On Tue, Apr 21, 2020 at 11:31 PM Pieter van Oostrum <piete...@vanoostrum.org> wrote: > > Chris Angelico <ros...@gmail.com> writes: > > > On Tue, Apr 21, 2020 at 6:07 AM Schachner, Joseph > > <joseph.schach...@teledyne.com> wrote: > >> > >> 16 base 10 digits / log base10( 2) = 53.1508495182 bits. Obviously, > >> fractional bits don't exist, so 53 bits. If you note that the first > >> non-zero digit as 4, and the first digit after the 15 zeroes was 2, > >> then you got an extra bit. 54 bits. Where did the extra bit come from? > >> It came from the IEEE format's assumption that the top bit of the > >> mantissa of a normalized floating point value must be 1. Since we know > >> what it must be, there is no reason to use an actual bit for it. The > >> 53 bits in the mantissa do not include the assumed top bit. > >> > >> Isn't floating point fun? > >> > > > > IEEE 64-bit packed floating point has 53 bits of mantissa, 11 scale > > bits (you've heard of "scale birds" in art? well, these are "scale > > bits"), and 1 sign bit. > > > > 53 + 11 + 1 == 64. > > I don't know if this was meant to be sarcasm, but in my universe 53 + 11 + 1 > == 65. > But the floating point standard uses 53 bits of precision where only 52 bits > are stored. > 52 + 11 + 1 == 64. >
Exactly - normally, if you want 53 bits plus 11 bits plus 1 bit, it's going to cost you 65 bits, but due to IEEE magic, it's possible to store 65 bits of data in 64 bits of memory. > > Yep, floating point is fun. > > > > That assumed top 1 bit is always there, except when it isn't. Because > > denormal numbers are a thing. They don't have that implied 1 bit. > > Yes, for subnormal numbers the implicit bit *is* stored. They are > characterized by the biased exponent being 0. So these have only 52 bits of > precision in the mantissa. > Right. Now try diagnosing weird bugs in floating-point calculations, and you aren't sure if it's because sometimes you have 53 bits of mantissa and sometimes only 52. Like I said, floating point is fun. Extremely USEFUL and PRACTICAL, but still quite magical. ChrisA -- https://mail.python.org/mailman/listinfo/python-list
