The hexadecimal representation of a floating point number from fprintf and snprintf seems to take the form (using A as a format specifier): 0XH.HHHHP[+-]d , where the number of hexadecimal digits to the right of the decimal point is variable.
I've found at least one manual explaining this format:
aA The double argument is rounded and converted to hexadecimal notation in the style [-]0xh.hhhp[+-]d, where the number of digits after the hexadecimal-point character is equal to the precision specification. If the precision is missing, it is taken as enough to represent the floating-point number exactly, and no rounding occurs. If the precision is zero, no hexadecimal-point character appears. The p is a literal character `p', and the exponent consists of a positive or negative sign followed by a decimal number representing an exponent of 2. The A conversion uses the prefix ``0X'' (rather than ``0x''), the letters ``ABCDEF'' (rather than ``abcdef'') to represent the hex digits, and the letter `P' (rather than `p') to separate the mantissa and exponent. Note that there may be multiple valid ways to represent floating-point point numbers in this hexadecimal format. For example, 0x3.24p+0, 0x6.48p-1 and 0xc.9p-2 are all equivalent. The format chosen depends on the internal representation of the number, but the implementation guarantees that the length of the mantissa will be minimized. Zeroes are always represented with a mantissa of 0 (preceded by a `-' if appropriate) and an exponent of +0.
I don't understand this explanation. I believe I understand the general approach to converting floating point numbers to binary: include a sign bit, an exponent bit that refers to the location of the first significant digit in binary, and use the rest of the bits to represent the number to the right of the decimal in binary.
What I don't understand is how this is done with this format, for example:
What does the character to the left of the decimal point represent?
What does it mean when the decimal point is missing?
What do the characters to the right of the decimal point represent?
Why is the letter P always there?
Can anyone show me some examples of converting to/from this representation and a floating point number?
I've found at least one manual explaining this format…
You should look to the C standard for information like this.
What does the character to the left of the decimal point represent?
In decimal, 34.5•100, 3.45•101, and .345•102, are the same number. In hexadecimal, 9A.B16•160, 9.AB16•161, .9AB16•162 are the same number. In the description of the style as [-]0xh.hhhhp±d, the h.hhhh part is just telling you that the formatting routine will format it with one hexadecimal digit to the left of the point (and adjust the exponent to match).
What does it mean when the decimal point is missing?
This is merely an aesthetic choice. When there are no digits in fraction positions, to the right of a decimal point, there is no need to have a decimal point to indicate where the fraction positions start. However, the %a format is mostly used to display the entire floating-point value, so we usually let all the digits appear and do not set the precision to zero.
What do the characters to the right of the decimal point represent?
They are digits in fraction positions, the same as in decimal notation. For example, the hexadecimal numeral 123.45616 represents 1•162 + 2•161 + 3•160 + 4•16−1 + 5•16−2 + 6•16−3.
Why is the letter P always there?
It stands for “power” and marks the start of the exponent. For output, we could suppress it if the exponent is zero. However, this output format is chosen to match the form of hexadecimal literals in source code, and, for those, we need “p” at least in integers to distinguish the hexadecimal integer literal 0x3 from the hexadecimal floating-point literal 0x3p0.