From 2cf04e455d8f087bd08cd1d43751007b5e41b3c5 Mon Sep 17 00:00:00 2001 From: Mark Dickinson Date: Fri, 26 May 2023 07:26:16 +0100 Subject: gh-104479: Update outdated tutorial floating-point reference (#104681) --- Doc/tutorial/floatingpoint.rst | 23 +++++++++++++---------- 1 file changed, 13 insertions(+), 10 deletions(-) diff --git a/Doc/tutorial/floatingpoint.rst b/Doc/tutorial/floatingpoint.rst index 306b1eb..b88055a 100644 --- a/Doc/tutorial/floatingpoint.rst +++ b/Doc/tutorial/floatingpoint.rst @@ -148,7 +148,7 @@ Binary floating-point arithmetic holds many surprises like this. The problem with "0.1" is explained in precise detail below, in the "Representation Error" section. See `Examples of Floating Point Problems `_ for -a pleasant summary of how binary floating point works and the kinds of +a pleasant summary of how binary floating-point works and the kinds of problems commonly encountered in practice. Also see `The Perils of Floating Point `_ for a more complete account of other common surprises. @@ -174,7 +174,7 @@ Another form of exact arithmetic is supported by the :mod:`fractions` module which implements arithmetic based on rational numbers (so the numbers like 1/3 can be represented exactly). -If you are a heavy user of floating point operations you should take a look +If you are a heavy user of floating-point operations you should take a look at the NumPy package and many other packages for mathematical and statistical operations supplied by the SciPy project. See . @@ -268,12 +268,14 @@ decimal fractions cannot be represented exactly as binary (base 2) fractions. This is the chief reason why Python (or Perl, C, C++, Java, Fortran, and many others) often won't display the exact decimal number you expect. -Why is that? 1/10 is not exactly representable as a binary fraction. Almost all -machines today (November 2000) use IEEE-754 floating point arithmetic, and -almost all platforms map Python floats to IEEE-754 "double precision". 754 -doubles contain 53 bits of precision, so on input the computer strives to -convert 0.1 to the closest fraction it can of the form *J*/2**\ *N* where *J* is -an integer containing exactly 53 bits. Rewriting :: +Why is that? 1/10 is not exactly representable as a binary fraction. Since at +least 2000, almost all machines use IEEE 754 binary floating-point arithmetic, +and almost all platforms map Python floats to IEEE 754 binary64 "double +precision" values. IEEE 754 binary64 values contain 53 bits of precision, so +on input the computer strives to convert 0.1 to the closest fraction it can of +the form *J*/2**\ *N* where *J* is an integer containing exactly 53 bits. +Rewriting +:: 1 / 10 ~= J / (2**N) @@ -308,7 +310,8 @@ by rounding up: >>> q+1 7205759403792794 -Therefore the best possible approximation to 1/10 in 754 double precision is:: +Therefore the best possible approximation to 1/10 in IEEE 754 double precision +is:: 7205759403792794 / 2 ** 56 @@ -321,7 +324,7 @@ if we had not rounded up, the quotient would have been a little bit smaller than 1/10. But in no case can it be *exactly* 1/10! So the computer never "sees" 1/10: what it sees is the exact fraction given -above, the best 754 double approximation it can get: +above, the best IEEE 754 double approximation it can get: .. doctest:: -- cgit v0.12