For some reasons I need to consider the following iterative process.
We have three positive numbers with constant total such that $x+y+z == t$
Consider the iterative algorithm:

• if the sum of numbers is not equal to $t$ - stop (with message "Total error")
• if one of the numbers is $t/2$ - stop (with message "Half")
• if one of the numbers is greater than $t/2$ (let it be $z$) - returns new numbers $2z-t,2x,2y$

• if all numbers are less than $t/2$ - return the new numbers $t-2z,t-2x,t-2y$

  ClearAll[genNumsReals]; genNumsReals[ tot_] :=
   Block[{a = RandomReal[{0., tot}], b},
    b = RandomReal[{0., tot - a}];
    {a, b, tot - a - b}];
  
  ClearAll[genNumsRationals]; genNumsRationals[e_, tot_] :=
   Block[{a = Rationalize[RandomReal[{e, tot - e}], e], b},
    b = Rationalize[RandomReal[{e, tot - a - e}], e];
    {a, b, tot - a - b}];
  
  Clear[nextNums]; nextNums[nums_, tot_] :=
   Which[
    Unequal[Total@nums, tot], {nums, "Error Total"},
    MemberQ[nums, tot/2], {nums, "Half Total"},
    AnyTrue[nums, GreaterThan[tot/2]], Mod[2 nums, tot],
    True, tot - 2 nums];
  
  numsNest[init_, tot_, iterMax_ : 100] :=
    NestWhileList[nextNums[#, tot] &, init,
     (UnsameQ[##] && Length@Last[{##}] == 3) &, All, iterMax];
  

Firstly, is it equal to Mod[-2 {x,y,z}, t]? Looks like not.
If the numbers are rational, the results are generally expected but interesting.
Either cycles of greater or lesser length are obtained,
either $t/2$ occurs.

But for reals I see some problems. For any initial data with machine precision we get "Half Total" sooner or later (here $t=1$):

{{0.84474, 0.117199, 0.0380613}, {0.68948, 0.234397, 
  0.0761226}, {0.37896, 0.468795, 0.152245}, {0.24208, 0.0624103, 
  0.69551}, {0.48416, 0.124821, 0.39102}, {0.0316802, 0.750359, 
  0.217961}, {0.0633604, 0.500718, 0.435922}, {0.126721, 0.00143543, 
  0.871844}, {0.253441, 0.00287087, 0.743688}, {0.506883, 0.00574174, 
  0.487375}, {0.0137658, 0.0114835, 0.974751}, {0.0275315, 0.0229669, 
  0.949502}, {0.0550631, 0.0459339, 0.899003}, {0.110126, 0.0918678, 
  0.798006}, {0.220252, 0.183736, 0.596012}, {0.440505, 0.367471, 
  0.192024}, {0.118991, 0.265058, 0.615952}, {0.237981, 0.530115, 
  0.231903}, {0.475963, 0.060231, 0.463806}, {0.0480747, 0.879538, 
  0.0723872}, {0.0961494, 0.759076, 0.144774}, {0.192299, 0.518152, 
  0.289549}, {0.384598, 0.0363047, 0.579098}, {0.769195, 0.0726095, 
  0.158195}, {0.538391, 0.145219, 0.31639}, {0.0767815, 0.290438, 
  0.632781}, {0.153563, 0.580876, 0.265561}, {0.307126, 0.161751, 
  0.531122}, {0.614252, 0.323503, 0.062245}, {0.228504, 0.647006, 
  0.12449}, {0.457009, 0.294012, 0.24898}, {0.0859828, 0.411977, 
  0.50204}, {0.171966, 0.823954, 0.00408077}, {0.343931, 0.647907, 
  0.00816154}, {0.687862, 0.295815, 0.0163231}, {0.375725, 0.591629, 
  0.0326462}, {0.75145, 0.183258, 0.0652924}, {0.502899, 0.366516, 
  0.130585}, {0.00579834, 0.733032, 0.261169}, {0.0115967, 0.466064, 
  0.522339}, {0.0231934, 0.932129, 0.0446777}, {0.0463867, 0.864258, 
  0.0893555}, {0.0927734, 0.728516, 0.178711}, {0.185547, 0.457031, 
  0.357422}, {0.628906, 0.0859375, 0.285156}, {0.257813, 0.171875, 
  0.570313}, {0.515625, 0.34375, 0.140625}, {0.03125, 0.6875, 
  0.28125}, {0.0625, 0.375, 0.5625}, {0.125, 0.75, 0.125}, {0.25, 0.5,
   0.25}, {{0.25, 0.5, 0.25}, "Half Total"}}

and for higher precision we get "Total error" (sooner or later too):

ClearAll[genNumsReals]; genNumsReals[tot_] :=
 Block[{a = RandomReal[{0., tot}, WorkingPrecision -> 30], b},
  b = RandomReal[{0., tot - a}, WorkingPrecision -> 30];
  {a, b, tot - a - b}];

Clear[nextNums]; nextNums[nums_, tot_] :=
 Which[
  Unequal[Total@nums, tot], {nums, "Error Total"},
  MemberQ[nums, tot/2], {nums, "Half Total"},
  AnyTrue[nums, GreaterThan[tot/2]],
  SetPrecision[Mod[2 nums, tot], 30],
  True, SetPrecision[tot - 2 nums, 30]];

{{0.659587099030496875713494880795, 0.185853135754329321586581016710,
  0.154559765215173802699924102495}, \
{0.319174198060993674630481109489, 0.371706271508658647739764546714,
  0.309119530430347622118603112540}, \
{0.361651603878012650739037781022, 0.256587456982682704520470906573,
  0.381760939139304755762793774920}, \
{0.276696792243974698521924437955, 0.486825086034634590959058186854,
  0.236478121721390488474412450159}, \
{0.446606415512050602956151124090, 0.0263498279307308180818836262915,
  0.527043756557219023051175099681}, \
{0.893212831024101205912302248180, 0.0526996558614616361637672525831,
  0.0540875131144380461023501993623}, \
{0.786425662048202411824604496360, 0.105399311722923272327534505166,
  0.108175026228876092204700398725}, \
{0.572851324096404823649208992720, 0.210798623445846544655069010332,
  0.216350052457752184409400797449}, \
{0.145702648192809647298417985439, 0.421597246891693089310138020664,
  0.432700104915504368818801594898}, \
{0.708594703614380705403164029121, 0.156805506216613821379723958671,
  0.134599790168991262362396810204}, \
{{0.708594703614380705403164029121, 0.156805506216613821379723958671,
   0.134599790168991262362396810204}, "Error Total"}}

I think the problems are in rounding, but how to understand it?