A 32-bit float can exactly represent about 7 decimal digits of mantissa. Your number requires more, and therefore cannot be represented exactly.

The mechanics of what happens are as follows:

A 32-bit float has a 24-bit mantissa. Your number requires 27 bits to be represented exactly, so the last three bits are getting truncated (set to zero). The three lowest bits of your number are 0112; these are getting set to 0002. Observe that 0112 is 310.

Let's see if I can address some of the confusion I'm seeing in the comments.

Make an array:

In [609]: x=np.arange(5)
In [610]: x
Out[610]: array([0, 1, 2, 3, 4])
In [611]: x.dtype
Out[611]: dtype('int32')

The default for arange is to make an int32.

astype is an array method; it can used on any array:

In [612]: x.astype(np.float32)
Out[612]: array([ 0.,  1.,  2.,  3.,  4.], dtype=float32)

arange also takes a dtype parameter

In [614]: np.arange(5, dtype=np.float32)
Out[614]: array([ 0.,  1.,  2.,  3.,  4.], dtype=float32)

whether it created the int array first and converted it, or made the float32 directly isn't any concern to me. This is a basic operation, done in compiled code.

I can also give it a float stop value, in which case it will give me a float array - the default float type.

In [615]: np.arange(5.0)
Out[615]: array([ 0.,  1.,  2.,  3.,  4.])
In [616]: _.dtype
Out[616]: dtype('float64')

zeros is similar; the default dtype is float64, but with a parameter I can change that. Since its primary task with to allocate memory, and it doesn't have to do any calculation, I'm sure it creates the desired dtype right away, without further conversion. But again, this is compiled code, and I shouldn't have to worry about what it is doing under the covers.

In [618]: np.zeros(5)
Out[618]: array([ 0.,  0.,  0.,  0.,  0.])
In [619]: _.dtype
Out[619]: dtype('float64')
In [620]: np.zeros(5,dtype=np.float32)
Out[620]: array([ 0.,  0.,  0.,  0.,  0.], dtype=float32)

randn involves a lot of calculation, and evidently it is compiled to work with the default float type. It does not take a dtype. But since the result is an array, it can be cast with astype.

In [623]: np.random.randn(3)
Out[623]: array([-0.64520949,  0.21554705,  2.16722514])
In [624]: _.dtype
Out[624]: dtype('float64')
In [625]: __.astype(np.float32)
Out[625]: array([-0.64520949,  0.21554704,  2.16722512], dtype=float32)

Let me stress that astype is a method of an array. It takes the values of the array and produces a new array with the desire dtype. It does not act retroactively (or in-place) on the array itself, or on the function that created that array.

The effect of astype is often (always?) the same as a dtype parameter, but the sequence of actions is different.

In https://stackoverflow.com/a/39625960/901925 I describe a sparse matrix creator that takes a dtype parameter, and implements it with an astype method call at the end.

When you do calculations such as dot or *, it tries to match the output dtype with inputs. In the case of mixed types it goes with the higher precision alternative.

In [642]: np.arange(5,dtype=np.float32)*np.arange(5,dtype=np.float64)
Out[642]: array([  0.,   1.,   4.,   9.,  16.])
In [643]: _.dtype
Out[643]: dtype('float64')
In [644]: np.arange(5,dtype=np.float32)*np.arange(5,dtype=np.float32)
Out[644]: array([  0.,   1.,   4.,   9.,  16.], dtype=float32)

There are casting rules. One way to look those up is with can_cast function:

In [649]: np.can_cast(np.float64,np.float32)
Out[649]: False
In [650]: np.can_cast(np.float32,np.float64)
Out[650]: True

It is possible in some calculations that it will cast the 32 to 64, do the calculation, and then cast back to 32. The purpose would be to avoid rounding errors. But I don't know how you find that out from the documentation or tests.

The IEEE 754-2008 16-bit base 2 format, aka binary16, doesn't give you a lot of precision. What do you expect from 16 bits? :) 1 bit is the sign bit, 5 bits are used for the exponent, and that leaves 10 bits to store the normalised 11 bit mantissa, so anything > 2**11 == 2048 has to be quantized.

According to Wikipedia, integers between 4097 and 8192 round to a multiple of 4, and integers between 8193 and 16384 round to a multiple of 8.

1. If the dataframe (say df) wholly consists of float64 dtypes, you can do:

df = df.astype('float32')

2. Only if some columns are float64, then you'd have to select those columns and change their dtype:

# Select columns with 'float64' dtype  
float64_cols = list(df.select_dtypes(include='float64'))

# The same code again calling the columns
df[float64_cols] = df[float64_cols].astype('float32')