Comments on: Anatomy of a floating point number

By: Javier

Javier — Sat, 14 Jul 2012 16:31:41 +0000

Hi Jack,

I am trying to explain why 1.f = (2-2^(-52))

1.f = 2⁰+2^(-1) + 2^(-2) + … + 2^(-52)
so,
2^(52) (1.f) = 2^(52) + 2^(51) + … + 2⁰ = 2^(53) – 1 (this is the key)

And finally,

1.f= (2^(53) – 1) / 2^(52) = 2 – 2^(-52)

Best regards,
Javier

By: Barnaby

Barnaby — Tue, 21 Feb 2012 17:56:55 +0000

Be great to add a diagram showing the bit structure of a float. The first time I learned about them there was a diagram and I found it incredibly clear in one glance.

By: Jack

Jack — Mon, 02 Jan 2012 15:16:51 +0000

Thanks for the more detailed article on floating point; they’re hard to find. I was wondering, my math is a little weak. How do you get the expression 2^1023(2 – 2^-52) from 1.f * 2^1023, where f is a string of 52 1′s. I assume the 2^1023 remains untouched, so the question then is how do you get (2 – 2^-52) from 1.f? I tried finding a common denominator, so (2^52 + 2^51 + … 2^0) / 2^52, but didn’t seem to get me anywhere.
Thanks.

By: What I’ve been reading this week: Low level links, Legos for adults and a free AI course by Peter Norvig.

Sun, 14 Aug 2011 01:39:02 +0000

[...] floating point arithmetic. On John D. Cook’s blog, Numbers are a Leaky Abstraction and Anatomy of a Floating Point Number are two interesting posts that deal with the various gotchas about working with floating point [...]

By: Antonio

Antonio — Thu, 07 Jul 2011 16:41:00 +0000

IEEE should hurry up with that floating point interval arithmetic standard http://en.wikipedia.org/wiki/Interval_arithmetic#IEEE_Interval_Standard_.E2.80.93_P1788

By: Nemo

Nemo — Wed, 13 Oct 2010 20:33:24 +0000

In C++, the standard way to deal with these constants is via the parameterized numeric_limits class:

#include

double x = std::numeric_limits::max();
double y = std::numeric_limits::epsilon();
double nan = std::numeric_limits::quiet_nan();
[etc.]

This has the advantage of working for float, double, long double, and whatever is in vogue tomorrow… It also works for the integer types (well, not NaN, but you know what I mean). It also works for typedefs where you neither know nor care what the underlying type is:

typedef long double real_t;

real_t x = std::numeric_limits::max();

The header is part of the C++98 standard, so this also has the advantage of being perfectly portable across 21st-century compilers.

By: Manjunath

Manjunath — Tue, 23 Jun 2009 08:10:55 +0000

of course i did similar thing for test this function by generating NaN and in VC6 its failed. Then i’ve tried with quiet NaN for verifying it.

By: John

John — Tue, 23 Jun 2009 07:44:39 +0000

In my test code, I made my own NaNs manually rather than using quiet_NaN(). For example, I'd have code like x = -1.0; z = sqrt(x); assert(!IsNumber(z)); or x = 1.0; y = 0.0; x /= y; z = y*x; assert(!IsNumber(z)); I also have some more elaborate code to generate a quiet NaN. Maybe VC6 has a bug in quiet_NaN? I wouldn't think so, but that seems more plausible than a violation of IEEE arithmetic.

By: Manjunath

Manjunath — Tue, 23 Jun 2009 07:32:04 +0000

John, In gcc its working fine, but VC6 its not working as expected.
Here is my x,
x = numeric_limits::quiet_NaN();

By: John

John — Tue, 23 Jun 2009 07:23:49 +0000

Manjunath, I’m surprised to hear that. I’ve tested that function on several versions of Visual C++, though I can’t recall whether I tested VC6. I’ve tested it on several versions of gcc as well.

By: Manjunath

Manjunath — Tue, 23 Jun 2009 05:54:52 +0000

bool IsNumber(double x)
{
// This looks like it should always be true,
// but it’s false if x is a NaN.
return (x == x);
}
mentioned in “IEEE floating-point exceptions in C++”
This is not working at least in VC++ 6.0 compiler

By: Math Teachers at Play #5 « Let’s Play Math!

Math Teachers at Play #5 « Let’s Play Math! — Fri, 17 Apr 2009 12:41:08 +0000

[...] John Cook presents Floating point numbers are a leaky abstraction. This is the first of two blog posts about how computer arithmetic works and what problems to look out for. (Part 2 here.) [...]

By: John

John — Tue, 07 Apr 2009 15:39:32 +0000

Karl, thanks for pointing out the mistake. I updated the post after reading your correction.

By: Karl Ove Hufthammer

Karl Ove Hufthammer — Tue, 07 Apr 2009 15:21:13 +0000

Thanks for the nice summary of floating point numbers. BTW, I think you made a small mistake in the third paragraph: 2043 should be 2047.