real_Big_Integer_Conversion
is
   I  : Integer;
   BI : Big_Integer;
   BR : Big_Real;

begin
   I  := 12345;
   BR := To_Real (I);
   Put_Line ("BR (from I):  " & BR'Image);

   BI := 123456;
   BR := To_Big_Real (BI);
   Put_Line ("BR (from BI): " & BR'Image);

end Show_Big_Real_Big_Integer_Conversion;

Here, we use the To_Real and the To_Big_Real and functions for the conversions.

String conversions

In addition to that, we can use string conversions:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Number_String_Conversion
is
   BI : Big_Integer;
   BR : Big_Real;
begin
   BI := From_String ("12345678901234567890");
   BR := From_String ("12345678901234567890.0");

   Put_Line ("BI: "
             & To_String (Arg   => BI,
                          Width => 5,
                          Base => 2));
   Put_Line ("BR: "
             & To_String (Arg   => BR,
                          Fore  => 2,
                          Aft   => 6,
                          Exp   => 18));
end Show_Big_Number_String_Conversion;

In this example, we use the From_String to convert a string to a big number. Note that the From_String function is actually called when converting a literal — because of the corresponding aspect for user-defined literals in the definitions of the Big_Integer and the Big_Real types.

For further reading...

Big numbers are implemented using user-defined literals, which we discussed previously. In fact, these are the corresponding type declarations:

--  Declaration from
--  Ada.Numerics.Big_Numbers.Big_Integers;

type Big_Integer is private
  with Integer_Literal => From_Universal_Image,
       Put_Image       => Put_Image;

function From_Universal_Image
  (Arg : String)
  return Valid_Big_Integer
    renames From_String;

--  Declaration from
--  Ada.Numerics.Big_Numbers.Big_Reals;

type Big_Real is private
  with Real_Literal => From_Universal_Image,
       Put_Image    => Put_Image;

function From_Universal_Image
  (Arg : String)
   return Valid_Big_Real
     renames From_String;

As we can see in these declarations, the From_String function renames the From_Universal_Image function, which is being used for the user-defined literals.

Also, we call the To_String function to get a string for the big numbers. Naturally, using the To_String function instead of the Image attribute — as we did in previous examples — allows us to customize the format of the string that we display in the user message.

Other features of big integers

Now, let's look at two additional features of big integers:

  • the natural and positive subtypes, and

  • other available operators and functions.

Big positive and natural subtypes

Similar to integer types, big integers have the Big_Natural and Big_Positive subtypes to indicate natural and positive numbers. However, in contrast to the Natural and Positive subtypes, the Big_Natural and Big_Positive subtypes are defined via predicates rather than the simple ranges of normal (ordinary) numeric types:

subtype Natural  is
  Integer range 0 .. Integer'Last;

subtype Positive is
  Integer range 1 .. Integer'Last;

subtype Big_Natural is Big_Integer
  with Dynamic_Predicate =>
         (if Is_Valid (Big_Natural)
            then Big_Natural >= 0),
       Predicate_Failure =>
         (raise Constraint_Error);

subtype Big_Positive is Big_Integer
  with Dynamic_Predicate =>
         (if Is_Valid (Big_Positive)
            then Big_Positive > 0),
       Predicate_Failure =>
         (raise Constraint_Error);

Therefore, we cannot simply use attributes such as Big_Natural'First. However, we can use the subtypes to ensure that a big integer is in the expected (natural or positive) range:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Big_Positive_Natural is
   BI, D, N : Big_Integer;
begin
   D  := 3;
   N  := 2;
   BI := Big_Natural (D / Big_Positive (N));

   Put_Line ("BI: " & BI'Image);
end Show_Big_Positive_Natural;

By using the Big_Natural and Big_Positive subtypes in the calculation above (in the assignment to BI), we ensure that we don't perform a division by zero, and that the result of the calculation is a natural number.

Other operators for big integers

We can use the mod and rem operators with big integers:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Big_Integer_Rem_Mod is
   BI : Big_Integer;
begin
   BI := 145 mod (-4);
   Put_Line ("BI (mod): " & BI'Image);

   BI := 145 rem (-4);
   Put_Line ("BI (rem): " & BI'Image);
end Show_Big_Integer_Rem_Mod;

In this example, we use the mod and rem operators in the assignments to BI.

Moreover, there's a Greatest_Common_Divisor function for big integers which, as the name suggests, calculates the greatest common divisor of two big integer values:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

procedure Show_Big_Integer_Greatest_Common_Divisor
is
   BI : Big_Integer;
begin
   BI := Greatest_Common_Divisor (145, 25);
   Put_Line ("BI: " & BI'Image);

end Show_Big_Integer_Greatest_Common_Divisor;

In this example, we retrieve the greatest common divisor of 145 and 25 (i.e.: 5).

Big real and quotients

An interesting feature of big reals is that they support quotients. In fact, we can simply assign 2/3 to a big real variable. (Note that we're able to omit the decimal points, as we write 2/3 instead of 2.0 / 3.0.) For example:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Real_Quotient_Conversion
is
   BR   : Big_Real;
begin
   BR := 2 / 3;
   --  Same as:
   --  BR := From_Quotient_String ("2 / 3");

   Put_Line ("BR:   " & BR'Image);

   Put_Line ("Q:    "
             & To_Quotient_String (BR));

   Put_Line ("Q numerator:    "
             & Numerator (BR)'Image);
   Put_Line ("Q denominator:  "
             & Denominator (BR)'Image);
end Show_Big_Real_Quotient_Conversion;

In this example, we assign 2 / 3 to BR — we could have used the From_Quotient_String function as well. Also, we use the To_Quotient_String to get a string that represents the quotient. Finally, we use the Numerator and Denominator functions to retrieve the values, respectively, of the numerator and denominator of the quotient (as big integers) of the big real variable.

Range checks

Previously, we've talked about the Big_Natural and Big_Positive subtypes. In addition to those subtypes, we have the In_Range function for big numbers:

    
    
    
        
pragma Ada_2022;

with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Big_Numbers.Big_Integers;
use  Ada.Numerics.Big_Numbers.Big_Integers;

with Ada.Numerics.Big_Numbers.Big_Reals;
use  Ada.Numerics.Big_Numbers.Big_Reals;

procedure Show_Big_Numbers_In_Range is

   BI : Big_Integer;
   BR : Big_Real;

   BI_From : constant Big_Integer := 0;
   BI_To   : constant Big_Integer := 1024;

   BR_From : constant Big_Real := 0.0;
   BR_To   : constant Big_Real := 1024.0;

begin
   BI := 1023;
   BR := 1023.9;

   if In_Range (BI, BI_From, BI_To) then
      Put_Line ("BI ("
                & BI'Image
                & ") is in the "
                & BI_From'Image
                & " .. "
                & BI_To'Image
                & " range");
   end if;

   if In_Range (BR, BR_From, BR_To) then
      Put_Line ("BR ("
                & BR'Image
                & ") is in the "
                & BR_From'Image
                & " .. "
                & BR_To'Image
                & " range");
   end if;

end Show_Big_Numbers_In_Range;

In this example, we call the In_Range function to check whether the big integer number (BI) and the big real number (BR) are in the range between 0 and 1024.