@@ -148,7 +148,7 @@ Binary floating-point arithmetic holds many surprises like this. The problem
148148with "0.1" is explained in precise detail below, in the "Representation Error"
149149section. See `Examples of Floating Point Problems
150150<https://jvns.ca/blog/2023/01/13/examples-of-floating-point-problems/> `_ for
151- a pleasant summary of how binary floating point works and the kinds of
151+ a pleasant summary of how binary floating- point works and the kinds of
152152problems commonly encountered in practice. Also see
153153`The Perils of Floating Point <https://www.lahey.com/float.htm >`_
154154for a more complete account of other common surprises.
@@ -174,7 +174,7 @@ Another form of exact arithmetic is supported by the :mod:`fractions` module
174174which implements arithmetic based on rational numbers (so the numbers like
1751751/3 can be represented exactly).
176176
177- If you are a heavy user of floating point operations you should take a look
177+ If you are a heavy user of floating- point operations you should take a look
178178at the NumPy package and many other packages for mathematical and
179179statistical operations supplied by the SciPy project. See <https://scipy.org>.
180180
@@ -268,12 +268,14 @@ decimal fractions cannot be represented exactly as binary (base 2) fractions.
268268This is the chief reason why Python (or Perl, C, C++, Java, Fortran, and many
269269others) often won't display the exact decimal number you expect.
270270
271- Why is that? 1/10 is not exactly representable as a binary fraction. Almost all
272- machines today (November 2000) use IEEE-754 floating point arithmetic, and
273- almost all platforms map Python floats to IEEE-754 "double precision". 754
274- doubles contain 53 bits of precision, so on input the computer strives to
275- convert 0.1 to the closest fraction it can of the form *J */2**\ *N * where *J * is
276- an integer containing exactly 53 bits. Rewriting ::
271+ Why is that? 1/10 is not exactly representable as a binary fraction. Since at
272+ least 2000, almost all machines use IEEE 754 binary floating-point arithmetic,
273+ and almost all platforms map Python floats to IEEE 754 binary64 "double
274+ precision" values. IEEE 754 binary64 values contain 53 bits of precision, so
275+ on input the computer strives to convert 0.1 to the closest fraction it can of
276+ the form *J */2**\ *N * where *J * is an integer containing exactly 53 bits.
277+ Rewriting
278+ ::
277279
278280 1 / 10 ~= J / (2**N)
279281
@@ -308,7 +310,8 @@ by rounding up:
308310 >>> q+ 1
309311 7205759403792794
310312
311- Therefore the best possible approximation to 1/10 in 754 double precision is::
313+ Therefore the best possible approximation to 1/10 in IEEE 754 double precision
314+ is::
312315
313316 7205759403792794 / 2 ** 56
314317
@@ -321,7 +324,7 @@ if we had not rounded up, the quotient would have been a little bit smaller than
3213241/10. But in no case can it be *exactly * 1/10!
322325
323326So the computer never "sees" 1/10: what it sees is the exact fraction given
324- above, the best 754 double approximation it can get:
327+ above, the best IEEE 754 double approximation it can get:
325328
326329.. doctest ::
327330
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