Bug introduced in 8.0 or earlier and persisting through 11.0.1 or later

While answering another question I discovered that TriangleWave does not automatically evaluate when given certain fractional values, specifically fractions with a denominator of 20:

TriangleWave[ Range[8] / 20 ]
{TriangleWave[1/20], 2/5, TriangleWave[3/20], 4/5, 1, 4/5, TriangleWave[7/20], 2/5}

These are reduced by Simplify:

TriangleWave[ Range[8] / 20 ] // Simplify
{1/5, 2/5, 3/5, 4/5, 1, 4/5, 3/5, 2/5}

This appears to be the only denominator under 10,000 that does not automatically evaluate:

Cases[TriangleWave[1/Range[1*^5]], _TriangleWave]

Is there a reason to believe that this behavior is anything other than a bug in TriangleWave?

  • $\begingroup$ It only seems to evaluate when (2 ArcSin[Sin[2 Pi t]])/Pi is an explicit NumberQ $\endgroup$
    – Rojo
    Jan 9, 2014 at 3:27
  • $\begingroup$ Interesting question - I've wondered sometimes how MM decides (and why) to not simplify. E.G., 4/Pi (ArcSin[Sin[2/20 Pi]]) results in a trig form, while 4/Pi (ArcSin[Sin[3/20 Pi]]) simplifies, yet both are rationals... $\endgroup$
    – ciao
    Jan 9, 2014 at 4:00
  • $\begingroup$ @rasher I have raised a question about that here. $\endgroup$
    – Mr.Wizard
    Jan 9, 2014 at 9:11
  • $\begingroup$ @Mr.Wizard I have added the bug header. Is the introduction version correct? I don't have access to earlier version than 9.0 at the moment. $\endgroup$
    – user31159
    Oct 8, 2016 at 12:00
  • $\begingroup$ @Xavier, version 8 has the bug; since TriangleWave[] was introduced in version 7, testing that as well (by somebody other than you or me) should make for an accurate record. $\endgroup$ Oct 8, 2016 at 12:09

1 Answer 1


Rojo's comment is spot on. If you look at the implementation of TriangleWave, you'll find something like this:

TriangleWave[t_?NumberQ] := With[{r = 2 ArcSin[Sin[2 π t]]/π}, r /; NumberQ[r]] /; Im[t] == 0

Note the use of NumberQ in the definition, which only checks if the argument is explicitly a number. For certain values of your input, ArcSin[Sin[2 π t]]/π is False for NumberQ:

{#, NumberQ@#}& /@ (ArcSin[Sin[2 π Range[8]/20]]/π) // TableForm

They should've used NumericQ instead of NumberQ.

  • $\begingroup$ How do you find the implementation? $\endgroup$
    – s0rce
    Jan 9, 2014 at 4:12
  • $\begingroup$ @s0rce You can clear the attributes and look at the definitions. $\endgroup$
    – rm -rf
    Jan 9, 2014 at 4:13
  • 1
    $\begingroup$ @s0rce Something like this peep[s_] := Module[{at}, at = Attributes[s]; ClearAttributes[s, at]; Print@Definition@s; SetAttributes[s, at]; ]; peep[TriangleWave] $\endgroup$ Jan 9, 2014 at 5:34
  • $\begingroup$ Thanks for the exploration. It sounds like you agree that this is a bug; is that correct? $\endgroup$
    – Mr.Wizard
    Jan 9, 2014 at 8:48
  • $\begingroup$ @Mr.Wizard Yes, I agree that it is a bug. $\endgroup$
    – rm -rf
    Jan 9, 2014 at 14:49

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