Quick high-precision calculation method and system for variable bottom exponential function

Abstract

The invention discloses a rapid high-precision calculation method and system for a variable bottom exponential function. The method comprises the following steps: scaling one of three initial values for subsequent calculation; the exponential function of any fixed base number meeting the convergence range is calculated through multiple iterations, the number of iterations is reduced based on the optimization of the CORDIC exponential function of the radix-8 hyperbolic rotation mode, and the efficiency is improved; and obtaining a final result from the converged variable value. According to the method, on the basis of keeping the flexibility of the generalized CORDIC, any fixed base number exponential function meeting the convergence condition can be directly calculated; besides, single-iteration multi-bit parallel processing is realized by adopting a radix-8 basis decomposition technology, so that the total number of iterations is reduced by more than 50%, meanwhile, the calculation precision is kept unchanged, and the hardware implementation efficiency is effectively optimized.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of integrated circuit function calculation, in particular to a fast and high-precision calculation method and system of variable base exponential function, especially suitable for high real-time requirement application scenarios such as artificial intelligence acceleration and high-precision floating point operation. BACKGROUND

[0002] Function value approximation algorithms play an important role in scientific computing, signal processing and engineering applications, and are mainly used for efficient solving of transcendental functions (such as exponential, logarithmic, trigonometric functions, etc.) and complex mathematical operations. Traditional methods include Taylor series expansion, lookup table (LUT), polynomial approximation and iterative algorithm, etc. Among them, CORDIC (Coordinated Rotation Digital Computer) as a function value approximation algorithm based on rotation iteration, is widely used in the calculation of technical functions such as DSP, FFT, DCT, etc. due to its hardware-friendly nature.

[0003] In the aspect of exponential and logarithmic function value approximation calculation, the traditional hyperbolic CORDIC algorithm has two main limitations: one is that it only supports natural base calculation, and needs additional multipliers to convert the base, resulting in waste of hardware resources; the second is that the radix-2 iteration results in high delay. To solve these problems, generalized hyperbolic CORDIC is proposed, which designs parameters such as radix 2, radix 10, etc., eliminates the radix conversion multiplier, directly supports the calculation of exponential and logarithmic functions with any fixed base, and supports the implementation of more complex functions. In addition, radix-2 hyperbolic CORDIC shows potential in floating point operation, which can be used for efficient calculation of exponential function and Nth root.

[0004] However, the radix-2 hyperbolic CORDIC still has the problem of high calculation delay, which is difficult to meet the real-time calculation requirement. The limitations of existing function value approximation algorithms include: low iteration efficiency, radix-2 method only processes 1 bit at a time, long calculation period under high precision requirement; large hardware resource occupation, improving the radix, such as radix-4 or higher, needs complex direction selection logic, which is difficult to balance speed and area efficiency; limited application scenarios, existing methods are difficult to meet the requirements of high precision and low delay at the same time.

[0005] Therefore, researching more efficient function value approximation algorithms, such as high-radix CORDIC, has become an important direction to meet the requirements of high precision and low delay, and to increase the range of base selection to adapt to different environmental requirements. SUMMARY

[0006] The present application aims at a fast and high-precision calculation method and system of variable base exponential function, which can directly calculate any fixed base exponential function meeting the convergence condition on the basis of preserving the flexibility of generalized CORDIC; in addition, the base-8 base decomposition technology is adopted to realize single iteration multi-bit parallel processing, so that the total iteration times are reduced by more than 50%, while the calculation precision is kept unchanged, and the hardware implementation efficiency is effectively optimized.

[0007] To achieve the above technical purposes, the technical scheme adopted by the present application is:

[0008] In the first aspect, the present application discloses a fast and high-precision calculation method of variable base exponential function, which comprises the following steps:

[0009] For the exponential function to be solved , the initial values of the horizontal axis component, the vertical axis component and the residual angle component are inputted respectively as ;

[0010] A scaling variable is introduced , and the base-8 hyperbolic rotation mode CORDIC algorithm is used for N times of iteration of the three variables, until converges to 0, , , and and are added to obtain the exponential function to be solved ; the formula of the horizontal axis variable , the vertical axis variable and the scaling variable of the i+1th iteration is:

[0011] ;

[0012] ;

[0013] ;

[0014] wherein , N is the maximum iteration times; before each iteration, the interval boundary is simplified based on the hardware condition, so that is a known quantity, and the value of the iteration direction factor is confirmed according to the following conditions:

[0015] ;

[0016] wherein, the calculation formula of the interval boundary and is:

[0017] ;​

[0018] ;

[0019] .

[0020] Further, at the beginning of each iteration, the inverse scaling factor of the i-th iteration of the radix-8 hyperbolic rotation mode CORDIC algorithm is calculated by the following formula is:

[0021] ;

[0022] The inverse scaling factor of the i-th iteration is multiplied with the inverse scaling factor generated in the previous iteration in turn, and the obtained accumulated inverse scaling factor is multiplied to the convergent value of the iteration variable to correct the data system error caused by the rotation iteration.

[0023] Further, the solving process of the inverse scaling factor is:

[0024] The Taylor expansion is performed on the inverse scaling factor , and the following expansion is obtained: ;

[0025] If , the inverse scaling factor is approximated to ;

[0026] If , the inverse scaling factor is approximated to the first two terms of the expansion , and the expansion is expanded to the required n-bit precision; n is a natural number;

[0027] If , the inverse scaling factor is approximated to the first three terms of the expansion , and the expansion is expanded to the required n-bit precision;

[0028] For the remaining iteration cases, the approximate value of the inverse scaling factor calculated in advance is directly referenced according to the requirement using the first lookup table.

[0029] Further, the solving process of the inverse hyperbolic tangent function includes:

[0030] The Taylor expansion is performed on the inverse hyperbolic tangent function , and the expansion is ;

[0031] For the case of , the inverse hyperbolic tangent function ​Approximately , while simplifying , is as follows:

[0032] ;

[0033] wherein is a constant; if is not , the result is , the constant part is stored in binary form, and other numerical values are quickly obtained through a shift operation, if , skip this iteration;

[0034] For the case of , the number of retained terms of the Taylor expansion is selected according to the accuracy requirement, and the approximate value of the inverse hyperbolic tangent function calculated in advance is directly applied by using a second lookup table.

[0035] In a second aspect, the application discloses a fast and high-precision calculation system of a variable-base exponential function, which comprises a preprocessing module, a base-8 hyperbolic rotation mode CORDIC iteration module and a post-processing module.

[0036] The preprocessing module is used for solving an exponential function , and initial values of horizontal axis components, vertical axis components and residual angle components are input as , respectively. ;

[0037] The base-8 hyperbolic rotation mode CORDIC iteration module introduces a scaling variable , and performs iteration of three variables , and for N times until converges to 0; the iteration formula is as follows:

[0038] ;

[0039] ;

[0040] ;

[0041] wherein , N is the maximum number of iterations; before each iteration, the interval boundary is simplified based on the hardware condition, so that is a known quantity, and the value of an iteration direction factor is determined according to the following conditions:

[0042] ;

[0043] wherein the interval boundary and The calculation formula is:

[0044] ;

[0045] ;

[0046] ;

[0047] The post-processing module will and Adding the two together yields the exponential function to be solved. ,in, , .

[0048] Furthermore, the radix-8 hyperbolic rotation mode CORDIC iterative module includes an inverse scaling factor calculation unit and an error elimination unit;

[0049] The inverse scaling factor calculation unit calculates the inverse scaling factor for the i-th iteration of the CORDIC algorithm in the radix-8 hyperbolic rotation mode using the following formula. for:

[0050] ;

[0051] The error elimination unit multiplies the inverse scaling factor of the i-th iteration with the inverse scaling factors generated in previous iterations, and then calculates the cumulative inverse scaling factor. Multiply by the convergence value of the iteration variable to correct for data system errors caused by rotation iteration.

[0052] Furthermore, the process of solving for the inverse scaling factor is as follows:

[0053] For the reverse scaling factor Performing a Taylor expansion, we obtain the following expansion: ;

[0054] Among them, if Inverse scaling factor Approximately ;

[0055] if Inverse scaling factor The first two terms of the approximate expansion Expand to Bit precision;

[0056] if Inverse scaling factor Approximate to the first three terms of the expansion Expanded to n-digit precision;

[0057] For the rest of the iteration cases, the first lookup table is used to directly reference the pre-computed approximate value of the inverse scaling factor according to the requirements.

[0058] Further, the solution process of the inverse hyperbolic tangent function includes:

[0059] The inverse hyperbolic tangent function is Taylor expanded to obtain the expansion formula ;

[0060] For the case of , the is approximated as , and the is simplified as follows:

[0061] ;

[0062] where is a constant; if is not , the result is , and the constant part is stored in binary form, and then the other numerical values are quickly obtained through a shift operation, and if , the iteration is skipped.

[0063] For the case of , the number of retained terms of the Taylor expansion is selected according to the accuracy requirement, and the second lookup table is used to directly apply the pre-computed approximate value of the inverse hyperbolic tangent function.

[0064] Compared with the prior art, the beneficial effects of the present application are as follows:

[0065] First, the fast and high-precision calculation method and system of the variable base exponential function of the present application can reduce the iteration number by at least 50% under the premise of maintaining the same accuracy as the traditional base-2 hyperbolic CORDIC, for example, from 18 times to 8 times at 16-bit precision, significantly improving the calculation speed.

[0066] Second, the fast and high-precision calculation method and system of the variable base exponential function of the present application support the calculation of exponential functions with any fixed base number that meets the convergence range, without the need for additional multipliers or lookup table conversion, expanding the application range of the traditional hyperbolic CORDIC. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 is the architecture diagram of the fast and high-precision calculation system of the variable base exponential function of the present application;

[0068] Figure 2For Q=0.5, the absolute value of error corresponding to r8b2 and r2b2 is compared with the schematic diagram;

[0069] Figure 3 For Q=0.6, the absolute value of error corresponding to r8b2 and r2b2 is compared with the schematic diagram;

[0070] Figure 4 For Q=0.7, the absolute value of error corresponding to r8b2 and r2b2 is compared with the schematic diagram. DETAILED DESCRIPTION

[0071] The embodiments of the present application are further described in detail below with reference to the accompanying drawings.

[0072] The present application discloses a fast and high-precision calculation method of variable base exponential function, which comprises the following steps:

[0073] For the exponential function to be solved , the initial values of the horizontal axis component, the vertical axis component and the residual angle component are inputted respectively ;

[0074] A scaling variable is introduced , and the CORDIC algorithm of base-8 hyperbolic rotation mode is used for N times of iteration of the three variables until converges to 0, at this time also converges to 0, , , and and are added to obtain the result; the base-8 hyperbolic rotation mode CORDIC iteration module of the present application uses function, and the iteration formula is:

[0075] ;

[0076] ;

[0077] ;

[0078] wherein, N is the maximum iteration number; before each iteration, the interval boundary is simplified based on the hardware condition, and the known quantity is taken as , and the value of the iteration direction factor is confirmed according to the following conditions:

[0079] ;

[0080] wherein, the calculation formula of the interval boundary and is:

[0081] ;

[0082] ;

[0083] .

[0084] Accordingly, as shown in Figure 1 , the application discloses a fast high-precision calculation system of variable base exponential function, which comprises a preprocessing module, a base-8 hyperbolic rotation mode CORDIC iteration module and a post-processing module.

[0085] The preprocessing module inputs initial values for the exponential function to be solved , respectively ;

[0086] The base-8 hyperbolic rotation mode CORDIC iteration module introduces a scaling variable , and performs N times of iteration of three variables , and until converges to 0; the iteration formula is as follows:

[0087] ;

[0088] ;

[0089] ;

[0090] wherein, x0, y0 and z0 are initial values of x, y and z respectively, N is the maximum number of iterations; before each iteration, the interval boundaries are simplified based on the hardware condition, and the known quantity is taken as , and the value of the iteration direction factor is confirmed according to the following conditions:

[0091] ;

[0092] wherein, the calculation formulae of the interval boundaries and are as follows:

[0093] ;

[0094] ;

[0095] ;

[0096] The post-processing module adds and to obtain the result, wherein,​ , .

[0097] The present application defines a scaling variable , wherein the initial value of the scaling variable is substituted into the iterative calculation, and the iteration process is as follows: introducing the scaling variable , confirming the iteration direction factor before each iteration , and calculating the interval boundary according to the scaling variable , bringing it into the iteration formula to iterate the three variables, and finally converging to , at this time , and the sum of the two is the result. The present application uses the base-8 hyperbolic rotation mode CORDIC iteration module to calculate the exponential function result of any fixed base that meets the convergence range through multiple iterations, and then uses the post-processing module to further process the convergence values of each branch of the base-8 hyperbolic rotation mode CORDIC iteration module, and finally obtains the operation result of the exponential function.

[0098] (I) Iteration direction factor

[0099] In the present application, the process of processing the exponential calculation algorithm is as follows: separating the integer part and the decimal part of the exponent, making the target exponential function value into the product of the integer power of the base and the decimal power of the base, calculating the value of the decimal power of the base through the base-8 hyperbolic rotation mode CORDIC iteration module in the system, and then multiplying it with the former to obtain the target value. Experiments have proved that the convergence range of the base-8 hyperbolic rotation mode CORDIC iteration module changes with the base, and when the base is 2, the convergence range of the algorithm involved includes , so the target exponential function value can be calculated while meeting the convergence range, and the base can be flexibly selected.

[0100] Compared with the base-2 generalized hyperbolic rotation mode CORDIC, the base-8 generalized hyperbolic rotation mode CORDIC iteration module of the present application does not need to repeat iteration to ensure convergence. In the base-2 generalized hyperbolic rotation mode CORDIC, in order to ensure the convergence of the function value, the change value of the iteration variable at each iteration should be less than twice the change value at the next iteration. When i takes certain values, , which is slightly greater than 2. Therefore, in order to make the base-2 generalized hyperbolic rotation mode CORDIC converge, when the iteration sequence number i is equal to 4, 13, 40 and other values, it should be repeated once. The base-8 generalized hyperbolic rotation mode CORDIC iteration module of the present application depends on the selection of the iteration direction factor , which determines the convergence of the function value and affects the complexity of its hardware implementation, and does not need to repeat iteration.

[0101] To ensure the convergence of the CORDIC iterative module for the radix-8 generalized hyperbolic rotation mode, the iterative direction factor is determined through the convergence equation. The selection criteria and scaling variables It must be strictly limited to the interval specified by the iteration conditions. However, for different... value The selection ranges overlap, therefore not all criteria are applicable. To address this, this invention designs a hardware-friendly judgment criterion that significantly simplifies the implementation architecture while ensuring convergence. The formula is:

[0102] ;

[0103] ;

[0104] ;

[0105] Simplify the interval boundaries to adapt to hardware conditions, and compare with fixed interval boundaries. satisfy It is worth noting that when At that time, there exists a zero-skip technique, that is, when If no changes occur in this iteration, the algorithm will skip it.

[0106] In the CORDIC iterative module of the radix-8 hyperbolic rotation mode, the iterative direction factor is... The selection criteria depend on variables , Store according to demand.

[0107] (ii) Inverse scaling factor

[0108] Through continuous iteration based on specific accuracy requirements, the radix-8 hyperbolic rotation mode CORDIC iterative module can achieve the same convergence function as the radix-2 hyperbolic CORDIC. The key difference lies in the scaling factor: in the radix-8 configuration, the scaling factor is a real-time calculated value rather than a deterministic constant. We redefine it as the cumulative inverse scaling factor. The formula is as follows:

[0109] ;

[0110] Iteration direction factor The value will change depending on the iteration, therefore It needs to be calculated in real time to maintain accuracy.

[0111] The inverse scaling factor in the i-th iteration of this invention After Taylor expansion, it expands to the following form: ,in, the reverse scaling factor is approximated as ; in the case of , the reverse scaling factor is approximated as the first two terms of the above expansion, in the case of , the reverse scaling factor is approximated to the first three terms of the expansion, still expanded to n-bit precision; for the rest of the iteration cases, pre-computed value storage using LUT is used as needed.

[0112] The meaning of calculating the above i-th reverse scaling factor at each iteration is that only its successive multiplication is needed, that is , and multiplying it to the convergence value of the iteration variable can correct the data system error caused by the rotation iteration.

[0113] (Three) inverse hyperbolic tangent function

[0114] The inverse hyperbolic tangent function is the inverse hyperbolic tangent function of replacing the natural exponential with the exponential with base , and the Taylor expansion form of the inverse hyperbolic tangent function is . In the present application, for the case of , can be approximated as , and will be simplified as follows:

[0115] ;

[0116] where is a constant, and if is not , the result is , and the constant part is stored in binary form, and then other numerical values are quickly obtained through a shift operation. For the case of , the higher the precision requirement, the more terms the Taylor expansion retains, at which time a lookup table can be used to directly store and reference the calculated data.

[0117] The present application starts from the initial value of three input variables, performs iteration in a pipeline architecture to improve the throughput of hardware, and after the first iteration, uses a parallel comparator to process the scaling variable before each subsequent iteration to obtain an iteration direction factor , which participates in subsequent calculations. It is worth noting that the present application does not require repeated iterations like the traditional generalized hyperbolic rotation mode CORDIC, so the number of iterations can be reduced by more than 50%. After all iterations are completed, the and The result is obtained by addition.

[0118] As shown in Figures 2 to 4 the absolute value of the difference between the accurate value and the calculated value of the conventional base-2 generalized hyperbolic rotation mode CORDIC and the base-8 generalized hyperbolic rotation mode CORDIC iteration module of the present application with base 2 and Q values of 0.5, 0.6, and 0.7 respectively increases with the iteration number. According to the experimental results in Figure 2 , the iteration number is significantly reduced and the calculation speed is greatly improved under the premise of maintaining the same accuracy as the conventional base-2.

[0119] Table 1 is the error comparison of different Q values generated by the base-8 generalized hyperbolic rotation mode CORDIC iteration module of the present application with base 2 and 3 within the convergence range. According to the experimental results, the base can be flexibly selected according to specific requirements when the convergence range is met. The calculation accuracy can still be maintained within an acceptable range after the base changes, realizing the demand for base change in some cases and further expanding the application range of the present technology.

[0120] Table 1 Error comparison of different Q values Q b Calculated value Exact value Error 0.13010 3 1.15366590 1.15364842 -0.00001748 0.11884 3 1.13932431 1.13946526 0.00014096 0.10000 3 1.11615694 1.11615694 -0.00003376 0.09990 3 1.11600113 1.11600056 -0.00000057 0.13010 2 1.09436953 1.09436955 0.00000002 0.11884 2 1.08586144 1.08586142 -0.00000002 0.10000 2 1.07177365 1.07177346 -0.00000019 0.09990 2 1.07169926 1.07169918 -0.00000009

[0121] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROMs, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of the present application can be implemented in various computer languages, such as object-oriented programming languages Java and interpreted scripting language JavaScript, etc.

[0122] The present application is described with reference to flowcharts and / or block diagrams according to the methods, devices (systems), and computer program products of the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, as well as the combination of flows and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices produce a device that implements the functions specified in the flowcharts and / or block diagrams. Figure 1 The device that implements the functions specified in one flow or multiple flows and / or blocks Figure 1 one block or multiple blocks.

[0123] These computer program instructions can also be stored in a computer readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer readable memory produce an article of manufacture including instructions which implement the Figure 1 function specified in the flow or flows and / or blocks Figure 1 of the block or blocks.

[0124] These computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions that are executed on the computer or other programmable apparatus provide steps for implementing the Figure 1 function specified in the flow or flows and / or blocks Figure 1 of the block or blocks.

[0125] Although preferred embodiments of the application have been described herein, substitutions and alterations are possible in view of the disclosure of this application without departing from the spirit and scope of the present application. Therefore, it is the intent of the appended claims to cover all such variations and modifications as come within the scope of the present application. 1

[0126] It will be apparent to those skilled in the art that various modifications and variations can be made to the present application without departing from the spirit or scope of the application. Thus, it is intended that the present application cover modifications and variations of this application provided they come within the scope of the appended claims and their equivalents.

Claims

1. A fast and high-precision calculation method for a variable-base exponential function, characterized in that, The method includes the following steps: For the exponential function to be solved Input the initial values ​​for the horizontal axis component, vertical axis component, and residual angle component. They are respectively ; Introducing scaling variables The CORDIC algorithm for the radix-8 hyperbolic rotation mode is used to iterate through the three variables N times until... Converges to 0, , ,Will and Adding the two together yields the exponential function to be solved. The horizontal axis variable in the (i+1)th iteration Vertical axis variables and scaling variables The formula is: ； ； ； in, N is the maximum number of iterations; before each iteration, the interval boundaries are simplified based on hardware conditions, so as to... Given a quantity, determine the iteration direction factor based on the following conditions. The possible values ​​of: ； Among them, the interval boundary and The calculation formula is: ； ； 。 2. The fast and high-precision calculation method for the variable-base exponential function according to claim 1, characterized in that, At the beginning of each iteration, the inverse scaling factor for the i-th iteration of the radix-8 hyperbolic rotation mode CORDIC algorithm is calculated using the following formula. for: ； The inverse scaling factor of the i-th iteration is multiplied sequentially by the inverse scaling factors generated in previous iterations, and then the cumulative inverse scaling factor is calculated. Multiply by the convergence value of the iteration variable to correct for data system errors caused by rotation iteration.

3. The fast and high-precision calculation method for the variable-base exponential function according to claim 2, characterized in that, The process of solving for the inverse scaling factor is as follows: For the reverse scaling factor Performing a Taylor expansion, we obtain the following expansion: ; Among them, if Inverse scaling factor Approximately ; if Inverse scaling factor The first two terms of the approximate expansion Expand to the required number of bits; if Inverse scaling factor Approximate to the first three terms of the expansion Expand to the required number of bits; For the remaining iterations, the approximate value of the inverse scaling factor, calculated in advance, is directly referenced using the first lookup table as needed.

4. The fast and high-precision calculation method for the variable-base exponential function according to claim 1, characterized in that, Inverse hyperbolic tangent function The solution process includes: The inverse hyperbolic tangent function Performing a Taylor expansion, we obtain the following expansion: ; for In this situation, Approximately At the same time, simplify It is in the following form: ； in It is a constant; if Not for The result is The constant part is stored as a binary number, and other values ​​are quickly obtained through shift operations. Skip this iteration; for In this case, the number of terms to retain in the Taylor expansion is selected according to the accuracy requirements, and the approximate value of the inverse hyperbolic tangent function is directly applied using the second lookup table.

5. A fast and high-precision calculation system for a variable-base exponential function, characterized in that, The system includes a preprocessing module, a radix-8 hyperbolic rotation mode CORDIC iteration module, and a postprocessing module; The preprocessing module is for the exponential function to be solved. Input the initial values ​​for the horizontal axis component, vertical axis component, and residual angle component. They are respectively ; The CORDIC iterative module for the base-8 hyperbolic rotation mode introduces a scaling variable. Perform N times on three variables , and The iteration continues until... Converging to 0; the horizontal axis variable in the (i+1)th iteration , Vertical axis component and scaling variables The formula is: ； ； ； in, Before each iteration, the interval boundaries are simplified based on hardware conditions. Given a quantity, determine the iteration direction factor based on the following conditions. The possible values ​​of: ； Among them, the interval boundary and The calculation formula is: ； ； ； The post-processing module will and Adding the two together yields the exponential function to be solved. ,in, , .

6. The fast and high-precision calculation system for the variable-base exponential function according to claim 5, characterized in that, The CORDIC iterative module of the radix-8 hyperbolic rotation mode includes an inverse scaling factor calculation unit and an error elimination unit; The inverse scaling factor calculation unit calculates the inverse scaling factor for the i-th iteration of the CORDIC algorithm in the radix-8 hyperbolic rotation mode using the following formula. for: ； The error elimination unit multiplies the inverse scaling factor of the i-th iteration with the inverse scaling factors generated in previous iterations, and then calculates the cumulative inverse scaling factor. Multiply by the convergence value of the iteration variable to correct for data system errors caused by rotation iteration.

7. The fast and high-precision calculation system for the variable-base exponential function according to claim 6, characterized in that, The process of solving for the inverse scaling factor is as follows: For the reverse scaling factor Performing a Taylor expansion, we obtain the following expansion: ; Among them, if Inverse scaling factor Approximately ; if Inverse scaling factor The first two terms of the approximate expansion Expand to the required number of bits; if Inverse scaling factor Approximate to the first three terms of the expansion Expand to the required number of bits; For the remaining iterations, the approximate value of the inverse scaling factor, calculated in advance, is directly referenced using the first lookup table as needed.

8. The fast and high-precision calculation system for the variable-base exponential function according to claim 5, characterized in that, Inverse hyperbolic tangent function The solution process includes: The inverse hyperbolic tangent function Performing a Taylor expansion, we obtain the following expansion: ; for In this situation, Approximately At the same time, simplify It is in the following form: ； in It is a constant; if Not for The result is The constant part is stored as a binary number, and other values ​​are quickly obtained through shift operations. Skip this iteration; for In this case, the number of terms to retain in the Taylor expansion is selected according to the accuracy requirements, and the approximate value of the inverse hyperbolic tangent function is directly applied using the second lookup table.

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