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Optimise modulo code with various divisors aside from 2

I know that we can optimise "find even numbers" code by using bitwise operator &. The following program:
if(i%2==0) sout("even")
else sout("odd")
can be optimised to:
if(i&1==0) sout("even")
else sout("odd")
The above approach works only for 2 as a divisor. What if we have to optimise the code when we have multiple divisors like 4, 9, 20, 56 and so on? Is there a way to further optimise this code?

You obviously didn't even try what you posted because it doesn't compile (even with a reasonable sout added). First expression statements in Java end in semicolon, and second i&1==0 parses as i & (1==0) -> i & true and the & operator doesn't take an int and a boolean.
If i is negative and odd, i%2 is -1 while i&1 is 1 = +1. That's because % is remainder not modulo.
In the limited cases where i%n and (i&(n-1)) are the same -- i nonnegative and n a power of two -- as the commenters said the Java runtime compiler (JIT) will actually produce the same code for both and obfuscating the source will only make your program more likely to be or become wrong without providing any benefit.
Fifty years ago when people were writing in assembler for machines with microsecond clocks (i.e. not even megaHertz much less gigaHertz) this sometimes helped -- only sometimes because usually only a small fraction of the code matters to execution time. In this century it's at best a waste and often harmful.

Java getChars method in Integer class, why is it using bitwise operations instead of arithmetic?

So I was examining the Integer's class source code (JDK 8) to understand how an int get converted to a String. It seems to be using a package private method called getChars (line 433) to convert an int to char array.
While the code is not that difficult to understand, however, there are multiple lines of code where they use bitwise shift operations instead of simple arithmetic multiplication/division, such as the following lines of code:
// really: r = i - (q * 100);
r = i - ((q << 6) + (q << 5) + (q << 2));
and
q = (i * 52429) >>> (16+3);
r = i - ((q << 3) + (q << 1)); // r = i-(q*10) ...
I just do not understand the point of doing that, is this actually an optimization and does it affect the runtime of the algorithm?
Edit:
To put it in a another way, since the compiler does this type of optimization internally, is this manual optimization necessary?

I don't know the reason for this specific change and unless you find the original author, it's unlikely you'll find an authoritative answer to that anyway.
But I'd like to respond to the wider point, which is that a lot of code in the runtime library (java.* and many internal packages) is optimized to a degree that would be very unusual (and I dare say irresponsible) to apply to "normal" application code.
And that has basically two reasons:
It's called a lot and in many different environment. Optimizing a method in your server to take 0.1% less CPU time when it's only executed 50 times per day on 3 servers each won't be worth the effort you put into it. If, however, you can make Integer.toString 0.1% faster for everyone who will ever execute it, then this can turn into a very big change indeed.
If you optimize your application code on a specific VM then updating that VM to a newer version can easily undo your optimization, when the compiler decides to optimize differently. With code in java.* this is far less of an issue, because it is always shipped with the runtime that will run it. So if they introduce a compiler change that makes a given optimization no longer optimal, then they can change the code to match this.
tl;dr java.* code is often optimized to an insane degree because it's worth it and they can know that it will actually work.

There are a couple reasons that this is done. Being a long-time embedded developer, using tiny microcontrollers that sometimes didn't even have a multiplication and division instruction, I can tell you that this is significantly faster. The key here is that the multiplier is a constant. If you were multiplying two variables, you'd either need to use the slower multiply and divide operators or, if they didn't exist, perform multiplication using a loop with the add operator.

What Crossover Method should I use for crossing Postfix expressions in Genetic Algorithm?

I'm building a project whose main objective is to find a given number (if possible, otherwise closest possible) using 6 given numbers and main operators (+, -, *, /). Idea is to randomly generate expressions, using the numbers given and the operators, in reverse polish (postfix) notation, because I found it the easiest to generate and compute later. Those expressions are Individuals in Population of my Genetic Algorithm. Those expressions have the form of an ArrayList of Strings in Java, where Strings are both the operators and operands (the numbers given).
The main question here is, what would be the best method to crossover these individuals (postfix expressions actually)? Right now I'm thinking about crossing expressions that are made out of all the six operands that are given (and 5 operators with them). Later I'll probably also cross the expressions that would be made out of less operands (5, 4, 3, 2 and also only 1), but I guess that I should figure this out first, as the most complex case (if you think it might be a better idea to start differently, I'm open to any suggestions). So, the thing is that every expression is made from all the operands given, and also the child expression should have all the operands included, too. I understand that this requires some sort of ordered crossover (often used in problems like TSP), and I read a lot about it (for example here where multiple methods are described), but I didn't quite figure out which one would be best in my case (I'm also aware that in Genetic Algorithms there is a lot of 'trial and error' process, but I'm talking about something else here).
What I'm saying is bothering me, are operators. If I had only a list of operands, then it wouldn't be a problem to cross 2 such lists, for example taking a random subarray of half elements from 1 parent, and fill the rest with remaining elements from parent 2 keeping the order like it was. But here, if I, say, take first half of an expression from first parent expression, I would definitely have to fill the child expression with remaining operands, but what should I do with operators? Take them from parent 2 like the remaining operands (but then I would have to watch out because in order to use an operator in postfix expression, I need to have at least 1 operand more, and checking that all the time might be time consuming, or not?), or maybe I could generate random operators for the rest of the child expression (but that wouldn't be a pure crossover then, would it)?
When talking about crossover, there is also mutation, but I guess I have that worked out. I can take an expression and perform a mutation where I'll just switch 2 operands, or take an expression and randomly change 1 or more operators. For that, I have some ideas, but the crossover is what really bothers me.
So, that pretty much sums my problem. Like I said, the main question is how to crossover, but if you have any other suggestions or questions about the program (maybe easier representation of expressions - other then list of strings - which may be easier/faster to crossover, maybe something I didn't mention here, it doesn't matter, maybe even a whole new approach to the problem?), I'd love to hear them. I didn't give any code here because I don't think it's needed to answer this question, but if you think it would help, I'll definitely edit in order to solve this. One more time, main question is to answer how to crossover, this specific part of the problem (idea or pseudocode expected, although the code itself would be great, too :D), but if you think that I need to change something more, or you know some other solutions to my whole problem, feel free to say.
Thanks in advance!

There are two approaches that come to mind:
Approach #1
Encode each genome as a fixed length expression where odd indices are numbers and even indices are the operators. For mutation, you could slightly change the numbers and/or change the operators.
Pros:
Very simple to code
Cons:
Would have to create an infix parser
Fixed length expressions
Approach #2
Encode each genome as a syntax tree. For instance, 4 + 3 / 2 - 1 is equivalent to Add(4, Subtract(Divide(3, 2), 1)) which looks like:
_____+_____
| |
4 ____-____
| |
__/__ 1
| |
3 2
Then when crossing over, pick a random node from each tree and swap them. For mutation, you could add, remove, and/or modify random nodes.
Pros:
Might find better results
Variable length expressions
Cons:
Adds time complexity
Adds programming complexity
Here is an example of the second approach:
Source

Quickly square a double

I am looking for the fastest way to square a double (double d). So far I came up with two approaches:
1. d*d
2. Math.pow(d, 2)
To test the performance I set up three test cases, in each I generate random numbers using the same seed for the three cases and just calculate the squared number in a loop 100 000 000 times.
In the first test case numbers are generated using random.nextDouble(), in the second case using random.nextDouble()*Double.MAX_VALUE and in the third one using random.nextDouble()*Double.MIN_VALUE.
The results of a couple of runs (approximate results, theres always some variation, run using java 1.8, compiled for java 1.6 on Mac OSX Mavericks)
Approach | Case 1 | Case 2 | Case 3
---------•--------•--------•-------
1 | ~2.16s | ~2.16s | ~2.16s
2 | ~9s | ~30s | ~60s
The conclusion seems to be that approach 1 is way faster but also that Math.pow seems to behave kind of weird.
So I have two questions:
Why is Math.pow so slow, and why does it cope badly with > 1 and even worse with < -1 numbers?
Is there a way to improve the performance over what I suggested as approach 1? I was thinking about something like:
long l = Double.doubleToRawLongBits(d);
long sign = (l & (1 << 63));
Double.longBitsToDouble((l<<1)&sign);
But that is a) wrong, and b) about the same speed as approach 1.

The fastest way to square a number is to multiply it by itself.
Why is Math.pow so slow?
It's really not, but it is performing exponentiation instead of simple multiplication.
and why does it cope badly with > 1 and even worse with < -1 numbers
First, because it does the math. From the Javadoc it also contains tests for many corner cases. Finally, I would not rely too much on your micro-benchmark.

Squaring by multipling with self is the fastest. Because that approch can be directly translated into simple, non-branching bytecode (and thus, indirectly, machine code).
Math.pow() is a quite complex function that comes with various guarantees for edge cases. And it need to be called instead of being inlined.

Math.pow() is slow because it has to deal with the generic case or raising a number to any given power.
As for why it is slower with negative numbers, it is because it has to test if the power is positive or negative in order to give the sign, so it is one more operation to do.

How to find function range?

I have an arbitrary function or inequality (consisting of a number of trigonometrical, logarithmical, exponential, and arithmetic terms) that takes several arguments and I want to get its range knowing the domains of all the arguments. Are there any Java libraries that can help to solve a problem? What are the best practices to do that? Am I right that for an arbitrary function the only thing can be done is a brute-force approximation? Also, I'm interested in functions that can build intersections and complements for given domains.
Upd. The functions are entered by the user so the complexity cannot be predicted. However, if the library will treat at least simple cases (1-2 variables, 1-2 terms) it will be OK. I suggest the functions will mostly define the intervals and contain at most 2 independent variables. For instance, definitions like
y > (x+3), x ∈ [-7;8]
y <= 2x, x ∈ [-∞; ∞]
y = x, x ∈ {1,2,3}
will be treated in 99% of cases and covering them will be enough for now.
Well, maybe it's faster to write a simple brute-force for treating such cases. Probably it will be satisfactory for my case but if there are better options I would like to learn them.

Notational remark: I assume you want to find the range of the function, i.e. the set of values that the function can take.
I think this problem is not simple. I don't think that "brute force" is a solution at all, what does "brute force" even mean when we have continuous intervals (i.e infinitely many points!).
However, there might be some special cases where this is actually possible. For example, when you take a sin(F(x)) function, you know that its range is [-1,1], regardless of the inner function F(x) or when you take Exp(x) you know the range is (0,+inf).
You could try constructing a syntax tree with information about the ranges associated to each node. Then you could try going bottom-up through the tree to try to compute the information about the actual intervals in which the function values lie.
For example, for the function Sin(x)+Exp(x) and x in (-inf, +inf) you would get a tree
+ range: [left range] union [right range]
/ \
sin exp range [-1, 1] , range: (0,+inf)
| |
x x
so here the result would be [-1, 1] union (0, +inf) = [-1, +inf).
Of course there are many problems with this approach, for example the operation on ranges for + is not always union. Say you have two functions F(x) = Sin(x) and G(x) = 1-Sin(x). Both have ranges [-1,1], but their sum collapses to {1}. You need to detect and take care of such behaviour, otherwise you will get only an upper bound on the possible range (So sort of codomain).
If you provide more examples, maybe someone can propose a better solution, I guess a lot depends on the details of the functions.
#High Performance Mark: I had a look at JAS and it seems that its main purpose is to deal with multivariate polynomial rings, but the question mentioned trigonometric, logarithmic and other transcendental functions so pure polynomial arithmetic will not be sufficient.

Here's another approach and depending on how crazy your function can be (see EDIT) it might give you the universal solution to your problem.
Compose the final expression, which might be rather complex.
After that use numerical methods to find minimum and maximum of the function - this should give you the resulting range.
EDIT: Only in the case that your final expression is not continuous the above would not work and you would have to divide into continuous sections for each you would have to find min and max. At the end you would have to union those.

I would have thought that this is a natural problem to tackle with a Computer Algebra System. I googled around and JAS seems to be the most-cited Java CAS.
If I had to confine myelf to numeric approaches, then I'd probably tackle it with some variety of interval computations. So: the codomain of sin is [-1,1], the codomain of exp is (0,+Inf), and the codomain of exp(sin(expression)) is ...
over to you, this is where I'd reach for Mathematica (which is callable from Java).