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A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23 ) × 2 127 ≈ 3.4028235 ...
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient.
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).
This gives from 33 to 36 significant decimal digits precision. 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.
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 decimal128 format supports 34 decimal digits of significand and an exponent range of −6143 to +6144, i.e. ±0.000 000 000 000 000 000 000 000 000 000 000 × 10 ^ −6143 to ±9.999 999 999 999 999 999 999 999 999 999 999 × 10 ^ 6144.
Like the binary floating-point formats, the number is divided into a sign, an exponent, and a significand. Unlike binary floating-point, numbers are not necessarily normalized; values with few significant digits have multiple possible representations: 1×10 2 =0.1×10 3 =0.01×10 4, etc. When the significand is zero, the exponent can be any ...
decimal32 supports 'normal' values, which can have 7 digit precision from ±1.000 000 × 10 ^ −95 up to ±9.999 999 × 10 ^ +96, plus 'subnormal' values with ramp-down relative precision down to ±1. × 10 ^ −101 (one digit), signed zeros, signed infinities and NaN (Not a Number). The encoding is somewhat complex, see below.