Understanding of float, double, long double precision and numerical range in C language

IEEE754 the representation of floating-point numbers. The expression range of float type data in C language is -3.4*10^38~+3.4*10^38. Double is -1.7*10^-308~1.7*10^308,long double for -1.2*10^-4932~1.2*10^4932.

Type	Number of bits (bits)	Valid numbers	Range of values
Float	32	6~7	-3.4*10^38~+3.4*10^38
Double	64	15~16	-1.7*10^-308~1.7*10^308
Long double	128/	18~19	-1.2*10^-4932~1.2*10^4932

Exactly how the range is calculated is analyzed as follows:

For single-precision floating-point numbers (float), the sign bit one, which refers to the digit 8 bits, the mantissa 23 bits. The index can be expressed in the range of -128~127. The mantissa is 23 bits.

The precision of float and double is determined by the number of bits in the mantissa. Floating-point numbers are stored in memory by scientific notation, and the integer part is always an implied "1", because it is invariant and therefore cannot affect precision. float:2^23 = 8388608, a total of seven bits, which means that there can be up to 7 valid digits, but it is guaranteed to be 6 bits, that is, the precision of float is 6~7 bit valid number; double:2^52 = 4503599627370496, altogether 16 bits, The precision of the double is 15~16 bit.

The negative exponent determines the absolute minimum non-zero that the floating point can express, and the positive exponent determines the maximum number of absolute value that the floating point can express, which determines the range of the floating-point value. The range of float is -2^128 ~ +2^128, that is, the range of -3.40E+38 ~ +3.40e+38;double is -2^1024 ~ +2^1024, or -1.79E+308 ~+1.79e+308.

Take float for example, as shown in the table below

Symbol	Tail	Index
1	23	8
Number (+ +)	Decimal parts (Determination of accuracy)	-127~128 Index (decision Range)

For example:

+1.1111111111111111111111*2^127 (23 1 after the decimal point, because the range of the mantissa is one or two, its highest bit is always 1, so just access the fractional part, so the decimal is 23 bits 1), approximately equal to 2*2^127=3.4*10^38. The 3.4*10^38 negative number is the same.

A double is similar to this, with a double with a sign bit of 63 bits and an exponent of 62~52 bits and a total of 11 bits. The range represented is -1024~1023. The mantissa is 51~0. The range represented is +1.1111111111111111..11111*2^1023 (52 1 after the decimal point) to 1.7*10^308. Negative numbers are the same.

Understanding of float, double, long double precision and numerical range in C language