I have a very large number I need to calculate, and none of the inbuilt datatypes in C# can handle such a large number.
Basicly I want to solve this:
Project Euler 16:
2^15 = 32768 and the sum of its digits
is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the
number 2^1000?
I have already written the code, but, as said before, the number is too large for c# datatypes. The code has been tested and verified with small numbers (such as 2^15) and it works perfectly.
using System;
namespace _16_2E1000
{
class Program
{
static void Main(string[] args)
{
ulong sum = 0;
ulong i = 1 << 1000;
string s = i.ToString();
foreach (char c in s)
{
sum += (ulong) Convert.ToInt64(c.ToString());
}
Console.WriteLine(sum);
Console.ReadLine();
}
}
}
You can use BigInteger from the J# classes. First question in this article tells you how. It's a bit of pain b/c then you have to provide the J# redistributable when you roll out tho.
First to answerer you exact question, look for a BigInt or BigNum type
Second, from what I know of Project Euler, there will be a cool, tricky way to do it that is much easier.
As a first guess I'd compute the answerer for 2^1 -> 2^n (for whatever n you can get to work) and look for patterns. Also look for patterns in the sequences
V(0) = 2^p
V(n) = floor(V(n - 1) / 10)
D(n) = V(n) % 10
I hope this is not a homework problem, but to get to the answer of 2^1000, you'll have to divide it into smaller chunks,
try something like,
2^1000 = 2 * 2^999 = 2^999 + 2^999 = 2^ 998 + 2^ 998 + 2^ 998 + 2^ 998
breaking into smaller bits till you get to solvable a problem,
complete solution to project Euler is on following links.
http://blog.functionalfun.net/2008/07/project-euler-problem-16-calculating.html
http://code.msdn.microsoft.com/projecteuler
It is not necessary to have Big Integer capabilities in order to solve this problem.
One could just use the property that:
2^n = 2^(n-1) + 2^(n-1)
If Big Integer is really necessary for other tasks, I have been using the BigInt class from F# in my C# programs and am happy with it.
The necessary steps:
Install the F# CTP
In your C# (or other .NET language) application add a reference to the FSharp.Core dll.
Add: using Microsoft.FSharp.Math;
In the "Class View" window familiarize yourself with the members of the two classes: BigInt and BigNum
After executing these steps one is basically ready to use the BigInt class.
One last hint:
To avoid declaring variables with improper names to hold constants that makes the code unreadable, I am using a name that starts with _ (underscore), followed by the integer constant. In this way one will have expressions like:
N = _2 * N;
clearly much more readable than:
N = Two * N;
Here's a BigInteger (source code is available) that you can use; though, as already mentioned, there are more efficient ways to do this than brute force.
BigInteger on codeplex
Actually, while a biginteger utility might be of interest here, you don't need it, even for this. Yes, it looks like it does, but you don't. In fact, use of a biginteger form may even slow things down.
Since I don't want to solve the problem for you, I'll just suggest you think about this in a modular way.
Related
I'm trying to solve a simple question on leetcode.com (https://leetcode.com/problems/number-of-1-bits/) and I encounter a strange behavior which is probably my lack of understanding...
My solution to the question in the link is the following:
public int HammingWeight(uint n) {
int sum = 0;
while (n > 0) {
uint t = n % 10;
sum += t == 0 ? 0 : 1;
n /= 10;
}
return sum;
}
My solution was to isolate each number and if it's one increase the sum. When I ran this on my PC it worked (yes - I know it's not the optimal solution and there are more elegant solutions considering it's binary representation).
But when I tried running in the leetcode editor it returned a wrong answer for the following input (00000000000000000000000000001011).
No real easy way to debug other then printing to the console so I printed the value of n when entering the method and got the result of 11 instead of 1011 - on my PC I got 11. If I take a different solution - one that uses bitwise right shift or calculating mod by 2 then it works even when the printed n is still 11. And I would have expected those solutions to fail as well considering that n is "wrong" (different from my PC and the site as described).
Am I missing some knowledge regarding the representation of uint? Or binary number in a uint variable?
Your code appears to be processing it as base 10 (decimal), but hamming weight is about base 2 (i.e. binary). So: instead if doing % 10 and /= 10, you should be looking at % 2 and /= 2.
As for what uint looks like as binary: essentially like this, but ... the CPU is allowed to lie about where each of the octets actually is (aka "endianness"). The good news is: it doesn't usually expose that lie to you unless you cheat and look under the covers by looking at raw memory. As long as you use regular operators (include bitwise operators): the lie will remain undiscovered.
Side note: for binary work that is about checking a bit and shuffling the data down, & 1 and >> 1 would usually be preferable to % 2 and / 2. But as canton7 notes: there are also inbuilt operations for this specific scenario which uses the CPU intrinsic instruction when possible (however: using the built-in function doesn't help you increase your understanding!).
This Kata has a poor writing, in the examples the Inputs are printed in binary representation while the Outputs are in printed in decimal representation. And there is no clues to help understand that.
00000000000000000000000000001011b is 11 (in decimal, 8 + 2 + 1). That is why you get 11 as input for the first test case.
There is no numbers made of 0s and 1s in base 10 you have to decode as base 2 stuff here.
To solve the Kata, you just need to work in base 2 as you succeed to do and like #MarcGravell explained.
Please check below code, it will work for you.
Its very simple way to solve.
var result = 0;
for(var i = 0; i < 32; i++)
{
if ((n & 1) == 1) result++;
n = n >> 1;
}
return result;
I'm trying to compute the cosine of 4203708359 radians in C#:
var x = (double)4203708359;
var c = Math.Cos(x);
(4203708359 can be exactly represented in double precision.)
I'm getting
c = -0.57977754519440394
Windows' calculator gives
c = -0.579777545198813380788467070278
PHP's cos(double) function (which internally just uses cos(double) from the C standard library) on Linux gives:
c = -0.57977754519881
C's cos(double) function in a simple C program compiled with Visual Studio 2017 gives
c = -0.57977754519881342
Here is the definition of Math.cos() in C#: https://github.com/dotnet/coreclr/blob/master/src/mscorlib/src/System/Math.cs#L57-L58
It appears to be a built-in function. I didn't dig (yet) in the C# compiler to check what this effectively compiles to but this is probably the next step.
In the meantime:
Why is the precision so poor in my C# example, and what can I do about it?
Is it simply that the cosine implementation in the C# compiler deals poorly with large integer inputs?
Edit 1: Wolfram Mathematica 11.0:
In[1] := N[Cos[4203708359], 50]
Out[1] := -0.57977754519881338078846707027800171954257546099993
Edit 2: I do need that level precision, and I'm ready to go pretty far in order to obtain it. I'd be happy to use an arbitrary precision library if there exists a good one that supports cosine (my efforts haven't led to one so far).
Edit 3: I posted the question on coreclr's issue tracker: https://github.com/dotnet/coreclr/issues/12737
I think I might know the answer. I'm pretty sure the sin/cos libraries don't take arbitrarily large numbers and calculate the sin/cos of them - they instead reduce them down to low numbers (between 0-2xpi?) and calculate them there. I mean, cos(x) = cos(x + 2xpi) = cos(x + 4xpi) = ...
Problem is, how is the program supposed to reduce your 10-digit number down? Realistically, it should figure out how many times it needs to multiply (2xpi) to get a value just below your number, then subtract that out. In your case, that's about 670 million.
So it's multiplying (2xpi) by this 9 digit value - so it's effectively losing 9 digits worth of significance from the math library's version of pi.
I ended up writing a little function to test what was going on:
private double reduceDown(double start)
{
decimal startDec = (decimal)start;
decimal pi = decimal.Parse("3.1415926535897932384626433832795");
decimal tau = pi * 2;
int num = (int)(startDec / tau);
decimal x = startDec - (num * tau);
double retVal;
double.TryParse(x.ToString(), out retVal);
return retVal;
//return start - (num * tau);
}
All this is doing is using decimal data type as a way of reducing down the value without losing digits of precision from pi - it still returns back a double. When I call it with a modification of your code:
var x = (double)4203708359;
var c = Math.Cos(x);
double y = reduceDown(x);
double c2 = Math.Cos(y);
MessageBox.Show(c.ToString() + Environment.NewLine + c2);
return;
... sure enough, the second one is accurate.
So my advice is - if you really need radians that high, and you really need the accuracy? Do something like that function above, and reduce the number down on your end in a way that you don't lose digits of precision.
Presumably, the salts are stored along with each password. You could use the PHP code to calculate that cosine, and store that also with the password. I would then also add a password version number and default all those older passwords to be version 1. Then, in your C# code, for any new passwords, you implement a new hashing algorithm, and store those password hashes as passwords version 2. For any version 1 passwords, to authenticate, you do not have to calculate the cosine, you simply use the one stored along with the password hash and the salt.
The programmer of that PHP code was probably wanting to do a clever version of pepper. By storing that cosine, or pepper along with the salt and the password hashes, you basically change that pepper into a salt2. So, another versionless way of doing this would be to use two salts in your C# hashing code. For new passwords you could leave the second salt blank or assign it some other way. For old passwords, it would be that cosine, but it is already calculated.
Regarding this part of my question: "Why is the precision so poor in my C# example", coreclr developers answered here: https://github.com/dotnet/coreclr/issues/12737
In a nutshell, .NET Framework 4.6.2 (x86 and x64) and .NET Core (x86) appear to use Intel's x87 FP unit (i.e. fcos or fsincos) that gives inaccurate results while .NET Core on x64 (and PHP, Visual Studio 2017 and gcc) use more accurate, presumably SSE2-based implementations that give correctly rounded results.
The Cobra programming language has a useful feature where you can use underscores in numeric literals to improve readability. For example, the following are equivalent, but the second line is easier to read:
x = 1000000
x = 1_000_000 # obviously 1 million
Is there anything equivalent for C#?
Answer as of C# 7
Yes, this is supported in C# 7. But be aware that there's no validation that you've put the underscores in the right place:
// At a glance, this may look like a billion, but we accidentally missed a 0.
int x = 1_00_000_000;
Answer from 2011
No, there's nothing like that in C#. You could do:
const int x = 1000 * 1000;
but that's about as nice as it gets.
(Note that this enhancement went into Java 7 as well... maybe one day it will be introduced in C#.)
Yes you can do this with C # 7.0 as shown here
public const long BillionsAndBillions = 100_000_000_000;
I'm currently writing a quick custom encoding method where I take a stamp a key with a number to verify that it is a valid key.
Basically I was taking whatever number that comes out of the encoding and multiplying it by a key.
I would then multiply those numbers to the deploy to the user/customer who purchases the key. I wanted to simply use (Code % Key == 0) to verify that the key is valid, but for large values the mod function does not seem to function as expected.
Number = 468721387;
Key = 12345678;
Code = Number * Key;
Using the numbers above:
Code % Key == 11418772
And for smaller numbers it would correctly return 0. Is there a reliable way to check divisibility for a long in .NET?
Thanks!
EDIT:
Ok, tell me if I'm special and missing something...
long a = DateTime.Now.Ticks;
long b = 12345;
long c = a * b;
long d = c % b;
d == 10001 (Bad)
and
long a = DateTime.Now.Ticks;
long b = 12;
long c = a * b;
long d = c % b;
d == 0 (Good)
What am I doing wrong?
As others have said, your problem is integer overflow. You can make this more obvious by checking "Check for arithmetic overflow/underflow" in the "Advanced Build Settings" dialog. When you do so, you'll get an OverflowException when you perform *DateTime.Now.Ticks * 12345*.
One simple solution is just to change "long" to "decimal" (or "double") in your code.
In .NET 4.0, there is a new BigInteger class.
Finally, you say you're "... writing a quick custom encoding method ...", so a simple homebrew solution may be satisfactory for your needs. However, if this is production code, you might consider more robust solutions involving cryptography or something from a third-party who specializes in software licensing.
The answers that say that integer overflow is the likely culprit are almost certainly correct; you can verify that by putting a "checked" block around the multiplication and seeing if it throws an exception.
But there is a much larger problem here that everyone seems to be ignoring.
The best thing to do is to take a large step back and reconsider the wisdom of this entire scheme. It appears that you are attempting to design a crypto-based security system but you are clearly not an expert on cryptographic arithmetic. That is a huge red warning flag. If you need a crypto-based security system DO NOT ATTEMPT TO ROLL YOUR OWN. There are plenty of off-the-shelf crypto systems that are built by experts, heavily tested, and readily available. Use one of them.
If you are in fact hell-bent on rolling your own crypto, getting the math right in 64 bits is the least of your worries. 64 bit integers are way too small for this crypto application. You need to be using a much larger integer size; otherwise, finding a key that matches the code is trivial.
Again, I cannot emphasize strongly enough how difficult it is to construct correct crypto-based security code that actually protects real users from real threats.
Integer Overflow...see my comment.
The value of the multiplication you're doing overflows the int data type and causes it to wrap (int values fall between +/-2147483647).
Pick a more appropriate data type to hold a value as large as 5786683315615386 (the result of your multiplication).
UPDATE
Your new example changes things a little.
You're using long, but now you're using System.DateTime.Ticks which on Mono (not sure about the MS platform) is returning 633909674610619350.
When you multiply that by a large number, you are now overflowing a long just like you were overflowing an int previously. At that point, you'll probably need to use a double to work with the values you want (decimal may work as well, depending on how large your multiplier gets).
Apparently, your Code fails to fit in the int data type. Try using long instead:
long code = (long)number * key;
The (long) cast is necessary. Without the cast, the multiplication will be done in 32-bit integer form (assuming number and key variables are typed int) and the result will be casted to long which is not what you want. By casting one of the operands to long, you tell the compiler to perform the multiplication on two long numbers.
I'm trying to compute 100! and there doesn't seem to be a built-in factorial function. So, I've written:
Protected Sub ComputeFactorial(ByVal n As ULong)
Dim factorial As ULong = 1
Dim i As Integer
For i = 1 To n
factorial = factorial * i
Next
lblAnswer.Text = factorial
End Sub
Unfortunately, running this with the value of 100 for n rseults in
Value was either too large or too
small for a UInt64.
So, is there a larger data type for holding numbers? Am i mistaken in my methods? Am I helpless?
Sounds like Project Euler.
.NET 4.0 has System.Numerics.BigInteger, or you can pick up a pretty sweet implementation here:
C# BigInteger Class
Edit: treed :(
I'll add - the version at CodeProject has additional features like integer square root, a primality test, Lucas sequence generation. Also, you don't have direct access to the buffer in the .NET implementation which was annoying for a couple things I was trying.
Until you can use System.Numerics.BigInteger you are going to be stuck using a non-Microsoft implementation like BigInteger on Code Project.
Hint: use an array to store the digits of the number. You can tell by inspection that the result will not have more than 200 digits.
You need an implementation of "BigNums". These are integers that dynamically allocate memory so that they can hold their value.
A version was actually cut from the BCL.
The J# library has an implementation of java.math.BigInteger that you can use from any language.
Alternatively, if precision/accuracy are not a concern (you only care about order of magnitude), you can just use 64-bit floats.
decimal will handle 0 through +/-79,228,162,514,264,337,593,543,950,335 with no decimal point(scale of zero)