Floating Point Visually Explained (2017)
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Floating Point Visually Explained (2017)
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Re: Floating Point Visually Explained (2017)
#2Wow, that's a much easier way to convert from decimal to floating point than I had ever seen. He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order.
Re: Floating Point Visually Explained (2017)
#3I saw a very simple visualisation of floating point by simply plotting points along the number line and zooming out. That gets the idea across very quickly!
Re: Floating Point Visually Explained (2017)
#4Wow, that's a much easier way to convert from decimal to floating point than I had ever seen. He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order.
>"He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order."
Could you elaborate on this? Maybe an example? This sounds interesting but I'm failing to grasp it.
Re: Floating Point Visually Explained (2017)
#5I always found the wikipedia examples for 16bit floating point helpful since the numbers are smaller. You can really see how the exponent and fraction affect each other in a very simple way.
https://en.wikipedia.org/wiki/Half-precision_floating-point_...
Re: Floating Point Visually Explained (2017)
#6Related, https://float.exposed/ is a great resource: both when trying to see how floating point is laid out, as well as when having to convert between the bit representation and the number for "actual work" ;)
Re: Floating Point Visually Explained (2017)
#7I saw a very simple visualisation of floating point by simply plotting points along the number line and zooming out. That gets the idea across very quickly!
Yes, I came across something like that in my numerical computation course. This was for me the most compelling visualization of what a floating point number was.
The floating number line is actually an (unevenly spaced in reals, uniformly spaced in binary) discrete number line that was used to approximate continuous infinite real numbers, in finite precision e.g.
Reals
0 __________________________________________________ 1
Floats
0 .. .... .. ... .. ... ... ... .. ... ... ... .. .... ... ... ... . ... .... ... .. . ... 1
Re: Floating Point Visually Explained (2017)
#8Wow, that's a much easier way to convert from decimal to floating point than I had ever seen. He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order.
IOW the "windows" are just labeled in order from left to right, starting with 0 for the leftmost window.
Re: Floating Point Visually Explained (2017)
#9I believe that this "window and offset" intuition, while indeed true and useful in the radix-2 ("binary") case, does not cleanly extend to the general case where no hidden bit is used even for non-"subnormal" numbers, and some numbers may thus have multiple representations. This shows up perhaps most clearly in the case of decimal floating point, but ISTR that a non-2 radix was also used in some mainframes.
Re: Floating Point Visually Explained (2017)
#10Wow, that's a much easier way to convert from decimal to floating point than I had ever seen. He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order.
>"He doesn't mention why biased notation is used (i.e. why the exponent is stored as 127+E): it's used so that if you sort positive numbers as if they were integers, they'll still end up in the right order." Could you elaborate on this? Maybe an example? This sounds interesting but I'm failing to grasp it.
The first thing to notice is that a (not subnormal) floating point number can be compared as such: you look at the exponent bits, and pick the one that has the greater exponent. If they are equal, then look at the mantissa, which is always between 1 and 2, and pick whichever one is larger. Since the exponent comes before the mantissa bits in the bit representation of a floating point number, it has "higher significance" when interpreted in fixed point. If we used two's complement to represent the exponent negative exponents would "look larger" than positive ones, despite them being smaller. But if we used a biased representation, where we shift the range so that there can only be a positive number in that field, and the smaller negative exponents can actually be small and the positive exponents are be larger than those so we can compare them directly.