JavaScript (ES6), 257 252 249 235 bytes

Saved 3 bytes thanks to @Shaggy

s=>eval(s.split` `.map(w=>(i='zeonwohrr44fx3n5t54nn3leel8tou7fn7n98etetwthfofisiseeinihuplmitidiby'.match(/../g).findIndex(x=>~(w+w.length+w).search(x)))>28?n+' '+'+-*/ '[n='',i-29]:(n=+n+(i<28?i<20?i:i*10-180:n*99),''),n='').join``+n)

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How?

Once the input string has been split into words, each word $W$ of length $L$ is converted to an index $i$ by looking for the first 2-character key substring that can be found in $W+L+W$.

Numbers

 index | word              | word + length + word         | key substring

-------+-------------------+------------------------------+---------------

    0  | "zero"            | "zero4zero"                  | "ze"

    1  | "one"             | "one3one"                    | "on"

    2  | "two"             | "two3two"                    | "wo"

    3  | "three"           | "three5three"                | "hr"

    4  | "four"            | "four4four"                  | "r4"

    5  | "five"            | "five4five"                  | "4f"

    6  | "six"             | "six3six"                    | "x3"

    7  | "seven"           | "seven5seven"                | "n5"

    8  | "eight"           | "eight5eight"                | "t5"

    9  | "nine"            | "nine4nine"                  | "4n"

   10  | "ten"             | "ten3ten"                    | "n3"

   11  | "eleven"          | "eleven6eleven"              | "le"

   12  | "twelve"          | "twelve6twelve"              | "el"

   13  | "thirteen"        | "thirteen8thirteen"          | "8t"

   14  | "fourteen"        | "fourteen8fourteen"          | "ou"

   15  | "fifteen"         | "fifteen7fifteen"            | "7f"

   16  | "sixteen"         | "sixteen7sixteen"            | "n7"

   17  | "seventeen"       | "seventeen9seventeen"        | "n9"

   18  | "eighteen"        | "eighteen8eighteen"          | "8e"

   19  | "nineteen"        | "nineteen8nineteen"          | "te"

   20  | "twenty"          | "twenty6twenty"              | "tw"

   21  | "thirty"          | "thirty6thirty"              | "th"

   22  | "forty"           | "forty5forty"                | "fo"

   23  | "fifty"           | "fifty5fifty"                | "fi"

   24  | "sixty"           | "sixty5sixty"                | "si"

   25  | "seventy"         | "seventy7seventy"            | "se"

   26  | "eighty"          | "eighty6eighty"              | "ei"

   27  | "ninety"          | "ninety6ninety"              | "ni"

   28  | "hundred"         | "hundred7hundred"            | "hu"

Operators

 index | word              | word + length + word         | key substring

-------+-------------------+------------------------------+---------------

   29  | "plus"            | "plus4plus"                  | "pl"

   30  | "minus"           | "minus5minus"                | "mi"

   31  | "times" or "time" | "times5times" or "time4time" | "ti"

   32  | "divided"         | "divided7divided"            | "di"

   33  | "by"              | "by2by"                      | "by"

Interpretation

The current number is stored in $n$, which is initialized to an empty string. Given the index $i$ of the next word, the following logic applies:

i > 28 ?                  // if the word is an operator:

  n +                     //   append n (which is either an empty string or a number)

  ' ' +                   //   append a space

  '+-*/ '[n = '', i - 29] //   reset n to an empty string and append the operator

                          //   the useless keyword 'by' is translated into a harmless space

: (                       // else:

    n =                   //   update n:

      +n + (              //     force the coercion of the current value of n to a number

        i < 28 ?          //     if the word is not 'hundred':

          i < 20 ?        //       if the value of the word is less than 'twenty':

            i             //         add i

          :               //       else:

            i * 10 - 180  //         add i * 10 - 180 (e.g. 'fifty' -> 23 * 10 - 180 = 50)

        :                 //     else:

          n * 99          //       multiply n by 100 by adding 99 * n to itself

      ),                  //

    ''                    //   remove this word from the original string

  )                       //

Perl 6, 170 139 129 128 124 bytes

-13 bytes thanks to nwellnhof!

{S:g/(<:N>+)+%s/({'+'X$0})/.EVAL}o{S:g/" ҈"/00/}o{TR"⁢ʼn؊⟠"*/൰ "}o*.words>>.&{first *.uniname.comb(.uc),' '..'�'}

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unival to the rescue again! This is (currently) even beating golfing languages like 05AB1E!

Explanation:

*.words     # Split by word boundaries (in this case spaces)

       >>.{                            }  # Map each word to

           first             ,' '..'�'   # The first character where:

                 *.uniname      # The unicode name of that character

                                # e.g. DIGIT FIVE

                  .comb(.uc)    # Contains the uppercase of the word

{             }o  # Pass this list to another function

                  # That converts the list to a string

 TR"⁢ʼn؊⟠"*/൰ "    #"# And parse out the garbage characters that were wrong

                  # INVISIBLE TIMES => "*"

                  # LOZENGE DIVIDED BY HORIZONTAL RULE => "/"

                  # ARABIC-INDIC PER TEN THOUSAND SIGN => "൰" (value ten)

                  # LATIN SMALL LETTER N PRECEDED BY APOSTROPHE => " "

{S:g/" ҈"/00/}o   # Replace the character for "hundred" with 00



{                                }o   # And finally combine with

 S:g/(<:N>+)+%s/   # Substitute each number-like character separated by spaces

                /({'+'X$0})/   # With the numbers prefixed by '+'s, in brackets

               # This works because Perl 6 supports numeric unicode literals, like

               # ፳ => 20, ፴ => 30, ፺ => 90, etc.

                            .EVAL   # And finally evaluate the whole expression

Python 2, 333 ... 284 277 275 bytes

lambda s:eval(''.join((list('+-/*')+[`N(w,0)*100+N(w,2)`])['pmdt'.find(w)]for w in re.split(' *(?:(p|m)|(t|d)i|by).*? ',s)if w))

N=lambda x,y:sum(10*(w[-3:]in'lveen')+'zeontwthfofisiseeiniteel'.find(w[:2])/2*10**('y'in w)for w in x.rpartition('hundred')[y].split())

import re

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Wolfram Language 95 94 bytes

Interpreter["SemanticExpression"][#~StringReplace~{"time "->"times ","plus minus"->"minus"}]&

# represents the input for the pure function.

If necessary,StringReplace replaces "time " with times ", "plus minus" with "minus". Interpreter["SemanticExpression"] does all the rest.

05AB1E, 166 147 141 bytes

„byK“¡×€µ‚•„í†ìˆÈŒšï¿Ÿ¯¥Š—¿áÓÁÏ“#vyN:}'…§D¦‚'+T«:'°¡'*т«:.•4º»Ÿć'Rþн•ŽH$S£Ž9oS:„tyD¦‚„ *T«:©“‰´Øè„Æ€ººß“#UX¡εð¡õK2ôJ.EO}®áX"+-**/"S:S‚ζJJ„--'+:.E

Way too long.. Will try to golf it down from here.

-4 byte thanks to @Emigna.

-2 bytes thanks to @JoKing.

Try it online or verify all test cases.

Explanation:

„byK                 # Remove "by" from the (implicit) input-string

“¡×€µ‚•„í†ìˆÈŒšï¿Ÿ¯¥Š—¿áÓÁÏ“

                     # Push string "zero one two three four five six seven eight nine ten eleven twelve"

 #                   # Split by spaces

  v   }              # Loop over each of these items:

   yN:               #  Replace the item with its 0-indexed index

'tK                 '# Remove all "t"

'…§¦                '# Push string "teen", and remove the first character: "een"

    '+              '# Push string "+"

      T«             # Append 10: "+10"

        :            # Replace all "een" with "+10"

'°¡                 '# Push string "hundred"

   '*               '# Push string "*"

     т«              # Append 100: "*100"

       :             # Replace all "hundred" with "*100"

.•4º»Ÿć'Rþн•        '# Push string "wenhirforfif"

            3ô       # Split the string into parts of size 3: ["wen","hir","for","fif"]

              Ž9o    # Push integer 2345

                 S   # Split to digits: [2,3,4,5]

                  :  # Replace each

'y                  '# Push string "y"

  „ *                # Push string " *"

     T«              # Append 10: " *10"

       :             # Replace all "y" with " *10"

©                    # Save the current string in the register (without popping)

 “‰´Øè„Æ€ººß“        # Push string "plus minus times time divided"

             'tK    '# Remove all "t": "plus minus imes ime divided"

                #    # Split by spaces: ["plus","minus","imes","ime","divided"]

                 U   # Pop and save it in variable `X`

                  X  # And push variable `X` back again

                   ¡ # Split the string by those operator-strings

ε          }         # Map each substring to:

 ð¡                  #  Split by spaces

   õK                #  Remove empty strings from the list

     2ô              #  Split the list into parts of two

       J             #  Join each pair together

        .E           #  Evaluate each as a Python `eval` expression

          O          #  Sum them

®                    # Put the string from the register to the stack again

 á                   # Remove everything except for letters

  X                  # Push variable `X`: ["plus","minus","imes","ime","divided"]

   "+-**/"           # Push string "+-**/"

          S          # Split to characters: ["+","-","*","*","/"]

           :         # Replace each

S                    # Split to characters

 ‚                   # Pair it with the eval sums

  ζ                  # Zip, swapping rows and columns

   JJ                # Join everything together to a single string again

„--                  # Push string "--"

   '+               '# Push string "+"

     :               # Replace all "--" with "+"

      .E             # Evaluate each as a Python `eval` expression (and output implicitly)

See this 05AB1E tip of mine (sections How to use the dictionary?, How to compress strings not part of the dictionary?, and How to compress large integers?) to understand how “¡×€µ‚•„í†ìˆÈŒšï¿Ÿ¯¥Š—¿áÓÁÏ“, '…§, '°¡, .•4º»Ÿć'Rþн•, Ž9o, and “‰´Øè„Æ€ººß“ work.

Step by step example:

• 
  Input: two hundred twenty two divided by two times minus fifty seven plus three hundred eighteen minus minus ten
  


• Step 1: Remove "by": two hundred twenty two divided two times minus fifty seven plus three hundred eighteen minus minus ten
  


• Step 2: Convert "zero" through "twelve" with the correct number: 2 hundred twenty 2 divided 2 times minus fifty 7 plus 3 hundred 8een minus minus 10
  


• Step 3: Remove all "t": 2 hundred weny 2 divided 2 imes minus fify 7 plus 3 hundred 8een minus minus 10
  


• Step 4: Replace all "een" with "+10": 2 hundred weny 2 divided 2 imes minus fify 7 plus 3 hundred 8+10 minus minus 10
  


• Step 5: Replace all "hundred" with "*100": 2 *100 weny 2 divided 2 imes minus fify 7 plus 3 *100 8+10 minus minus 10
  


• Step 6: Replace all ["wen","hir","for","fif"] with the correct digit: 2 *100 2y 2 divided 2 imes minus 5y 7 plus 3 *100 8+10 minus minus 10
  


• Step 7: Replace all "y" with " *10": 2 *100 2 *10 2 divided 2 imes minus 5 *10 7 plus 3 *100 8+10 minus minus 10
  


• Step 8: Split by ["plus","minus","ime","imes","divided"]: ["2 *100 2 *10 2 "," 2 "," "," 5 *10 7 "," 3 *100 8+10 "," "," 10"]
  


• Step 9: Split each on spaces: [["2","*100","2","*10","2",""],["","","2",""],["",""],["","5","*10","7",""],["","3","*100","8+10",""],["",""],["","10"]]
  


• Step 10: Remove empty items: [["2","*100","2","*10","2"],["2"],,["5","*10","7"],["3","*100","8+10"],,["10"]]
  


• Step 11: Split into parts of size 2 and join: [["2*100","2*10","2"],["2"],"",["5*10","7"],["3*100","8+10"],"",["10"]]
  


• Step 12: Python eval each: [[200,20,2],[2],"",[50,7],[300,18],"",[10]]
  


• Step 13: Sum each: [222,2,"",57,318,"",10]
  


• Step 14: Re-push the string from the register, and remove everything except letters: dividedimesminusplusminusminus
  


• Step 15: Replace "plus","minus","imes","ime","divided" with the operation characters: ["/","*","-","+","-","-"]
  


• Step 16: Zip and join both together: 222/2*-57+318--10
  


• Step 17: Replace any "--" with "+": 222/2*-57+318+10
  


• 
  Output: Python eval the string, and output implicitly: -5999.0
  

sfk, 572 449 423 bytes

This could all be one line, but for the purposes of reading it I used newlines instead of spaces.

xed -i

_plus_+_

_minus_-_

_times[ortext]time_*_

_divided?by_/_

+xed

"_[white][2 chars of a-z][chars of a-z]ty_[parts 1,2]0_"

"_[white][2 chars of a-z][chars of a-z]een_[part1]1[part2]_"

_ten_10_

_lev_11_

_twe_12_

+xed

_ze_0_

_on_1_

_tw_2_

_th_3_

_fo_4_

_fi_5_

_si_6_

_se_7_

_ei_8_

_ni_9_

+xed

_0[white][keep][digit]__

"_[chars of e-z ]__"

+xed

"_?dd[keep][2 digits]_[part1]_"

_?dd[keep][digit]_[part1]0_

_dd_00_

+calc #text

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