Abstract:
In a computing system, evaluating a mathematical expression in presented hierarchically according to the rules of precedence of operations, initial operations at the bottom of the hierarchy may yield values too large to be calculated conventionally, even if the ultimate value of the expression may represent a calculable value. The mathematical expression is evaluated top down to determine if portions of the mathematical expression are re-expressible to simplify the expression and avoid initial or intermediate calculations that would prevent calculation of the mathematical expression. Portions of the original mathematical expression are re-expressible by substituting known mathematical identities, applying arithmetic rules, or treating a portion of the mathematical expression as a variable on which other operations are performed. Once the mathematical expression has been re-expressed, the mathematical expression is simplified, and, if possible, its value calculated.

Description:
FIELD OF THE INVENTION  
       [0001]     The present invention generally pertains to evaluation of mathematical expressions performed by a computer, and more specifically, to a method and system for performing calculations on expressions involving or potentially yielding very large numbers.  
       BACKGROUND OF THE INVENTION  
       [0002]     One of the many benefits of computers is the ability quickly to evaluate complex or lengthy mathematical expressions. What may be very time consuming or impractical for a person to compute may be rather simple for a computer to calculate. As a result, a number of computer algebra systems have been developed to allow scientists, engineers, mathematicians, students, and others to use computers to more easily calculate and solve even complex mathematical equations.  
         [0003]     As increasingly powerful as computers have become, however, computers have their limits in terms of the magnitude of the expressions they are able to accommodate. For a personal computer, a number on the order of magnitude of 10 9 , or a number raised to the power of one billion, is too large for a typical personal computer to accommodate. A number on the order of magnitude of 10 15 , or a number raised to the power of a million billion, is too large for even a supercomputer to accommodate.  
         [0004]     In addition, the capacity of a computer may limit the precision with which mathematical expressions are calculated. In most computer programs and computing environments, the precision of any calculation is limited by the word size of the computer, because the word size of the computer determines largest number that can be stored in one of the processor&#39;s registers. Arbitrary-precision arithmetic consists of a set of algorithms, functions, and data structures designed specifically to deal with numbers that can be of arbitrary size. Arbitrary-precision arithmetic is a common feature in computer algebra systems and some specific math and engineering software packages, but is rarely included in other software  
         [0005]     Unfortunately, sometimes mathematical expressions that are too large for the computer to accommodate, or that may be calculated only to an arbitrary level of precision, may represent only intermediate values that are part of a calculation. The calculation ultimately may result in a manageable number, but if an intermediate calculation is too large for the computing system to accommodate, the calculation cannot be performed.  
         [0006]      FIGS. 1A, 1B , and  2  further illustrate the concern by illustrating how a computing system calculates a mathematical expression. In  FIG. 1A , mathematical expression  100  represents how a computing system evaluates the expression 30*2+5. Rules of precedence of operations dictate the form of mathematic expression  100 , such that the product of 30*2 is calculated, then the sum of that product is added to 5. More particularly, as regarded in a tree structure, from top down, the addition operator  102 , representing the last operation, is at the top of tree structure  100 , over children including multiplication operator  104  and the value 5  106 . In turn, multiplication operator has two children, value 30  108  and value 2  110 . Values 30  108  and 2  110  also can be thought of as the “grandchildren” of addition operator  102 , because values 30  108  and 2  110  are two steps removed from addition operator  102 , in contrast to multiplication operator  104  and value 5  106  which are once removed from addition operator  102 .  
         [0007]     The value of mathematical expression  100  is calculated from the bottom up to give effect to the rules of precedence of operations. Thus, value 30  108  and value 2  110  are received and, moving up tree structure  100 , are multiplied together upon reaching multiplication operator  104 . Moving further up tree structure  100 , the product determined from the multiplication of values 30  108  and 2  110  is received along with value 5  106 . Moving up the hierarchical structure of mathematical expression  100 , upon reaching addition operator  102 , the product of values 30  108  and 2  110  are added to value 5  106 .  
         [0008]     Shown another way, once value 30  108  and value 2  110  are multiplied together, tree structure 100 of  FIG. 1A  becomes mathematical expression  120  of  FIG. 1B . As the expression is calculated from the bottom up, mathematical expression  100  becomes simplified to mathematical expression  120 . Thus, from the perspective of addition operator  102 , only the children of addition operator  102 , product  122  and value 5  106  are relevant; the children of each operator only are ever relevant, regardless of what the grandchildren of the term might be. Again, the hierarchical tree structure is created based on rules of precedence of operations, and the tree is calculated from the bottom up.  
         [0009]     Unfortunately, sometimes the conventional approach cannot calculate an expression that, although manageable on the whole, includes one or more steps that the computing system is unable to calculate. For example,  FIG. 2  illustrates a mathematical expression  200  representing how a computing system would calculate the expression 12345 100000000000000 % 18970907, where “%” represents the modulus division operator. In tree structure  200 , the modulus division operator  202  has children that include the exponential operator  204 , which in turn has children base 12345  208  and exponent 1000000000000000  210 , and quotient 18970907  204 . The result of modulus division yields a result which, at most, is one less than the quotient or, in the example of  FIG. 2 , is quotient 18970906  206 . This is a value that is manageable within even a simple computing system. By contrast, however, calculating the expression 12345 1000000000000000  would overwhelm even most supercomputers. Accordingly, even though the overall result of the expression represented by mathematical expression  200  is manageable, processing of mathematical expression  200  ends at the calculation of 12345 100000000000000 .  
         [0010]     It would therefore be desirable to provide a method and system for a computing system to automatically calculate the result of mathematical expressions when the overall result of the mathematical expression is manageable by the computing system, even when the mathematical expression includes a calculation the computing system is not able to accommodate.  
       SUMMARY OF THE INVENTION  
       [0011]     One advantage of the present invention is that it provides a method and a system for re-expressing a presented mathematical expression such that calculations included in the mathematical expression that may be unmanageable or impossible may be simplified to enable the mathematical expression to be calculated or simplified. Conventional computing methods and systems, to give effect to rules of precedence of operations, conceive of mathematical expressions in a hierarchical tree structure, where the first operations to be performed are situated at the bottom of the tree structure, and the value of the tree structure is calculated from the bottom up. Initial calculations might present calculations that are very lengthy or possibly too large to be calculated by conventional computing means. However, according to an embodiment of the present invention, the mathematical expression is reconsidered from the top down, allowing for the possibility that the mathematical expression may be simplified to avoid complex or impossible calculations. Applying known mathematical simplifications and identities, or treating incalculable portions of expressions as variables, it becomes possible to calculate or, at least, simplify the mathematical expression, even when the first operation to be performed might prove too large for the computing system to calculate.  
         [0012]     One aspect of the present invention is thus directed to a method of calculating a presented mathematical expression including at least one first operation and a second operation where the first operation is calculated before the second expression according to rules of precedence of operators. The second operation is evaluated to determine if a combination of the second operation acting on the at least one first operation is replaceable with a mathematically identical expression that involves reduced computational effort to calculate than the presented mathematical expression. The combination of the second operation acting on the at least one first combination is re-expressed with the mathematically identical expression. The mathematically identical expression is then calculated.  
         [0013]     The at least one first operation may be evaluated to determine if the first operation will result in a computational effort exceeding a predetermined computational effort threshold before evaluating the second operation.  
         [0014]     Re-expressing the combination with the mathematically identical expression includes replacing the combination with a mathematically identical expression that replaces computation of values with computation of smaller values. For example, re-expressing the combination with the mathematically identical expression may include re-expressing nˆk % m with ((a % m)*(b % m)) % m, where % represents a modulus division operator, nˆk represents the first operation, % m represents the second operation, a and b are factors of nˆk. The combination includes n! % m may be re-expressed as zero, where % represents a modulus division operator, n! represents the first operation, % m represents the second operation, and m&lt;=n. The combination (n−1)! % n may be re-expressed as n−1, where % represents the modulus division operator, (n−1)! represents the first operation, % n represents the second operation, and n is a prime number. The combination (n−1)! % n may be re-expressed as zero, where % represents the modulus division operator, (n−1)! represents the first operation, % n represents the second operation, and n is neither a prime number nor equal to four. The combination log(b, bˆn) may be re-expressed as n, where bˆn represents the first operation, log(b, bˆn) represents the second operation, b is a positive real number and n is an integer. The combination log(b, bˆn) may be re-expressed as −1, where bˆn represents the first operation, log(b, bˆn) represents the second operation, and n is negative one. The combination log(bˆn, b) may be re-expressed as 1/n, where bˆn represents the first operation, log(bˆn, b) represents the second operation, b is a positive real number and n is an integer. The combination log(bˆm, bˆn) may be re-expressed as n/m, where b{circumflex over ( m)}and bˆn represent the first operation, log(bˆm, bˆn) represents the second operation, and m is nonzero. The combination gcd(m, n) may be re-expressed as gcd(m, n % m), where gcd represents a greatest common divisor operation, n represents the first operation, and gcd(m, n) represents the second operation). The combination n!/m! may be re-expressed as n*(n−1)*(m+1), where n&gt;m, n! and m! represent the first operation, and n!/m! represents the second operation. The combination of permutation(m, n) may be re-expressed as n!/(n−m)!, where m and n represent the first operation, and permutation(m, n) represents the second operation.  
         [0015]     In addition, the combination may be re-expressed according to arithmetic rules, including re-expressing n−n as zero and n/n as one, when n is nonzero. Alternatively, re-expressing the combination may include cascading the second operation down to redistribute the first operation to replaces computation of values with computation of smaller values.  
         [0016]     Re-expressing the combination with the mathematically identical expression also may include includes treating the first operation as a variable and computing a remainder of the mathematical identical expression to yield a simplified expression in as applied to the first operation. Once the simplified expression is reached, an attempt may be made to calculate the value of the simplified expression as applied to the first operation.  
         [0017]     Re-expressing the mathematical expression may include iteratively evaluating each additional operation that is presented by the presented mathematical expression, or the mathematically identical expression resulting from the re-expressing of the presented mathematical expression. 
     
    
     BRIEF DESCRIPTION OF THE DRAWING FIGURES  
       [0018]     The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same becomes better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:  
         [0019]      FIGS. 1A and 1B  (Prior Art) are diagrams of a mathematical expression presented in a hierarchical tree structure representing how a computing system might approach the expression according to rules of precedence of operations;  
         [0020]      FIG. 2  (Prior Art) is a diagram of a mathematical expression that yields an intermediate product that is too large to be handled by most computing systems;  
         [0021]      FIG. 3  is a functional block diagram of a computing device or personal computer (PC) adaptable to use an embodiment of the present invention;  
         [0022]      FIGS. 4A, 5A , and  6 A are mathematical expressions represented in a hierarchical form suitable for processing by a computing system where each of the mathematical expressions would yield an intermediate result too large to be calculated by conventional computing systems;  
         [0023]      FIGS. 4B, 5B , and  6 B are the mathematical expressions of  FIGS. 4A, 5A , and  6 A, respectively, re-expressed according to embodiments of the present invention allowing the mathematical expressions to be simplified or calculated by typical computing systems; and  
         [0024]      FIG. 7  is a flow diagram illustrating the logical steps for adapting mathematical expressions according to an embodiment of the present invention allowing the expressions to be processed by a typical computing system. 
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0000]     Exemplary Computing System for Implementing Present Invention  
         [0025]     With reference to  FIG. 3 , an exemplary conventional computing system suitable for use with an embodiment of the present invention is shown. The system includes a general purpose computing device in the form of a PC  320   a , provided with a processing unit  321 , a system memory  322 , and a system bus  323 . The system bus couples various system components including the system memory to processing unit  321  and may be any of several types of bus structures, including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The system memory includes read only memory (ROM)  324  and random access memory (RAM)  325 . A basic input/output system  326  (BIOS), containing the basic routines that help to transfer information between elements within the PC  320 , such as during start up, is stored in ROM  324 . PC  320  further includes a hard disk drive  327  for reading from and writing to a hard disk (not shown), a magnetic disk drive  328  for reading from or writing to a removable magnetic disk  329 , and an optical disk drive  330  for reading from or writing to a removable optical disk  331 , such as a compact disk-read only memory (CD-ROM) or other optical media. Hard disk drive  327 , magnetic disk drive  328 , and optical disk drive  330  are connected to system bus  323  by a hard disk drive interface  332 , a magnetic disk drive interface  333 , and an optical disk drive interface  334 , respectively. The drives and their associated computer readable media provide nonvolatile storage of computer readable machine instructions, data structures, program modules, and other data for PC  320   a . Although the exemplary environment described herein employs a hard disk, removable magnetic disk  329 , and removable optical disk  331 , it will be appreciated by those skilled in the art that other types of computer readable media, which can store data and machine instructions that are accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks (DVDs), Bernoulli cartridges, RAMs, ROMs, and the like, may also be used in the exemplary operating environment.  
         [0026]     A number of program modules may be stored on the hard disk, magnetic disk  329 , optical disk  331 , ROM  324 , or RAM  325 , including an operating system  335 , one or more application programs  336 , other program modules  337 , and program data  338 . A user may enter commands and information in PC  320  and provide control input through input devices, such as a keyboard  340  and a pointing device  342  that communicate with system bus  323  via I/O device interface  346 . Pointing device  342  may include a mouse, stylus, wireless remote control, or other pointer, but in connection with the present invention, such conventional pointing devices may be omitted, since the user can employ the interactive display for input and control. As used hereinafter, the term “mouse” is intended to encompass virtually any pointing device that is useful for controlling the position of a cursor on the screen. One or more audio input/output devices  343 , including headsets, speakers, and microphones, also engage personal computer  320  via I/O device interface  346 . Still further input devices (not shown) may include a joystick, haptic joystick, yoke, foot pedals, game pad, satellite dish, scanner, or the like. These and other input/output (I/O) devices are often connected to processing unit  321  through an I/O interface  346  that is coupled to the system bus  323 . The term I/O interface is intended to encompass each interface specifically used for a serial port, a parallel port, a game port, a keyboard port, and/or a universal serial bus (USB). A monitor  347  is connected to system bus  323  via an appropriate interface, such as a video adapter  348 . It will be appreciated that PCs are often coupled to other peripheral output devices (not shown), such as speakers (through a sound card or other audio interface—not shown) and printers.  
         [0027]     PC  320  can also operate in a networked environment using logical connections to one or more remote computers, such as a remote computer  349 . Remote computer  349  may be another PC, a server (which is typically generally configured much like PC  320   a ), a router, a network PC, a peer device, or a satellite or other common network node, and typically includes many or all of the elements described above in connection with PC  320   a , although only an external memory storage device  350  has been illustrated in  FIG. 3 . The logical connections depicted in  FIG. 3  include a local area network (LAN)  351  and a wide area network (WAN)  352 . Such networking environments are common in offices, enterprise wide computer networks, intranets, and the Internet.  
         [0028]     When used in a LAN networking environment, PC  320  is connected to LAN  351  through a network interface or adapter  353 . When used in a WAN networking environment, PC  320  typically includes a modem  354 , or other means such as a cable modem, Digital Subscriber Line (DSL) interface, or an Integrated Service Digital Network (ISDN) interface for establishing communications over WAN  352 , such as the Internet. Modem  354 , which may be internal or external, is connected to the system bus  323  or coupled to the bus via I/O device interface  346 , i.e., through a serial port. In a networked environment, program modules, or portions thereof, used by PC  320  may be stored in the remote memory storage device. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used, such as wireless communication and wide band network links.  
         [0000]     Re-expressing Mathematical Expressions to Avoid Excessive Computational Demands  
         [0029]      FIG. 4A  shows a block diagram of a mathematical expression  400 . Mathematical expression  400  yields a relatively small number, or at least a number small enough such that it would not overload a personal computer. Mathematical expression  400  performs modulus division by a quotient of 18970907  406 . Because modulus division yields the remainder of division by the quotient, the result will be a value between zero and one less than the quotient, or, in this case, one less than 18970907  406 . A value of up to one less than quotient 1897907  406  is a manageable number for a calculator, let alone a computer.  
         [0030]     However, even though the result of mathematical expression  400  is manageable, mathematical expression is not because of the large values that would have to be calculated to calculate mathematical expression  400 . Modulus division operator  402  is performed on a sum resulting from application of addition operator  404  to exponential operator  408  raising base 12345 to exponent 1000000000000000  414  and value 1  410 . Calculation of base 12345  412  raised to exponent 1000000000000000  414  will overflow personal computers as well as larger computers, even if the ultimate result of mathematical expression  400  is a manageable value.  
         [0031]     According to an embodiment of the present invention, upon encountering a value that is too large to be calculated by the computing system, the computing system further evaluates mathematical expression  400  to determine whether mathematical expression may be re-expressed in a way that allows for mathematical expression to be computed. As shown in  FIG. 4A , mathematical expressions are logically represented in a top-down, tree fashion where operations to be performed first according to the rules of precedence of operators are presented at the bottom of the tree. (Alternatively, in other words, in mathematical expression of  FIG. 4A , modulo operator  402  is a parent of which addition operator  404  and value 18970907  406  are children, of which exponential operator  408  and value 1  410  are grandchildren, and of which base 12345  412  and exponent 1000000000000000  414  are great-grandchildren.) In mathematical expression  400 , considering the tree structure of mathematical expression  400  from the bottom up, base 12345  412  raised by exponential operator  408  to exponent 1000000000000000  414  is recognized as yielding a value too large to be computed. Thus, according to an embodiment of the present invention, another way to express the overly computation is sought. In particular, because a bottom-up analysis of the tree of mathematical expression  400  is not calculable, mathematical expression is reconsidered from a top-down approach.  
         [0032]     Despite the fact that mathematical expression  400  is not calculable using a conventional bottom-up approach, using a top-down approach, mathematical expression  400  is calculable. Modulus division operator  402  is “cascadable,” such that it can be distributed down the tree structure of mathematical expression  400 . Cascading modulus division operator  402  down through the tree structure allows for values otherwise too large to be calculated to be replaced with calculable expressions that yield the same result according to algebraic identities, thereby rendering mathematical expression  400  calculable. Thus, as will be described in connection with  FIG. 4B , subsidiary portion  420  of mathematical expression  400  may be expressed in a form allowing mathematical expression  400  to be calculated.  
         [0033]     In  FIG. 4B , mathematical expression  400  ( FIG. 4A ) is rewritten in an algebraically identical form  450 . More specifically, because modulus division operator  402  is cascadable beyond addition operator  404  so that subsidiary portion  420  of mathematical expression  400  is replaced by subsidiary expression  452 . Modulus division operator  402  is cascadable down to children of addition operator  404  to replace subsidiary portion  420  of mathematical expression with two modulus division operations, because of identity (1): 
 
( a   b   +c )% d =(( a   b % d )+( c % d ))% d   (1) 
 
         [0034]     Furthermore, where a b  may yield a large value, as in the case of base 12345  466  raised by exponential operator  458  to exponent 468, calculation of that potentially large value is avoidable according to identity (2): 
 
 x   y % z =( x % z ) y % z   (2) 
 
         [0035]     Thus, where z is a considerably smaller number than x, the result of x % z will be no more than one less than z, thus (x % z) y  will be a manageable number to calculate.  
         [0036]     Moreover, even if x or y are large numbers such that x y  yields a large number, calculation of the mathematical expression can be simplified according to identity (3) where the sum of a and b is equal to y: 
 
 x   y   =x   (a+b)   =x   a   *x   b   (3) 
 
         [0037]     Furthermore, where m and n are factors of x y , x y  is re-expressible as: 
 
 x   y =(mn)% z =(( m % z )*( n %  z ))% z   (4) 
 
         [0038]     Thus, combining identities (2), (3) and (4), for even large values of x or y, the expression x y  % z can be expressed according to identity (5): 
 
 x   y % z=x   (a+b) % z =(( x   a % z )*( x   b % z ))% z =((( x % z ) a % z ))*(( x % z ) b %  z )))%  z   (5) 
 
         [0039]     As a result of successively applying this identity multiple times, the expression is re-expressible in a form that is manageable for the computer to calculate. Using identity (2) and other “smart evaluation functions,” even for large values of x and y, the expression x y  % z is calculable as long as z includes a figure not having more than a few hundred decimal digits.  
         [0040]     In the case of mathematical expression  400  ending upon a computing system encountering the untenable prospect of having to calculate base 12345  466  raised by exponential operator  458  to exponent 1000000000000000  468 , the cascading of modulus division operator  454  allows for reformulating portion  420  ( FIG. 4A ) of mathematical expression  400  as portion  452  ( FIG. 4B ) of mathematical expression  450 .  
         [0041]     Thus, according to an embodiment of the present invention, instead of calculating mathematical expressions from the bottom up and stopping upon encountering a portion of the expression, the mathematical expression is reevaluated from the top down to determine if any portion of the mathematical expression is rewritable in a manageable form calculable by the computing system.  
         [0042]     There are a number of additional algebraic expressions including multiple, sequential operations that, when evaluated from the bottom up, yield a result too large to be calculated by the computing system. However, according to embodiments of the present invention, reevaluating these multiple operator expressions other than strictly from the bottom up allows these expressions to be reformed in a manageable, calculable form, as listed in Table (1):  
                   TABLE 1                       Function   Simplification and Reduction in Computational Burden                   n! % m   When m &lt;= n, the result of the expression will           always be zero; result of the expression also will           always be zero if all prime factors of m are less           than n. n! need not be calculated.       (n − 1)! % n   Result of expression will be n − 1 if n is a prime           number, and 0 if n is neither 4 nor a prime number.           n! need not be calculated.       log(b, b{circumflex over ( )}n)   Result of expression is n where b is a positive real           number and n is any integer; alternatively, if n = −1,           the expression becomes log(b, 1/b) = −1; Neither           b{circumflex over ( )}n nor log(b, b{circumflex over ( )}n) need be calculated.       log(b{circumflex over ( )}n, b)   Result of expression is 1/n where b is a positive           real number and n is any integer. Neither b{circumflex over ( )}n nor           log(b{circumflex over ( )}n, b) need be calculated.       log(b{circumflex over ( )}m, b{circumflex over ( )}n)   Result of expression is n/m where m is nonzero. None           of the expressions b{circumflex over ( )}m, b{circumflex over ( )}n, or log(b{circumflex over ( )}m, b{circumflex over ( )}n) need be           calculated.       gcd(m, n)   Result of greatest common divisor (gcd) of m and n is           equivalent to gcd(m, n % m). Expression represented           by n need not be calculated, and thus is useful for           large n but m includes a manageable value.       n!/m!   Result of n!/m! is n*(n − 1)*(m + 1) if n &gt; m, n − m           is a small integer; useful when n! will result in an           unmanageably long calculation       permutation   Result of permutation (m, n) = n!/(n − m)!. Permutation       (m, n)   (m, n) need not be calculated.       common   Some expressions in common arithmetic, such as n − n,       arithmetic   n/n (for nonzero values of n), are replaceable with           simple identities because, for example, n − n = 0, n/n =           1, etc. Value of n, however expressed, need not be           calculated.                  
 
         [0043]     In sum, if an unmanageable expression is detected in calculating an expression from the bottom up, reevaluating the expression from the top down may allow the mathematical expression to be re-expressed. Re-expression of the mathematical expression may result in the unmanageable calculation being replaced with a simpler, more manageable calculation, or render the computation of the unmanageable portion of the calculation unnecessary.  
         [0000]     Treating Values as Variables to Avoid Excessive Computational Demands  
         [0044]      FIG. 5A  shows a mathematical expression  500 , formatted for processing by a computing system, of (12345 1000000000000000 ) 2 . Mathematical expression  500  includes a base 12345  502  raised by exponential operator  504  to exponent 1000000000000000  506 , the result of which is raised by exponential operator  508  to exponent 2  510 . As in the case of the preceding example of mathematical expression  400  ( FIG. 4A ), calculating mathematical expression  500  from the bottom up may result in a value too large to be calculated by a computing system. Conventional computer algebraic systems typically would stop the calculation, and return mathematical expression  500  in the original form (12345 1000000000000000 ) 2  presented because the computing system will not be able to calculate the value of mathematical expression.  
         [0045]     However, according to an embodiment of the present invention, upon encountering an incalculable expression, if some portion or all of the incalculable expression is replaced with a variable, the remaining steps in the calculation may be performable. As a result, although the entire calculation may not be performable, a partially calculated expression is presented to the user.  
         [0046]      FIG. 5B  illustrates how an embodiment of the present invention may re-express mathematical expression  500  ( FIG. 5A ) as mathematical expression  550 . Because calculation of base 12345  502  ( FIG. 5A ) raised by exponential operator  504  to exponent 1000000000000000  506  will overwhelm even significant computing systems, according to an embodiment of the present invention, base  502  is replaced with a variable x  552  instead of uselessly attempting to calculate 12345 1000000000000000 . Accordingly, replacing (12345 1000000000000000 ) 2  with (x 1000000000000000 ) 2 . Base x  552 , raised by exponential operator  504  to exponent 100000000000000  506 , in turn raised by exponential operator  508  to exponent 2  510 , yields the result x 2000000000000000 . In turn, replacing base x  552  with original base 12345  502  yields a final result of 12345 2000000000000000 . Thus, although a final, quantitative value of original mathematical expression  500  is not yielded, a partially calculated result is provided.  
         [0047]     Depending on the expression being calculated, a final quantitative result may be obtainable.  FIG. 6A  shows a mathematical expression  600 , formatted for processing by a computing system, of (12345 1000000000000000 ) 0.00000000000001 . Mathematical expression  600  includes a base 12345  602  raised by exponential operator  604  to exponent 1000000000000000  606 , the result of which is raised by exponential operator  608  to exponent 0.00000000000001  610 . As in the case of the preceding example of mathematical expressions  400  ( FIG. 4A ) and  500  ( FIG. 5A ), calculating mathematical expression  600  from the bottom up may result in a value too large to be calculated by a computing system. Conventional computer algebraic systems typically would stop the calculation, and return mathematical expression  600  in the original form (12345 1000000000000000 ) 0.00000000000001  presented because the computing system will not be able to calculate the value of mathematical expression.  
         [0048]      FIG. 6B , comparable to  FIG. 5B , illustrates how an embodiment of the present invention may re-express mathematical expression  600  ( FIG. 6A ) as mathematical expression  650 . Because calculation of base 12345  602  ( FIG. 5A ) raised by exponential operator  604  to exponent 1000000000000000  606  will overwhelm even significant computing systems, according to an embodiment of the present invention, base  602  is replaced with a variable x  652  instead of uselessly attempting to calculate 12345 1000000000000000 ) 0.000000000000001 . Accordingly, replacing (12345 1000000000000000 ) 0.00000000000001  with (x  1000000000000000 ) 0.00000000000001 . Base x  652 , raised by exponential operator  604  to exponent 1000000000000000  606 , in turn raised by exponential operator  608  to exponent 0.00000000000001  610 , yields the result x 10 . In turn, replacing base x  652  with original base 12345  602  yields the simplified expression 12345 10 . The expression 12345 10  is a very manageable calculation, yielding the result 82207405646327461794954634291560556640625. Thus, replacing a portion of a mathematical expression that may yield a large, unmanageable value with a variable, the remainder of the calculation can be completed, potentially yielding a simple quantitative result.  
         [0049]     In sum, by replacing a portion of a mathematical expression that may yield a very large, incalculable result may allow a computing system to calculate the remaining portions of the expression to simplify the expression or yield a quantitative result.  
         [0000]     Process of Re-expressing Mathematical Expressions to Permit Computation  
         [0050]      FIG. 7  is a flow diagram  700  illustrating the logical steps of an embodiment of the present invention for evaluating and, if necessary, re-expressing a mathematical expression so that the expression may be calculated when conventional computation of the expression would generate values too large to be processed by a conventional computing system. At step  702 , evaluation of the expression begins. At step  704 , instead of evaluating expressions from the bottom up, as is the practice with conventional computer algebra systems, the expression is evaluated from the top down. At decision step  706 , for each operator, children of the operator are evaluated to determine if the operator has any grandchildren. For example, referring to  FIG. 4A , for modulus division operator  402 , addition operator  404  and quotient 18970907  406  are children of modulus division operator  402 , while exponential operator  408  and value 1  410  are grandchildren of modulus division operator  402 .  
         [0051]     If it is determined at decision step  706  that the operator has grandchildren, at decision step  708 , it is determined if the operator includes a smart evaluation operator, such as those previously discussed in connection with  FIG. 4A  and Table (1). If so, at decision step  710 , it is determined if smart evaluation is applicable to the operator. If it is determined at decision step  710  that smart evaluation is applicable to the operator, at step  712 , smart evaluation is applied to the operator to eliminate potentially incalculable terms. At step  714 , the result is computed, and at step  716 , the result is presented to the user. At step  718 , flow diagram  700  ends.  
         [0052]     Alternatively, if it is determined at decision step  710  that a smart evaluation function is not applicable to the operator, at decision step  720 , it is determined if the operator can be cascaded downward through expression, as modulus division operator  402  ( FIG. 4A ) was cascaded down through the mathematical expression  450  in  FIG. 4B , thereby simplifying the task of computing the expression. If so, at step  722 , the operator is cascaded down through the expression, and flow diagram  700  loops to step  704  to evaluate the next operator potentially to further simplify the expression. It should be appreciated that the logical steps of flow diagram  700  are performed recursively as needed to fully simplify the entire mathematical expression presented for calculation.  
         [0053]     On the other hand, if it is determined at decision step  720  that the operator cannot be cascaded down through the expression, at decision step  728 , it is determined if the expression including the operator can be represented as an operation on a variable, as described in connection with  FIGS. 5A-6B . If so, at step  730 , the expression is re-expressed as an operation performed on a variable, and at step  714 , the result is computed by simplifying or, if possible, completely calculating the expression as previously described. At step  716 , the result is presented to the user, and at step  718 , flow diagram ends. Alternatively, if at decision step  728  it is determined that the expression cannot be represented as an operation performed on a variable, at step  732 , the expression is returned in its original form, and, at step  718 , flow diagram  700  ends.  
         [0054]     If it is determined at decision step  706  that the operator does not have grandchildren, at decision step  720 , it is determined if the expression potentially will produce results that are too large to be calculated. If not, flow diagram  700  proceeds to step  714 , where the result of the expression is calculated. If so, at step  724 , an attempt to estimate the size of the result is made. At decision step  726 , it is determined if the estimated result is too large to be computed. If so, flow diagram  700  progresses to decision step  728 , where it is determined if the expression can be represented as an operation performed on a variable, as previously described. On the other hand, it is determined that the estimated result is not to large to be calculated, the result is calculated at step  714 .  
         [0055]     Although the present invention has been described in connection with the preferred form of practicing it and modifications thereto, those of ordinary skill in the art will understand that many other modifications can be made to the present invention within the scope of the claims that follow. Accordingly, it is not intended that the scope of the invention in any way be limited by the above description, but instead be determined entirely by reference to the claims that follow.