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rpn[str_] :=
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StringRiffle[
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ToString /@
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Module[{in = StringSplit[str], stack = {}, out = {}, next},
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While[in != {}, next = in[[1]]; in = in[[2 ;;]];
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Which[DigitQ[next], AppendTo[out, next], LetterQ[next],
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AppendTo[stack, next], next == ",",
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While[stack[[-1]] != "(", AppendTo[out, stack[[-1]]];
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stack = stack[[;; -2]]], next == "^", AppendTo[stack, "^"],
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next == "*",
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While[stack != {} && MatchQ[stack[[-1]], "^" | "*" | "/"],
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AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]];
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AppendTo[stack, "*"], next == "/",
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While[stack != {} && MatchQ[stack[[-1]], "^" | "*" | "/"],
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AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]];
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AppendTo[stack, "/"], next == "+",
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While[stack != {} &&
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MatchQ[stack[[-1]], "^" | "*" | "/" | "+" | "-"],
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AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]];
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AppendTo[stack, "+"], next == "-",
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While[stack != {} &&
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MatchQ[stack[[-1]], "^" | "*" | "/" | "+" | "-"],
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AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]];
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AppendTo[stack, "-"], next == "(", AppendTo[stack, "("],
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next == ")",
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While[stack[[-1]] =!= "(", AppendTo[out, stack[[-1]]];
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stack = stack[[;; -2]]]; stack = stack[[;; -2]];
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If[StringQ[stack[[-1]]], AppendTo[out, stack[[-1]]];
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stack = stack[[;; -2]]]]];
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While[stack != {}, AppendTo[out, stack[[-1]]];
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stack = stack[[;; -2]]]; out]];
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Print[rpn["3 + 4 * 2 / ( 1 - 5 ) ^ 2 ^ 3"]];
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