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
