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If a decimal string with at most 6 significant digits is converted to the IEEE 754 single-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 single-precision number is converted to a decimal string with at least 9 ...
If a decimal string with at most 15 significant digits is converted to the IEEE 754 double-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 double-precision number is converted to a decimal string with at least ...
If a decimal string with at most 33 significant digits is converted to the IEEE 754 quadruple-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 quadruple-precision number is converted to a decimal string with at ...
Since 2 10 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2 −1022 ≈ 2 × 10 −308 to approximately 2 1024 ≈ 2 × 10 308. The number of normal floating-point numbers in a system (B, P, L, U) where B is the base of the system, P is the precision of the significand (in base B),
The 80-bit floating-point format has a range (including subnormals) from approximately 3.65 × 10 −4951 to 1.18 × 10 +4932. Although log 10 ( 2 64) ≈ 19.266, this format is usually described as giving approximately eighteen significant digits of precision (the floor of log 10 ( 2 63), the minimum guaranteed precision).
Because floating-point numbers have limited precision, only a subset of real or rational numbers are exactly representable; other numbers can be represented only approximately. Many languages have both a single precision (often called float ) and a double precision type (often called double ).
Bfloat16 is designed to maintain the number range from the 32-bit IEEE 754 single-precision floating-point format (binary32), while reducing the precision from 24 bits to 8 bits. This means that the precision is between two and three decimal digits, and bfloat16 can represent finite values up to about 3.4 × 10 38.
For floating-point arithmetic, the mantissa was restricted to a hundred digits or fewer, and the exponent was restricted to two digits only. The largest memory supplied offered 60 000 digits, however Fortran compilers for the 1620 settled on fixed sizes such as 10, though it could be specified on a control card if the default was not satisfactory.