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. > 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. > Yep, floating point is fun. > > ChrisA -- Pieter van Oostrum www: http://pieter.vanoostrum.org/ PGP key: [8DAE142BE17999C4] -- https://mail.python.org/mailman/listinfo/python-list
