Maths trick: doing fewer comparisons

Note: this is not an optimisation. It is just one more tool you should have in your toolbox when looking for optimisations. It may be useful.

This is the trick:

\[\min(x,y) = \dfrac{x + y - |x - y|}{2}\]
\[\max(x,y) = \dfrac{x + y + |x - y|}{2}\]

You can check for yourself that it is always true: when x > y, |x - y| is the same as x - y, etc.

What good is it for? There is often an implicit comparison in min or max. It might be interesting to replace it with a call to the branchless fabs.

Example usage

Consider the following code:

float a, b, c, d;
/* ... */
return (a > b) && (c > d);

That kind of code is often used eg. in collision checks, where a lot of tests can be done. This code does two comparisons. On some architectures, this means two branches. Not always something you want.

The test condition is equivalent to:

(a - b > 0) && (c - d > 0)

Now when are two given numbers both positive? That is if and only if the smallest is positive:

min(a - b, c - d) > 0

We may now use our trick:

(a - b) + (c - d) - |(a - b) - (c + d)| > 0

And so the code could be rewritten as such:

float a, b, c, d;
/* ... */
return (a - b) + (c - d) > fabsf((a - b) - (c - d));

We basically replaced the additional test with a call to fabsf and some additions/subtractions. It may be possible to reorganise the input data so that this second version performs better.

C++ trick: selectively restrict implicit conversions

TL;DR: given a class Foo with an implicit constructor from int, how to allow the implicit conversion in f(42); but not in g(42); where both f and g take a Foo const & argument?


So I have this real class that performs numeric operations that I want use just like any other C++ numeric type. For instance, I can write the following:

float f = 15, g = 3.5;
int x = f / g;

If I decide that I need double precision, I can write:

double f = 15, g = 3.5;
int x = f / g;

And of course, using my real class for even higher precision:

real f = 15, g = 3.5;
int x = f / g;

I like that. I can just write code as usual, and when I need higher precision, I use real instead of double. It's transparent and convenient.

Implementation example

Here is a highly simplified example of a real class:

struct real
    inline real(double d) : m_value(d) {}
    inline operator int() const { return (int)m_value; }
    /* ... */
    long double m_value;

It is possible to write real f = 15 because of the implicit constructor. Actually, C++ constructors are always implicit unless specified otherwise.

It is possible to write int x = f / g because of the conversion operator.

So far, so good.

The problem with implicit promotion

Here is how fabs could be implemented:

real fabs(real const &r)
    return real(r.m_value < 0 ? -r.m_value : r.m_value);

But now we have a problem. A subtle problem. Consider the following code:

double x = fabs(-5.0);

What does this do? Well, it depends. It depends whether <cmath> was included or not. Because if <cmath> wasn’t included, then that code is going to automatically promote -5.0 to a real and call our custom function instead of the one provided by the math library! With no compile-time warning!

This is confusing. It should not happen. But it is a well known problem and there are several obvious workarounds:

  1. What most professional C++ programmers will tell you: use namespaces
  2. Mark the real(int) constructor explicit

The problem with 1. is that I am not a professional C++ programmer. I am a C programmer who uses C++. I use preprocessor macros and printf and memalign and goto. Try and stop me!

The problem with 2. is that I can no longer write real f = 15, I would need real f(15) or real f = real(15) instead. This is not acceptable, I want real to behave exactly like float and others, to the fullest extent of what the language allows.

Another solution

Fortunately, the C++ standard has a solution for us: “Implicit conversions will be performed [...] if the parameter type contains no template-parameters that participate in template argument deduction” (ISO/IEC 14882:1998, section You cannot have implicit conversion and template argument deduction at the same time.

It means we just have to make fabs a template function! Which means making real a template class, too.

A quick way to fix real would be:

/* N is unused */
template<int N> struct real_base
    inline real_base(double d) : m_value(d) {}
    inline operator int() const { return (int)m_value; }
    /* ... */
    long double m_value;
typedef real_base<0> real;

The template argument is useless, unfortunately. It will just have to be here, forever. But who knows, you might find a use for it one day.

And to fix fabs:

/* A generic template declaration is needed */
template<int N> real_base<N> fabs(real_base<N> const &r);
/* Here we just add template<> to the previous version */
real fabs(real const &r)
    return real(r.m_value < 0 ? -r.m_value : r.m_value);

So, what happens with double x = fabs(-5.0); when we forget to include <cmath> now? Well, here is what GCC says:

In function ‘int main()’:
error: no matching function for call to ‘fabs(double)’
note: candidate is:
note: template<int N> real_base<N> fabs(const real_base<N>&)

It seems we’ve successfully managed to avoid the problematic implicit conversion, yet still allow it in places where it was useful!

So what is the rule? It’s simple: where implicit conversion should not be allowed, make the function a specialised template function.