Thanks, Sturla, you've confirmed what I thought was the case (and explained more thoroughly the answer others gave more succinctly, but also more opaquely). :-)
DG On Sat, Nov 7, 2009 at 11:40 AM, Sturla Molden <stu...@molden.no> wrote: > David Cournapeau wrote: > > On Fri, Nov 6, 2009 at 6:54 AM, David Goldsmith <d.l.goldsm...@gmail.com> > wrote: > > > >> Interesting thread, which leaves me wondering two things: is it > documented > >> somewhere (e.g., at the IEEE site) precisely how many *decimal* > mantissae > >> are representable using the 64-bit IEEE standard for float > representation > >> (if that makes sense); and are such decimal mantissae uniformly > distributed > >> > > > > They are definitely not uniformly distributed: that's why two numbers > > are close around 1 when they have only a few EPS difference, but > > around 1e100, you have to add quite a few EPS to even get a different > > number at all. > > > > That may be my audio processing background, but I like to think about > > float as numbers which have the same relative precision at any "level" > > - a kind of dB scale. If you want numbers with a fixed number of > > decimals, you need a fixed point representation. > > > > David Godsmith was asking about the mantissae. For a double, that is a > 53 bit signed integer. I.e. you have 52 bit fractional part (bit 0-51), > 11 bit exponent (bit 52-62), and one sign bit (bit 63). The mantissae is > uniformly distributed like any signed integer. The mantissae of a double > have 2**53 different integer values: -2**52 to 2**52-1. > > But the value of a floating point number is > > value = (-1)**signbit * 2**(exponent - bias) * (1 - fraction) > > with bias = 1023 for a double. Thus, floating point numbers are not > uniformly distributed, but the mantissae is. > > For numerical illiterates this might come as a surprise. But in > numerical mathematics, the resolution is in the number of "significant > digits", not in "the number of decimals". 101 and .00201 have the same > numerical precision. > > A decimal, on the other hand, can be thought of as a floating point > number using base-10 instead of base-2 for the exponent: > > value = (-1)**signbit * 10**(exponent - bias) * (1 - fraction) > > Decimals and floats are not fundamentally different. There are number > exactly representable with a decimal that cannot be exactly represented > with a float. But numerical computation do not become more precise with > a decimal than a float. > > > > > > > > > > > > > > > > > > _______________________________________________ > NumPy-Discussion mailing list > NumPy-Discussion@scipy.org > http://mail.scipy.org/mailman/listinfo/numpy-discussion >
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