What is the actual lowest possible positive REAL number

Question

MSDN says that the range of REAL numbers is - 3.40E + 38 to -1.18E - 38, 0 and 1.18E - 38 to 3.40E + 38. Apparently the true lower limit is much lower. The following script populates a REAL column with 1.401298E-45:

CREATE TABLE a
  (
    r1 REAL NULL ,
    r2 REAL NULL ,
    r3 REAL NULL
  ) ;
GO
INSERT  INTO a
        ( r1, r2 )
VALUES  ( 1.18E-37, 10 ) ;
GO
DECLARE @i INT ;
SET @i = 1 ;

WHILE @i < 20 
  BEGIN ;

    UPDATE  a
    SET     r1 = r1 / r2 ;

    SELECT  r1 ,
            r2
    FROM    a ;

    SET @i = @i + 1 ;

  END ;
GO
DROP TABLE a ;

r1            r2
------------- -------------
1.18E-38      10

(snip)

r1            r2
------------- -------------
1.401298E-45  10

Can anyone tell me what is the actual lowest possible positive number?

Answer 1

The minimum positive (subnormal) single-precision floating-point value is 2⁻¹⁴⁹ ≈ 1.4 × 10⁻⁴⁵. The minimum positive normal value is 2⁻¹²⁶ ≈ 1.18 × 10⁻³⁸ (reference).

DECLARE
    @r1 real = POWER(2e0, -126),
    @r2 real = POWER(2e0, -23)

SELECT
    @r1,
    @r2,
    @r1 * @r2,
    CONVERT(binary(4), @r1 * @r2);

For double-precision, the minimum positive subnormal is:

DECLARE @r1 float = POWER(2e0, -1075);

SELECT @r1, CONVERT(binary(8), @r1);