Floating point problem

[Chris Angelico]

On Tue, Apr 21, 2020 at 11:31 PM Pieter van Oostrum
<pieter-l at vanoostrum.org> wrote:
>
> Chris Angelico <rosuav at gmail.com> writes:
>
> > On Tue, Apr 21, 2020 at 6:07 AM Schachner, Joseph
> > <Joseph.Schachner at 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