Difference between rem2pi and mod2pi?

General Usage
1

I’m trying to understand the difference between rem2pi(x, RoundNearest) and mod2pi (or more precisely mod2pi - π) from Base, Math functions. Is that correct that roughly speaking both function return the same thing, that is the principal value of an angle between -π and +π?

I understand from the doc that mod(x, y) is the same as rem(x, y, RoundDown), but things gets complicated when it says that mod2pi(x) is different from mod(x, 2π)…

2

When it says mod2pi(x) is different from mod(x, 2π), it just means that mod(x, 2π) uses a floating point representation of 2pi, while mod2pi uses the mathematical number. This makes a difference for large inputs, since the rounding error in 2pi will get multiplied by floor(x/2pi).

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3

Thanks. So is it correct that mod2pi(x) is the same as rem2pi(x, RoundDown)?

And then, since the doc of rem2pi(x, RoundDown) says it return a number in [0, 2π], how is mod2pi - π different from rem2pi(x, RoundNearest) which return a number in [-π, π]? Are those the same except for the extraneous substraction (which of course brings some error)? Or are there some input values which return different output values?

4 
julia> rem2pi(-1.0, RoundNearest)
-1.0

julia> mod2pi(-1.0)
5.283185307179586

negative numbers strike again.

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5

Good catch! Also, I now realize that my reasoning about substracting π to mod2pi(x) is silly.

So it seems that for my usecase, I should stick with rem2pi(x, RoundNearest).

I’m not sure what you mean by “negative numbers strike again.”, but my first implementation, before discovering rem2pi and friends was (x+π) % 2π - π which at first seems to work but indeed breaks when (x+π) is negative. This lead me to the surprising discovery that rem(x, y) output range depends on the sign of x by default (i.e. when not specifying the rounding mode argument which I didn’t know it existed!). Is it this kind of “surprise for new users” that you were thinking of?

7

What’s wrong with it? -1.0+2pi=5.283185307179586 so it’s in the correct range…

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9

@pierre-haessig This is the same as the difference between the rem and mod functions (as specified by ieee). rem returns a result with the same sign as the divisor. mod is periodic. These definitions correspond when the divisor and dividend have the same sign.

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10

Thanks for the context. In fact, I think the suprise for me was mostly about coming from Python where -1 % 3 returns 2 (I just check to be sure!) while Julia returns -1.

Digging further in Python, I just found out about the relatively new (Python 3.7+) function math.remainder which claims to be the “IEEE 754-style remainder”. Since the doc says it returns r such that |r| ≤ y/2, this maps to Julia’s rem used with RoundNearest if I’m not mistaken.

About realizing one of my proposition was silly: it’s just that I thought mod2pi(x) - π would give the principal angle in [-π, π]. But in fact, it should be mod2pi(x+π) - π. Otherwise, the output range is correct, but the value is wrong by 180°…

11

By the way, I can access the doc of rem(x, y, r::RoundingMode=RoundToZero) through the REPL, but oddly enough I don’t see it in the doc Base.rem which is only about rem(x, y). Why is that?