/ C++, OPTIMIZATION

Basic Floating Point Optimizations

Ever seen some people write f * 0.5 when they mean f / 2?

Or if the compiler is able to optimize the f * 1.0 that you added for clarity?

Maybe you wrote f + f instead of f * 2 as a clever optimization?

Modern compilers are basically magic, but do they actually perform these optimizations? And, more importantly, why is f + 0.0 slower than f - 0.0?

Preliminaries

Many posts have been written about elaborate magic involving IEEE 754 floating point numbers. Some of my favorites include the fast inverse square root (commonly attributed to John Carmack though the method is much older), basically everything from the Random ASCII blog, and the reverse-Z depth test for rendering.

In this post I want to take a look at less flashy, more foundational things. We’re going over a couple small optimizations that we probably expected from our compilers. Some of them are performed but others are not actually legal or only work due to slightly arcane rules.

While this might apply to other languages as well, I’m most familiar with C++ which I’ll be using for this post. We will assume that our C++ implementation uses:

  • Two’s complement for integers
  • IEEE 754 for floating points

(This is currently implementation-defined behavior, but at least the first might change)

As always, it is highly instructive to look at the generated assembly for which we will use the excellent page by Matt Godbolt.

The Curios Case of f + 0.0

Most programmers are taught that floating point math is not associative (for example -1e9f + 1e9f + 1 == 1 vs. -1e9f + (1e9f + 1) == 0). The same goes for integers if overflows might be involved.

Sometimes one might come across trivial computations like i + 0, f * 1.0, i * 1, etc.. We might want to spell them out for clarity or consistency or because it’s part of a generic, templated algorithm that happens to be instantiated for these values. It is very tempting to think “my compiler is smart and will do the-right-thing (tm)”.

Does it though?

As any good lawyer will tell you: it depends.

Integers are easier to reason about (for humans and computers alike) and have less surprises than floats, so i + 0, i - 0, i * 1, and i / 1 will all be optimized to just i. Sometimes, the compiler can do a strength reduction and for example replace i * 2 with i + i, which is semantically identical but faster. i * -1 can be replaced with -i but only because signed integer overflows are undefined behavior. std::numeric_limits<int>::min() * -1 is not representable but because it’s undefined behavior the compiler is free to assume that this case will not happen.

In contrast, nothing is ever easy in floating points. The typical gang that will ruin your special-case-free reasoning is ±Inf, NaN, and ±0. While f - 0.0, f * 1.0, and f / 1.0 are optimized to f, you might be surprised to see that f + 0.0 is not. It turns out that IEEE 754 has a special rule for sums of values with equal magnitude but different signs:

6.3 The sign bit

When the sum of two operands with opposite signs (or the difference of two operands with like signs) is exactly zero, the sign of that sum (or difference) shall be +0 in all rounding-direction attributes except roundTowardNegative; under that attribute, the sign of an exact zero sum (or difference) shall be −0.

However, x + x == x − (−x) retains the same sign as x even when x is zero.

Let’s make a small table what this means for different operations:

f f + 0.0 f - 0.0 f + -0.0 f - -0.0
+0.0 +0.0 +0.0 +0.0 +0.0
-0.0 +0.0 -0.0 -0.0 +0.0

(Cases where the expression cannot be legally replaced by f are bold.)

Yes, f + 0.0 is slower than f - 0.0.

Even better, f - -0.0 is as “slow” as f + 0.0 while f - -0 is optimized to f. (Because -0 is an integer which is converted to +0.0, not -0.0.)

One might wonder if this whole +0.0 vs -0.0 business makes any difference. Remember that 1.0 / +0.0 == Inf and 1.0 / -0.0 == -Inf and you can easily construct pathological cases:

double clamp(double vmin, double vmax, double v) { 
    return std::max(vmin, std::min(vmax, v)); 
}

std::cout << clamp(2.0, 5.0, 1.0 / (-1e-200 / 1e200 + 0.0)) << std::endl; // 5
std::cout << clamp(2.0, 5.0, 1.0 / (-1e-200 / 1e200 - 0.0)) << std::endl; // 2

Though overall, the usefulness of these special values can be doubted and if you are not actively expecting Infs or NaNs you might be well advised to sprinkle a few assert(isfinite(f)) across your code.

Wall of Fame and Shame

Here are the results of a few common “trivial” operations: (godbolt)

expression compiled notes
f + 0.0 f + 0.0 optimized if -fno-signed-zeros
f - 0.0 f  
f + -0.0 f  
f - -0.0 f + 0.0 optimized if -fno-signed-zeros
f * 1.0 f  
f / 1.0 f  
f / 2.0 f * 0.5  
f / 3.0 f / 3.0 cannot guarantee same rounding if f * 0.333.. were used
f * 0.0 f * 0.0 optimized if -fno-signed-zeros and -ffinite-math-only
f * -1.0 -f actually implemented via flipping the sign bit with xor
f - f f - f optimized if -ffinite-math-only

Conclusion

At some point people tend to give up trying to understand floats and accept that they are “strange” or even “unreliable”. And it’s not exactly false. But they might be throwing the baby out with the bathwater. It is true that floats involve a good amount of complexity but most rules are actually designed to help making numerical algorithms reliable.

Don’t forget that IEEE 754 has a few useful guarantees. For example, the five basic operations +. -, *, /, and sqrt are guaranteed to give “exact” results, which means the closest representable number corresponding to the current rounding mode. This especially implies that everything that is representable will be computed exactly:

  • 1.0 + 2.0 == 3.0
  • 3.0 / 4.0 == 0.75
  • sqrt(5.0625) == 2.25

floats have 23 bit mantissa, meaning integer computations with float will be exact if input and output are at most a few millions. double has enough precision that any int32_t computation will be exact.