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Python function doing with `&` operator

I got a code like this:
def f(x):
return 1 if x & (x-1) == 0 else 0
print(f(20))
Can anyone explain what this function is doing?
i need to understand x & x-1 meaning.

First up, go and have a look at this answer to see how bitwise operators work. Once, you understand that, just be aware that the expression x & x-1 (with unsigned x, at least) will give you zero if and only if x is a power of two or zero (I consider zero to be a special case simply because zero and'ed with anything gives you zero).
The reason x & (x - 1) gives you zero for a power of two is because a power of two takes the binary form on the first line below (a single 1 bit followed by zero or more 0 bits) and subtracting one from it gives you the second form (with the bits all inverted from that single 1 bit onward):
10000000 = 128 (128)
01111111 = 127 (64 + 32 + 16 + 8 + 4 + 2 + 1)
-------- AND
00000000
So, if you and them, all bits become zero. Note that no non-power-of-two other than zero has this property, for example 42:
00101010 = 42 (32 + 8 + 2)
00101001 = 41 (32 + 8 + 1)
-------- AND
00101000 = 40 (32 + 8)

Python round to next highest power of 10

How would I manage to perform math.ceil such that a number is assigned to the next highest power of 10?
# 0.04 -> 0.1
# 0.7 -> 1
# 1.1 -> 10
# 90 -> 100
# ...
My current solution is a dictionary that checks the range of the input number, but it's hardcoded and I would prefer a one-liner solution. Maybe I am missing a simple mathematical trick or a corresponding numpy function here?

You can use math.ceil with math.log10 to do this:
>>> 10 ** math.ceil(math.log10(0.04))
0.1
>>> 10 ** math.ceil(math.log10(0.7))
1
>>> 10 ** math.ceil(math.log10(1.1))
10
>>> 10 ** math.ceil(math.log10(90))
100
log10(n) gives you the solution x that satisfies 10 ** x == n, so if you round up x it gives you the exponent for the next highest power of 10.
Note that for a value n where x is already an integer, the "next highest power of 10" will be n:
>>> 10 ** math.ceil(math.log10(0.1))
0.1
>>> 10 ** math.ceil(math.log10(1))
1
>>> 10 ** math.ceil(math.log10(10))
10

Your problem is under-specified, you need to step back and ask some questions.
What type(s) are your inputs?
What type(s) do you want for your outputs?
For results less than 1, what exactly do you want to round to? Do you want actual powers of 10 or floating point approximations of powers of 10? You are aware that negative powers of 10 can't be expressed exactly in floating point right? Let's assume for now that you want floating point approximations of powers of 10.
If the input is exactly a power of 10 (or the closest floating point approximation of a power of 10), should the output be the same as the input? Or should it be the next power of 10 up? "10 -> 10" or "10 -> 100"? Let's assume the former for now.
Can your input values be any possible value of the types in question? or are they more constrained.
In another answer it was proposed to take the logarithm, then round up (ceiling function), then exponentiate.
def nextpow10(n):
return 10 ** math.ceil(math.log10(n))
Unfortunately this suffers from rounding errors. First of all n is converted from whatever data type it happens to have into a double precision floating point number, potentially introducing rounding errors, then the logarithm is calculated potentially introducing more rounding errors both in its internal calculations and in its result.
As such it did not take me long to find an example where it gave an incorrect result.
>>> import math
>>> from numpy import nextafter
>>> n = 1
>>> while (10 ** math.ceil(math.log10(nextafter(n,math.inf)))) > n:
... n *= 10
...
>>> n
10
>>> nextafter(n,math.inf)
10.000000000000002
>>> 10 ** math.ceil(math.log10(10.000000000000002))
10
It is also theoretically possible for it to fail in the other direction, though this seems to be much harder to provoke.
So for a robust solution for floats and ints we need to assume that the value of our logarithm is only approximate, and we must therefore test a couple of possibilities. Something along the lines of
def nextpow10(n):
p = round(math.log10(n))
r = 10 ** p
if r < n:
r = 10 ** (p+1)
return r;
I believe this code should give correct results for all arguments in a sensible real-world range of magnitudes. It will break for very small or very large numbers of non integer and non-floating point types because of issues converting them to floating point. Python special cases integer arguments to the log10 function in an attempt to prevent overflow, but still with a sufficiently massive integer it may be possible to force incorrect results due to rounding errors.
To test the two implementations I used the following test program.
n = -323 # 10**-324 == 0
while n < 1000:
v = 10 ** n
if v != nextpow10(v): print(str(v)+" bad")
try:
v = min(nextafter(v,math.inf),v+1)
except:
v += 1
if v > nextpow10(v): print(str(v)+" bad")
n += 1
This finds lots of failures in the naive implementation, but none in the improved implementation.

It seems you want rather the lowest next power of 10...
Here is a way using pure maths and no log, but recursion.
def ceiling10(x):
if (x > 10):
return ceiling10(x / 10) * 10
else:
if (x <= 1):
return ceiling10(10 * x) / 10
else:
return 10
for x in [1 / 1235, 0.5, 1, 3, 10, 125, 12345]:
print(x, ceiling10(x))

Check this out!
>>> i = 0.04123
>>> print i, 10 ** len(str(int(i))) if int(i) > 1 else 10 if i > 1.0 else 1 if i > 0.1 else 10 ** (1 - min([("%.100f" % i).replace('.', '').index(k) for k in [str(j) for j in xrange(1, 10) if str(j) in "%.100f" % i]]))
0.04123 0.1
>>> i = 0.712
>>> print i, 10 ** len(str(int(i))) if int(i) > 1 else 10 if i > 1.0 else 1 if i > 0.1 else 10 ** (1 - min([("%.100f" % i).replace('.', '').index(k) for k in [str(j) for j in xrange(1, 10) if str(j) in "%.100f" % i]]))
0.712 1
>>> i = 1.1
>>> print i, 10 ** len(str(int(i))) if int(i) > 1 else 10 if i > 1.0 else 1 if i > 0.1 else 10 ** (1 - min([("%.100f" % i).replace('.', '').index(k) for k in [str(j) for j in xrange(1, 10) if str(j) in "%.100f" % i]]))
1.1 10
>>> i = 90
>>> print i, 10 ** len(str(int(i))) if int(i) > 1 else 10 if i > 1.0 else 1 if i > 0.1 else 10 ** (1 - min([("%.100f" % i).replace('.', '').index(k) for k in [str(j) for j in xrange(1, 10) if str(j) in "%.100f" % i]]))
90 100
This code based on principle of ten's power in len(str(int(float_number))).
There are 4 cases:
int(i) > 1.
Float number - converted to int, thereafter string str() from it, will give us a string with length which is we are looking exactly. So, first part, for input i > 1.0 - it is ten 10 in power of this length.
& 3. Little branching: i > 1.0 and i > 0.1 <=> it is 10 and 1 respectively.
And last case, when i < 0.1: Here, ten shall be in negative power. To get first non zero element after comma, I've used such construction ("%.100f" % i).replace('.', '').index(k), where k run over [1:10] interval. Thereafter, take minimum of result list. And decrease by one, it is first zero, which shall be counted. Also, here standard python's index() may crash, if it will not find at least one of non-zero element from [1:10] interval, that is why in the end I must "filter" listing by occurrence: if str(j) in "%.100f" % i.
Additionally, to get deeper precise - %.100f may be taken differ.

I think the simplest way is:
import math
number = int(input('Enter a number: '))
next_pow_ten = round(10 ** math.ceil(math.log10(number)))
print(str(10) + ' power ' + str(round(math.log10(number))) + ' = '\
+ str(next_pow_ten))
I hope this help you.

a specific shortcut works for big-integers that are already coming in as string-format :
instead of having to first convert it to integer, or running it through the log()/ceiling() function, or perform any sort of modulo math, the next largest power-of-10 is simply :
10 ** length(big_int_str_var)
—- below : 1st one generates a string formatted power-of-10, the 2nd one is numeric
echo 23958699683561808518065081866850688652086158016508618152865101851111111111111 |
tee >( gpaste | gcat -n >&2; ) | gcat - |
python3 -c '\
import sys; [ print("1"+"0"*len(_.strip("\n"))) for _ in sys.stdin ]'
or '... [ print( 10 ** len(_.strip("\n"))) for _ in sys.stdin ]'
1 23958699683561808518065081866850688652086158016508618152865101851111111111111
1 100000000000000000000000000000000000000000000000000000000000000000000000000000

y = math.ceil(x)
z = y + (10 - (y % 10))
Something like this maybe? It's just off the top of my head but it worked when I tried a few numbers in terminal.

bitwise operation and Two's complement in python

I have an array of (short) data which is shifted 4 to left and it is also a signed number. I need to plot it around zero.
for instance: if the number in the array is 0b0000000011111111 and I shift it to left by 4, I will get 0b000000001111. it is fine.
for instance: if the number in the array is 0b100011111111 and I shift it to left by 4, I will get 0b000010001111. It is fine, but now it is not a negative number.
can someone help ?

You have to write your own implementation of the arithmetic right shift of a 16-bit value, if this is what you need. I suggest you this one, very easy to understand:
def arithmetic_right_shift_on_16_bits(val, n):
# Get the sign bit
s = val & 0x8000
# Perform the shifts, padding with the sign bit
for _ in range(n):
val >>= 1
val |= s
return val
a = arithmetic_right_shift_on_16_bits(0b0000000011111111, 4)
print(bin(a)) # 0b1111
b = arithmetic_right_shift_on_16_bits(0b1000000011111111, 4)
print(bin(b)) # 0b1111100000001111

This does not happen in Python:
>>> bin(0b0000000011111111 >> 4) # 15
'0b1111'
>>> bin(0b100011111111 >> 4) # 143
'0b10001111'

BTW, the number you show as an example of a negative short integer is not one, because you gave only 12 bits. So I will assume that the number of interest is 0b1000000011111111. As Python does not use 2's complement, it is shown as 33023. But you can do the 2's complement by hand: for a short int (16 bits), you just substract 0x10000 to the number.
You can then write:
def short_signed_right_shift(x, i):
if x < 0 or x >= 0x10000: raise ValueError # only accept 16 bits numbers
return x >> i if (x < 0x7FFF) else 0x10000 + ((x - 0x10000) >> i)
You can then use it:
>>> bin(short_signed_right_shift(0b0000000011111111, 4))
'0b1111'
>>> bin(short_signed_right_shift(0b1000000011111111, 4))
'0b1111100000001111'

Trouble with signs on zero-exponents equations

Python seems to have trouble returning the correct value for numbers to the power of zero.
When I give it a literal equation, it works properly, but it always returns positive 1 for anything more complex than a raw number to the zeroeth.
Here are some tests:
>>> -40 ** 0 # this is the correct result
-1
>>> (0 - 40) ** 0 # you'd expect this to give the same thing, but...
1
>>> a = -40 # let's try something else...
>>> a ** 0
1
>>> int(-40) ** 0 # this oughtn't to change anything, yet...
1
>>> -66.6 ** 0 # raw floats are fine.
-1.0
>>> (0 - 66.6) ** 0.0 # ...until you try and do something with them.
1.0
UPDATE: pow() gives this result, too, so probably the first result is exceptional...
>>> pow(-60, 0)
1
Could it be some problem with signed integers? I need this for a trinary switch with values 1, -1, or 0, depending on whether an input is any positive or negative value, or zero. I could accomplish the same thing with something like:
if val > 0: switch = 1
elif val < 0: switch = -1
else: switch = 0
...and then using the variable switch for my purposes.
But that wouldn't answer the question I have about how Python deals with zero-powers.
(I will also accept that -40 ** 0 only returns -1 by accident (phenomenally), but I doubt this is the case...)

Python is correct and doing what you would expect it to do. It is a matter of order of operations. ANY number (negative or positive) to zeroth power is equal to 1. But keep in mind also that multiplication comes before subtraction in order of operations. So in more detail, what python sees is this:
1st case:
-40 ** 0 = -(40 ** 0) = -(1) = -1
2nd case:
(0 - 40) ** 0 = (-40) ** 0 = 1
In the 5th case as well it has to do with the parentheses
int(-40) ** 0 = (-40) ** 0 = 1

Just stumpled upon this question, I don't get the syntax.
But I don't think that (-40)^0 = -40^0.
On the left side, the exponential is the last operation. This is why the left side should equal 1.
On the right side, the minus sign is the last operation. This is why the result should be -1.

There is not any problem , here . every number with power 0 is 1 .
In python signs like -.+,... have less precedences to power (**) so when you put 0 - 40 inside the parenthesize you have (-1)**0 that is 1 but when you do -1**0 first you have 1**0 then -.
>>> (0-4)**0 == (-1)**0 == 1
>>> -1**0 == -(1**0) == -1

Generating digits of square root of 2

I want to generate the digits of the square root of two to 3 million digits.
I am aware of Newton-Raphson but I don't have much clue how to implement it in C or C++ due to lack of biginteger support. Can somebody point me in the right direction?
Also, if anybody knows how to do it in python (I'm a beginner), I would also appreciate it.

You could try using the mapping:
a/b -> (a+2b)/(a+b) starting with a= 1, b= 1. This converges to sqrt(2) (in fact gives the continued fraction representations of it).
Now the key point: This can be represented as a matrix multiplication (similar to fibonacci)
If a_n and b_n are the nth numbers in the steps then
[1 2] [a_n b_n]T = [a_(n+1) b_(n+1)]T
[1 1]
which now gives us
[1 2]n [a_1 b_1]T = [a_(n+1) b_(n+1)]T
[1 1]
Thus if the 2x2 matrix is A, we need to compute An which can be done by repeated squaring and only uses integer arithmetic (so you don't have to worry about precision issues).
Also note that the a/b you get will always be in reduced form (as gcd(a,b) = gcd(a+2b, a+b)), so if you are thinking of using a fraction class to represent the intermediate results, don't!
Since the nth denominators is like (1+sqrt(2))^n, to get 3 million digits you would likely need to compute till the 3671656th term.
Note, even though you are looking for the ~3.6 millionth term, repeated squaring will allow you to compute the nth term in O(Log n) multiplications and additions.
Also, this can easily be made parallel, unlike the iterative ones like Newton-Raphson etc.

EDIT: I like this version better than the previous. It's a general solution that accepts both integers and decimal fractions; with n = 2 and precision = 100000, it takes about two minutes. Thanks to Paul McGuire for his suggestions & other suggestions welcome!
def sqrt_list(n, precision):
ndigits = [] # break n into list of digits
n_int = int(n)
n_fraction = n - n_int
while n_int: # generate list of digits of integral part
ndigits.append(n_int % 10)
n_int /= 10
if len(ndigits) % 2: ndigits.append(0) # ndigits will be processed in groups of 2
decimal_point_index = len(ndigits) / 2 # remember decimal point position
while n_fraction: # insert digits from fractional part
n_fraction *= 10
ndigits.insert(0, int(n_fraction))
n_fraction -= int(n_fraction)
if len(ndigits) % 2: ndigits.insert(0, 0) # ndigits will be processed in groups of 2
rootlist = []
root = carry = 0 # the algorithm
while root == 0 or (len(rootlist) < precision and (ndigits or carry != 0)):
carry = carry * 100
if ndigits: carry += ndigits.pop() * 10 + ndigits.pop()
x = 9
while (20 * root + x) * x > carry:
x -= 1
carry -= (20 * root + x) * x
root = root * 10 + x
rootlist.append(x)
return rootlist, decimal_point_index

As for arbitrary big numbers you could have a look at The GNU Multiple Precision Arithmetic Library (for C/C++).

For work? Use a library!
For fun? Good for you :)
Write a program to imitate what you would do with pencil and paper. Start with 1 digit, then 2 digits, then 3, ..., ...
Don't worry about Newton or anybody else. Just do it your way.

Here is a short version for calculating the square root of an integer a to digits of precision. It works by finding the integer square root of a after multiplying by 10 raised to the 2 x digits.
def sqroot(a, digits):
a = a * (10**(2*digits))
x_prev = 0
x_next = 1 * (10**digits)
while x_prev != x_next:
x_prev = x_next
x_next = (x_prev + (a // x_prev)) >> 1
return x_next
Just a few caveats.
You'll need to convert the result to a string and add the decimal point at the correct location (if you want the decimal point printed).
Converting a very large integer to a string isn't very fast.
Dividing very large integers isn't very fast (in Python) either.
Depending on the performance of your system, it may take an hour or longer to calculate the square root of 2 to 3 million decimal places.
I haven't proven the loop will always terminate. It may oscillate between two values differing in the last digit. Or it may not.

The nicest way is probably using the continued fraction expansion [1; 2, 2, ...] the square root of two.
def root_two_cf_expansion():
yield 1
while True:
yield 2
def z(a,b,c,d, contfrac):
for x in contfrac:
while a > 0 and b > 0 and c > 0 and d > 0:
t = a // c
t2 = b // d
if not t == t2:
break
yield t
a = (10 * (a - c*t))
b = (10 * (b - d*t))
# continue with same fraction, don't pull new x
a, b = x*a+b, a
c, d = x*c+d, c
for digit in rdigits(a, c):
yield digit
def rdigits(p, q):
while p > 0:
if p > q:
d = p // q
p = p - q * d
else:
d = (10 * p) // q
p = 10 * p - q * d
yield d
def decimal(contfrac):
return z(1,0,0,1,contfrac)
decimal((root_two_cf_expansion()) returns an iterator of all the decimal digits. t1 and t2 in the algorithm are minimum and maximum values of the next digit. When they are equal, we output that digit.
Note that this does not handle certain exceptional cases such as negative numbers in the continued fraction.
(This code is an adaptation of Haskell code for handling continued fractions that has been floating around.)

Well, the following is the code that I wrote. It generated a million digits after the decimal for the square root of 2 in about 60800 seconds for me, but my laptop was sleeping when it was running the program, it should be faster that. You can try to generate 3 million digits, but it might take a couple days to get it.
def sqrt(number,digits_after_decimal=20):
import time
start=time.time()
original_number=number
number=str(number)
list=[]
for a in range(len(number)):
if number[a]=='.':
decimal_point_locaiton=a
break
if a==len(number)-1:
number+='.'
decimal_point_locaiton=a+1
if decimal_point_locaiton/2!=round(decimal_point_locaiton/2):
number='0'+number
decimal_point_locaiton+=1
if len(number)/2!=round(len(number)/2):
number+='0'
number=number[:decimal_point_locaiton]+number[decimal_point_locaiton+1:]
decimal_point_ans=int((decimal_point_locaiton-2)/2)+1
for a in range(0,len(number),2):
if number[a]!='0':
list.append(eval(number[a:a+2]))
else:
try:
list.append(eval(number[a+1]))
except IndexError:
pass
p=0
c=list[0]
x=0
ans=''
for a in range(len(list)):
while c>=(20*p+x)*(x):
x+=1
y=(20*p+x-1)*(x-1)
p=p*10+x-1
ans+=str(x-1)
c-=y
try:
c=c*100+list[a+1]
except IndexError:
c=c*100
while c!=0:
x=0
while c>=(20*p+x)*(x):
x+=1
y=(20*p+x-1)*(x-1)
p=p*10+x-1
ans+=str(x-1)
c-=y
c=c*100
if len(ans)-decimal_point_ans>=digits_after_decimal:
break
ans=ans[:decimal_point_ans]+'.'+ans[decimal_point_ans:]
total=time.time()-start
return ans,total

Python already supports big integers out of the box, and if that's the only thing holding you back in C/C++ you can always write a quick container class yourself.
The only problem you've mentioned is a lack of big integers. If you don't want to use a library for that, then are you looking for help writing such a class?

Here's a more efficient integer square root function (in Python 3.x) that should terminate in all cases. It starts with a number much closer to the square root, so it takes fewer steps. Note that int.bit_length requires Python 3.1+. Error checking left out for brevity.
def isqrt(n):
x = (n >> n.bit_length() // 2) + 1
result = (x + n // x) // 2
while abs(result - x) > 1:
x = result
result = (x + n // x) // 2
while result * result > n:
result -= 1
return result