

Eval_simplify = ->
  push(cadr(p1))
  runUserDefinedSimplifications()
  Eval()
  simplify()

runUserDefinedSimplifications = ->
  # -----------------------
  # unfortunately for the time being user
  # specified simplifications are only
  # run in things which don't contain
  # integrals. 
  # Doesn't work yet, could be because of
  # some clobbering as "transform" is called
  # recursively? 
  if userSimplificationsInListForm.length ! = 0 and ! Find(cadr(p1), symbol(INTEGRAL))
    originalexpanding = expanding
    expanding = false
    if DEBUG then console.log("runUserDefinedSimplifications passed: " + stack[tos-1].toString())
    Eval()
    if DEBUG then console.log("runUserDefinedSimplifications after eval no expanding: " + stack[tos-1].toString())
    expanding = originalexpanding


    p1 = stack[tos-1]

    if DEBUG then console.log "patterns to be checked: "
    for eachSimplification in userSimplificationsInListForm
      if DEBUG then console.log "..." + eachSimplification

    atLeastOneSuccessInRouldOfRulesApplications = true
    numberOfRulesApplications = 0

    
    while atLeastOneSuccessInRouldOfRulesApplications and numberOfRulesApplications < MAX_CONSECUTIVE_APPLICATIONS_OF_ALL_RULES
      atLeastOneSuccessInRouldOfRulesApplications = false
      numberOfRulesApplications++
      for eachSimplification in userSimplificationsInListForm
        success = true
        eachConsecutiveRuleApplication = 0
        while success and eachConsecutiveRuleApplication < MAX_CONSECUTIVE_APPLICATIONS_OF_SINGLE_RULE
          eachConsecutiveRuleApplication++
          if DEBUG then console.log "simplify - tos: " + tos + " checking pattern: " + eachSimplification + " on: " + p1
          push_symbol(NIL)
          success = transform(eachSimplification, true)
          if success
            atLeastOneSuccessInRouldOfRulesApplications = true
          p1 = stack[tos-1]
          if DEBUG then console.log "p1 at this stage of simplification: " + p1
        if eachConsecutiveRuleApplication == MAX_CONSECUTIVE_APPLICATIONS_OF_SINGLE_RULE
          stop("maximum application of single transformation rule exceeded: " + eachSimplification)

    if numberOfRulesApplications == MAX_CONSECUTIVE_APPLICATIONS_OF_ALL_RULES
      stop("maximum application of all transformation rules exceeded ")

    if DEBUG
      console.log "METAX = " + get_binding(symbol(METAX))
      console.log "METAA = " + get_binding(symbol(METAA))
      console.log "METAB = " + get_binding(symbol(METAB))

  # ------------------------

simplifyForCodeGeneration = ->
  save()
  runUserDefinedSimplifications()
  codeGen = true
  # in "codeGen" mode we completely
  # eval and simplify the function bodies
  # because we really want to resolve all
  # the variables indirections and apply
  # all the simplifications we can. 
  simplify_main()
  codeGen = false
  restore()

simplify = ->
  save()
  simplify_main()
  restore()

simplify_main = ->
  p1 = pop()

  # when we do code generation, we proceed to
  # fully evaluate and simplify the body of
  # a function, so we resolve all variables
  # indirections and we simplify everything
  # we can given the current assignments. 
  if codeGen and car(p1) == symbol(FUNCTION)
    fbody = cadr(p1)    
    push fbody
    # let's simplify the body so we give it a
    # compact form
    eval()
    simplify()
    p3 = pop()

    # replace the evaled body
    args = caddr(p1); # p5 is B

    push_symbol(FUNCTION)
    push p3
    push args
    list(3)
    p1 = pop()

  if (istensor(p1))
    simplify_tensor()
    return

  if (Find(p1, symbol(FACTORIAL)))
    push(p1)
    simfac()
    p2 = pop()
    push(p1)
    rationalize()
    simfac()
    p3 = pop()
    if (count(p2) < count(p3))
      p1 = p2
    else
      p1 = p3

  f10()
  f1()
  f2()
  f3()
  f4()
  f5()
  f9()
  simplify_polarRect()
  if do_simplify_nested_radicals
    # if there is some de-nesting then
    # re-run a simplification because
    # the shape of the expression might
    # have changed significantly. 
    # e.g. simplify(14^(1/2) - (16 - 4*7^(1/2))^(1/2))
    # needs some more semplification after the de-nesting. 
    if simplify_nested_radicals()
      if DEBUG then console.log("de-nesting successful into: " + p1.toString())
      push(p1)
      simplify()
      return

  simplify_rectToClock()
  simplify_rational_expressions()

  push(p1)

simplify_tensor = ->
  i = 0
  p2 = alloc_tensor(p1.tensor.nelem)
  p2.tensor.ndim = p1.tensor.ndim
  for i in [0...p1.tensor.ndim]
    p2.tensor.dim[i] = p1.tensor.dim[i]
  for i in [0...p1.tensor.nelem]
    push(p1.tensor.elem[i])
    simplify()
    p2.tensor.elem[i] = pop()

  check_tensor_dimensions p2

  if (isZeroAtomOrTensor(p2))
    p2 = zero; # null tensor becomes scalar zero
  push(p2)


# try rationalizing

f1 = ->
  if (car(p1) ! = symbol(ADD))
    return
  push(p1)
  rationalize()
  p2 = pop()
  if (count(p2) < count(p1))
    p1 = p2

# try condensing

f2 = ->
  if (car(p1) ! = symbol(ADD))
    return
  push(p1)
  Condense()
  p2 = pop()
  if (count(p2) <= count(p1))
    p1 = p2

# this simplifies forms like (A-B) / (B-A)

f3 = ->
  push(p1)
  rationalize()
  negate()
  rationalize()
  negate()
  rationalize()
  p2 = pop()
  if (count(p2) < count(p1))
    p1 = p2


f10 = ->

  carp1 = car(p1)
  miao = cdr(p1)
  if ( carp1 == symbol(MULTIPLY) || isinnerordot(p1))
    # both operands a transpose? 

    if (car(car(cdr(p1))) == symbol(TRANSPOSE)) and (car(car(cdr(cdr(p1)))) == symbol(TRANSPOSE))
      if DEBUG then console.log "maybe collecting a transpose " + p1
      a = cadr(car(cdr(p1)))
      b = cadr(car(cdr(cdr(p1))))
      if carp1 == symbol(MULTIPLY)
        push(a)
        push(b)
        multiply()
      else if isinnerordot(p1)
        push(b)
        push(a)
        inner()
      push_integer(1)
      push_integer(2)
      originalexpanding = expanding
      expanding = false
      transpose()
      expanding = originalexpanding

      p2 = pop()
      if (count(p2) < count(p1))
        p1 = p2
      if DEBUG then console.log "collecting a transpose " + p2

# try expanding denominators

f4 = ->
  if (isZeroAtomOrTensor(p1))
    return
  push(p1)
  rationalize()
  inverse()
  rationalize()
  inverse()
  rationalize()
  p2 = pop()
  if (count(p2) < count(p1))
    p1 = p2

# simplifies trig forms

simplify_trig = ->
  save()
  p1 = pop()
  f5()
  push(p1)
  restore()

f5 = ->
  if (Find(p1, symbol(SIN)) == 0 && Find(p1, symbol(COS)) == 0)
    return

  p2 = p1

  trigmode = 1
  push(p2)
  Eval()
  p3 = pop()

  trigmode = 2
  push(p2)
  Eval()
  p4 = pop()

  trigmode = 0

  if (count(p4) < count(p3) || nterms(p4) < nterms(p3))
    p3 = p4

  if (count(p3) < count(p1) || nterms(p3) < nterms(p1))
    p1 = p3

# if it's a sum then try to simplify each term

f9 = ->
  if (car(p1) ! = symbol(ADD))
    return
  push_integer(0)
  p2 = cdr(p1)
  while (iscons(p2))
    push(car(p2))
    simplify()
    add()

    oldp1 = p1
    oldp2 = p2
    p1 = pop()
    simplify_rational_expressions()
    push(p1)
    p1 = oldp1
    p2 = oldp2

    p2 = cdr(p2)
  p2 = pop()
  if (count(p2) < count(p1))
    p1 = p2


simplify_rational_expressions = ->
  push p1
  denominator()
  denom = pop()

  if isone(denom)
    return

  push p1
  numerator()
  num = pop()

  if isone(num)
    return

  if !( polyVar = areunivarpolysfactoredorexpandedform(num, denom))
    return

  push num
  push denom
  gcd()

  push(polyVar)
  factor()

  theGCD = pop()

  # if there are no common factors then
  # bail
  if isone(theGCD)
    return

  push num
  push(polyVar)
  factor()
  push theGCD
  #divide()
  inverse()
  multiply_noexpand()
  simplify()
  sasa = stack[tos-1].toString()

  push denom
  push(polyVar)
  factor()
  push theGCD
  #divide()
  inverse()
  multiply_noexpand()
  simplify()
  sasa = stack[tos-1].toString()

  divide()
  #simplify()
  Condense()
  sasa = stack[tos-1].toString()

  p2 = pop()
  if (count(p2) < count(p1))
    p1 = p2


# things like 6*(cos(2/9*pi)+i*sin(2/9*pi))
# where we have sin and cos, those might start to
# look better in clock form i.e.  6*(-1)^(2/9) 
simplify_rectToClock = ->
  #debugger

  if (Find(p1, symbol(SIN)) == 0 && Find(p1, symbol(COS)) == 0)
    return

  push(p1)
  Eval()
  clockform()

  p2 = pop(); # put new (hopefully simplified expr) in p2
  if DEBUG then console.log "before simplification clockform: " + p1 + " after: " + p2

  if (count(p2) < count(p1))
    p1 = p2

simplify_polarRect = ->
  push(p1)

  polarRectAMinusOneBase()
  Eval()

  p2 = pop(); # put new (hopefully simplified expr) in p2

  if (count(p2) < count(p1))
    p1 = p2

polarRectAMinusOneBase = ->
  save()
  p1 = pop()
  
  if isimaginaryunit(p1)
    push(p1)
    restore()
    return

  if (equal(car(p1), symbol(POWER)) and isminusone(cadr(p1)) )

    # base we just said is minus 1
    push(one)
    negate()

    # exponent
    push(caddr(p1))
    polarRectAMinusOneBase()
    
    power()
    # try to simplify it using polar and rect
    polar()
    rect()

  else if (iscons(p1))
    h = tos
    while (iscons(p1))
      #console.log("recursing on: " + car(p1).toString())
      push(car(p1))
      polarRectAMinusOneBase()
      #console.log("...transformed into: " + stack[tos-1].toString())
      p1 = cdr(p1)
    list(tos - h)
  else
    push(p1)

  restore()
  return

nterms = (p) ->
  if (car(p) ! = symbol(ADD))
    return 1
  else
    return length(p) - 1

simplify_nested_radicals = ->
  if recursionLevelNestedRadicalsRemoval > 0
    if DEBUG then console.log("denesting bailing out because of too much recursion")
    return false

  push(p1)
  somethingSimplified = take_care_of_nested_radicals()


  # in this paragraph we check whether we can collect
  # common factors without complicating the expression
  # in particular we want to avoid 
  # collecting radicals like in this case where
  # we collect sqrt(2):
  #   2-2^(1/2) into 2^(1/2)*(-1+2^(1/2))
  # but we do like to collect other non-radicals e.g.
  #   17/2+3/2*5^(1/2) into 1/2*(17+3*5^(1/2))
  # so what we do is we count the powers and we check
  # which version has the least number of them. 
  simplificationWithoutCondense = stack[tos-1]

  prev_expanding = expanding
  expanding = 0
  yycondense()
  expanding = prev_expanding

  simplificationWithCondense = pop()
  #console.log("occurrences of powers in " + simplificationWithoutCondense + " :" + countOccurrencesOfSymbol(symbol(POWER),simplificationWithoutCondense))
  #console.log("occurrences of powers in " + simplificationWithCondense + " :" + countOccurrencesOfSymbol(symbol(POWER),simplificationWithCondense))

  if (countOccurrencesOfSymbol(symbol(POWER),simplificationWithoutCondense) < countOccurrencesOfSymbol(symbol(POWER),simplificationWithCondense))
    push(simplificationWithoutCondense)
  else
    push(simplificationWithCondense)


  # we got out result, wrap up
  p1 = pop()
  return somethingSimplified

take_care_of_nested_radicals = ->
  if recursionLevelNestedRadicalsRemoval > 0
    if DEBUG then console.log("denesting bailing out because of too much recursion")
    return false


  save()
  p1 = pop()
  #console.log("take_care_of_nested_radicals p1: " + p1.toString())
  
  if equal(car(p1), symbol(POWER))

    #console.log("ok it's a power ")
    base = cadr(p1)
    exponent = caddr(p1)
    #console.log("possible double radical base: " + base)
    #console.log("possible double radical exponent: " + exponent)

    if ! isminusone(exponent) and equal(car(base), symbol(ADD)) and isfraction(exponent) and (equalq(exponent,1,3) or equalq(exponent,1,2))

      #console.log("ok there is a radix with a term inside")
      firstTerm = cadr(base)
      push(firstTerm)
      take_care_of_nested_radicals()
      pop()
      secondTerm = caddr(base)
      push(secondTerm)
      take_care_of_nested_radicals()
      pop()

      #console.log("possible double radical term1: " + firstTerm)
      #console.log("possible double radical term2: " + secondTerm)

      numberOfTerms = 0
      countingTerms = base
      while (cdr(countingTerms) ! = symbol(NIL))
        numberOfTerms++
        countingTerms = cdr(countingTerms)
      #console.log("number of terms: " + numberOfTerms)
      if numberOfTerms > 2
        #console.log("too many terms under outer radix ")
        push(p1)
        restore()
        return false


      # list here all the factors
      commonInnerExponent = null
      commonBases = []
      termsThatAreNotPowers = []
      if (car(secondTerm) == symbol(MULTIPLY))
        # product of factors
        secondTermFactor = cdr(secondTerm)
        if iscons(secondTermFactor)
          while (iscons(secondTermFactor))
            #console.log("second term factor BIS: " + car(secondTermFactor).toString())
            potentialPower = car(secondTermFactor)
            if (car(potentialPower) == symbol(POWER)) 
              innerbase = cadr(potentialPower)
              innerexponent = caddr(potentialPower)
              if equalq(innerexponent,1,2)
                #console.log("tackling double radical 1: " + p1.toString())
                if ! commonInnerExponent? 
                  commonInnerExponent = innerexponent
                  commonBases.push(innerbase)
                else
                  if equal(innerexponent, commonInnerExponent)
                    #console.log("common base: " + innerbase.toString())
                    commonBases.push(innerbase)
                  else
                    #console.log("no common bases here ")
                #console.log("this one is a power base: " + innerbase + " , exponent: " + innerexponent)
            else
              termsThatAreNotPowers.push(potentialPower)
            secondTermFactor = cdr(secondTermFactor)
      else if (car(secondTerm) == symbol(POWER)) 
        innerbase = cadr(secondTerm)
        innerexponent = caddr(secondTerm)
        if ! commonInnerExponent?  and equalq(innerexponent,1,2)
          #console.log("tackling double radical 2: " + p1.toString())
          commonInnerExponent = innerexponent
          commonBases.push(innerbase)

      
      if commonBases.length == 0
        push(p1)
        restore()
        return false

      A = firstTerm
      #console.log("A: " + A.toString())

      push_integer(1)
      for i in commonBases
        push(i)
        multiply()
        #console.log("basis with common exponent: " + i.toString())
      C = pop()
      #console.log("C: " + C.toString())
      
      push_integer(1)
      for i in termsThatAreNotPowers
        push(i)
        multiply()
        #console.log("terms that are not powers: " + i.toString())
      B = pop()
      #console.log("B: " + B.toString())


      if equalq(exponent,1,3)
        push(A)
        negate()
        push(C)
        multiply()
        push(B)
        divide()    # 4th coeff
        #console.log("constant coeff " + stack[tos-1].toString())
        checkSize = pop()
        push(checkSize)
        real()
        yyfloat()
        if Math.abs(pop().d) > Math.pow(2, 32)
          push(p1)
          restore()
          return false
        push(checkSize)

        push_integer(3)
        push(C)
        multiply()   # 3rd coeff
        #console.log("next coeff " + stack[tos-1].toString())
        checkSize = pop()
        push(checkSize)
        real()
        yyfloat()
        if Math.abs(pop().d) > Math.pow(2, 32)
          pop()
          push(p1)
          restore()
          return false
        push(checkSize)

        push(symbol(SECRETX))
        multiply()


        push_integer(-3)
        push(A)
        multiply()
        push(B)
        divide()  # 2nd coeff
        checkSize = pop()
        push(checkSize)
        real()
        yyfloat()
        if Math.abs(pop().d) > Math.pow(2, 32)
          pop()
          pop()
          push(p1)
          restore()
          return false
        push(checkSize)

        #console.log("next coeff " + stack[tos-1].toString())
        push(symbol(SECRETX))
        push_integer(2)
        power()
        multiply()

        push_integer(1) # 1st coeff
        #console.log("next coeff " + stack[tos-1].toString())
        push(symbol(SECRETX))
        push_integer(3)
        power()
        multiply()

        add()
        add()
        add()
      
      else if equalq(exponent,1,2)
        push(C) # 3th coeff
        checkSize = pop()
        push(checkSize)
        real()
        yyfloat()
        if Math.abs(pop().d) > Math.pow(2, 32)
          push(p1)
          restore()
          return false
        push(checkSize)
        #console.log("constant coeff " + stack[tos-1].toString())

        push_integer(-2)
        push(A)
        multiply()
        push(B)
        divide()  # 2nd coeff
        checkSize = pop()
        push(checkSize)
        real()
        yyfloat()
        if Math.abs(pop().d) > Math.pow(2, 32)
          pop()
          push(p1)
          restore()
          return false
        push(checkSize)

        #console.log("next coeff " + stack[tos-1].toString())
        push(symbol(SECRETX))
        multiply()


        push_integer(1) # 1st coeff
        #console.log("next coeff " + stack[tos-1].toString())
        push(symbol(SECRETX))
        push_integer(2)
        power()
        multiply()

        add()
        add()

      #console.log("whole polynomial: " + stack[tos-1].toString())

      push(symbol(SECRETX))


      recursionLevelNestedRadicalsRemoval++
      #console.log("invoking roots at recursion level: " + recursionLevelNestedRadicalsRemoval)
      roots()
      recursionLevelNestedRadicalsRemoval--
      if equal(stack[tos-1], symbol(NIL))
        if DEBUG then console.log("roots bailed out because of too much recursion")
        pop()
        push(p1)
        restore()
        return false

      #console.log("all solutions: " + stack[tos-1].toString())

      # exclude the solutions with radicals
      possibleSolutions = []
      for eachSolution in stack[tos-1].tensor.elem
        if ! Find(eachSolution, symbol(POWER))
          possibleSolutions.push(eachSolution)

      pop() # popping the tensor with the solutions

      #console.log("possible solutions: " + possibleSolutions.toString())
      if possibleSolutions.length == 0
        push(p1)
        restore()
        return false

      possibleRationalSolutions = []
      realOfpossibleRationalSolutions = []
      #console.log("checking the one with maximum real part ")
      for i in possibleSolutions
        push(i)
        real()
        yyfloat()
        possibleRationalSolutions.push(i)
        realOfpossibleRationalSolutions.push(pop().d)

      whichRationalSolution = realOfpossibleRationalSolutions.indexOf(Math.max.apply(Math, realOfpossibleRationalSolutions))
      SOLUTION = possibleRationalSolutions[whichRationalSolution]
      #console.log("picked solution: " + SOLUTION)

      
      ###
      #possibleNewExpressions = []
      #realOfPossibleNewExpressions = []
      # pick the solution which cubic root has no radicals
      lowercase_b = null
      for SOLUTION in possibleSolutions
        console.log("testing solution: " + SOLUTION.toString())

        debugger
        if equalq(exponent,1,3)
          push(A)
          push(SOLUTION)
          push_integer(3)
          power()
          push_integer(3)
          push(C)
          multiply()
          push(SOLUTION)
          multiply()
          add()
          divide()
          console.log("argument of cubic root: " + stack[tos-1].toString())
          push_rational(1,3)
          power()
        else if equalq(exponent,1,2)
          push(A)
          push(SOLUTION)
          push_integer(2)
          power()
          push(C)
          add()
          divide()
          console.log("argument of cubic root: " + stack[tos-1].toString())
          push_rational(1,2)
          power()
        console.log("b is: " + stack[tos-1].toString())

        lowercase_b = pop()

        if ! Find(lowercase_b, symbol(POWER))
          break
      ###

      if equalq(exponent,1,3)
        push(A)
        push(SOLUTION)
        push_integer(3)
        power()
        push_integer(3)
        push(C)
        multiply()
        push(SOLUTION)
        multiply()
        add()
        divide()
        #console.log("argument of cubic root: " + stack[tos-1].toString())
        push_rational(1,3)
        power()
      else if equalq(exponent,1,2)
        push(A)
        push(SOLUTION)
        push_integer(2)
        power()
        push(C)
        add()
        divide()
        #console.log("argument of cubic root: " + stack[tos-1].toString())
        push_rational(1,2)
        power()
      #console.log("b is: " + stack[tos-1].toString())

      lowercase_b = pop()


      if ! lowercase_b? 
        push(p1)
        restore()
        return false

      push(lowercase_b)
      push(SOLUTION)
      multiply()
      

      if equalq(exponent,1,3)
        #console.log("a is: " + stack[tos-1].toString())

        lowercase_a = pop()

        push(lowercase_b)
        push(C)
        push_rational(1,2)
        power()
        multiply()

        push(lowercase_a)
        add()
        simplify()
      else if equalq(exponent,1,2)
        #console.log("a could be: " + stack[tos-1].toString())

        lowercase_a = pop()

        push(lowercase_b)
        push(C)
        push_rational(1,2)
        power()
        multiply()

        push(lowercase_a)
        add()
        simplify()
        possibleNewExpression = pop()
        #console.log("verifying if  " + possibleNewExpression + " is positive")
        push(possibleNewExpression)
        real()
        yyfloat()
        possibleNewExpressionValue = pop()
        if ! isnegativenumber(possibleNewExpressionValue)
          #console.log("... it is positive")
          push(possibleNewExpression)
        else
          #console.log("... it is NOT positive")
          push(lowercase_b)
          negate()
          lowercase_b = pop()

          push(lowercase_a)
          negate()
          lowercase_a = pop()


          push(lowercase_b)
          push(C)
          push_rational(1,2)
          power()
          multiply()

          push(lowercase_a)
          add()
          simplify()
          # possibleNewExpression is now at top of stack


      #console.log("potential new expression: " + stack[tos-1].toString())
      p1 = pop()
      #newExpression = pop()
      #debugger
      #push(newExpression)
      #real()
      #yyfloat()
      #possibleNewExpressions.push(newExpression)
      #realOfPossibleNewExpressions.push(pop().d)

      #whichExpression = realOfPossibleNewExpressions.indexOf(Math.max.apply(Math, realOfPossibleNewExpressions))
      #p1 = possibleNewExpressions[whichExpression]
      #console.log("final new expression: " + p1.toString())

      push(p1)
      restore()
      return true



    else
      push(p1)
      restore()
      return false

  else if (iscons(p1))
    h = tos
    anyRadicalSimplificationWorked = false
    while (iscons(p1))
      #console.log("recursing on: " + car(p1).toString())
      push(car(p1))
      anyRadicalSimplificationWorked = anyRadicalSimplificationWorked or take_care_of_nested_radicals()
      #console.log("...transformed into: " + stack[tos-1].toString())
      p1 = cdr(p1)
    list(tos - h)
    restore()
    return anyRadicalSimplificationWorked
  else
    push(p1)
    restore()
    return false

  throw new Error("control flow should never reach here")
