Python -- floating point arithmetic
Dear Python-User, today i create some slides about floating point arithmetic. I used an example from http://docs.python.org/tutorial/floatingpoint.html so i start the python shell on my linux machine: d...@maxwell $ python Python 2.6.5 (release26-maint, May 25 2010, 12:37:06) [GCC 4.3.4] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> >>> sum = 0.0 >>> >>> for i in range(10): ... sum += 0.1 ... >>> >>> sum 0.99989 >>> >>> But thats looks a little bit wrong for me ... i must be a number greater then 1.0 because 0.1 = 0.155511151231257827021181583404541015625000 in python ... if i print it. So i create an example program: sum = 0.0 n = 10 d = 1.0 / n print "%.60f" % ( d ) for i in range(n): print "%.60f" % ( sum ) sum += d print sum print "%.60f" % ( sum ) RESULTs -- 0.1555111512312578270211815834045410156250 0. 0.1555111512312578270211815834045410156250 0.20001110223024625156540423631668090820312500 0.3000444089209850062616169452667236328125 0.40002220446049250313080847263336181640625000 0.5000 0.59997779553950749686919152736663818359375000 0.6999555910790149937383830547332763671875 0.79993338661852249060757458209991455078125000 0.8999111821580299874767661094665527343750 1.0 0.99988897769753748434595763683319091796875000 and the jump from 0.50*** to 0.5999* looks wrong for me ... do i a mistake or is there something wrong in the representation of the floating points in python? my next question, why could i run print "%.66f" % ( sum ) but not print "%.67f" % ( sum ) can anybody tell me how python internal represent a float number?? Best and many thanks in advanced, David -- http://mail.python.org/mailman/listinfo/python-list
Python -- floating point arithmetic
-BEGIN PGP SIGNED MESSAGE- Hash: SHA384 Dear Python-User, today i create some slides about floating point arithmetic. I used an example from http://docs.python.org/tutorial/floatingpoint.html so i start the python shell on my linux machine: d...@maxwell $ python Python 2.6.5 (release26-maint, May 25 2010, 12:37:06) [GCC 4.3.4] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> sum = 0.0 >>> for i in range(10): ... sum += 0.1 ... >>> sum 0.99989 >>> But thats looks a little bit wrong for me ... i must be a number greater then 1.0 because 0.1 = 0.155511151231257827021181583404541015625000 in python ... if i print it. So i create an example program: sum = 0.0 n = 10 d = 1.0 / n print "%.60f" % ( d ) for i in range(n): print "%.60f" % ( sum ) sum += d print sum print "%.60f" % ( sum ) - RESULTs -- 0.1555111512312578270211815834045410156250 0. 0.1555111512312578270211815834045410156250 0.20001110223024625156540423631668090820312500 0.3000444089209850062616169452667236328125 0.40002220446049250313080847263336181640625000 0.5000 0.59997779553950749686919152736663818359375000 0.6999555910790149937383830547332763671875 0.79993338661852249060757458209991455078125000 0.8999111821580299874767661094665527343750 1.0 0.99988897769753748434595763683319091796875000 and the jump from 0.50*** to 0.5999* looks wrong for me ... do i a mistake or is there something wrong in the representation of the floating points in python? my next question, why could i run print "%.66f" % ( sum ) but not print "%.67f" % ( sum ) can anybody tell me how python internal represent a float number?? Best and many thanks in advanced, David -BEGIN PGP SIGNATURE- Version: GnuPG v2.0.15 (GNU/Linux) iQIcBAEBCQAGBQJMNGpsAAoJEJ82b5xvAlAYYMYP/RTaRcB2NCawBQ25V463+pkO /YtTqsamrFqENrljqpsPrwOOqR02TQrOvZrV72snDPkkcN36tZHKwbmcnOS0tOAc kjX0/oNQOvvEMyJUuHJt9kdNjxMEvUUlENEZHLtpxbypIAo3Waf0FBxKo9F4fJJS PaDIuDgXiLOiaTTUCwa5kKowzjUe8BJOczpUulvGiIvbUac4cxwkPiGvv6L4wzmI /4x1mG5FSibEzcI2zFRsQNHfTqxuKC3a49DuSPtXZo4YWdqVeSXLntQk70uTa78K q4cBVEIDqETQyG0mcRABJpcTMPsnGgbgJD74uhDSTARuyHh405XbjKlic7pe1M12 AhuN71QjpGFl80OOXxCja4OKJCAhPEkfhNsJjrlFnSXAoFWg5YvhDbSVkjW6ftt0 tzyGZEoBpRSjGSIeAojYaqt1Xdxwldz2qFjRLX03io3Fgr8PKRbcLRgg1FXaMFPc gYaY0UEh4dEjt/5afp3ET0LkOhZVMbbi2oSkkFQpRMAzik63zSJRwRxiQXWunlSi HhKqL4sAICf6MODzquuNzJO9wH8Fkpwi+JPEAwnQm/S6Ty00dN22GezG4TWGfepH AhKOIEtRIcMgI7kyBfUdOQe6sifKGGuTEeXK12Td9znZN+wKfqG+Ch1mq4Rwy6P7 p2xwF7ZUWNgRU+5Y/bLG =aS+I -END PGP SIGNATURE- -- http://mail.python.org/mailman/listinfo/python-list
Re: Python -- floating point arithmetic
On 07/07/2010 08:08 PM, Raymond Hettinger wrote: > On Jul 7, 5:55 am, Mark Dickinson wrote: >> On Jul 7, 1:05 pm, david mainzer wrote: >> >> >> >>> Dear Python-User, >> >>> today i create some slides about floating point arithmetic. I used an >>> example from >> >>> http://docs.python.org/tutorial/floatingpoint.html >> >>> so i start the python shell on my linux machine: >> >>> d...@maxwell $ python >>> Python 2.6.5 (release26-maint, May 25 2010, 12:37:06) >>> [GCC 4.3.4] on linux2 >>> Type "help", "copyright", "credits" or "license" for more information.>>> >>> >>> sum = 0.0 >>>>>>>>> for i in range(10): >> >>> ... sum += 0.1 >>> ...>>> >>> sum >>> 0.99989 >> >>> But thats looks a little bit wrong for me ... i must be a number greater >>> then 1.0 because 0.1 = >>> 0.155511151231257827021181583404541015625000 >>> in python ... if i print it. > > [Mark Dickinson] >> So you've identified one source of error here, namely that 0.1 isn't >> exactly representable (and you're correct that the value stored >> internally is actually a little greater than 0.1). But you're >> forgetting about the other source of error in your example: when you >> do 'sum += 0.1', the result typically isn't exactly representable, so >> there's another rounding step going on. That rounding step might >> produce a number that's smaller than the actual exact sum, and if >> enough of your 'sum += 0.1' results are rounded down instead of up, >> that would easily explain why the total is still less than 1.0. > > One key for understanding floating point mysteries is to look at the > actual binary sums rather that their approximate representation as a > decimal string. The hex() method can make it easier to visualize > Mark's explanation: > >>>> s = 0.0 >>>> for i in range(10): > ... s += 0.1 > ... print s.hex(), repr(s) > > > 0x1.ap-4 0.10001 > 0x1.ap-3 0.20001 > 0x1.4p-2 0.30004 > 0x1.ap-2 0.40002 > 0x1.0p-1 0.5 > 0x1.3p-1 0.59998 > 0x1.6p-1 0.69996 > 0x1.9p-1 0.79993 > 0x1.cp-1 0.89991 > 0x1.fp-1 0.99989 > > Having used hex() to understand representation error (how the binary > partial sums are displayed), you can use the Fractions module to gain > a better understanding of rounding error introduced by each addition: > >>>> s = 0.0 >>>> for i in range(10): > exact = Fraction.from_float(s) + Fraction.from_float(0.1) > s += 0.1 > actual = Fraction.from_float(s) > error = actual - exact > print '%-35s%-35s\t%s' % (actual, exact, error) > > > 3602879701896397/36028797018963968 3602879701896397/36028797018963968 > 0 > 3602879701896397/18014398509481984 3602879701896397/18014398509481984 > 0 > 1351079888211149/4503599627370496 10808639105689191/36028797018963968 > 1/36028797018963968 > 3602879701896397/9007199254740992 > 14411518807585589/36028797018963968 -1/36028797018963968 > 1/2 > 18014398509481985/36028797018963968 -1/36028797018963968 > 5404319552844595/9007199254740992 > 21617278211378381/36028797018963968 -1/36028797018963968 > 3152519739159347/4503599627370496 > 25220157913274777/36028797018963968 -1/36028797018963968 > 7205759403792793/9007199254740992 > 28823037615171173/36028797018963968 -1/36028797018963968 > 2026619832316723/2251799813685248 > 32425917317067569/36028797018963968 -1/36028797018963968 > 9007199254740991/9007199254740992 > 36028797018963965/36028797018963968 -1/36028797018963968 > > Hope this helps your slides, > > > Raymond Thanks to all for your help :) All your comments helped me and now i know how it works in python ! Best David -- http://mail.python.org/mailman/listinfo/python-list
