2012-04-26 13:41:12 8 Comments

I know `UIKit`

uses `CGFloat`

because of the resolution independent coordinate system.

But every time I want to check if for example `frame.origin.x`

is `0`

it makes me feel sick:

```
if (theView.frame.origin.x == 0) {
// do important operation
}
```

Isn't `CGFloat`

vulnerable to false positives when comparing with `==`

, `<=`

, `>=`

, `<`

, `>`

?
It is a floating point and they have unprecision problems: `0.0000000000041`

for example.

Is `Objective-C`

handling this internally when comparing or can it happen that a `origin.x`

which reads as zero does not compare to `0`

as true?

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## 9 comments

## @Igor 2017-01-22 18:43:13

You can use such code for compare float with zero:

This will compare with 0.1 accuracy, that enough for CGFloat in this case.

## @chux 2017-04-16 06:09:41

Casting to

`int`

without insuring`theView.frame.origin.x`

is in/near that range of`int`

leads to undefined behavior (UB) - or in this case, 1/100th the range of`int`

.## @Sneftel 2018-12-13 16:00:25

There's absolutely no reason to convert to integer like this. As chux said, there's the potential for UB from out-of-range values; and on some architectures this will be significantly slower than just doing the computation in floating point. Lastly, multiplying by 100 like that will compare with 0.01 precision, not 0.1.

## @GoZoner 2012-04-26 13:51:15

[The 'right answer' glosses over selecting

`K`

. Selecting`K`

ends up being just as ad-hoc as selecting`VISIBLE_SHIFT`

but selecting`K`

is less obvious because unlike`VISIBLE_SHIFT`

it is not grounded on any display property. Thus pick your poison - select`K`

or select`VISIBLE_SHIFT`

. This answer advocates selecting`VISIBLE_SHIFT`

and then demonstrates the difficulty in selecting`K`

]Precisely because of round errors, you should not use comparison of 'exact' values for logical operations. In your specific case of a position on a visual display, it can't possibly matter if the position is 0.0 or 0.0000000003 - the difference is invisible to the eye. So your logic should be something like:

However, in the end, 'invisible to the eye' will depend on your display properties. If you can upper bound the display (you should be able to); then choose

`VISIBLE_SHIFT`

to be a fraction of that upper bound.Now, the 'right answer' rests upon

`K`

so let's explore picking`K`

. The 'right answer' above says:So we need

`K`

. If getting`K`

is more difficult, less intuitive than selecting my`VISIBLE_SHIFT`

then you'll decide what works for you. To find`K`

we are going to write a test program that looks at a bunch of`K`

values so we can see how it behaves. Ought to be obvious how to choose`K`

, if the 'right answer' is usable. No?We are going to use, as the 'right answer' details:

Let's just try all values of K:

Ah, so K should be 1e16 or larger if I want 1e-13 to be 'zero'.

So, I'd say you have two options:

for the value of 'epsilon', as I've suggested. If you are doing graphics and 'zero' is meant to be a 'visible change' than examine your visual assets (images, etc) and judge what epsilon can be.engineering judgement`K`

.## @Romain 2012-04-26 13:53:17

One aspect of resolution-independence is that you cannot tell for sure what a "visible shift" is at compile-time. What is invisible on a super-HD screen might very well be obvious on a tiny-ass screen. One should at least make it a function of screen size. Or name it something else.

## @GoZoner 2016-08-28 13:56:13

But at least selecting 'visible shift' is based on easily understood display (or frame) properties - unlike the <correct answer's>

`K`

which is difficult and non-intuitive to select.## @R.. 2012-04-26 14:33:33

First of all, floating point values are not "random" in their behavior. Exact comparison can and does make sense in plenty of real-world usages. But if you're going to use floating point you need to be aware of how it works. Erring on the side of assuming floating point works like real numbers will get you code that quickly breaks. Erring on the side of assuming floating point results have large random fuzz associated with them (like most of the answers here suggest) will get you code that appears to work at first but ends up having large-magnitude errors and broken corner cases.

First of all, if you want to program with floating point, you should read this:

What Every Computer Scientist Should Know About Floating-Point Arithmetic

Yes, read all of it. If that's too much of a burden, you should use integers/fixed point for your calculations until you have time to read it. :-)

Now, with that said, the biggest issues with exact floating point comparisons come down to:

The fact that lots of values you may write in the source, or read in with

`scanf`

or`strtod`

,do not existas floating point values and get silently converted to the nearest approximation. This is what demon9733's answer was talking about.The fact that many results get rounded due to not having enough precision to represent the actual result. An easy example where you can see this is adding

`x = 0x1fffffe`

and`y = 1`

as floats. Here,`x`

has 24 bits of precision in the mantissa (ok) and`y`

has just 1 bit, but when you add them, their bits are not in overlapping places, and the result would need 25 bits of precision. Instead, it gets rounded (to`0x2000000`

in the default rounding mode).The fact that many results get rounded due to needing infinitely many places for the correct value. This includes both rational results like 1/3 (which you're familiar with from decimal where it takes infinitely many places) but also 1/10 (which also takes infinitely many places in binary, since 5 is not a power of 2), as well as irrational results like the square root of anything that's not a perfect square.

Double rounding. On some systems (particularly x86), floating point expressions are evaluated in higher precision than their nominal types. This means that when one of the above types of rounding happens, you'll get two rounding steps, first a rounding of the result to the higher-precision type, then a rounding to the final type. As an example, consider what happens in decimal if you round 1.49 to an integer (1), versus what happens if you first round it to one decimal place (1.5) then round that result to an integer (2). This is actually one of the nastiest areas to deal with in floating point, since the behaviour of the compiler (especially for buggy, non-conforming compilers like GCC) is unpredictable.

Transcendental functions (

`trig`

,`exp`

,`log`

, etc.) are not specified to have correctly rounded results; the result is just specified to be correct within one unit in the last place of precision (usually referred to as1ulp).When you're writing floating point code, you need to keep in mind what you're doing with the numbers that could cause the results to be inexact, and make comparisons accordingly. Often times it will make sense to compare with an "epsilon", but that epsilon should be based on the

magnitude of the numbers you are comparing, not an absolute constant. (In cases where an absolute constant epsilon would work, that's strongly indicative that fixed point, not floating point, is the right tool for the job!)Edit:In particular, a magnitude-relative epsilon check should look something like:Where

`FLT_EPSILON`

is the constant from`float.h`

(replace it with`DBL_EPSILON`

for`double`

s or`LDBL_EPSILON`

for`long double`

s) and`K`

is a constant you choose such that the accumulated error of your computations is definitely bounded by`K`

units in the last place (and if you're not sure you got the error bound calculation right, make`K`

a few times bigger than what your calculations say it should be).Finally, note that if you use this, some special care may be needed near zero, since

`FLT_EPSILON`

does not make sense for denormals. A quick fix would be to make it:and likewise substitute

`DBL_MIN`

if using doubles.## @Daniel Fischer 2012-04-26 15:20:14

`fabs(x+y)`

is problematic if`x`

and`y`

(can) have different sign. Still, a good answer against the tide of cargo-cult comparisons.## @R.. 2012-04-26 15:41:09

If

`x`

and`y`

have different sign, it's no problem. The righthand side will be "too small", but since`x`

and`y`

have different sign, they should not compare equal anyway. (Unless they are so small as to be denormal, but then the second case catches it)## @R.. 2012-04-26 15:41:45

@Daniel: +1 to your comment anyway for invoking the word "cargo-cult". :-)

## @Nicolás 2012-04-26 21:04:06

I'm curious about your statement: "especially for buggy, non-conformant compilers like GCC". Is really GCC buggy and also non-conformant?

## @Nicolás 2012-04-27 23:09:14

Thanks @R.. for your complete explanations! It's good to know that the option

`-std=c99`

is useful to circumvent this problem.## @Stephen Canon 2012-05-11 14:46:07

Since the question is tagged iOS, it's worth noting that Apple's compilers (both clang and Apple's gcc builds) have always used FLT_EVAL_METHOD = 0, and attempt to be completely strict about not carrying excess precision. If you find any violations of that, please file bug reports.

## @Stephen Canon 2012-05-11 16:07:06

@R..: Apple never shipped an x86 processor without SSE2. All arithmetic other than

`long double`

is done on SSE, even on 32-bit.## @R.. 2012-05-11 16:15:34

OK that makes sense, but I think your usage of "Apple's compilers" was unclear. I assumed it applied to clang (Apple's compiler) on any target rather than just on Apple systems.

## @Zarel 2012-05-14 05:46:55

This answer doesn't seem to explain how "erring on the side of assuming floating point results have large random fuzz associated with them will get you code that appears to work at first but ends up having large-magnitude errors and broken corner cases."

## @Nemo 2012-12-19 01:57:08

@R.: You might find this question interesting. I would certainly like to know your opinion on it, if any.

## @Christian Rau 2013-07-05 16:17:54

"First of all, floating point values are not "random" in their behavior. Exact comparison can and does make sense in plenty of real-world usages."- Just two sentences and already earned a +1! That's one of the most disturbing misassumptions people make when working with floating points.## @hfossli 2014-05-14 11:25:20

@R.. Would you approve this? gist.github.com/hfossli/4616c778bea3a334f034 I replaced "K" with "accuracy" if that makes sense.

## @supercat 2014-07-09 23:40:29

An issue not yet mentioned in this answer's list is that if a computation yields a "not a number" result, that answer will compare unequal

to itself. That's not an issue when comparing a variable to a constant that isn't a NaN, but can be an issue when e.g. testing whether a floating-point value is in a table.## @Vaddadi Kartick 2016-04-08 06:55:48

What's the absolute error? In other words, what's the smallest value of ERR for which fabs(x-y) < ERR is valid for all x and y?

## @CyberMew 2018-03-23 06:18:23

So

Kshould be a value more than 1 if we were to make it less accurate? Meaning more margin for error.## @Albert Renshaw 2018-12-13 07:23:25

Note

`__FLT_EPSILON__`

can be used if you do not desire to import float.h## @R.. 2018-12-13 15:36:18

@AlbertRenshaw: That's not C. It's a gcc feature meant to be used only for implementation internals.

`float.h`

is a standard freestanding header and there's no reason not to include it.## @High Performance Mark 2012-04-26 13:55:23

Since 0 is exactly representable as an IEEE754 floating-point number (or using any other implementation of f-p numbers I've ever worked with) comparison with 0 is probably safe. You might get bitten, however, if your program computes a value (such as

`theView.frame.origin.x`

) which you have reason to believe ought to be 0 but which your computation cannot guarantee to be 0.To clarify a little, a computation such as :

will (unless your language or system is broken) create a value such that (areal==0.0) returns true but another computation such as

may not.

If you can assure yourself that your computations produce values which are 0 (and not just that they produce values which ought to be 0) then you can go ahead and compare f-p values with 0. If you can't assure yourself to the required degree, best stick to the usual approach of 'toleranced equality'.

In the worst cases the careless comparison of f-p values can be extremely dangerous: think avionics, weapons-guidance, power-plant operations, vehicle navigation, almost any application in which computation meets the real world.

For Angry Birds, not so dangerous.

## @Almo 2012-04-26 14:12:56

Re: Angry Birds: Financially, a bug the stops 10 million gamers from squishing the Pigs could have repurcussions.

## @Pascal Cuoq 2013-08-06 17:04:37

Actually,

`1.30 - 2*(0.65)`

is a perfect example of an expression that obviously evaluates to 0.0 if your compiler implements IEEE 754, because the doubles represented as`0.65`

and`1.30`

have the same significands, and multiplication by two is obviously exact.## @High Performance Mark 2013-12-16 10:00:15

Still getting rep from this one, so I changed the second example snippet.

## @Michael 2013-05-07 12:09:19

The correct question: how does one compare points in Cocoa Touch?

The correct answer: CGPointEqualToPoint().

A different question: Are two calculated values are the same?

The answer posted here: They are not.

How to check if they are close? If you want to check if they are close, then don't use CGPointEqualToPoint(). But, don't check to see if they are close. Do something that makes sense in the real world, like checking to see if a point is beyond a line or if a point is inside a sphere.

## @starmole 2012-05-14 04:19:50

I want to give a bit of a different answer than the others. They are great for answering your question as stated but probably not for what you need to know or what your real problem is.

Floating point in graphics is fine! But there is almost no need to ever compare floats directly. Why would you need to do that? Graphics uses floats to define intervals. And comparing if a float is within an interval also defined by floats is always well defined and merely needs to be consistent, not accurate or precise! As long as a pixel (which is also an interval!) can be assigned that's all graphics needs.

So if you want to test if your point is outside a [0..width[ range this is just fine. Just make sure you define inclusion consistently. For example always define inside is (x>=0 && x < width). The same goes for intersection or hit tests.

However, if you are abusing a graphics coordinate as some kind of flag, like for example to see if a window is docked or not, you should not do this. Use a boolean flag that is separate from the graphics presentation layer instead.

## @JHumphrey 2012-05-14 03:48:20

Comparing to zero

canbe a safe operation, as long as the zero wasn't a calculated value (as noted in an above answer). The reason for this is that zero is a perfectly representable number in floating point.Talking perfectly representable values, you get 24 bits of range in a power-of-two notion (single precision). So 1, 2, 4 are perfectly representable, as are .5, .25, and .125. As long as all your important bits are in 24-bits, you are golden. So 10.625 can be repsented precisely.

This is great, but will quickly fall apart under pressure. Two scenarios spring to mind: 1) When a calculation is involved. Don't trust that sqrt(3)*sqrt(3) == 3. It just won't be that way. And it probably won't be within an epsilon, as some of the other answers suggest. 2) When any non-power-of-2 (NPOT) is involved. So it may sound odd, but 0.1 is an infinite series in binary and therefore any calculation involving a number like this will be imprecise from the start.

(Oh and the original question mentioned comparisons to zero. Don't forget that -0.0 is also a perfectly valid floating-point value.)

## @Lucas Membrane 2012-05-14 01:41:29

The last time I checked the C standard, there was no requirement for floating point operations on doubles (64 bits total, 53 bit mantissa) to be accurate to more than that precision. However, some hardware might do the operations in registers of greater precision, and the requirement was interpreted to mean no requirement to clear lower order bits (beyond the precision of the numbers being loaded into the registers). So you could get unexpected results of comparisons like this depending on what was left over in the registers from whoever slept there last.

That said, and despite my efforts to expunge it whenever I see it, the outfit where I work has lots of C code that is compiled using gcc and run on linux, and we have not noticed any of these unexpected results in a very long time. I have no idea whether this is because gcc is clearing the low-order bits for us, the 80-bit registers are not used for these operations on modern computers, the standard has been changed, or what. I'd like to know if anyone can quote chapter and verse.

## @John White 2012-05-02 22:52:19

I'd say the right thing is to declare each number as an object, and then define three things in that object: 1) an equality operator. 2) a setAcceptableDifference method. 3)the value itself. The equality operator returns true if the absolute difference of two values is less than the value set as acceptable.

You can subclass the object to suit the problem. For example, round bars of metal between 1 and 2 inches might be considered of equal diameter if their diameters differed by less than 0.0001 inches. So you'd call setAcceptableDifference with parameter 0.0001, and then use the equality operator with confidence.

## @Tom Swirly 2012-05-14 18:53:49

This is Not A Good Answer. First, the whole "object thing" does nothing whatsoever to solve your issue. And second, your actual implementation of "equality" isn't in fact the correct one.

## @John White 2012-08-16 17:53:43

Tom, maybe you'd think again about the "object thing". With real numbers, represented to high precision, equality rarely happens. But one's

ideaof equality may be tailored if it suits you. It would be nicer if there was an overridable 'approximately equal' operator, but there ain't.