What is the best way to compare floats for almost-equality in Python

Question

It s well known that comparing floats for equality is a little fiddly due to rounding and precision issues   For example  https   randomascii wordpress com 2012 02 25 comparing-floating-point-numbers-2012-edition   What is the recommended way to deal with this in Python   Surely there is a standard library function for this somewhere

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I liked  Sesquipedal  s suggestion but with modification  a special use case when both values are 0 returns False    In my case I was on Python 2 7 and just used a simple function   if f1   0 and f2    0      return True else      return abs f1-f2   lt  tol max abs f1  abs f2

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Useful for the case where you want to make sure 2 numbers are the same  up to precision   no need to specify the tolerance    Find minimum precision of the 2 numbers Round both of them to minimum precision and compare   def isclose a b                                              astr str a                                               aprec len astr split      1   if     in astr else 0      bstr str b                                               bprec len bstr split      1   if     in bstr else 0      prec min aprec bprec                                            return round a prec   round b prec                                   As written  only works for numbers without the  e  in their string representation   meaning 0 9999999999995e-4  lt  number  lt   0 9999999999995e11    Example    gt  gt  gt  isclose 10 0 10 049  True  gt  gt  gt  isclose 10 0 10 05  False

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For some of the cases where you can affect the source number representation  you can represent them as fractions instead of floats  using integer numerator and denominator  That way you can have exact comparisons   See Fraction from fractions module for details

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I found the following comparison helpful   str f1     str f2

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Use Python s decimal module  which provides the Decimal class   From the comments      It is worth noting that if you re   doing math-heavy work and you don t   absolutely need the precision from   decimal  this can really bog things   down  Floats are way  way faster to   deal with  but imprecise  Decimals are   extremely precise but slow

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Python 3 5 adds the math isclose and cmath isclose functions as described in PEP 485   If you re using an earlier version of Python  the equivalent function is given in the documentation   def isclose a  b  rel tol 1e-09  abs tol 0 0       return abs a-b   lt   max rel tol   max abs a   abs b    abs tol    rel tol is a relative tolerance  it is multiplied by the greater of the magnitudes of the two arguments  as the values get larger  so does the allowed difference between them while still considering them equal   abs tol is an absolute tolerance that is applied as-is in all cases  If the difference is less than either of those tolerances  the values are considered equal

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The common wisdom that floating-point numbers cannot be compared for equality is inaccurate  Floating-point numbers are no different from integers  If you evaluate  a    b   you will get true if they are identical numbers and false otherwise  with the understanding that two NaNs are of course not identical numbers    The actual problem is this  If I have done some calculations and am not sure the two numbers I have to compare are exactly correct  then what  This problem is the same for floating-point as it is for integers  If you evaluate the integer expression  7 3 3   it will not compare equal to  7 3 3    So suppose we asked  How do I compare integers for equality   in such a situation  There is no single answer  what you should do depends on the specific situation  notably what sort of errors you have and what you want to achieve   Here are some possible choices   If you want to get a  true  result if the mathematically exact numbers would be equal  then you might try to use the properties of the calculations you perform to prove that you get the same errors in the two numbers  If that is feasible  and you compare two numbers that result from expressions that would give equal numbers if computed exactly  then you will get  true  from the comparison  Another approach is that you might analyze the properties of the calculations and prove that the error never exceeds a certain amount  perhaps an absolute amount or an amount relative to one of the inputs or one of the outputs  In that case  you can ask whether the two calculated numbers differ by at most that amount  and return  true  if they are within the interval  If you cannot prove an error bound  you might guess and hope for the best  One way of guessing is to evaluate many random samples and see what sort of distribution you get in the results   Of course  since we only set the requirement that you get  true  if the mathematically exact results are equal  we left open the possibility that you get  true  even if they are unequal   In fact  we can satisfy the requirement by always returning  true   This makes the calculation simple but is generally undesirable  so I will discuss improving the situation below    If you want to get a  false  result if the mathematically exact numbers would be unequal  you need to prove that your evaluation of the numbers yields different numbers if the mathematically exact numbers would be unequal  This may be impossible for practical purposes in many common situations  So let us consider an alternative   A useful requirement might be that we get a  false  result if the mathematically exact numbers differ by more than a certain amount  For example  perhaps we are going to calculate where a ball thrown in a computer game traveled  and we want to know whether it struck a bat  In this case  we certainly want to get  true  if the ball strikes the bat  and we want to get  false  if the ball is far from the bat  and we can accept an incorrect  true  answer if the ball in a mathematically exact simulation missed the bat but is within a millimeter of hitting the bat  In that case  we need to prove  or guess estimate  that our calculation of the ball s position and the bat s position have a combined error of at most one millimeter  for all positions of interest   This would allow us to always return  false  if the ball and bat are more than a millimeter apart  to return  true  if they touch  and to return  true  if they are close enough to be acceptable   So  how you decide what to return when comparing floating-point numbers depends very much on your specific situation   As to how you go about proving error bounds for calculations  that can be a complicated subject  Any floating-point implementation using the IEEE 754 standard in round-to-nearest mode returns the floating-point number nearest to the exact result for any basic operation  notably multiplication  division  addition  subtraction  square root    In case of tie  round so the low bit is even    Be particularly careful about square root and division  your language implementation might use methods that do not conform to IEEE 754 for those   Because of this requirement  we know the error in a single result is at most 1 2 of the value of the least significant bit   If it were more  the rounding would have gone to a different number that is within 1 2 the value    Going on from there gets substantially more complicated  the next step is performing an operation where one of the inputs already has some error  For simple expressions  these errors can be followed through the calculations to reach a bound on the final error  In practice  this is only done in a few situations  such as working on a high-quality mathematics library  And  of course  you need precise control over exactly which operations are performed  High-level languages often give the compiler a lot of slack  so you might not know in which order operations are performed   There is much more that could be  and is  written about this topic  but I have to stop there  In summary  the answer is  There is no library routine for this comparison because there is no single solution that fits most needs that is worth putting into a library routine   If comparing with a relative or absolute error interval suffices for you  you can do it simply without a library routine

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If you want to use it in testing TDD context  I d say this is a standard way   from nose tools import assert almost equals  assert almost equals x  y  places 7   default is 7

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Is something as simple as the following not good enough   return abs f1 - f2   lt   allowed error

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This maybe is a bit ugly hack  but it works pretty well when you don t need more than the default float precision  about 11 decimals    The round to function uses the format method from the built-in str class to round up the float to a string that represents the float with the number of decimals needed  and then applies the eval built-in function to the rounded float string to get back to the float numeric type   The is close function just applies a simple conditional to the rounded up float   def round to float num  prec       return eval          str int prec      f   format     str float num          def is close float a  float b  prec       if round to float a  prec     round to float b  prec           return True     return False   gt  gt  gt a   10 0 10 0  gt  gt  gt b   10 0001 10 0001  gt  gt  gt print is close a  b  prec 3  True  gt  gt  gt print is close a  b  prec 4  False   Update   As suggested by  stepehjfox  a cleaner way to build a rount to function avoiding  eval  is using nested formatting   def round to float num  prec       return      precision f   format float num  precision prec    Following the same idea  the code can be even simpler using the great new f-strings  Python 3 6     def round to float num  prec       return f  float num   prec f     So  we could even wrap it up all in one simple and clean  is close  function   def is close a  b  prec       return f  a   prec f      f  b   prec f

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I m not aware of anything in the Python standard library  or elsewhere  that implements Dawson s AlmostEqual2sComplement function  If that s the sort of behaviour you want  you ll have to implement it yourself   In which case  rather than using Dawson s clever bitwise hacks you d probably do better to use more conventional tests of the form if abs a-b   lt   eps1  abs a  abs b     eps2 or similar  To get Dawson-like behaviour you might say something like if abs a-b   lt   eps max EPS abs a  abs b   for some small fixed EPS  this isn t exactly the same as Dawson  but it s similar in spirit

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math isclose   has been added to Python 3 5 for that  source code   Here is a port of it to Python 2  It s difference from one-liner of Mark Ransom is that it can handle  inf  and  -inf  properly   def isclose a  b  rel tol 1e-09  abs tol 0 0               Python 2 implementation of Python 3 5 math isclose       https   hg python org cpython file tip Modules mathmodule c l1993               sanity check on the inputs     if rel tol  lt  0 or abs tol  lt  0          raise ValueError  tolerances must be non-negative          short circuit exact equality -- needed to catch two infinities of       the same sign  And perhaps speeds things up a bit sometimes      if a    b          return True        This catches the case of two infinities of opposite sign  or       one infinity and one finite number  Two infinities of opposite       sign would otherwise have an infinite relative tolerance        Two infinities of the same sign are caught by the equality check       above      if math isinf a  or math isinf b           return False        now do the regular computation       this is essentially the  weak  test from the Boost library     diff   math fabs b - a      result      diff  lt   math fabs rel tol   b   or                 diff  lt   math fabs rel tol   a    or                diff  lt   abs tol       return result

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In terms of absolute error  you can just check  if abs a - b   lt   error      print  Almost equal     Some information of why float act weird in Python https   youtu be v4HhvoNLILk t 1129  You can also use math isclose for relative errors

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I would agree that Gareth s answer is probably most appropriate as a lightweight function solution   But I thought it would be helpful to note that if you are using NumPy or are considering it  there is a packaged function for this   numpy isclose a  b  rtol 1e-05  atol 1e-08  equal nan False    A little disclaimer though  installing NumPy can be a non-trivial experience depending on your platform

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To compare up to a given decimal without atol rtol   def almost equal a  b  decimal 6       return   0   1 f   format a  decimal       0   1 f   format b  decimal   print almost equal 0 0  0 0001  decimal 5     False print almost equal 0 0  0 0001  decimal 4     True

[python] What is the best way to compare floats for almost-equality in Python?

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