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Custom implementation of Math.Exp with Big Decimal in Java

I have an Algorithmic implementation which deal in extremely small and extremely large values.
I am using
BigDecimalMath result = BigDecimalMath.exp(a)
//where a is any bigdecimal value
BigDecimalMath library can be found here
https://arxiv.org/src/0908.3030v2/anc
According to my best knowledge this function calculate only upto E9 (i.e -3.44E9) but my smallest value is -3.47E14 (for which it give overflow error)
I am implementing this Algorithm in JAVA as it already implemented in other programming languages so I have to find the solution for this problem.
Can anyone help in this with or without using this library.

How are keywords represented in binary form?

How are keywords represented in binary form?
For ex:: In java, how is the sin() represented in binary? How is sqrt() and other functions represented.
If not only in java, in any language how is it represented?? because ultimately everything is translated into binary and then into on and off signals.
Thanks in advance.

Firstly, sin is not a keyword in Java. It is an identifier. Keywords are things like if, class, and so on.
It depends on when you are asking about.
In the source code, the sin identifier is represented as characters, and those characters are represented as bits (i.e. binary) .... if you want to look at it that way.
In the classfile that is output by the javac compiler, the word sin is represented as string in the Constant Pool. (The JVM spec specifies the format of classfiles in great detail.)
When the classfile is first loaded by a JVM, the word sin becomes a Java String object.
When the code is linked by the JVM, the reference to the String is resolved to some kind of reference to a method. (The details are implementation specific. You'd need to read the JVM source code to find out more.)
When the code is JIT compiler, the reference to the method (typically) turns into the address in memory of the first native instruction of the JIT compiled method. (Strictly speaking, this is not "assembly language". But the native instructions could be represented as assembly language. Assembly language is really just a "human friendly" textual representation of the instructions.)
so how does the computer know that when sin is written it has to do the sine of a number.
What happens is that the Java runtime loads that class containing the method. Then it looks for the sin(double) method in the class that it loaded. What typically happens is that the named method resolves to some bytecodes that are the instructions that tell the runtime what the method should do. But in the case of sin, the method is a native method, and the instructions are actually native instructions that are part of one of the JVM's native libraries.
If not of methods, Can we have binary representation of Keywords?? Like int, and float etc??
It depends on the actual keywords. But generally speaking, genuine Java keywords are transformed by the compiler into a form that doesn't have a distinct / discrete representation for the individual keywords.

If not only in java, in any language how is it represented?? because ultimately everything is translated into binary and then into on and off signals.
This tells me that you probably have a fundamental misunderstanding of how programming languages are implemented. So instead of answering this question (it doesn't really have a proper answer other than "well they're not represented at all"), I will try to help you understand why this question is the wrong one to ask.
Your computer runs machine code, and only machine code. You can feed it any random sequence of bytes, it doesn't matter what they were intended to be, as soon as you point the program counter to it it will be interpreted as if it is machine code (of course giving it bytes that were not intended to be machine code is probably a bad idea). As a running example, I'll use this x64 code:
48 01 F7 48 89 F8 C3
If you have no idea what's going on, that's normal at this level. Most people don't read machine code (but they could if they learned it, it's not magic). This is where the zeroes and ones are, to the processor it's not even in hexadecimal, that's just what humans like to read.
At a level above that there is assembly, which is in most cases really just a different way of looking at machine code, in such a way that humans find it easier to read. The example from earlier looks more sensible in assembly:
add rdi, rsi
mov rax, rdi
ret
Still not very clear what's going on to someone who doesn't know x64 assembly, but at least it gives some sort of clue: there's an add in it. It probably adds things.
At a yet higher level, you could have java bytecode or java, but I think the java aspect of this question misses the point, it's probably there because OP doesn't realize that java is different from "the classic picture". Java just complicates matters without explaining the big picture. Let's use C instead. The example in C could look like:
int64_t foo_or_whatever(int64_t x, int64_t y)
{
return x + y;
}
If you don't know C but you do know Java, the only strange thing here is int64_t, which is roughly the equivalent of a long in Java.
So yes, things were added, as the assembly code suggested. Now where did the keywords go?
That question doesn't make as much sense as you thought it did. The compiler understands keywords, and uses them to create machine code that implements your program. After that point they stop being relevant. They only mean something in the context of the high level language that you wrote the code in, you could say that at that level, they are stored as ASCII or UTF8 string in a file. They have nothing to do with machine code, they do not appear in any form there, and you can write machine code without having translated it from a high level language that has keywords. That return and ret looks vaguely similar is a bit of a red herring, they have something to do with each other but the relation is far from simple (that it worked out simply in the example I'm using is of course no accident).
The int64_t has perhaps not entirely disappeared (mostly it has, though). The fact that the addition operates on 64bit integers is encoded in the instruction 48 01 F7. Not the keyword int64_t (which isn't even a keyword, but let's not get into that), "the fact that what you have there is an addition between 64bit integers", which is an conceptually different thing though caused here by the use of int64_t. To split that instruction out while skipping some of the detail (because this is a beginner question), there's
48 = 01001000 encoding REX.W, meaning this instruction is 64bit
01 = 00000001 encoding add rm64, r64 in this case
D1 = 11010001 encoding the operands rdi and rsi
To learn more about what the processor does with machine code (in case your follow-up question is "but how does it know what to do with something like 48 01 F7"), study computer architecture. If you want a book, I recommend Computer Architecture, Fifth Edition: A Quantitative Approach, which is quite accessible to beginners and commonly used in first-year courses about computer architectures.
To learn more about the journey from high level language to machine code, study compiler construction. If you want a book, I recommend Compilers: Principles, Techniques, and Tools, but it may be hard to get through it as a beginner. If you want a free course, you could follow Compilers on Coursera (the first few lectures especially will give you an overview of what compilers do without getting too technical yet).
Incidentally, if you give the example C code to GCC, it makes
lea rax, [rdi + rsi]
ret
It's still doing the same thing, but in a way that didn't fit my story, so I took the liberty of doing it in a slightly different way.

sin() is a function so it's represented as a memory address where its code block is.
Keywords (like for) aren't represented as binary, for for example is converted to a list of byte code jump instructions which are compiled into assembly instructions which are represented as binary.
My point is that you cannot convert most keywords directly into binary. You can unroll them into bytecode which you could then convert to native machine code and binary but not directly to binary.
Here, read this then after you understand it move onto how bytecode is converted to native code.
Keywords and Functions
That said, a keyword in Java (and most languages) is a reserved word like for, while or return but your examples are not keywords, they are function names like sin() and sqrt()

Not really sure what you want to know here; so let's go "bytecode"...
Both the .sin() and .sqrt() methods are static methods from the Math class; therefore, the compiler will generate a call site with both arguments, a reference to the method and then call invokestatic.
Other than invokestatic, you have invokevirtual, invokespecial, invokeinterface and (since Java 7) invokedynamic.
Now, at runtime, the JIT will kick in; and the JIT may end up producing pure native code, but this is not a guarantee. In any event, the code will be fast enough.
And the same goes for the JDK libraries themselves; the JIT will kick in and maybe turn the byte code into native code given a sufficient time to analyze it (escape analysis, inlining etc).
And since the JIT does "whatever it wants", you reliably cannot have a "binary" representation of any method from any class.

How to make a simple public-key cryptographic algorithm? [closed]

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I want to make a simple public-key(asymmetric) encryption. It doesn't have the be secure, I just want to understand the concepts behind them. For instance, I know simple symmetric ciphers can be made with an XOR. I saw in a thread on stackexchange that you need to use trapdoor functions, but I can't find much about them. I want to say, take a group of bytes, and be able to split them someway to get a public/private key. I get the ideas of a shared secret. Say, I generate the random number of 256(not random at all :P), and I split it into 200 and 56. If I do an XOR with 200, I can only decrypt with 200. I want to be able to split numbers random and such to be able to do it asymmetrically.

OK, just a simple demo-idea, based on adding/modulo operation.
Lets say we have a modulo value, for our example 256. This is a public-known, common value.
Let's say you generate a random secret private key in the interval [1-255], for example, pri=133.
Keep secret key in the pocket.
Generate a public key, pub = 256 - pri = 123. This public key (123)
you can share to the world.
Imagine, 3rd party does not know, how to compute the private key from a public. So, they know only public key (123).
Someone from the public wants to send you an encrypted ASCII-byte. He gets his byte, and adds to it the public key by modulo 256 operation:
encrypted = (input_value + pub) % modulto;
For example, I want to send you the letter "X", ASCII code = 88 in encrypted form.
So, I compute:
(88 + 123) % 256 = 211;
I am sending you the value 211 - encrypted byte.
You decrypt it by the same scheme with your private key:
decrypted = (input_value + pri) % 256 = (211 + 133) % 256 = 88;
Of course, using the simple generation pair in this example is weak, because of
the well-known algorithm for generating the private key from the public, and anybody can easily recover the private using the modulo and public.
But, in real cryptography, this algorithm is not known. But, theoretically,
it can be discovered in future.

This is an area of pure mathematics, there's a book called "the mathematics of cyphers" it's quite short but a good introduction. I do suggest you stay away from implementing your own though, especially in Java (you want a compiler that targets a real machine for the kind of maths involved, and optimises accordingly). You should ask about this on the math or computer-science stack-exchanges.
I did get a downvote, so I want to clarify. I'm not being heartless but cyphers are firmly in the domain of mathematics, not programming (even if it is discreet maths, or the mathsy side of comp-sci) it requires a good understanding of algebraic structures, some statistics, it's certainly a fascinating area and I encourage you to read. I do mean the above though, don't use anything you make, the people who "invent" these cyphers have forgotten more than you or I know, implement exactly what they say at most. In Java you ought to expect a really poor throughput btw. Optimisations involving register pressure and allocation pay huge dividends in cypher throughput. Java is stack-based for starters.
Addendum (circa 6 years on)
Java has improved in some areas now (I have a compiler fetish, it's proper weird) however looking back I was right but for the sort-of wrong reasons, Java is much easier to attack through timing, I've seen some great use of relying on tracing compiling techniques to work out what version of software is being used for example. It's also really hard to deal with Spectre which isn't going away any time soon (I like caches.... I feel dirty saying that now)
HOWEVER: above all, don't do this yourself! Toy with it AT MOST - it's very much in the domain of mathematics, and I must say it's probably better done on paper, unless you like admiring a terminal with digits spewn all over it.

http://en.wikipedia.org/wiki/RSA_(algorithm)
Is the standard one on which the (whole) internet is based

Using int flags in lieu of booleans

So, for example, Notification has the following flag:
public static final int FLAG_AUTO_CANCEL = 0x00000010;
This is hexadecimal for the number 16. There are other flags with values:
0x00000020
0x00000040
0x00000080
Each time, it goes up by a power of 2. Converting this to binary, we get:
00010000
00100000
01000000
10000000
Hence, we can use a bitwise operators to determine which of the flags are present, etc, since each flag contains only one 1 and they are all in different locations.
Question:
This all makes perfect sense, but why not just use booleans? Is this merely stylistic, or are there memory or efficiency benefits?
EDIT:
I understand that by combining them, we can store a lot of information in a single int. Is this used solely so we can pass a lot of boolean type values in a single int instead of having to pass a ton of parameters? I don't mean to trivialize that, it's very convenient, but are there any other benefits?

What you're talking about is called a Bit Field. One advantage is that all the information can be contained in a single variable (with no overhead like that of an ArrayList). This is useful for keeping function signatures tidy, and will have some minor benefits with efficiency because of fewer stack operations, but probably this will be offset by additional bitshift operations. Additionally, you can use (for example) one byte to store 8 fields rather than wasting 7 additional bytes. You can also, if you're clever with it, perform several flag checks in a single operation.
Having said that, personal preference may see the list of booleans as cleaner or preferable. Bitfields are most common in embedded systems where space is limited or something of that nature.
In reference to your edit: it's storing the values of the flags in ints, but those are just reference constants-- you aren't editing those, you're sticking those bits into (or out of) the flags field, which is a single int. I don't really know why they chose a bitfield for this application; perhaps someone that grew up programming space-limited microcontrollers coded that specific class. The general consensus seems to be that bitfields shouldn't be included in new code.

This is a common idiom in C, where resource constraints are a much larger concern, and you usually see it in Java where the Java API is directly mapping an underlying well-known C API. However, it's not a great idea in Java for a wide number of reasons.
As of Java 5, most of the uses for one-bit bit fields are taken care of very nicely by EnumSet, which is internally implemented using a bit field (so it's extremely fast) but is type-safe, easy to read, and Iterable.

Random Number Generation within range with different distribution in Java

I want to generate Random number in different range. For example range 10^14 in Java with different distribution like log, normal, binomial etc. Is there any particular library for the same. I found discussion on colt and math uncommon library. But is it safe enough to generate values as int and then multiply by the corresponding range suffix. What is best practice for the same.

Apache Commons Math has a RandomDataImpl class that does nextBinomial, nextExponential and some other types (above my head unfortunately).
Hopefully that gets you everything you need. You might need to check some of the other classes in the library.