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Notes on floating-point in Swift

Steve Canon (@stephentyrone), July 2016

Frequently asked questions

Q: Why is .pi special? Why isn't tau/e/oneOverPi also available on FloatingPoint?

A: There are lots of important constants in mathematics. POSIX defines the following macros in C:

Macro Value
M_E e, the base of the natural logarithm
M_LOG2E log2(e)
M_LOG10E log10(e)
M_LN2 log(2) = 1/log2(e)
M_LN10 log(10) = 1/log10(e)
M_PI π
M_PI_2 π/2
M_PI_4 π/4
M_1_PI 1/π
M_2_PI 2/π
M_2_SQRTPI 2/sqrt(π)
M_SQRT2 sqrt(2)
M_SQRT1_2 sqrt(1/2)

It would certainly be possible to add all of these and more as static properties on the FloatingPoint protocol. However, there is a big drawback to this approach to keep in mind. The C macros are in the global namespace and don't pollute autocomplete lists; static properties on protocols appear in the autocomplete list when you write Float.<TAB>. That list is already pretty large for floating point types; adding 12 or more new values with short, cryptic names would only add noise.

So why is .pi included? We talked about this when SE-0067 was proposed; it turns out that in real code, M_PI and its variants like M_PI_2 are used about an order of magnitude more frequently than all other constants combined. So it makes sense to carve out an exception for .pi. If you want M_PI_2, just write (Float.pi / 2), which is always computed exactly for binary floating-point types, so you're not losing any accuracy.

All of this isn't to say that an expanded set of constants won't be available in the future; they're simply unlikely to be approved as top-level members of the FloatingPoint protocol. I can definitely imagine everyone agreeing to Float.Constant.e, for example.

Q: Why isn't there a member function that rounds to a specified number of decimal digits?

A: This is a common request. Folk want something like:

public func rounded(_ rule: RoundingRule, decimalPlaces digits: Int = 0) -> Self

Many languages have responded to user demand and provided such a function. This is a source of bug reports and user confusion, because it is not possible to implement such a function on binary floating-point types. The only number of decimal digits that you can correctly round to is zero. To illustrate this, let's look at an example from a confused Python user on Stack Overflow:

I want a to be rounded to 13.95.

>>> a
13.949999999999999
>>> round(a, 2)
13.949999999999999

What went wrong here? Well, nothing did, except for the library providing an interface that cannot possibly work as users expect it to. The trouble is that 13.95 is not representable as a binary floating-point number; in general, very few numbers with one or more digits after the decimal point are representable. So if you try to "round a to two decimal digits", you get the closest representable value to 13.95, which is 13.949999999999999289457264239899814128875732421875. If subsequent computations assume you have an exact two-digit decimal value, they will misbehave and produce incorrect results.

Here's an example of that in action:

>>> a = .291
>>> b = int(100*round(a,2))

What should b be? Surely a rounded to two digits is .29, so b should be exactly 29. What value does b actually have?

>>> b
28

OK, so what should users do instead? Rounding to some number of decimal digits is a well-defined operation only on decimal representations. So if you need to perform this operation, there are two options:

  • Use a decimal number type.
  • Since you usually want to perform this rounding for the purposes of display, round as part of formatting the number, rather than as a separate operation:
let toDisplay = String(format: "%.2f", a)

This rounds correctly to two digits, and is simpler as well.

Q: Why aren't math functions like sin/log/pow included?

A: The FloatingPoint and BinaryFloatingPoint protocols stick tightly to the requirements of IEEE 754, with a small number of additional convenience operations added. This makes them useful for performing arithmetic, and also for implementing a math library, but they don't have all the math operations that folks want. The trouble with throwing in everything that everyone could want is that the protocols become huge and a major burden for implementors who want to conform to them. My goal in speccing these was to require only what's necessary to be a useful IEEE 754 floating-point type.

There absolutely should be a "math" protocol added at some future point with all the usual math library operations.

Q: Why doesn't rounded return an Int?

A: Short answer: because IEEE 754 requires rounding functions that return Self.

Longer answer: because returning Self is more correct and more useful. Either operation can be constructed from the other. If you have the version that returns Self, then the operation returning Int is just:

let integerResult = Int(x.rounded(.up))                      [1]

Simple, easy to write, and hard to screw up. If you only have the version that returns Int, the situation is more complicated. If you're lucky and Int is bigger than the significand of the floating-point type, you can do something along the following lines (this is a little bit fast and loose, intended only for illustration):

if x <= Self(Int.min) || x >= Self(Int.max) || x.isNaN {
  return x
}
return Self(x.rounded(.up))

This works because of the assumption that Int was wider than the significand; thus any value outside the range of Int is necessarily already an integer.

This is already obviously more complicated than what we saw in [1], but the situation where Int is smaller than the significand of the floating-point type is much worse. In order to handle such cases correctly, you need to re-implement all of the rounding logic; why bother having a rounding function in the standard library at all?