I have just spent a while trying to find a bug in my code which has turned out to be an unusual (at least to me) parameterisation for the R::dexp function. For example:
cppFunction("
double my_dexp(double x, double lambda, double is_log) {
return R::dexp(x, lambda, is_log);
}
")
> my_dexp(4.5, 2.5, FALSE)
[1] 0.06611956
> dexp(4.5, 2.5, FALSE)
[1] 3.251824e-05
Looking here I can see that they use the definition:
double R::dexp(double x, double sl, int lg)
but I haven't been able to find out what sl stands for. I'm not sure if this is documented anywhere - so hopefully this post stands as a warning to others who have used the function like me, and also if anyone can help as to what parameterisation has been used, and why.
If you look at the function definition for dexp,
R> dexp
function (x, rate = 1, log = FALSE)
.Call(C_dexp, x, 1/rate, log)
you'll see that dexp calls the C function C_dexp with parameter 1/rate. This is what R::dexp is mirroring. In Rcp, they always use the same parameterisation as R itself does at the C level which may be different than the R level.
That means
R> my_dexp(4.5, 1/2.5, FALSE) - dexp(4.5, 2.5, FALSE)
[1] 0
If you look at the Wikipedia page on the exponential function, you'll see the alternative parameterisation based on the reciprocal of the rate parameter, lambda. In this parameterisation, the parameter beta=1/lambda takes the role of a survival parameter. So the expected duration of survival of the system is beta units of time.
Related
I'm trying to write a function which can take in functions as its arguments in Rcpp. I have written an example function in R that shows the kind of functionality that I'm aiming for:
simulate_and_evaluate <- function(simulate, evaluate) {
y <- simulate(1)
eval <- evaluate(y)
return(eval)
}
simulate_fun <- function(n) rnorm(n, 0, 1)
evaluate_fun <- function(x) dnorm(x, 0, 1)
simulate_and_evaluate(simulate = simulate_fun,
evaluate = evaluate_fun)
In this function simulate_and_evaluate, this takes in two arguments which are both functions, one that simulates a number and one that evaluates a function with this simualted number. So as an example, we can simulate a value from a standard normal and evaluate the density of a standard normal at that point. Does anyone know if there's a way to do this in Rcpp?
Rcpp aims for seamless interfacing of R and C++ objects. As functions are first class R objects represented internally as a type a SEXP can take, we can of course also ship them with Rcpp. There are numerous examples.
So here we simply rewrite your function as a C++ function:
Rcpp::cppFunction("double simAndEval(Function sim, Function eval) {
double y = as<double>(sim(1));
double ev = as<double>(eval(y));
return(ev);
}")
And we can then set the RNG to the same value, run your R function and this C++ function and get the same value. Which is awesome.
R> set.seed(123)
R> simulate_and_evaluate(simulate = simulate_fun,
+ evaluate = evaluate_fun)
[1] 0.341
R> set.seed(123) # reset RNG
R> simAndEval(simulate_fun, evaluate_fun)
[1] 0.341
R>
But as #MrFlick warned you, this will not run any faster because we added no compiled execution of the actual functions we are merely calling them from C++ rathern than R.
The topic has been discussed before. Please search StackOverflow, maybe with a string [rcpp] Function to get some meaningful hits.
Anyone know how to use Boost to solve simple definite integrals?
E.g. -x^2 + 1 from -1 to 1?
I have tried reading the boost documentation, but I can't seem to figure out how to properly pass the function.
Thanks
Edit: My attempt so far
using namespace boost::math;
typename function_type; // this is probably wrong
function_type f // and this
{
return -x*x+1;
};
int main(int, char**)
{
const double val =
integral(0.0,
1,
0.001,
f); // my question is, what do I put in here? How do I format f.
}
The first thing to observe is that the Boost library you've shown doesn't actually have a function to calculate integrals. That might have set you on the wrong track.
The library is used for multi-precision floating point operations, and one of the examples happens to be a simple approximation of integrals, per Riemann. The point of the example is that Riemann integrals are so simple that you can use them to demonstrate a fancy library.
In your case, you wouldn't even need to bother with passing a function. You can just write out the Riemann method substituting -x^2 + 1 directly.
That said, the typical C++ way to pass it as an argument would be [](double x) { return -x*x+1.0;}. That's an unnamed function or lambda. It doesn't need a name of its own, since the parameter already has a name.
My goal is to be able to model signal-dependent Gaussian noise in Halide. I have a model built in OpenCV which I am now porting to Halide. The challenge is that Halide's random number generator is not normally distributed, so I need to use an external function to produce the noise values.
The implementation attempt uses the C++ random number generator to produced normally distributed noise, a Halide Func to produce the signal-depended standard deviations of the noise at each pixel, and then the noise is added to the pixels in renoise. Below I show the layout of the functions.
// Note: This is an implementation of the noise model found in the paper below:
// "Noise measurement for raw-data of digital imaging sensors by
// automatic segmentation of non-uniform targets"
float get_normal_dist_rand( float mean, float std_dev ) {
std::default_random_engine generator;
std::normal_distribution<float> distribution(mean,std_dev);
float out = distribution(generator);
return out;
}
Func make_get_std_dev( Func *in_func ) {
Var x, y, c;
float q = 0.0060;
float p = 0.0500;
// std_dev = q * sqrt(unnoised_pixel - p)
Func get_std_dev("get_std_dev");
get_std_dev(x,y,c) = q * sqrt( (*in_func)(x,y,c) - p );
return get_std_dev;
}
Func make_renoise( Func *in_func, Func *std_dev ) {
Var x, y, c;
// Noise parameters
// noised_pixel = unnoised_pixel +
// gaussian_rand( std_dev = q * sqrt(unnoised_pixel - p) )
// q and p values do not vary between channels
Func renoise("renoise");
renoise(x,y,c) = (*in_func)(x,y,c) +
get_normal_dist_rand(0,(*std_dev)(x,y,c));
return renoise;
}
This makes sense to me, but unfortunately I receive the following error when I try to compile:
../common/pipe_stages.cpp: In function 'Halide::Func make_renoise(Halide::Func*, Halide::Func*)':
../common/pipe_stages.cpp:223:64: error: cannot convert 'std::enable_if<true, Halide::FuncRef>::type {aka Halide::FuncRef}' to 'float' for argument '2' to 'float get_normal_dist_rand(float, float)'
get_normal_dist_rand(0,(*std_dev)(x,y,c));
^
So it seems that the output of a Func cannot be provided to a C++ function. I guess this makes sense as a limitation of Halide, but I don't really see an alternative to implement the signal dependent normally distributed noise. Is there another way to use external C++ functions in Halide? I have seen folks talking about using "extern" but unfortunately documentation on that functionality seems to be quite light, and I am unable to find what I need.
You'll need to use one of our extern mechanisms to bind to C++ code. HalideExtern_* is the easier of the two and will let you make a call to get random numbers one at a time. Alas test/correctness/c_function.cpp is the immediate example for this, which will help, but could be clearer.
I expect you'll want to request a buffer of random numbers at a time for efficiency reasons. This can be done via the define_extern mechanism. The C++ function has to participate in bounds inference so it is a little more involved. The test for this is correctness/extern_producer.cpp.
I'd expect either transforming our random numbers to be appropriately distributed or implementing the random number generation algorithm in Halide is the right way to go for really fast production code, but that is likely more work than you want to do to get this working initially.
You could also use Halide's RNG along with a binomial approximation to the Gaussian:
Expr gaussian_random(Expr sigma) {
return (random_float() + random_float() + random_float() - 1.5f) * 2 * sigma;
}
Add more instances of randomFloat to get closer and closer to a true normal distribution.
I have the following expression:
A = cos(5x),
where x is a letter indicating a generic parameter.
In my program I have to work on A, and after some calculations I must have a result that must still be a function of x , explicitly.
In order to do that, what kind of variable should A (and I guess all the other variables that I use for my calculations) be?
Many thanks to whom will answer
I'm guessing you need precision. In which case, double is probably what you want.
You can also use float if you need to operate on a lot of floating-point numbers (think in the order of thousands or more) and analysis of the algorithm has shown that the reduced range and accuracy don't pose a problem.
If you need more range or accuracy than double, long double can also be used.
To define function A(x) = cos(5 * x)
You may do:
Regular function:
double A(double x) { return std::cos(5 * x); }
Lambda:
auto A = [](double x) { return std::cos(5 * x); };
And then just call it as any callable object.
A(4.); // cos(20.)
It sounds like you're trying to do a symbolic calculation, ie
A = magic(cos(5 x))
B = acos(A)
print B
> 5 x
If so, there isn't a simple datatype that will do this for you, unless you're programming in Mathematica.
The most general answer is "A will be an Expression in some AST representation for which you have a general algebraic solver."
However, if you really want to end up with a C++ function you can call (instead of a symbolic representation you can print as well as evaluating), you can just use function composition. In that case, A would be a
std::function<double (double )>
or something similar.
I've just finished second year at Uni doing a games course, this is always been bugging me how math and game programming are related. Up until now I've been using Vectors, Matrices, and Quaternions in games, I can under stand how these fit into games.
This is a General Question about the relationship between Maths and Programming for Real Time Graphics, I'm curious on how dynamic the maths is. Is it a case where all the formulas and derivatives are predefined(semi defined)?
Is it even feasible to calculate derivatives/integrals in realtime?
These are some of things I don't see how they fit inside programming/maths As an example.
MacLaurin/Talor Series I can see this is useful, but is it the case that you must pass your function and its derivatives, or can you pass it a single function and have it work out the derivatives for you?
MacLaurin(sin(X)); or MacLaurin(sin(x), cos(x), -sin(x));
Derivatives /Integrals This is related to the first point. Calculating the y' of a function done dynamically at run time or is this something that is statically done perhaps with variables inside a set function.
f = derive(x); or f = derivedX;
Bilnear Patches We learned this as a way to possible generate landscapes in small chunks that could be 'sewen' together, is this something that happens in games? I've never heard of this (granted my knowlages is very limited) being used with procedural methods or otherwise. What I've done so far involves arrays for vertex information being processesed.
Sorry if this is off topic, but the community here seems spot on, on this kinda thing.
Thanks.
Skizz's answer is true when taken literally, but only a small change is required to make it possible to compute the derivative of a C++ function. We modify skizz's function f to
template<class Float> f (Float x)
{
return x * x + Float(4.0f) * x + Float(6.0f); // f(x) = x^2 + 4x + 6
}
It is now possible to write a C++ function to compute the derivative of f with respect to x. Here is a complete self-contained program to compute the derivative of f. It is exact (to machine precision) as it's not using an inaccurate method like finite differences. I explain how it works in a paper I wrote. It generalises to higher derivatives. Note that much of the work is done statically by the compiler. If you turn up optimization, and your compiler inlines decently, it should be as fast as anything you could write by hand for simple functions. (Sometimes faster! In particular, it's quite good at amortising the cost of computing f and f' simultaneously because it makes common subexpression elimination easier for the compiler to spot than if you write separate functions for f and f'.)
using namespace std;
template<class Float>
Float f(Float x)
{
return x * x + Float(4.0f) * x + Float(6.0f);
}
struct D
{
D(float x0, float dx0 = 0) : x(x0), dx(dx0) { }
float x, dx;
};
D operator+(const D &a, const D &b)
{
// The rule for the sum of two functions.
return D(a.x+b.x, a.dx+b.dx);
}
D operator*(const D &a, const D &b)
{
// The usual Leibniz product rule.
return D(a.x*b.x, a.x*b.dx+a.dx*b.x);
}
// Here's the function skizz said you couldn't write.
float d(D (*f)(D), float x) {
return f(D(x, 1.0f)).dx;
}
int main()
{
cout << f(0) << endl;
// We can't just take the address of f. We need to say which instance of the
// template we need. In this case, f<D>.
cout << d(&f<D>, 0.0f) << endl;
}
It prints the results 6 and 4 as you should expect. Try other functions f. A nice exercise is to try working out the rules to allow subtraction, division, trig functions etc.
2) Derivatives and integrals are usually not computed on large data sets in real time, its too expensive. Instead they are precomputed. For example (at the top of my head) to render a single scatter media Bo Sun et al. use their "airlight model" which consists of a lot of algebraic shortcuts to get a precomputed lookup table.
3) Streaming large data sets is a big topic, especially in terrain.
A lot of the maths you will encounter in games is to solve very specific problems, and is usually kept simple. Linear algebra is used far more than any calculus. In Graphics (I like this the most) a lot of the algorithms come from research done in academia, and then they are modified for speed by game programmers: although even academic research makes speed their goal these days.
I recommend the two books Real time collision detection and Real time rendering, which contain the guts of most of the maths and concepts used in game engine programming.
I think there's a fundamental problem with your understanding of the C++ language itself. Functions in C++ are not the same as mathmatical functions. So, in C++, you could define a function (which I will now call methods to avoid confusion) to implement a mathmatical function:
float f (float x)
{
return x * x + 4.0f * x + 6.0f; // f(x) = x^2 + 4x + 6
}
In C++, there is no way to do anything with the method f other than to get the value of f(x) for a given x. The mathmatical function f(x) can be transformed quite easily, f'(x) for example, which in the example above is f'(x) = 2x + 4. To do this in C++ you'd need to define a method df (x):
float df (float x)
{
return 2.0f * x + 4.0f; // f'(x) = 2x + 4
}
you can't do this:
get_derivative (f(x));
and have the method get_derivative transform the method f(x) for you.
Also, you would have to ensure that when you wanted the derivative of f that you call the method df. If you called the method for the derivative of g by accident, your results would be wrong.
We can, however, approximate the derivative of f(x) for a given x:
float d (float (*f) (float x), x) // pass a pointer to the method f and the value x
{
const float epsilon = a small value;
float dy = f(x+epsilon/2.0f) - f(x-epsilon/2.0f);
return epsilon / dy;
}
but this is very unstable and quite inaccurate.
Now, in C++ you can create a class to help here:
class Function
{
public:
virtual float f (float x) = 0; // f(x)
virtual float df (float x) = 0; // f'(x)
virtual float ddf (float x) = 0; // f''(x)
// if you wanted further transformations you'd need to add methods for them
};
and create our specific mathmatical function:
class ExampleFunction : Function
{
float f (float x) { return x * x + 4.0f * x + 6.0f; } // f(x) = x^2 + 4x + 6
float df (float x) { return 2.0f * x + 4.0f; } // f'(x) = 2x + 4
float ddf (float x) { return 2.0f; } // f''(x) = 2
};
and pass an instance of this class to a series expansion routine:
float Series (Function &f, float x)
{
return f.f (x) + f.df (x) + f.ddf (x); // series = f(x) + f'(x) + f''(x)
}
but, we're still having to create a method for the function's derivative ourselves, but at least we're not going to accidentally call the wrong one.
Now, as others have stated, games tend to favour speed, so a lot of the maths is simplified: interpolation, pre-computed tables, etc.
Most of the maths in games is designed to to as cheap to calculate as possible, trading speed over accuracy. For example, much of the number crunching uses integers or single-precision floats rather than doubles.
Not sure about your specific examples, but if you can define a cheap (to calculate) formula for a derivative beforehand, then that is preferable to calculating things on the fly.
In games, performance is paramount. You won't find anything that's done dynamically when it could be done statically, unless it leads to a notable increase in visual fidelity.
You might be interested in compile time symbolic differentiation. This can (in principle) be done with c++ templates. No idea as to whether games do this in practice (symbolic differentiation might be too expensive to program right and such extensive template use might be too expensive in compile time, I have no idea).
However, I thought that you might find the discussion of this topic interesting. Googling "c++ template symbolic derivative" gives a few articles.
There's many great answers if you are interested in symbolic calculation and computation of derivatives.
However, just as a sanity check, this kind of symbolic (analytical) calculus isn't practical to do at real time in the context of games.
In my experience (which is more 3D geometry in computer vision than games), most of the calculus and math in 3D geometry comes in by way of computing things offline ahead of time and then coding to implement this math. It's very seldom that you'll need to symbolically compute things on the fly and then get on-the-fly analytical formulae this way.
Can any game programmers verify?
1), 2)
MacLaurin/Taylor series (1) are constructed from derivatives (2) in any case.
Yes, you are unlikely to need to symbolically compute any of these at run-time - but for sure user207442's answer is great if you need it.
What you do find is that you need to perform a mathematical calculation and that you need to do it in reasonable time, or sometimes very fast. To do this, even if you re-use other's solutions, you will need to understand basic analysis.
If you do have to solve the problem yourself, the upside is that you often only need an approximate answer. This means that, for example, a series type expansion may well allow you to reduce a complex function to a simple linear or quadratic, which will be very fast.
For integrals, the you can often compute the result numerically, but it will always be much slower than an analytic solution. The difference may well be the difference between being practical or not.
In short: Yes, you need to learn the maths, but in order to write the program rather than have the program do it for you.