0

I'm trying to do a project that uses the trigonometric functions, but I'm having some trouble with them.

When I type in this code:

```
import math
print 0.5+math.cos((2*math.pi)/3)
```

I get the answer 2.22044604925e-016. That's obviously not right. cos((2*pi)/3) is -0.5, so I should get the answer 0. Can anybody help me?

This Question has been **Answered**

0

Floating point algorithms in all computer languages suffer from a small error as the floating point world meets the binary world. Representing -0.5 in just a sequence of 1 and 0 is not possible with true accuracy. Look at this ...

```
import math
# use a list to show the small binary error
mylist = [math.cos((2*math.pi)/3)]
print mylist # [-0.49999999999999978]
# you could use round() to 15 digits
mylist = [round(math.cos((2*math.pi)/3), 15)]
print mylist # [-0.5]
```

*Edited
by vegaseat*: round()

0

Floating point algorithms in all computer languages suffer from a small error as the floating point world meets the binary world. Representing -0.5 in just a sequence of 1 and 0 is not possible with true accuracy. Look at this ...

`import math # use a list to show the small binary error mylist = [math.cos((2*math.pi)/3)] print mylist # [-0.49999999999999978] # you could use round() to 15 digits mylist = [round(math.cos((2*math.pi)/3), 15)] print mylist # [-0.5]`

I don't exactly agree that -0.5 is not representable in binary because 0.5 is 2**(-1). On the other hand the binary 2*math.pi/3 is not exactly equal to the mathematical 2*pi/3 .

0

In fact you can obtain the binary representation of floating points numbers using the module *bitstring* available in pypi

```
>>> from bitstring import BitString
>>> bs = BitString(float=-0.5, length=64)
>>> bs.bin
'0b1011111111100000000000000000000000000000000000000000000000000000'
```

This representation of -0.5 is the same as the IEEE 754 representation of the

floating point number on 64 bits. To understand its meaning, go to this online

application http://babbage.cs.qc.edu/IEEE-754/Decimal.html and enter -0.5 in the first field, then click 'not rounded' and read the representation and its meaning below.

Also note that some native support exist in the python interpreter: sys.float_info contains constants read in your system's float.h header, and the float class has a method hex() which returns a hexadecimal representation of the number.

A nice exercise would be to write a class which displays the same information as the page http://babbage.cs.qc.edu/IEEE-754/Decimal.html about a floating point number and its binary representation :)

A more subtle question is how to make sure that you obtain the binary representation actually used by your processor ? Apparently, some processors use 80 bits and not 64 ...

*Edited
by Gribouillis*: n/a

0

Wit this:

```
import math
print(math.cos((2*math.pi)/3)) # -0.5
print(0.5 + math.cos((2*math.pi)/3)) # 2.22044604925e-16
print(-0.5 + math.cos((2*math.pi)/3)) # -1.0
x = math.cos((2*math.pi)/3)
print(x) # -0.5
print([x]) # [-0.4999999999999998]
print([-0.5 + x]) # [-0.9999999999999998]
```

Strange?

1

Wit this:

`import math print(math.cos((2*math.pi)/3)) # -0.5 print(0.5 + math.cos((2*math.pi)/3)) # 2.22044604925e-16 print(-0.5 + math.cos((2*math.pi)/3)) # -1.0 x = math.cos((2*math.pi)/3) print(x) # -0.5 print([x]) # [-0.4999999999999998] print([-0.5 + x]) # [-0.9999999999999998]`

Strange?

It is not strange, because printing a float x prints str(x) in fact, while printing a list invokes the repr() of the list items. The method float.__str__() yields less decimals than float.__repr__, so the floating number is rounded

```
>>> x = 1.0/3
>>> str(x)
'0.333333333333'
>>> repr(x)
'0.33333333333333331'
>>>
>>> x = -0.49999999999999
>>> str(x)
'-0.5'
>>> repr(x)
'-0.49999999999999001'
```

About the actual bits used by C python, the PyFloatObject structure from floatobject.h uses a C double to store the number, so the bits are the same as the bits of the C type double. I wrote a small functions to extract the actual bits using module ctypes

```
from ctypes import *
from binascii import hexlify
def double_to_bits(f):
cf = c_double(f)
n = sizeof(cf)
assert not n % 2
pc = cast(pointer(cf), POINTER(c_char))
h = ''.join(pc[i] for i in xrange(n))
return bin(int(b"1" + hexlify(h), 16))[3:]
print double_to_bits(-0.5)
""" my output (this should be machine dependent) -->
0000000000000000000000000000000000000000000000001110000010111111
"""
```

Interestingly, this yields the same result as using struct.pack() :

```
>>> from binascii import hexlify
>>> import struct
>>> p = struct.pack("d", -0.5)
>>> print bin(int(b"1" + hexlify(p), 16))[3:]
0000000000000000000000000000000000000000000000001110000010111111
```

Also notice that on my machine, this representation differs from the IEEE 754 representation referred to above. How is it coded ? It would be worth having a python module which recognizes the different floating point formats ...

*Edited
by Gribouillis*: n/a

0

To complete the previous post, the difference with the IEEE 754 representation is due to the endianness of the system. One can recover the standard representation by reverting the bytes:

```
from ctypes import *
from binascii import hexlify
import sys
def double_to_bytes(f):
cf = c_double(f)
n = sizeof(cf)
assert not n % 2
pc = cast(pointer(cf), POINTER(c_char))
h = ''.join(pc[i] for i in xrange(n))
s = bin(int(b"1" + hexlify(h), 16))[3:]
return [s[i:i+8] for i in xrange(0, len(s), 8)]
L = double_to_bytes(-0.5)
print "This system uses %s endian byteorder" % sys.byteorder
print " ".join(L)
print " ".join(reversed(L))
""" my output -->
This system uses little endian byteorder
00000000 00000000 00000000 00000000 00000000 00000000 11100000 10111111
10111111 11100000 00000000 00000000 00000000 00000000 00000000 00000000
"""
```

*Edited
by Gribouillis*: n/a

0

Sorry I haven't replied to this sooner. Thank you for all the help with this. My program is working excellently now.

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