Unicode polynomial equation

This task has been flagged for clarification due to it being believed to be too difficult to implement in a reasonable amount of effort in more than one (usually very specialised) language. It may need to be divided into multiple tasks or modified substantially so that multiple implementations are practical, and that may cause code on this page in its current state to be flagged incorrect afterwards. See this page's Talk page for discussion.

The objective of this task is to parse in a difficult polynomial, and generate a "pretty" representation of the polynomial in Unicode.

Unicode polynomial equation is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

In the target language define a "polynomial" object (or structure or record). Using this object also define the routines for parsing a polynomial as input, and generating a normalised Unicode representation of the polynomial as output.

Task details:

Given a string containing an untidy Unicode polynomial, e.g.

-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹

Coerce (or convert) the string into the "polynomial" object, at the same time normalise the polynomial to a canonical form. The ideal normalised output (in this example) would be:

x⁵ - 2x⁴ + 42x³ + 40x + 1
Specific examples of Unicode and polynomial texts to be parsed as test cases.
Description Input example test cases
"Zero" coefficients are removed x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1
The "0" polynomial case 0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵
"One" coefficients are normalised 1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰
Signs are normalised +x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1
ASCII representations are parsed x^5 - 2x**4 + 42x^3 + 40x + 1
Non-ASCII representations are parsed x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1 (c.f. & ·)
Specifically permit non-polynomials where terms have negative exponents x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰ (n.b. Unicode Fraction)
Spaces in numbers and between operators are ignored x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1
Single commas are ignored in numbers x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1
A coefficient may be duplicated, zero, or missing 0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1
Support Scientific notation and optionally
support Unicode Decimal Exponent Symbol U+23E8/⏨
1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1
Unicode characters that must be specifically supported are: ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁻ ⁺ · × ⁄ ↑ ⏨.
Where · & × are multiplication, and ⁄ is Unicode Fraction.
Support fractions for both input and output. x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2
On output round the decimal to appropriate fraction.
Optionally support Unicode Vulgar fractions for both input and output.
¼ ½ ¾ ⅐ ⅑ ⅒ ⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞ ↉
x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½
On output round the decimal to appropriate fraction.

There are (at least) three possible ways of achieving this task.

  • Using an external parsing library.
  • Using a built-in parsing/formatting library.
  • Coding a custom polynomial parsing routing.

Either one, or all of these approaches are accepted and appear as a subtitle.

Go

Although this program provides full support for Unicode vulgar fractions, note that there is no guarantee that they (or arithmetic on them) will successfully 'round trip' due to floating point arithmetic being used in the underlying calculations and some of them being recurring rather than exact decimals in any case.

package main

import (
    "fmt"
    "log"
    "math"
    "regexp"
    "strconv"
    "strings"
)

var powers = strings.NewReplacer(
    "0", "⁰",
    "1", "¹",
    "2", "²",
    "3", "³",
    "4", "⁴",
    "5", "⁵",
    "6", "⁶",
    "7", "⁷",
    "8", "⁸",
    "9", "⁹",
    "-", "⁻",
)

var fractions = [][2]string{
    {".25", "¼"},
    {".5", "½"},
    {".75", "¾"},
    {".14285714285714285", "⅐"},
    {".1111111111111111", "⅑"},
    {".1", "⅒"},
    {".3333333333333333", "⅓"},
    {".6666666666666666", "⅔"},
    {".2", "⅕"},
    {".4", "⅖"},
    {".6", "⅗"},
    {".8", "⅘"},
    {".16666666666666666", "⅙"},
    {".8333333333333334", "⅚"},
    {".125", "⅛"},
    {".375", "⅜"},
    {".625", "⅝"},
    {".875", "⅞"},
}

func printEquation(coefs map[int]float64) {
    fmt.Print("=> ")
    if len(coefs) == 0 {
        fmt.Println("0\n")
        return
    }
    max, min := math.MinInt32, math.MaxInt32
    for k := range coefs {
        if k > max {
            max = k
        }
        if k < min {
            min = k
        }
    }
    for p := max; p >= min; p-- {
        if c := coefs[p]; c != 0 {
            if p < max {
                sign := "+"
                if c < 0 {
                    sign = "-"
                    c = -c
                }
                fmt.Printf(" %s ", sign)
            }
            if c != 1 || (c == 1 && p == 0) {
                cs := fmt.Sprintf("%v", c)
                ix := strings.Index(cs, ".")
                if ix >= 0 {
                    dec := cs[ix:]
                    for _, frac := range fractions {
                        if dec == frac[0] {
                            cs = strings.Replace(cs, dec, frac[1], 1)
                            break
                        }
                    }
                }
                if cs[0] == '0' && len(cs) > 1 && cs[1] != '.' {
                    cs = cs[1:]
                }
                fmt.Print(cs)
            }
            if p != 0 {
                ps := strconv.Itoa(p)
                ps = powers.Replace(ps)
                if ps == "¹" {
                    ps = ""
                }
                fmt.Printf("x%s", ps)
            }
        }
    }
    fmt.Println("\n")
}

func check(err error) {
    if err != nil {
        log.Fatal(err)
    }
}

func main() {
    equs := []string{
        `-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹`,
        `x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1`,
        `0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵`,
        `1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰`,
        `+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1`,
        `x^5 - 2x**4 + 42x^3 + 40x + 1`,
        `x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1`,
        `x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰`,
        `x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1`,
        `x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1`,
        `0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1`,
        `1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1`,
        `x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2`,
        `x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5`,
        `x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x`,
        `⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝`,
    }
    rgx := regexp.MustCompile(`\s+(\+|-)\s+`)
    rep := strings.NewReplacer(
        ",", "",
        " ", "",
        "¼", ".25",
        "½", ".5",
        "¾", ".75",
        "⅐", ".14285714285714285",
        "⅑", ".1111111111111111",
        "⅒", ".1",
        "⅓", ".3333333333333333",
        "⅔", ".6666666666666666",
        "⅕", ".2",
        "⅖", ".4",
        "⅗", ".6",
        "⅘", ".8",
        "⅙", ".16666666666666666",
        "⅚", ".8333333333333334",
        "⅛", ".125",
        "⅜", ".375",
        "⅝", ".625",
        "⅞", ".875",
        "↉", ".0",
        "⏨", "e",
        "⁄", "/",
    )
    rep2 := strings.NewReplacer(
        "⁰", "0",
        "¹", "1",
        "²", "2",
        "³", "3",
        "⁴", "4",
        "⁵", "5",
        "⁶", "6",
        "⁷", "7",
        "⁸", "8",
        "⁹", "9",
        "⁻⁻", "",
        "⁻", "-",
        "⁺", "",
        "**", "",
        "^", "",
        "↑", "",
        "⁄", "/",
    )
    var err error
    for _, equ := range equs {
        fmt.Println(equ)
        terms := rgx.Split(equ, -1)
        ops := rgx.FindAllString(equ, -1)
        for i := 0; i < len(ops); i++ {
            ops[i] = strings.TrimSpace(ops[i])
        }
        coefs := make(map[int]float64)
        for i, term := range terms {
            s := strings.Split(term, "x")
            t := s[0]
            t = strings.TrimRight(t, "·× ")
            t = rep.Replace(t)
            c := 1.0
            inverse := false
            if t == "" || t == "+" || t == "-" {
                t += "1"
            }
            ix := strings.Index(t, "/")
            if ix == len(t)-1 {
                inverse = true
                t = t[0 : len(t)-1]
                c, err = strconv.ParseFloat(t, 64)
                check(err)
            } else if ix >= 0 {
                u := strings.Split(t, "/")
                m, err := strconv.ParseFloat(u[0], 64)
                check(err)
                n, err := strconv.ParseFloat(u[1], 64)
                check(err)
                c = m / n
            } else {
                c, err = strconv.ParseFloat(t, 64)
                check(err)
            }
            if i > 0 && ops[i-1] == "-" {
                c = -c
            }
            if c == -0.0 {
                c = 0
            }
            if len(s) == 1 {
                coefs[0] += c
                continue
            }
            u := s[1]
            u = strings.TrimSpace(u)
            if u == "" {
                p := 1
                if inverse {
                    p = -1
                }
                if c != 0 {
                    coefs[p] += c
                }
                continue
            }
            u = rep2.Replace(u)
            jx := strings.Index(u, "/")
            p := 1
            if jx >= 0 {
                v := strings.Split(u, "/")
                p, err = strconv.Atoi(v[0])
                if (err != nil) {
                    p = 1
                }
                d, err := strconv.ParseFloat(v[1], 64)
                check(err)
                c /= d
            } else {
                p, err = strconv.Atoi(strings.TrimSpace(u))
                if (err != nil) {
                    p = 1
                }
            }
            if inverse {
                p = -p
            }
            if c != 0 {
                coefs[p] += c
            }
        }
        printEquation(coefs)
    }
}
Output:
-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵
=> 0

1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x^5 - 2x**4 + 42x^3 + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰
=> 1 + 40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵

x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x
=> -40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵

⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝
=> ⅐x⁵ - x⁴ + 42⅕x³ + ⅑x - 40¾

Julia

The task allows the "polynomials" to be parsed to have negative exponents. This makes them Laurent polynomials, not ordinary polynomials.

import Base.print

struct LaurentPolynomial{T}
    powtocoef::Dict{Int,T}
    varname::Char
end

function tosuper(i)
    s = ""
    if i != 1
        if i < 0
            s = "\u207b"
            i = -i
        end
        s *= prod([reverseudi[x] for x in reverse(digits(i))])
    end
    return s
end

function print(io::IO, lp::LaurentPolynomial)
    if isempty(lp.powtocoef) || all(iszero, values(lp.powtocoef))
        print("0")
        return
    end
    firstterm = true
    dorev = minimum(keys(lp.powtocoef)) >= 0
    for p in sort!(collect(lp.powtocoef), lt =(a,b)->a[1]<b[1], rev=dorev)
        e, c = p[1], p[2]
        if c != 0
            if !firstterm
                print(c < 0 ? " - " : " + ")
                c = abs(c)
            end
            sfrac = get(tofracs, c - trunc(c), "")
            sint = string(Int(trunc(c)))
            s = round(c) == c ? sint : sfrac == "" ? string(c) : 
                sint == "0" ? sfrac : sint * sfrac
            print(c == 1 && e != 0 ? "" : s)
            e != 0 && print(lp.varname, tosuper(e))
            firstterm = false
        end
    end
end

const uniexp = [['\u2070', '\u00b9', '\u00b2', '\u00b3']; collect('\u2074':'\u2079')]
const allsuper = String(vcat(uniexp, ['\u207a', '\u207b']))
const udi = Dict([u => i-1 for (i, u) in enumerate(uniexp)])
const reverseudi = Dict([v => k for (k, v) in udi])
const uch = Dict(['\u207a' => '+', '\u207b' => '-', '\u00b7' => ' ',
    '\u00d7' => ' ', '\u2091' => '^', '\u23e8' => 'E'])
const fracs = Dict(['¼' => "$(1/4)", '½' => "$(1/2)", '¾' => "$(3/4)", '⅐' => "$(1/7)",
    '⅑' => "$(1/9)", '⅒' => "$(1/10)", '⅓' => "$(1/3)", '⅔' => "$(2/3)", '⅕' => "$(1/5)",
    '⅖' => "$(2/5)", '⅗' => "$(3/5)", '⅘' => "$(4/5)", '⅙' => "(1/6)", '⅚' => "$(5/6)",
    '⅛' => "$(1/8)", '⅜' => "$(3/8)", '⅝' => "$(5/8)", '⅞' => "$(7/8)", '↉' => "$(0/3)"])
const tofracs = Dict(0.25 => '¼', 0.5 => '½', 0.75 => '¾', 0.2 => '⅕')
allfrac = join(vcat(collect(keys(fracs))), "|")

utoascii(c) = (x = haskey(udi, c) ? Char('0' + udi[c]) : haskey(uch, c) ? uch[c] : c)

function fcoef(termstring, varname)
    m = match(r"(\d+)⁄(\d+)", termstring).captures
    return "$(parse(Float64, m[1])/parse(Float64, m[2]))$varname^0"
end

function xcoef(termstring, varname)
    m = match(Regex("(\\d+)$varname([^⁄\\+]*)⁄(\\d+)"), termstring).captures
    coe = parse(Float64, m[1])/parse(Float64, m[3])
    return "$(coe)$varname$(m[2])"
end

function fromvfrac(termstring)
    m = match(r"(\d*)(\D+)", termstring).captures
    s = string((m[1] == "" ? 0 : parse(Float64, m[1])) + parse(Float64, fracs[m[2][1]]))
end

function normalizeexpression(s, varname='x')
    s = replace(s, Regex("\\d*(" * allfrac * ")") => (x) -> fromvfrac(x))
    s = replace(replace(s, r"[^\+].+" => (x) -> "+$x"), r".+[^\+]$" => (x) -> "$x+")
    s = replace(replace(replace(s, r"\s+" => ""), r"\*\*|↑" => "^"), "\u23e8" => "e")
    s = replace(s, Regex(varname * "(?=[$allsuper]+)") => "$varname^")
    s = replace(prod([utoascii(c) for c in s]), r"\s+" => "")
    s = replace(s, r"[\-\+]+" => (x) -> isodd(count(y -> y == '-', x)) ? "+-" : "+")
    s = replace(replace(s, r"\^\+\-" => "^-"), r"\^\+" => "^")
    s = replace(s, Regex("(?<=[\\-\\+])$varname") => "1$varname")
    s = replace(s, r"(\d+)x([^⁄\+]*)⁄(\d+)" => (x) -> xcoef(x, varname))
    s = replace(s, r"(?<=[0-9])(,)(?=[0-9])" => "")
    s = replace(s, r"\d+⁄\d+" => (x) -> fcoef(x, varname))
    s = replace(s, r"\+([\d\.]+)\+" => SubstitutionString("+\\1$varname^0+"))
    return replace(s, Regex("(⁄|/)(" * varname * ")\\^?") =>
        (x) -> varname * (x[end] == '^' ? "^-" : "^-1"))
end

function topoly(s::String, varname='x')
    allcoef = Dict{Int,Float64}(0 => 0.0)
    s = normalizeexpression(s, varname)
    reg = Regex("[\\.\\-\\+\\deE\\/]*" * varname * "(?:\\^[\\d\\-]+)?(?=(?:[^\\+\\-]?(?:\\+|\\-)))")
    matched = collect(eachmatch(reg, s))
    pairs = [split(x.match, varname * "^") for x in matched]
    for p in pairs
        p[1] = replace(p[1], "+-" => "-")
        if length(p) == 1
            push!(p, "1")
            p[1] = replace(p[1], Regex(varname * "\$") => "")
        end
        coef, expo = tryparse(Float64, p[1]), tryparse(Int, p[2])
        coef = coef == nothing ? 1.0 : coef
        expo = expo == nothing ? 1 : expo
        if haskey(allcoef, expo)
            coef += allcoef[expo]
        end
        allcoef[expo] = coef
    end
    return LaurentPolynomial(allcoef, varname)
end

testcases = [
"1x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1",
"0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵",
"1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰",
"+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1",
"x^5 - 2x**4 + 42x^3 + 40x + 1",
"x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1",
"x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰",
"x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1",
"x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1",
"0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1",
"1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1",
"x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2",
"x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½",
]

for s in testcases
    println(lpad(s, 48), "  =>  ", topoly(s, 'x'))
end
Output:
                1x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵  =>  0
                    1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
              +x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
                   x^5 - 2x**4 + 42x^3 + 40x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
          x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
                x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰  =>  x⁻⁵ - 2x⁻⁴ + 42x⁻³ + 40x⁻¹ + 1
               x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
            x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
     0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
 1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1  =>  x⁵ - 2x⁴ + 42x³ + 40x + 1
              x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2  =>  x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½
                  x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½  =>  x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

Nim

Translation of: Go
import re, sequtils, strutils, tables
from unicode import toRunes

const

  Powers = [("0", "⁰"), ("1", "¹"), ("2", "²"), ("3", "³"), ("4", "⁴"),
            ("5", "⁵"), ("6", "⁶"), ("7", "⁷"), ("8", "⁸"), ("9", "⁹")]

  Fractions = [(".25", "¼"), (".5", "½"), (".75", "¾"), (".14285714285714", "⅐"),
               (".11111111111111", "⅑"), (".1", "⅒"), (".33333333333333", "⅓"),
               (".66666666666667", "⅔"), (".2", "⅕"), (".4", "⅖"), (".6", "⅗"),
               (".8", "⅘"), (".16666666666667", "⅙"), (".83333333333333", "⅚"),
               (".125", "⅛"), (".375", "⅜"), (".625", "⅝"), (".875", "⅞")]

  Reps1 = [(",", ""), (" ", ""), ("¼", ".25"), ("½", ".5"), ("¾", ".75"),
           ("⅐", ".14285714285714"), ("⅑", ".11111111111111"), ("⅒", ".1"),
           ("⅓", ".33333333333333"), ("⅔", ".66666666666667"), ("⅕", ".2"),
           ("⅖", ".4"), ("⅗", ".6"), ("⅘", ".8"), ("⅙", ".16666666666667"),
           ("⅚", ".83333333333333"), ("⅛", ".125"), ("⅜", ".375"), ("⅝", ".625"),
           ("⅞", ".875"), ("↉", ".0"), ("⏨", "e"), ("⁄", "/")]

  Reps2 = [("⁰", "0"), ("¹", "1"), ("²", "2"), ("³", "3"), ("⁴", "4"), ("⁵", "5"),
           ("⁶", "6"), ("⁷", "7"), ("⁸", "8"), ("⁹", "9"), ("⁻⁻", ""), ("⁻", "-"),
           ("⁺", ""), ("**", ""), ("^", ""), ("↑", ""), ("⁄", "/")]

  Eqs = [
    "-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹",
    "x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1",
    "0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵",
    "1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰",
    "+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1",
    "x^5 - 2x**4 + 42x^3 + 40x + 1",
    "x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1",
    "x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰",
    "x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1",
    "x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1",
    "0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1",
    "1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1",
    "x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2",
    "x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5",
    "x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x",
    "⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝"]

type Coefs = Table[int, float]

proc printEquation(coefs: Coefs) =
  stdout.write "=> "
  if coefs.len == 0:
    echo "0\n"
    return

  var max = int.low
  var min = int.high
  for k in coefs.keys:
    if k > max: max = k
    if k < min: min = k

  for p in countdown(max, min):
    var c = coefs.getOrDefault(p)
    if c != 0:
      if p < max:
        var sign = '+'
        if c < 0:
          sign = '-'
          c = -c
        stdout.write ' ', sign, ' '

      if c != 1 or (c == 1 and p == 0):
        var cs = $c
        cs.trimZeros()
        let ix = cs.find('.')
        if ix >= 0:
          let d = cs[ix..^1]
          for frac in Fractions:
            if d == frac[0]:
              cs = cs.replace(d, frac[1])
              break
        if cs[0] == '0' and cs.len > 1 and cs[1] != '.':
          cs = cs[1..^1]
        stdout.write cs

      if p != 0:
        var ps = ($p).multiReplace(Powers)
        if ps == "¹": ps = ""
        stdout.write 'x', ps

  echo '\n'


template getFloat(s: string): float =
  try:
    s.parseFloat()
  except ValueError:
    quit "Expected float, got: " & s, QuitFailure

let rgx = re"\s+(\+|-)\s+"

for eq in Eqs:
  echo eq
  let terms = eq.split(rgx)
  let ops = toSeq(eq.findAll(rgx)).mapIt(it.strip())
  var coefs: Coefs
  for i, term in terms:

    let s = term.split("x")
    var t = s[0]
    t = unicode.strip(t, runes = "·× ".toRunes)
    t = t.multiReplace(Reps1)
    var c = 1.0
    var inverse = false
    if t == "" or t == "+" or t == "-": t &= '1'

    let ix = t.find('/')
    if ix == t.high:
      inverse = true
      t.setLen(t.high)
      c = t.getFloat()
    elif ix >= 0:
      let u = t.split('/')
      let m = u[0].getFloat()
      let n = u[1].getFloat()
      c = m / n
    else:
      c = t.getFloat()
    if i > 0 and ops[i - 1] == "-": c = -c
    if c == -0.0: c = 0.0

    if s.len == 1:
      coefs[0] = coefs.getOrDefault(0) + c
      continue

    var u = s[1].strip()
    if u.len == 0:
      let p = if inverse: -1 else: 1
      if c != 0:
        coefs[p] = coefs.getOrDefault(p) + c
      continue

    u = u.multiReplace(Reps2)
    let jx = u.find('/')
    var p: int
    if jx >= 0:
      let v = u.split('/')
      p = try: v[0].parseInt() except ValueError: 1
      let d = v[1].getFloat()
      c /= d
    else:
      p = try: u.strip().parseInt() except ValueError: 1
    if inverse: p = -p
    if c != 0: coefs[p] = coefs.getOrDefault(p) + c

  printEquation(coefs)
Output:
-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵
=> 0

1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x^5 - 2x**4 + 42x^3 + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰
=> 1 + 40x-¹ + 42x-³ - 2x-⁴ + x-⁵

x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x
=> -40x-¹ + 42x-³ - 2x-⁴ + x-⁵

⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝
=> ⅐x⁵ - x⁴ + 42⅕x³ + ⅑x - 40¾

Phix

To simplify this task I first created a test file (save as utf8, Unicode_polynomial_equation.e):

constant raw_text = """
-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
===> x^5 - 2x^4 + 42x^3 + 40x + 1
x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵
==> 0
1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x^5 - 2x**4 + 42x^3 + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰
==> 1 + 40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵
===> 1 + 40x^-1 + 42x^-3 - 2x^-4 + x^-5
x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2
==> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½
===> x^5 - 0.5x^4 + 101.25x^3 + 100.75x + 2.5
x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½
==> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½
x<sup>5</sup> - 2x<sup>4</sup> + 42x<sup>3</sup> + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x^5 - 2x^4 + 42x^3 + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x⁵ - 2x⁴ + 42x³ + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
x5 - 2x4 + 42x3 + 40x + 1
==> x⁵ - 2x⁴ + 42x³ + 40x + 1
"""
global function get_lines()
    return split(raw_text,"\n")
end function

Explanation:
On finding a line beginning with "==> ", parse the previous line and check that the output matches the rest of the line.
Lines beginning "===> " contain alternative (valid) representations of the previous line, for use when uni_frac is false. Note these equivalences can then be used further on, eg the result on line 5 maps to line 3 via 2, not that it matters should the same equivalent be defined twice, and also note that "==> " should always be the unicode version and "===> " the ascii one.

Obviously this is not well tested and //will// fail on the next thing thrown at it, but it shows the basic approach. Error handling omitted using that good ol' standby excuse "for clarity".

Note that space is not skipped when getting exponents, otherwise it might wrongly treat say "x + 2" as "x^2", since we /are/ allowing exponentiation symbols to be omitted. At the start of parse() we strip spaces around '^' (etc) to help with that.

Processing the input in utf32 form is a complete no-brainer; whereas constructing utf8 output is a little more subtle, but avoids creating a utf32 "string" just because all bytes happen to be less than 255, which would not necessarily be valid utf8/32.

-- demo\rosetta\Unicode_polynomial_equation.exw
constant uni_frac = platform()!=WINDOWS -- if true output unicode superscripts and vulgar fractions

constant UTF8BOM = {#EF,#BB,#BF},
         SPTWO = #00B2,         -- superscript two
         STHRE = #00B3,         -- superscript three
         MIDOT = #00B7,         -- middle dot
         SPONE = #00B9,         -- superscript one
         QUART = #00BC,         -- one quarter
         AHALF = #00BD,         -- one half
         THQTR = #00BE,         -- three quarters
         MULTY = #00D7,         -- multiplication sign
         DIVDE = #2044,         -- division sign
         SZERO = #2070,         -- superscript zero
         SFOUR = #2074,         -- superscript four
         SFIVE = #2075,         -- superscript five
         SPSIX = #2076,         -- superscript six
         SSEVN = #2077,         -- superscript seven
         SEGHT = #2078,         -- superscript eight
         SNINE = #2079,         -- superscript nine
         SPLUS = #207A,         -- superscript plus
         SMNUS = #207B,         -- superscript minus
         SVNTH = #2150,         -- one seventh
         NINTH = #2151,         -- one ninth
         TENTH = #2152,         -- one tenth
         THIRD = #2153,         -- one third
         TWTHD = #2154,         -- two thirds
         FIFTH = #2155,         -- one fifth
         TWFTH = #2156,         -- two fifths
         THFTH = #2157,         -- three fifths
         FRFTH = #2158,         -- four fifths
         SIXTH = #2159,         -- one sixth
         FVSIX = #215A,         -- five sixths
         EIGTH = #215B,         -- one eigth
         THEGH = #215C,         -- three eigths
         FVEGH = #215D,         -- five eigths
         UPARW = #2191,         -- uparrow
         BASET = #23E8,         -- base 10
         SPACE = ' ',           -- space
             T = 10             -- align nxt tbl

constant {vulgar_fractions,unicode_vulgar_fractions} = columnize({{{1,4},QUART},
                                                                  {{1,2},AHALF},
                                                                  {{3,4},THQTR},
                                                                  {{1,7},SVNTH},
                                                                  {{1,9},NINTH},
                                                                  {{1,T},TENTH},
                                                                  {{1,3},THIRD},
                                                                  {{2,3},TWTHD},
                                                                  {{1,5},FIFTH},
                                                                  {{2,5},TWFTH},
                                                                  {{3,5},THFTH},
                                                                  {{4,5},FRFTH},
                                                                  {{1,6},SIXTH},
                                                                  {{5,6},FVSIX},
                                                                  {{1,8},EIGTH},
                                                                  {{3,8},THEGH},
                                                                  {{5,8},FVEGH}})

constant EXPONENTS = {SZERO,SPONE,SPTWO,STHRE,SFOUR,SFIVE,SPSIX,SSEVN,SEGHT,SNINE}

function skip(sequence s, integer sdx, sequence set)
    while sdx<=length(s) 
      and find(s[sdx],set) do
        sdx += 1
    end while
    return sdx
end function

function get_sign(sequence s, integer sdx, bool allow_superscripts)
    integer sgn = +1, ch
    for i=sdx to length(s) do
        ch = s[sdx]
        if allow_superscripts then
            ch = iff(ch=SPLUS?'+':
                 iff(ch=SMNUS?'-':
                 iff(ch=SPACE?'?':ch))) -- (do not skip spaces, see note)
        end if
        if ch!='+'
        and ch!=' ' then
            if ch!='-' then exit end if
            sgn *= -1
        end if
        sdx += 1
    end for
    return {sgn,sdx}
end function

function get_num(sequence s, integer sdx, atom n=0, bool allow_superscripts=false, as_fraction=false)
    integer sgn = +1, ch, f, e10, d = 1
    atom p10 = 10
    bool none = not as_fraction -- (cope with "x" == "1x^1" and != "0x^0")
                                -- (but not when processing the "34" of "12.34", obvs)
    if not as_fraction then
        {sgn,sdx} = get_sign(s,sdx,allow_superscripts)
    end if
    if not allow_superscripts then
        sdx = skip(s,sdx,{' '}) 
    end if
    for i=sdx to length(s) do
        ch = s[sdx]
        if ch>='0' and ch<='9' then
            if as_fraction then
                n += (ch-'0')/p10
                p10 *= 10
            else
                n = n*10+ch-'0'
            end if
            none = false
        elsif allow_superscripts then
            f = find(ch,EXPONENTS)
            if f=0 then exit end if
            n = n*10+f-1
            none = false
        elsif not find(ch," ,") then
            exit
        end if
        sdx += 1
    end for
    if not allow_superscripts then
        if find(ch,{'e','E',BASET}) then
            {e10,f,sdx} = get_num(s,sdx+1)
            if f!=1 then ?9/0 end if
            n *= power(10,e10)
        elsif ch='.' then
            if as_fraction then ?9/0 end if
            {n,f,sdx} = get_num(s,sdx+1,n,as_fraction:=true)
            if f!=1 then ?9/0 end if
            none = false
        else
            f = find(ch,unicode_vulgar_fractions)
            if f!=0 then
                if as_fraction then ?9/0 end if
                integer {vfn,vfd} = vulgar_fractions[f]
                if uni_frac then
                    n = n*vfd + vfn
                    if d!=1 then ?9/0 end if
                    d = vfd
                else
                    n += vfn/vfd
                end if
                sdx += 1
                none = false
            end if
        end if
    end if
    if none then n = 1 end if
    n *= sgn
    return {n,d,sdx}
end function

function get_term(sequence s, integer sdx)
    integer last_sdx = sdx, -- (infinite loop check/prevent)
            e = 0
    atom c, d = 1, f
    bool sdiv = false
    {c,d,sdx} = get_num(s,sdx)
    sdx = skip(s,sdx,{' ',MIDOT,MULTY})
    if sdx<=length(s) 
    and (s[sdx]='/' or
         s[sdx]=DIVDE) then
        sdx += 1
        if sdx<=length(s) and s[sdx]!='x' then
            {d,f,sdx} = get_num(s,sdx)
            if f!=1 then ?9/0 end if
            if not uni_frac then
                c /= d
                d = 1
            end if
        else
            sdiv = true
        end if
    end if
    if sdx<=length(s) and s[sdx]='x' then
        sdx = skip(s,sdx+1,{'^','*',SPLUS,UPARW})
        {e,f,sdx} = get_num(s,sdx,allow_superscripts:=true)
        if f!=1 then ?9/0 end if
        if sdiv then e = -e end if
    else
        if sdiv then ?9/0 end if
    end if
    if sdx<=length(s) 
    and (s[sdx]='/' or
         s[sdx]=DIVDE) then
        if d!=1 then ?9/0 end if
        {d,f,sdx} = get_num(s,sdx+1)
        if f!=1 then ?9/0 end if
        if not uni_frac then
            c /= d
            d = 1
        end if
    end if
    if sdx=last_sdx then ?9/0 end if
    return {e,c,d,sdx}
end function

function unicode_superscripts(integer e)
-- display helper
    string res = ""
    if e>9 then
        res = unicode_superscripts(floor(e/10))
        e = remainder(e,10)
    end if
    res &= utf32_to_utf8({EXPONENTS[e+1]})
    return res
end function

enum EXP, COEF, FRAC    -- contents of terms[i]

function poly(sequence terms)
-- display helper
    string r = ""
    for t=length(terms) to 1 by -1 do
        {integer e, atom c, integer f} = terms[t]
        if c!=0 then
            if c=1 and f=1 and e!=0 then
                r &= iff(r=""? "":" + ")
            elsif c=-1 and f=1 and e!=0 then
                r &= iff(r=""?"-":" - ")
            else
                if r!="" then
                    r &= iff(c<0?" - ":" + ")
                    c = abs(c)
                end if
                if f!=1 then    -- (hence/only when uni_frac==true)
                    integer k = find({remainder(c,f),f},vulgar_fractions)
                    if k then
                        c = floor(c/f)
                        if c!=0 then r &= sprintf("%d",c) end if
                        r &= utf32_to_utf8({unicode_vulgar_fractions[k]})
                    else
                        r &= sprintf("%g",c/f)
                    end if
                else
                    r &= sprintf("%g",c)
                end if
            end if
            if e!=0 then
                r &= 'x'
                if e!=1 then
                    if uni_frac then
                        if e<0 then
                            r &= utf32_to_utf8({SMNUS})
                            e = -e
                        end if
                        r &= unicode_superscripts(e)
                    else
                        r &= sprintf("^%d",e)
                    end if
                end if
            end if
        end if
    end for
    if r="" then r="0" end if
    return r
end function

function parse(sequence s)
    sequence terms = {}
    integer sdx = 1, e, f
    atom c
    s = match_replace("<sup>",s,"^")
    s = match_replace("</sup>",s,"")
    s = match_replace("**",s,"^")
    s = match_replace(" ^",s,"^")
    s = match_replace("^ ",s,"^")
    s = match_replace({' ',SPLUS},s,{SPLUS})
    s = match_replace({' ',SMNUS},s,{SMNUS})
    for i=1 to length(EXPONENTS) do
        e = EXPONENTS[i]
        s = match_replace({' ',e},s,{e})
    end for 
    while sdx<=length(s) do
        {e,c,f,sdx} = get_term(s,sdx)
        if c!=0 then -- (aside: +5 -5 may yet leave c==0)
            terms = append(terms,{e,c,f})
        end if
    end while
    terms = sort(terms)
    -- merge, eg "10x^2 + 10x^2" -> 20x^2
    for i=length(terms) to 2 by -1 do
        if terms[i][EXP] = terms[i-1][EXP] then
            if terms[i-1][FRAC]!=terms[i][FRAC] then
                ?9/0 -- placeholder for more code
            else
                terms[i-1][COEF] += terms[i][COEF]
            end if
            terms[i..i] = {} -- (delete 2nd)
        end if
    end for
    return poly(terms)
end function

include Unicode_polynomial_equation.e

sequence alts = {},     -- (unicode versions)
         altn = {},     -- (idx of ascii equivalents)
--       lines = read_lines("Unicode_polynomial_equation.txt")
         lines = get_lines()

if lines[1][1..3] = UTF8BOM then
    -- remove/ignore any utf8 byte order mark
    lines[1] = lines[1][4..$]
end if

for i=2 to length(lines) do
    if length(lines[i])>5 
    and lines[i][1..5] = "===> " then
        alts = append(alts,lines[i-1][5..$])
        altn = append(altn,i)
    end if
end for

for i=2 to length(lines) do
    if length(lines[i])>4 
    and lines[i][1..4] = "==> " then
        sequence line = utf8_to_utf32(lines[i-1]),
                 res = parse(line),
                 expected = lines[i][5..$]
        if res=expected then
            -- (res is the unicode version)
            if platform()!=WINDOWS or res="0" then
                printf(1,"%2d: %40s   ok\n",{i-1,res})
            else
                -- (unicode output on windows consoles is fiddly...)
                printf(1,"%2d: ok\n",i-1)
            end if
        else
            integer k = find(expected,alts)
            if k and res=lines[altn[k]][6..$] then
                -- (res is the ascii equivalent)
                printf(1,"%2d: %40s   ok\n",{i-1,res})
            else
                printf(1,"%d: error - %s\n",{i-1,res})
            end if
        end if
    end if
end for

?"done"
{} = wait_key()
Output:

uni_frac = false (ie windows)

 1:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
 4:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
 6:                                        0   ok
 8:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
10:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
12:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
14:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
16:       1 + 40x^-1 + 42x^-3 - 2x^-4 + x^-5   ok
19:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
21:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
23:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
25:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
27: x^5 - 0.5x^4 + 101.25x^3 + 100.75x + 2.5   ok
30: x^5 - 0.5x^4 + 101.25x^3 + 100.75x + 2.5   ok
32:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
34:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
36:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok
38:             x^5 - 2x^4 + 42x^3 + 40x + 1   ok

uni_frac = true (linux or pwa/p2js only, unless you have managed to find and install a decent windows console unicode font, which I haven't)

 1:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
 4:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
 6:                                        0   ok
 8:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
10:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
12:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
14:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
16:           1 + 40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵   ok
19:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
21:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
23:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
25:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
27:           x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½   ok
30:           x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½   ok
32:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
34:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
36:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok
38:                x⁵ - 2x⁴ + 42x³ + 40x + 1   ok

Wren

Translation of: Go
Library: Wren-pattern
import "./pattern" for Pattern

var powers = [
    ["0", "⁰"],
    ["1", "¹"],
    ["2", "²"],
    ["3", "³"],
    ["4", "⁴"],
    ["5", "⁵"],
    ["6", "⁶"],
    ["7", "⁷"],
    ["8", "⁸"],
    ["9", "⁹"],
    ["-", "⁻"]
]

var fractions = [
    [".25", "¼"],
    [".5", "½"],
    [".75", "¾"],
    [".14285714285714", "⅐"],
    [".11111111111111", "⅑"],
    [".1", "⅒"],
    [".33333333333333", "⅓"],
    [".66666666666667", "⅔"],
    [".2", "⅕"],
    [".4", "⅖"],
    [".6", "⅗"],
    [".8", "⅘"],
    [".16666666666667", "⅙"],
    [".83333333333333", "⅚"],
    [".125", "⅛"],
    [".375", "⅜"],
    [".625", "⅝"],
    [".875", "⅞"]
]

var printEquation = Fn.new { |coefs|
    System.write("=> ")
    if (coefs.isEmpty) {
        System.print("0\n")
        return
    }
    var keys = coefs.keys.toList
    var max = keys[0]
    var min = keys[0]
    for (k in keys.skip(1)) {
        if (k > max) max = k
        if (k < min) min = k
    }
    for (p in max..min) {
        var c = coefs[p] ? coefs[p] : 0
        if (c != 0) {
            if (p < max) {
                var sign = "+"
                if (c < 0) {
                    sign = "-"
                    c = -c
                }
                System.write(" %(sign) ")
            }
            if (c != 1 || (c == 1 && p == 0)) {
                var cs = c.toString
                var ix = cs.indexOf(".")
                if (ix >= 0) {
                    var dec = cs[ix..-1]
                    for (frac in fractions) {
                        if (dec == frac[0]) {
                            cs = cs[0...ix] + frac[1]
                            break
                        }
                    }
                }
                if (cs[0] == "0" && cs.count > 1 && cs[1] != ".") cs = cs[1..-1]
                System.write(cs)
            }
            if (p != 0) {
                var ps = p.toString
                for (power in powers) ps = ps.replace(power[0], power[1])
                if (ps == "¹") ps = ""
                System.write("x%(ps)")
            }
        }
    }
    System.print("\n")
}

var equs = [
    "-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹",
    "x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1",
    "0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵",
    "1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰",
    "+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1",
    "x^5 - 2x**4 + 42x^3 + 40x + 1",
    "x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1",
    "x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰",
    "x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1",
    "x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1",
    "0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1",
    "1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1",
    "x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2",
    "x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5",
    "x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x",
    "⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝"
]

var pattern = Pattern.new("+1/s/n+1/s")

var reps = [
    [",", ""],
    [" ", ""],
    ["¼", ".25"],
    ["½", ".5"],
    ["¾", ".75"],
    ["⅐", ".14285714285714"],
    ["⅑", ".11111111111111"],
    ["⅒", ".1"],
    ["⅓", ".33333333333333"],
    ["⅔", ".66666666666667"],
    ["⅕", ".2"],
    ["⅖", ".4"],
    ["⅗", ".6"],
    ["⅘", ".8"],
    ["⅙", ".16666666666667"],
    ["⅚", ".83333333333333"],
    ["⅛", ".125"],
    ["⅜", ".375"],
    ["⅝", ".625"],
    ["⅞", ".875"],
    ["↉", ".0"],
    ["⏨", "e"],
    ["⁄", "/"]
]

var reps2 = [
    ["⁰", "0"],
    ["¹", "1"],
    ["²", "2"],
    ["³", "3"],
    ["⁴", "4"],
    ["⁵", "5"],
    ["⁶", "6"],
    ["⁷", "7"],
    ["⁸", "8"],
    ["⁹", "9"],
    ["⁻⁻", ""],
    ["⁻", "-"],
    ["⁺", ""],
    ["**", ""],
    ["^", ""],
    ["↑", ""],
    ["⁄", "/"]
]

for (equ in equs) {
    System.print(equ)
    var terms = pattern.splitAll(equ)
    var ops = pattern.findAll(equ).map { |m| m.text.trim() }.toList
    var coefs = {}
    var i = 0
    for (term in terms) {
        var s = term.split("x")
        var t = s[0].trimEnd("·× ")
        for (rep in reps) t = t.replace(rep[0], rep[1])
        var c = 1
        var inverse = false
        if (t == "" || t == "+" || t == "-") t = t + "1"
        var ix = t.indexOf("/")
        if (ix == t.count - 1) {
            inverse = true
            t = t[0..-2]
            c = Num.fromString(t)
        } else if (ix >= 0) {
            var u = t.split("/")
            var m = Num.fromString(u[0])
            var n = Num.fromString(u[1])
            c = m / n
        } else {
            c = Num.fromString(t)
        }
        if (i > 0 && ops[i-1] == "-") c = -c
        if (c == -0) c = 0
        var cont = false
        if (s.count == 1) {
            coefs[0] = coefs[0] ? coefs[0] + c : c
            cont = true
        }
        if (!cont) {
            var u = s[1].trim()
            if (u == "") {
                var p = 1
                if (inverse) p = -1
                if (c != 0) coefs[p] = coefs[p] ? coefs[p] + c : c
                cont = true
            }
            if (!cont) {
                for (rep in reps2) u = u.replace(rep[0], rep[1])
                var jx = u.indexOf("/")
                var p = 1
                if (jx >= 0) {
                    var v = u.split("/")
                    p = Num.fromString(v[0])
                    if (p == null) p = 1
                    var d = Num.fromString(v[1])
                    c = c / d
                } else {
                    p = Num.fromString(u.trim())
                    if (p == null) p = 1
                }
                if (inverse) p = -p
                if (c != 0) coefs[p] = coefs[p] ? coefs[p] + c : c
            }
        }
        i = i + 1
    }
    printEquation.call(coefs)
}
Output:
-0.00x⁺¹⁰ + 1.0·x ** 5 + -2e0x^4 + +0,042.00 × x ⁺³ + +.0x² + 20.000 000 000x¹ - -1x⁺⁰ + .0x⁻¹ + 20.x¹
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 42x³ + 0x² + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0e+0x⁰⁰⁷ + 00e-00x + 0x + .0x⁰⁵ - 0.x⁴ + 0×x³ + 0x⁻⁰ + 0/x + 0/x³ + 0x⁻⁵
=> 0

1x⁵ - 2x⁴ + 42x³ + 40x + 1x⁰
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

+x⁺⁵ + -2x⁻⁻⁴ + 42x⁺⁺³ + +40x - -1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x^5 - 2x**4 + 42x^3 + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x↑5 - 2.00·x⁴ + 42.00·x³ + 40.00·x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁻⁵ - 2⁄x⁴ + 42x⁻³ + 40/x + 1x⁻⁰
=> 1 + 40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵

x⁵ - 2x⁴ + 42.000 000x³ + 40x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - 2x⁴ + 0,042x³ + 40.000,000x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

0x⁷ + 10x + 10x + x⁵ - 2x⁴ + 42x³ + 20x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

1E0x⁵ - 2,000,000.e-6x⁴ + 4.2⏨1x³ + .40e+2x + 1
=> x⁵ - 2x⁴ + 42x³ + 40x + 1

x⁵ - x⁴⁄2 + 405x³⁄4 + 403x⁄4 + 5⁄2
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁵ - 0.5x⁴ + 101.25x³ + 100.75x + 2.5
=> x⁵ - ½x⁴ + 101¼x³ + 100¾x + 2½

x⁻⁵ - 2⁄x⁴ + 42x⁻³ - 40/x
=> -40x⁻¹ + 42x⁻³ - 2x⁻⁴ + x⁻⁵

⅐x⁵ - ⅓x⁴ - ⅔x⁴ + 42⅕x³ + ⅑x - 40⅛ - ⅝
=> ⅐x⁵ - x⁴ + 42⅕x³ + ⅑x - 40¾