cube root estimation

sqrt3

(: 
   Estimate the cube root of a number.
   WARNINGS: 1) Does not work well with many digits due to xs:double size limitation.
             2) Does not generate complex numbers.
   based on http://www4.wittenberg.edu/academics/mathcomp/bjsdir/CubeRootTalk.pdf
:)
declare function local:sqrt3($number as xs:double) {

  let $answer := ()
  
  let $negative := $number lt 0 
  let $number := if ($negative) then $number * -1 else $number
  let $numLength := fn:string-length(fn:replace(fn:string($number), "[^0-9]", ""))

  let $hasDecimal := fn:matches(xs:string($number), "\.")
  let $beforeDecimal := fn:replace(xs:string($number), "^([0-9]+?)\.[0-9]*$", "$1")
  let $afterDecimal := if ($hasDecimal) then fn:replace(xs:string($number), "^[0-9]+?\.([0-9]*)$", "$1") else ()

  (: Group numbers by threes, starting before then after the decimal point :)
  let $beforeGroups := fn:reverse(
    for $i in (1 to xs:integer(math:ceil(fn:string-length($beforeDecimal) div 3)))
    return xs:double(fn:replace($beforeDecimal, "^.*?([0-9]{1,3})[0-9]{"||($i - 1) * 3||"}$", "$1"))
  )
  let $beforeGroups :=
    if (fn:count($beforeGroups) eq 1 and $beforeDecimal eq "0") then () else $beforeGroups
  let $afterGroups := 
    for $i in (1 to xs:integer(math:ceil(fn:string-length($afterDecimal) div 3)))
    return xs:double(fn:replace($afterDecimal, "^[0-9]{"||($i - 1) * 3||"}([0-9]{1,3}).*?$", "$1"))
  let $numPieces := ($beforeGroups, $afterGroups)
  let $newLast := 
    let $num := xs:string($numPieces[fn:last()])
    return 
      if (fn:string-length($num) eq 1) then xs:double($num || "00")
      else if (fn:string-length($num) eq 2) then xs:double($num || "0")
      else ()
  let $numPieces := if ($newLast) then ($numPieces[1 to (fn:last() - 1)], $newLast) else $numPieces
  let $numPieces := (
    $numPieces
    ,
    let $precisionDeficit := ($numLength - fn:count($numPieces))
    return 
      if ($precisionDeficit gt 0) then
        for $i in (1 to $precisionDeficit)
        return "000"
      else ()
  )
  
  (: Find the largest cube of a single digit number less than the first group :)
  let $largestCube := 1
  let $_ := 
    for $j in (2 to 9)
    return
      if (math:pow($j, 3) le $numPieces[1]) then 
        xdmp:set($largestCube, $j)
      else ()

  let $_ := xdmp:set($answer, ($answer, xs:string($largestCube)))
  let $diff := $numPieces[1] - math:pow($largestCube, 3)
  let $diffNext := ()
  let $numAnswer := ()
  let $combo := ()

  (: Find the largest factor of a single digit number less than all the rest of the groups :)
  let $_ :=
    try {
      for $i in (2 to fn:count($numPieces) )
      let $numAnswer := xs:double(fn:string-join($answer)) * 10
      let $_ := xdmp:set($diffNext, xs:double(xs:string($diff) || $numPieces[$i]))
      let $largestFactor := 1
      let $_ := 
        for $j in (1 to 9)
        let $_ := xdmp:set($combo, $j * ((3 * math:pow($numAnswer,2)) + ((3 * $numAnswer) * $j) + math:pow($j,2)))
        return
          if ($combo le $diffNext) then 
            xdmp:set($largestFactor, $j)
          else ()
    
      let $_ := xdmp:set($answer, ($answer, xs:string($largestFactor)))
      return xdmp:set($diff, ($diffNext - ($largestFactor * ((3 * math:pow($numAnswer,2)) + ((3 * $numAnswer) * $largestFactor) + math:pow($largestFactor,2)))) ) 
  } catch ($exception) {}
  
  return 
    xs:double(fn:string-join(
      for $num at $i in $answer
      return (
        if ($i - 1 eq fn:count($beforeGroups)) then 
          if (fn:count($beforeGroups) eq 1 and $beforeDecimal eq "0") then "0."
          else "." 
        else () 
        ,
        $num
      )
    )) * (if ($negative) then -1 else 1)

};