Why does the floating-point value of 4*0.1 look nice in Python 3 but 3*0.1 doesn't?

Question

I know that most decimals don't have an exact floating point representation (Is floating point math broken?).

But I don't see why 4*0.1 is printed nicely as 0.4, but 3*0.1 isn't, when both values actually have ugly decimal representations:

>>> 3*0.1
0.30000000000000004
>>> 4*0.1
0.4
>>> from decimal import Decimal
>>> Decimal(3*0.1)
Decimal('0.3000000000000000444089209850062616169452667236328125')
>>> Decimal(4*0.1)
Decimal('0.40000000000000002220446049250313080847263336181640625')

Answer 1

The simple answer is because 3*0.1 != 0.3 due to quantization (roundoff) error (whereas 4*0.1 == 0.4 because multiplying by a power of two is usually an "exact" operation).

You can use the .hex method in Python to view the internal representation of a number (basically, the exact binary floating point value, rather than the base-10 approximation). This can help to explain what's going on under the hood.

>>> (0.1).hex()
'0x1.999999999999ap-4'
>>> (0.3).hex()
'0x1.3333333333333p-2'
>>> (0.1*3).hex()
'0x1.3333333333334p-2'
>>> (0.4).hex()
'0x1.999999999999ap-2'
>>> (0.1*4).hex()
'0x1.999999999999ap-2'

0.1 is 0x1.999999999999a times 2^-4. The "a" at the end means the digit 10 - in other words, 0.1 in binary floating point is very slightly larger than the "exact" value of 0.1 (because the final 0x0.99 is rounded up to 0x0.a). When you multiply this by 4, a power of two, the exponent shifts up (from 2^-4 to 2^-2) but the number is otherwise unchanged, so 4*0.1 == 0.4.

However, when you multiply by 3, the little tiny difference between 0x0.99 and 0x0.a0 (0x0.07) magnifies into a 0x0.15 error, which shows up as a one-digit error in the last position. This causes 0.1*3 to be very slightly larger than the rounded value of 0.3.

Python 3's float repr is designed to be round-trippable, that is, the value shown should be exactly convertible into the original value. Therefore, it cannot display 0.3 and 0.1*3 exactly the same way, or the two different numbers would end up the same after round-tripping. Consequently, Python 3's repr engine chooses to display one with a slight apparent error.

Answer 2

repr (and str in Python 3) will put out as many digits as required to make the value unambiguous. In this case the result of the multiplication 3*0.1 isn't the closest value to 0.3 (0x1.3333333333333p-2 in hex), it's actually one LSB higher (0x1.3333333333334p-2) so it needs more digits to distinguish it from 0.3.

On the other hand, the multiplication 4*0.1 does get the closest value to 0.4 (0x1.999999999999ap-2 in hex), so it doesn't need any additional digits.

You can verify this quite easily:

>>> 3*0.1 == 0.3
False
>>> 4*0.1 == 0.4
True

I used hex notation above because it's nice and compact and shows the bit difference between the two values. You can do this yourself using e.g. (3*0.1).hex(). If you'd rather see them in all their decimal glory, here you go:

>>> Decimal(3*0.1)
Decimal('0.3000000000000000444089209850062616169452667236328125')
>>> Decimal(0.3)
Decimal('0.299999999999999988897769753748434595763683319091796875')
>>> Decimal(4*0.1)
Decimal('0.40000000000000002220446049250313080847263336181640625')
>>> Decimal(0.4)
Decimal('0.40000000000000002220446049250313080847263336181640625')

Answer 3

Here's a simplified conclusion from other answers.

If you check a float on Python's command line or print it, it goes through function repr which creates its string representation.

Starting with version 3.2, Python's str and repr use a complex rounding scheme, which prefers nice-looking decimals if possible, but uses more digits where necessary to guarantee bijective (one-to-one) mapping between floats and their string representations.

This scheme guarantees that value of repr(float(s)) looks nice for simple decimals, even if they can't be represented precisely as floats (eg. when s = "0.1").

At the same time it guarantees that float(repr(x)) == x holds for every float x

Answer 4

Not really specific to Python's implementation but should apply to any float to decimal string functions.

A floating point number is essentially a binary number, but in scientific notation with a fixed limit of significant figures.

The inverse of any number that has a prime number factor that is not shared with the base will always result in a recurring dot point representation. For example 1/7 has a prime factor, 7, that is not shared with 10, and therefore has a recurring decimal representation, and the same is true for 1/10 with prime factors 2 and 5, the latter not being shared with 2; this means that 0.1 cannot be exactly represented by a finite number of bits after the dot point.

Since 0.1 has no exact representation, a function that converts the approximation to a decimal point string will usually try to approximate certain values so that they don't get unintuitive results like 0.1000000000004121.

Since the floating point is in scientific notation, any multiplication by a power of the base only affects the exponent part of the number. For example 1.231e+2 * 100 = 1.231e+4 for decimal notation, and likewise, 1.00101010e11 * 100 = 1.00101010e101 in binary notation. If I multiply by a non-power of the base, the significant digits will also be affected. For example 1.2e1 * 3 = 3.6e1

Depending on the algorithm used, it may try to guess common decimals based on the significant figures only. Both 0.1 and 0.4 have the same significant figures in binary, because their floats are essentially truncations of (8/5)(2^-4) and (8/5)(2^-6) respectively. If the algorithm identifies the 8/5 sigfig pattern as the decimal 1.6, then it will work on 0.1, 0.2, 0.4, 0.8, etc. It may also have magic sigfig patterns for other combinations, such as the float 3 divided by float 10 and other magic patterns statistically likely to be formed by division by 10.

In the case of 3*0.1, the last few significant figures will likely be different from dividing a float 3 by float 10, causing the algorithm to fail to recognize the magic number for the 0.3 constant depending on its tolerance for precision loss.

Edit: https://docs.python.org/3.1/tutorial/floatingpoint.html

Interestingly, there are many different decimal numbers that share the same nearest approximate binary fraction. For example, the numbers 0.1 and 0.10000000000000001 and 0.1000000000000000055511151231257827021181583404541015625 are all approximated by 3602879701896397 / 2 ** 55. Since all of these decimal values share the same approximation, any one of them could be displayed while still preserving the invariant eval(repr(x)) == x.

There is no tolerance for precision loss, if float x (0.3) is not exactly equal to float y (0.1*3), then repr(x) is not exactly equal to repr(y).