How does the `%b` format specifier represent float64 values in Go\'s `fmt.Printf`?

Understanding "%b" for float64

The "%b" format specifier in fmt.Printf for float64 values represents the floating-point number in binary scientific notation with a two's complement exponent. In this notation, the number is expressed as a mantissa (significand) multiplied by a power of two raised to an exponent, both of which are represented in binary format.

For instance, when fmt.Printf("%bn", 1.0) is executed, it produces the output: 4503599627370496p-52. This indicates that the:

• Significand: 4503599627370496
• Exponent: -52

Decoding the Significand

The significand (or mantissa) is a 53-bit floating-point number. In binary, it can be represented as:

0.11111111111110000000000000000000000000000000000000000000000000

To convert this binary significand to decimal, we multiply it by 2^(1 - exponent).

In this case, the exponent is -52, so:

0.11111111111110000000000000000000000000000000000000000000000000 * 2^(1 - (-52))
= 0.11111111111110000000000000000000000000000000000000000000000000 * 2^(53)
= 1.0

Therefore, the significand represents the value 1.

Decoding the Exponent

The exponent is a 11-bit signed integer. The standard IEEE 754 binary representation for floating-point numbers uses a biased exponent, where a positive value represents the actual exponent, while a negative value indicates a subnormal number.

For the exponent -52, we calculate the unbiased exponent:

Unbiased exponent = Biased exponent - 1023
= -52 - 1023
= -1075

This negative value signifies a subnormal number. Subnormal numbers are used to represent numbers that are too small to be represented using a normalized exponent range.

Calculating the Float64 Value

Combining the significand and the exponent, we can calculate the float64 value:

value = significand * 2^(exponent)
= 1.0 * 2^(-1075)
= 5e-324

Understanding Min Subnormal Positive Double

The minimum subnormal positive double value is the smallest positive double value that is less than 1.0. Its hexadecimal representation is 0x0000000000000001.

Converting this hexadecimal value to binary:

0000000000000000000000000000000000000000000000000000000000000001

This binary representation can be decomposed as:

• Sign bit: 0 (positive)
• Exponent: -1022 (subnormal exponent)
• Significand: 1.0

Using the same calculation as before:

value = significand * 2^(exponent)
= 1.0 * 2^(-1022)
= 5e-324

Therefore, the minimum subnormal positive double value is 5e-324.

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