Chapter 5: Numbers & Bits (continued)¶
The float Type¶
In [1]:
b = 42.0
In [2]:
id(b)
Out[2]:
139923238853936
In [3]:
type(b)
Out[3]:
float
In [4]:
b
Out[4]:
42.0
In [6]:
42.
Out[6]:
42.0
In [7]:
float(42)
Out[7]:
42.0
In [8]:
float("42")
Out[8]:
42.0
In [11]:
1 / 3
Out[11]:
0.3333333333333333
In [12]:
40.0 + 2
Out[12]:
42.0
Scientific Notation¶
In [14]:
1.23e0
Out[14]:
1.23
In [19]:
1e0
Out[19]:
1.0
In [20]:
1e3 # = thousands
Out[20]:
1000.0
In [21]:
1e-3 # = milli
Out[21]:
0.001
Special Values¶
In [22]:
float("nan") # also float("NaN")
Out[22]:
nan
In [24]:
float("inf") # also float("+inf")
Out[24]:
inf
In [25]:
float("-inf")
Out[25]:
-inf
In [26]:
float("nan") == float("nan")
Out[26]:
False
In [27]:
42 + float("nan")
Out[27]:
nan
In [28]:
float("inf") == float("inf")
Out[28]:
True
In [30]:
float("inf") + 42 == float("inf")
Out[30]:
True
In [37]:
float("inf") + float("-inf")
Out[37]:
nan
In [38]:
float("inf") - float("inf")
Out[38]:
nan
Imprecision¶
In [39]:
1e15 + 1
Out[39]:
1000000000000001.0
In [40]:
1e16 + 1
Out[40]:
1e+16
In [41]:
from math import sqrt
In [42]:
sqrt(2) ** 2
Out[42]:
2.0000000000000004
In [43]:
0.1 + 0.2
Out[43]:
0.30000000000000004
In [44]:
sqrt(2) ** 2 == 2
Out[44]:
False
In [45]:
0.1 + 0.2 == 0.3
Out[45]:
False
In [46]:
threshold = 1e-15
In [47]:
abs((sqrt(2) ** 2) - 2) < threshold
Out[47]:
True
In [48]:
abs((0.1 + 0.2) - 0.3) < threshold
Out[48]:
True
In [49]:
format(0.1, ".50f")
Out[49]:
'0.10000000000000000555111512312578270211815834045410'
In [50]:
format(0.2, ".50f")
Out[50]:
'0.20000000000000001110223024625156540423631668090820'
In [51]:
format(0.3, ".50f")
Out[51]:
'0.29999999999999998889776975374843459576368331909180'
In [52]:
format(1 / 3, ".50f")
Out[52]:
'0.33333333333333331482961625624739099293947219848633'
In [53]:
roughly_a_third = round(1 / 3, 5)
In [54]:
roughly_a_third
Out[54]:
0.33333
In [55]:
format(roughly_a_third, ".50f")
Out[55]:
'0.33333000000000001517008740847813896834850311279297'
In [56]:
format(0.125, ".50f")
Out[56]:
'0.12500000000000000000000000000000000000000000000000'
In [57]:
format(0.25, ".50f")
Out[57]:
'0.25000000000000000000000000000000000000000000000000'
In [58]:
0.125 + 0.125 == 0.25
Out[58]:
True
Binary Representations¶
$$float = (-1)^{sign} * 1.fraction * 2^{exponent-1023}$$
In [59]:
one_eighth = 1 / 8
In [60]:
one_eighth.hex()
Out[60]:
'0x1.0000000000000p-3'
In [61]:
one_eighth.as_integer_ratio()
Out[61]:
(1, 8)
In [62]:
roughly_a_third.hex()
Out[62]:
'0x1.555475a31a4bep-2'
In [63]:
roughly_a_third.as_integer_ratio()
Out[63]:
(3002369727582815, 9007199254740992)