COMPUTING DEVICE PERFORMANCE OF LOW PRECISION ARITHMETIC FUNCTIONS WITH ARRAYS OF PRECALCULATED VALUES

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First Claim
1. A method of improving computing device performance of arithmetic functions comprising:
 generating, for a first set of numeric input values, a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to a first arithmetic function given a specific, unique numeric input value of the first set of numeric input values;
storing the first set of numeric solution values into an array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values;
receiving, subsequent to the generating and the storing, a request to perform the first arithmetic function for a first provided numeric input value;
identifying a cell of the array corresponding to the first provided numeric input value; and
providing, in response to the request, the numeric solution value, from the first set of numeric solution values, that is stored in the identified cell;
wherein each numeric input value, of the first set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format; and
wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values.
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Abstract
Reduced precision computer number formats inherently limit the quantity of discrete numeric values that can be represented. Therefore, the solution values of an arithmetic function, for each numeric value that is individually and uniquely expressible utilizing such a reduced precision computer number format, can be precomputed since the quantity of unique solution values can be limited to a quantity that can be conveniently stored, such as in an array. Subsequently, rather than computing the solution value of such an arithmetic function, for a given input value, the precomputed array can be referenced and a solution value corresponding to the given input value can be read from the array. Reading numeric values from an array can be substantially faster than computing solution values of a computationallyexpensive arithmetic function.
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20 Claims
 1. A method of improving computing device performance of arithmetic functions comprising:
generating, for a first set of numeric input values, a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to a first arithmetic function given a specific, unique numeric input value of the first set of numeric input values; storing the first set of numeric solution values into an array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; receiving, subsequent to the generating and the storing, a request to perform the first arithmetic function for a first provided numeric input value; identifying a cell of the array corresponding to the first provided numeric input value; and providing, in response to the request, the numeric solution value, from the first set of numeric solution values, that is stored in the identified cell; wherein each numeric input value, of the first set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
 12. A microprocessor comprising:
logic circuitry for detecting when a second set of numeric input values includes a first provided numeric input value and, in response, returning a first numeric solution value that is the solution to a first arithmetic function given any one of the second set of numeric input values; and storage circuitry having an array encoded thereon, the array comprising a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to the first arithmetic function given a specific, unique numeric input value of a first set of numeric input values, the first set of numeric solution values being stored in the array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; wherein each numeric input value, of the first and second set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format, the first set of numeric input values consisting of a first subset of all possible numeric values representable with the predefined quantity of bits by the computer number format and the second set of numeric input values consisting of a second subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first and second subsets being wholly exclusive of one another; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values; wherein the microprocessor performs the first arithmetic function by receiving the first provided numeric input value, returning the first numeric solution value if the second set of numeric input values includes the first provided numeric input value, identifying a cell of the array corresponding to the first provided numeric input value and returning the numeric solution value that is stored in the identified cell if the first set of numeric input values includes the first provided numeric input value.  View Dependent Claims (13)
 14. A computing device comprising:
one or more processing units; a first computerreadable storage medium having an array encoded thereon, the array comprising a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to a first arithmetic function given a specific, unique numeric input value of a first set of numeric input values, the first set of numeric solution values being stored in the array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; and a second computerreadable storage medium comprising computerexecutable instructions which, when executed by the one or more processing units, cause the computing device to; receive a request to perform the first arithmetic function for a first provided numeric input value; identify a cell of the array corresponding to the first provided numeric input value; and provide, in response to the request, the numeric solution value, from the first set of numeric solution values, that is stored in the identified cell; wherein each numeric input value, of the first set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values.  View Dependent Claims (15, 16, 17, 18, 19, 20)
1 Specification
Some arithmetic functions can be very expensive to compute on conventional computer processing units (CPUs), taking tens or sometimes hundreds of clock cycles to compute. For example, an exponential function, such as computing the numeric value of Euler'"'"'s number raised to a power specified by an input numeric value can require multiple processor cycles and can, thereby, take a substantially longer amount of time to compute than other mathematical functions, such as a multiplication function, which can often be accomplished using highly optimized circuits in Arithmetic and Logic Units (ALUs) on modern CPUs. Often times, such specialized arithmetic functions without optimizations in ALUs, significantly slow down the overall execution of programs, in accordance with Ahmdal'"'"'s Law. Speeding up the execution of such functions can therefore result in significant overall efficiency benefits in terms of program run time.
Computing devices express numeric values utilizing standardized computer number formats, either integer or floating point. For example, the INT8 format utilizes eight bits to represent one of 255 integers ranging from −128 to 127. The precision and dynamic range of the numeric representation depends upon the number of bits used in the representation. For example, the INT8 format cannot be utilized to represent a fractional number, nor can it be utilized to represent any number larger than 127 or smaller than −128. The precision and dynamic range needs of the application determine the numerical format used. For example, a common computer number format is the IEEE singleprecision floatingpoint format which utilizes 32 bits of binary data to represent integers and fractional numeric values. Because the singleprecision floatingpoint format can uniquely identify billions of different numeric values, it can more accurately identify numeric values than, for example the aforementioned INT8 format.
However, utilization of a singleprecision floatingpoint format can require the maintenance and processing of 32 bits of numeric value data at each point in a computation. Hardware requirements (in terms of transistors) in ALUs for computation and memory requirements while operating on numerical formats with higher precision are proportionally higher. Hence, it is a lot more efficient to operate with lower precision when the application can tolerate that. Deep neural networks is one such class of applications where the reduced precision of 16bit or even 8bit representations is sufficient to achieve the desired accuracy of results. Examples of such representation formats are the aforementioned INT8 format, a floatingpoint analog thereof, referred to as FP8, or, commonly, a halfprecision floatingpoint format which utilizes 16, instead of 32, bits of data to represent numeric values. Nevertheless, even with such reduced precision computer number formats, the processing of certain arithmetic functions, such as an exponential function, a logarithmic function, a hyperbolic tangent function, a sigmoid function, or other similar arithmetic functions can be timeconsuming and can require substantial processing resources.
Reduced precision computer number formats inherently limit the quantity of discrete numeric values that can be represented utilizing the format. In such instances, the output values of an arithmetic function, for each numeric value that is individually and uniquely expressible, can be precomputed since the quantity of unique output values can be limited to a quantity that can be conveniently stored, such as in an array. Subsequently, rather than computing the output value every time such an arithmetic function is invoked, the precomputed array can be referenced and an output value corresponding to the given input value can be read from the array. Reading numeric values from an array can be substantially faster than computing output values of a computationally expensive arithmetic function. Due to the limited precision and dynamic range of lower precision numeric formats, the output values of an arithmetic function for multiple different input values can be the same due to saturation or rounding effects. In such instances, multiple equal numeric values need not be separately stored and, instead, logic circuitry, or the execution of computerexecutable instructions, can identify ranges of input numeric values for which the resulting solution values will all be the same due to the limited precision and range of the computer number format, and, in such instances, that resulting solution value can be returned without looking up the entire array. Multiple different computationally intensive arithmetic functions can be precomputed, and their output values can be stored in separate cells of a single row or column, depending on the orientation, of the aforementioned array.
This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter.
Additional features and advantages will be made apparent from the following detailed description that proceeds with reference to the accompanying drawings.
The following detailed description may be best understood when taken in conjunction with the accompanying drawings, of which:
The following description relates to utilizing precomputed numeric output values to more quickly find the solution to an arithmetic function. Reduced precision computer number formats inherently limit the quantity of discrete numeric values that can be uniquely represented utilizing the format. In such instances, the output values of an arithmetic function, for each numeric value that is individually and uniquely expressible, can be precomputed since the quantity of unique output values can be limited to a quantity that can be conveniently stored, such as in an array. Subsequently, rather than computing the output value when such an arithmetic function is invoked, the precomputed array can be referenced and an output value corresponding to the given input value can be read from the array. Reading numeric values from an array can be substantially faster than computing output values of a computationally expensive arithmetic function. Due to the limited precision and dynamic range of lower precision numeric formats, the output values of an arithmetic function for multiple different input values can be the same due to saturation or rounding effects. In such instances, multiple equal numeric values need not be separately stored and, instead, logic circuitry, or the execution of computer executable instructions, can identify ranges of input numeric values for which the resulting solution values will all be the same due to the limited precision of the computer number format, and, in such instances, that resulting output value can be returned without looking up the entire array. Multiple different computationally intensive arithmetic functions can be precomputed, and their output values can be stored in separate cells of a single row or column, depending on the orientation, of the aforementioned array.
Although not required, the description below will be in the general context of computerexecutable instructions, such as program modules, being executed by a computing device. More specifically, the description will reference acts and symbolic representations of operations that are performed by one or more computing devices or peripherals, unless indicated otherwise. As such, it will be understood that such acts and operations, which are at times referred to as being computerexecuted, include the manipulation by a processing unit of electrical signals representing data in a structured form. This manipulation transforms the data or maintains it at locations in memory, which reconfigures or otherwise alters the operation of the computing device or peripherals in a manner well understood by those skilled in the art. The data structures where data is maintained are physical locations that have particular properties defined by the format of the data.
Generally, program modules include routines, programs, objects, components, data structures, and the like that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the computing devices need not be limited to conventional personal computers, and include other computing configurations, including servers, handheld devices, multiprocessor systems, microprocessor based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, and the like. Similarly, the computing devices need not be limited to standalone computing devices, as the mechanisms may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.
With reference to
In the INT8 and FP8 computer number formats, only 255 unique numeric values can be expressed, with the INT8 computer number format limiting those 255 unique numeric values to integer values, while the FP8 computer number format includes fractional values in those 255 numeric values. Thus, in either the INT8 or FP8 formats, the possible numeric input values to the arithmetic function 120 would be one of those 255 numeric values, since no other numeric values can be represented by those computer number formats. Accordingly, if the solutions to the arithmetic formula 120 for each of those 255 unique numeric values were precomputed, then no matter what input was provided, the processor 110 would already have access to the solution, and could obtain the solution through a simple lookup, which can be orders of magnitude faster than computationally solving the arithmetic function 120 in realtime depending on the underlying hardware used to perform the computation in real time. In a similar manner, the FP16 format is only capable of expressing 16,535 unique numeric values, with those 16,535 unique numeric values including fractional values. While the quantity of unique numeric values is substantially greater, the mechanisms described herein are equally applicable. For example, all 16,535 solutions to the arithmetic function 120, for all 16,535 possible numeric input values, can be stored in 128 kilobytes or less of memory, which is a sufficiently small amount of computerreadable storage that its utilization for the mechanisms described herein would have minimal impact and cost.
For purposes of illustration and description, references below will be made primarily to the INT8 computer number format strictly for conceptual simplicity and ease of graphical illustration. As indicated, the mechanisms described herein are applicable to any computer number format. Turning back to
According to one aspect, a numeric solution value for each of the numeric input values 130 can be precomputed. In such an instance, as indicated, every possible input will have an already precomputed solution to the arithmetic function 120, enabling the processor 110 to obtain a solution to any possible input, again, given the INT8 computer number format being utilized, by simply performing a lookup, as opposed to actually processing the arithmetic function 120. The numeric solution values 140, shown in
Once the numeric solution values 140 have been precomputed, they can be stored into an array, such as the exemplary array 150. Subsequently, should the processor 110 be requested to perform the arithmetic function 120, the processor 110 can do so by simply looking up the appropriate numeric solution value from the array 150, given the numeric input value for which the arithmetic function 120 was requested to be performed.
Turning to
Such a decoding of memory addresses of individual cells of the array 150 based on a numeric input value can be performed by the decoder 220. Thus, for example, upon receipt of a request to perform the arithmetic function 120 (shown in
As indicated previously, for certain arithmetic functions, given the limitations of precision offered by computer number formats, rounding or saturation effects can cause multiple different numeric input values to have the same numeric solution values. For example, as illustrated in
Turning to
Conversely, if the range detector 310 determines that the numeric input value 210 is in the third range, processing can proceed as detailed above. More specifically, the range detector 310 can trigger the invocation of the decoder 220, which can then identify a memory address of a cell, such as the exemplary cell 351, in the array 350, having stored therein the solution to the arithmetic function for the numeric input value 210. As can be seen from the exemplary system 300 of
Turning to
Another arithmetic function that can have solutions precomputed for it can be a logarithmic function, such as the natural logarithm of an input. A range of numeric input values 430 can, like the numeric input values 130, be all of the numeric values that are representable utilizing the computer number format. In the example illustrated in
The numeric solution values 440 can themselves be stored in an array, such as the exemplary array 450. According to one aspect, the numeric solution values 440 can be stored in a row of the array 450, such as the exemplary row 452. The storage of the numeric solution values 440 in the exemplary row 452 can be aligned with the storage of the numeric solution values 140, in the exemplary row 451, such that the cells of a single column, such as the exemplary column 461, comprise the solutions to various different arithmetic functions given the same numeric input value. For example, the exemplary column 461 can comprise a cell, from the row 451, containing the solution to the arithmetic function of Euler'"'"'s number raised to an exponent, where the input value of that function is “3”. Correspondingly, the exemplary column 461 can comprise a cell, from the row 452, containing the solution to the arithmetic function of the natural log of an input value, where that value is, again, “3”. Although described in terms of rows of numeric solution values and columns of the same input value, the twodimensional array 450 could be oriented in an orthogonal manner, where the numeric solution values are arranged in columns and individual rows contain the numeric solution values of different functions given a same input numeric value.
A decoder, such as that described previously, can similarly be utilized with a twodimensional array. For example, if each row of the array consists of 255 separate cells, corresponding to the 255 different numeric values expressible by, for example, the INT8 computer number format, then equivalent cells, such as the cells in a same column, can be 255 memory addresses apart, assuming the cells of the array are stored sequentially and are byte addressable. Thus, a decoder can identify an appropriate cell within a first row, and then increment the corresponding memory address based on a quantity of rows “down” the same column until reaching the row corresponding to the particular arithmetic function for which the result, given the numeric input value, is desired.
Turning to
The exemplary system 502, also shown in
The exemplary system 503, shown in
Turning to
Turning to
Turning to
The computing device 800 also typically includes computer readable media, which can include any available media that can be accessed by computing device 800 and includes both volatile and nonvolatile media and removable and nonremovable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes media implemented in any method or technology for storage of content such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CDROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired content and which can be accessed by the computing device 800. Computer storage media, however, does not include communication media. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any content delivery media. By way of example, and not limitation, communication media includes wired media such as a wired network or directwired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of the any of the above should also be included within the scope of computer readable media.
The system memory 830 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 831 and random access memory (RAM) 832. A basic input/output system 833 (BIOS), containing the basic routines that help to transfer content between elements within computing device 800, such as during startup, is typically stored in ROM 831. RAM 832 typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 820. By way of example, and not limitation,
The computing device 800 may also include other removable/nonremovable, volatile/nonvolatile computer storage media. By way of example only,
The drives and their associated computer storage media discussed above and illustrated in
The computing device 800 may operate in a networked environment using logical connections to one or more remote computers. The computing device 800 is illustrated as being connected to the general network connection 851 (to a network 852) through a network interface or adapter 850, which is, in turn, connected to the system bus 821. In a networked environment, program modules depicted relative to the computing device 800, or portions or peripherals thereof, may be stored in the memory of one or more other computing devices that are communicatively coupled to the computing device 800 through the general network connection 861. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between computing devices may be used.
Although described as a single physical device, the exemplary computing device 800 can be a virtual computing device, in which case the functionality of the abovedescribed physical components, such as the CPU 820, the system memory 830, the network interface 860, and other like components can be provided by computerexecutable instructions. Such computerexecutable instructions can execute on a single physical computing device, or can be distributed across multiple physical computing devices, including being distributed across multiple physical computing devices in a dynamic manner such that the specific, physical computing devices hosting such computerexecutable instructions can dynamically change over time depending upon need and availability. In the situation where the exemplary computing device 800 is a virtualized device, the underlying physical computing devices hosting such a virtualized computing device can, themselves, comprise physical components analogous to those described above, and operating in a like manner. Furthermore, virtual computing devices can be utilized in multiple layers with one virtual computing device executing within the construct of another virtual computing device. The term “computing device”, therefore, as utilized herein, means either a physical computing device or a virtualized computing environment, including a virtual computing device, within which computerexecutable instructions can be executed in a manner consistent with their execution by a physical computing device. Similarly, terms referring to physical components of the computing device, as utilized herein, mean either those physical components or virtualizations thereof performing the same or equivalent functions.
The descriptions above include, as a first example a method of improving computing device performance of arithmetic functions comprising: generating, for a first set of numeric input values, a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to a first arithmetic function given a specific, unique numeric input value of the first set of numeric input values; storing the first set of numeric solution values into an array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; receiving, subsequent to the generating and the storing, a request to perform the first arithmetic function for a first provided numeric input value; identifying a cell of the array corresponding to the first provided numeric input value; and providing, in response to the request, the numeric solution value, from the first set of numeric solution values, that is stored in the identified cell; wherein each numeric input value, of the first set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values.
A second example is the method of the first example, wherein the first set of numeric input values consists of all possible numeric values representable with the predefined quantity of bits by the computer number format.
A third example is the method of the first example, wherein the first set of numeric input values consists of a first subset of all possible numeric values representable with the predefined quantity of bits by the computer number format, the determining only being performed if the first set of numeric input values includes the first provided numeric input value.
A fourth example is the method of the third example, further comprising: providing, in response to the request, a first numeric solution value if a second set of numeric input values includes the first provided numeric input value; wherein the second set of numeric input values consists of a second subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first and second subsets being wholly exclusive of one another; and wherein the provided first numeric solution value is the solution to the first arithmetic function given any one of the second set of numeric input values.
A fifth example is the method of the fourth example, wherein the provided first numeric solution value is one of: a maximum value representable with the predefined quantity of bits by the computer number format, a minimum value representable with the predefined quantity of bits by the computer number format, or zero.
A sixth example is the method of the fourth example, further comprising: providing, in response to the request, a second numeric solution value if a third set of numeric input values includes the first provided numeric input value; wherein the third set of numeric input values consists of a third subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first, second and third subsets all being wholly exclusive of one another; wherein the provided second numeric solution value is the solution to the first arithmetic function given any one of the third set of numeric input values; wherein each of the first set of numeric input values is numericly larger than any of the second set of numeric input values; and wherein each of the first set of numeric input values is numericly smaller than any of the third set of numeric input values.
A seventh example is the method of the first example, wherein the storing the first set of numeric solution values into the array comprises correlating a first memory addresses of a first cell of the array to a first numeric input value whose solution to the first arithmetic function is a first numeric solution value stored in the first cell.
An eighth example is the method of the first example, wherein each numeric solution value, of the first set of numeric solution values, is represented by a second predefined quantity of bits differing from the predefined quantity of bits utilized to represent each numeric input value of the first set of numeric input values.
A ninth example is the method of the first example, further comprising: generating, for the first set of numeric input values, a second set of numeric solution values, wherein each numeric solution value, in the second set of numeric solution values, is a solution to a second arithmetic function given a specific, unique numeric input value of the first set of numeric input values; storing the second set of numeric solution values into the array such that each cell of a portion of the array comprising the second set of numeric solution values corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the second set of numeric solution values, which is the solution to the second arithmetic function given the corresponding one of the input numeric values; receiving, subsequent to the generating and the storing, a request to perform the second arithmetic function for a second provided numeric input value; identifying a second cell of the array corresponding to the second provided numeric input value; and providing, in response to the request, the numeric solution value, from the second set of numeric solution values, that is stored in the identified second cell.
A tenth example is the method of the ninth example, wherein either: (1) each column or (2) each row of the array comprises: a first cell having stored therein a first numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given a first input numeric value, from the first set of numeric input values; and a second cell having stored therein a second numeric solution value, from the second set of numeric solution values, which is the solution to the second arithmetic function given the first input numeric value, from the first set of numeric input values.
An eleventh example is the method of the first example, wherein the first arithmetic function is one of: a logarithm of a numeric input value, an exponentiation of the numeric input value, a hyperbolic of the numeric input value or a sigmoid of the numeric input value.
A twelfth example is a microprocessor comprising: logic circuitry for detecting when a second set of numeric input values includes a first provided numeric input value and, in response, returning a first numeric solution value that is the solution to a first arithmetic function given any one of the second set of numeric input values; and storage circuitry having an array encoded thereon, the array comprising a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to the first arithmetic function given a specific, unique numeric input value of a first set of numeric input values, the first set of numeric solution values being stored in the array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; wherein each numeric input value, of the first and second set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format, the first set of numeric input values consisting of a first subset of all possible numeric values representable with the predefined quantity of bits by the computer number format and the second set of numeric input values consisting of a second subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first and second subsets being wholly exclusive of one another; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values; wherein the microprocessor performs the first arithmetic function by receiving the first provided numeric input value, returning the first numeric solution value if the second set of numeric input values includes the first provided numeric input value, identifying a cell of the array corresponding to the first provided numeric input value and returning the numeric solution value that is stored in the identified cell if the first set of numeric input values includes the first provided numeric input value.
A thirteenth example is the microprocessor of the twelfth example, wherein the array is encoded onto the storage circuitry during an initialization of the microprocessor by the microprocessor precalculating each numeric solution value, in the first set of numeric solution values.
A fourteenth example is a computing device comprising: one or more processing units; a first computerreadable storage medium having an array encoded thereon, the array comprising a first set of numeric solution values, wherein each numeric solution value, in the first set of numeric solution values, is a solution to a first arithmetic function given a specific, unique numeric input value of a first set of numeric input values, the first set of numeric solution values being stored in the array such that each cell of the array corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given the corresponding one of the input numeric values; and a second computerreadable storage medium comprising computerexecutable instructions which, when executed by the one or more processing units, cause the computing device to: receive a request to perform the first arithmetic function for a first provided numeric input value; identify a cell of the array corresponding to the first provided numeric input value; and provide, in response to the request, the numeric solution value, from the first set of numeric solution values, that is stored in the identified cell; wherein each numeric input value, of the first set of numeric input values, is represented by a predefined quantity of bits in accordance with a predefined computer number format; and wherein the first set of numeric input values is contiguous such that, given the predefined quantity of bits, the predefined computer number format cannot represent another numeric value between two adjacent input numeric values of the first set of numeric input values.
A fifteenth example is the computing device of the fourteenth example, wherein the first computerreadable storage medium is a ROM.
A sixteenth example is the computing device of the fourteenth example, wherein the first computerreadable storage medium is a RAM; and wherein further the second computerreadable storage medium comprises further computerexecutable instructions which, when executed by the one or more processing units, cause the computing device to read the array from the second computerreadable storage medium and load it into the RAM.
A seventeenth example is the computing device of the fourteenth example, wherein the second computerreadable storage medium comprises further computerexecutable instructions which, when executed by the one or more processing units, cause the computing device to: generate, for the first set of numeric input values, the first set of numeric solution values; and store the first set of numeric solution values into the array.
An eighteenth example is the computing device of the fourteenth example, wherein the second computerreadable storage medium comprises further computerexecutable instructions which, when executed by the one or more processing units, cause the computing device to: provide, in response to the request, a first numeric solution value if a second set of numeric input values includes the first provided numeric input value; wherein the first set of numeric input values consists of a first subset of all possible numeric values representable with the predefined quantity of bits by the computer number format, the determining only being performed if the first set of numeric input values includes the first provided numeric input value; wherein the second set of numeric input values consists of a second subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first and second subsets being wholly exclusive of one another; and wherein the provided first numeric solution value is the solution to the first arithmetic function given any one of the second set of numeric input values.
A nineteenth example is the computing device of the fourteenth example, wherein one of the processing units comprises logic circuitry for detecting when a second set of numeric input values includes the first provided numeric input value and, in response, returning a provided first numeric solution value that is the solution to a first arithmetic function given any one of the second set of numeric input values; wherein the first set of numeric input values consists of a first subset of all possible numeric values representable with the predefined quantity of bits by the computer number format, the determining only being performed if the first set of numeric input values includes the first provided numeric input value; and wherein the second set of numeric input values consists of a second subset of the all possible numeric values representable with the predefined quantity of bits by the computer number format, the first and second subsets being wholly exclusive of one another.
A twentieth example is the computing device of the fourteenth example, wherein either: (1) each column or (2) each row of the array comprises: a first cell having stored therein a first numeric solution value, from the first set of numeric solution values, which is the solution to the first arithmetic function given a first input numeric value, from the first set of numeric input values; and a second cell having stored therein a second numeric solution value, from a second set of numeric solution values, which is a solution to a second arithmetic function given the first input numeric value, from the first set of numeric input values; wherein each numeric solution value, in the second set of numeric solution values, is the solution to the second arithmetic function given a specific, unique numeric input value of the first set of numeric input values, the second set of numeric solution values also having been stored into the array such that each cell of a portion of the array comprising the second set of numeric solution values corresponds to one of the input numeric values, from the first set of numeric input values, and has stored therein a numeric solution value, from the second set of numeric solution values, which is the solution to the second arithmetic function given the corresponding one of the input numeric values.
As can be seen from the above descriptions, mechanisms for improving the performance of a computing device in generating a solution to an arithmetic function by utilizing an array of precomputed numeric solution values have been presented. In view of the many possible variations of the subject matter described herein, we claim as our invention all such embodiments as may come within the scope of the following claims and equivalents thereto.