FORM procedure to approximates some zeta values by rational numbers. #form

MakeRationalApproximation.wls

chunkifyString[s_String, n_Integer] :=
  StringJoin @@@ Partition[Characters[s], UpTo[n]];

writeRationalApproximation[
  stream_, x_, n_, width_:72
] := Module[{s, lines, i},
  s = Rationalize[x, Abs[x * 10^(-n)]] // InputForm // ToString;
  lines = chunkifyString[s, width - 1];
  Do[
    s = lines[[i]];
    If[i < Length[lines],
      s = s <> "\\";
    ];
    WriteLine[stream, s];
  , {i, Length[lines]}];
];

n = 200;

constants = Table[
  {"z" <> ToString[i], Zeta[i]}, {i, 2, 12}
] ~ Join ~ {
  {"z5z3", SpecialFunctions`MultipleZetaValue[{5, 3}]},
  {"z7z3", SpecialFunctions`MultipleZetaValue[{7, 3}]},
  {"z9z3", SpecialFunctions`MultipleZetaValue[{9, 3}]},
  {"z5z3z3", SpecialFunctions`MultipleZetaValue[{5, 3, 3}]},
  {"z6z4z1z1", SpecialFunctions`MultipleZetaValue[{6, 4, 1, 1}]}
};

out = OpenWrite["RationalApproximation.prc", BinaryFormat -> True];

WriteLine[out, "#procedure RationalApproximation()"];
WriteLine[out, "* Approximates z2, z3 etc. by rational numbers."];
WriteLine[out, "* The relative errors are < 10^-" <> ToString[n] <> "."];
Do[
  WriteLine[out, "  id " <> ToString[c[[1]]] <> " = \\"];
  writeRationalApproximation[out, c[[2]], n + 0.1];
  WriteLine[out, "  ;"];
, {c, constants}];
WriteLine[out, "#endprocedure"];

Close[out];

RationalApproximation.prc

#procedure RationalApproximation()
* Approximates z2, z3 etc. by rational numbers.
* The relative errors are < 10^-200.
  id z2 = \
16020477561421692952487130706964323462700663188794863242830247036589271\
925412774344912016394073406219/9739282494232553676474274187877549323464\
250480060655692338461020488417652652504432571071309828391142
  ;
  id z3 = \
22900205778039760633800755309246091627637015525974362585845671818945337\
998208464555618946329453959543/1905085002036659311006241247931009390825\
2520945932649399230934116604827261300902939472785087063645452
  ;
  id z4 = \
17927734279097906568458659053518841157857583903382328100393844711661041\
467432741678775380083105798569/1656412217783236542503825391890589682291\
6627018594003017360269229526707715002782448866141025907259289
  ;
  id z5 = \
10019631834006845890915735791912045131384046751981242656968347612651846\
448223293908329722968096388543/9662806096478235451137463161039974764749\
743657192120723046983388695058417315384212275765047706362024
  ;
  id z6 = \
14240641642786716444406574948352270741732048853413689815750283209079013\
893480541071019260227090412743/1399787561828813597633806805340887330928\
2222272530255724787274916767158942132278303064033244949560181
  ;
  id z7 = \
10003775654518472179328462365950159181212595201836124848608408319553129\
804033065087933012109479477027/9920942949939198104836171480706809555019\
041486786646171017569865621882118462058545244827663928586931
  ;
  id z8 = \
40787740933520804515988056999909137902613590142734931586603469098589930\
33580703072666515126002722549/40622110121045183458862471061540533150382\
79603013821433408369245232515380041628372813952473048006811
  ;
  id z9 = \
35629138685937025281785238317553599485296383911212889161231536005734139\
63737417328850389628804016709/35557724806523797291661250307333621655163\
09852506844145165356285153613171732908871153981398558666212
  ;
  id z10 = \
10791366655691279491343243491206331133838755390218747293907250378819341\
603403302914895537344599292451/1078064449481488763644454456534692274399\
6296892467632778925978994566359335057413324403497449086719253
  ;
  id z11 = \
48296103505229214266995002237392658539603735632773200590801362287462237\
51290180335542983021938814105/48272247910416656719558231429793909488284\
79394896726908094492792787474704290755983101767106789629277
  ;
  id z12 = \
14075090131725826273206160379596285211344069192703488269414746274255543\
349538576223914619675182147945/1407162729346574182852916958016101369854\
2704194906842595006978562243797391307266040075785806985331596
  ;
  id z5z3 = \
17617859193452652888737471212731697034210877756678661676726111627328593\
46467889183638222478707153921/46722212745751284127190723160910640951146\
361432029184668839885198760539552151501994585476261341926016
  ;
  id z7z3 = \
10363620777973984600800316131162869684341364170204288961183796600616180\
68791604872882737435669379635/12308822816673558919790317128254611397840\
9707583552095062008265500468081845815157192763461160793286239
  ;
  id z9z3 = \
92893839306146309479613867913681705129990452985321002587374056192530643\
9200756779487824176169779507/460902271345016416070850924778860878264196\
349176248351919498172888040717261975507898069724815173433821
  ;
  id z5z3z3 = \
39858636711025994206772381295241219153251689127897521436352920955834054\
6025645276162926255866485537/505044876843925870839119622379459191622892\
994874325382816468723492843903919722828483385746072214454316
  ;
  id z6z4z1z1 = \
22755073682775958710309660864070582291154393234059293222863179447779599\
875436932363342236819763954/8545109335266554688533007993952857015793148\
989052715942152138563147259592817714358626670354013813021071
  ;
#endprocedure