HIGH SPEED FUNCTION GENERATING APPARATUS AND METHOD

Abstract: A method and apparatus (201) for general-purpose generation of digital mathematical functions in connection with a computer processor enabling computation of such functions as exponential, logarithmic, hyperbolic and trigonometric functions, as well as numerous special-purpose engineering functions. The invention employs compact look-up tables (206, 207, 208) accessed by look-up techniques and value interpolation to determine values not found in the tables. In specific embodiments, specific techniques are employed for table generation and value interpolation to attain results at desired levels of speed, precision, economy, and reliability. Applications include alternatives to conventional hardware multipliers and numerical processors. Individual embodiments may be realized either directly in hardware or in software. Multiple embodiments may be realized as software or hardware function libraries. Coupled multiple embodiments may be realized as general-purpose hardware or software mathematical processors.

Description: HIGH SPEED FUNCTION GENERATING APPARATUS AND METHOD Reference to a Related Application Reference is made to the U.S.A. patent application Serial No. 07/623,238 filed January 30, 1991 in the name of the present inventor and entitled "Method and Apparatus for Generating Mathematical Functions" which application is a continuation in part of U.S.A. patent application Serial No. 07/366.376 (now abandoned) filed June 14. 1989 in the name of the present inventor and entitled "Method and Apparatus for Generating Mathematical Functions. 1" These applications and the present application describe generation of restricted-range fixed-point mathematical functions of restricted-range arguments with the functions possessing derivatives of restricted magnitude. Background of the Invention A portion of the disclosure of this patent document contains material to which a claim of copyright protection is made. The owner has no objection to the facsimile reproduction of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but reserves all other rights whatsoever. The present invention is related to numerical processing and more particularly to a method and computer-based apparatus involv¬ ing the use of a new fundamental architecture using look-up tables for particularly efficient generation of general purpose mathematical transfor¬ mations (i.e.. mappings) of numerical data. In particular, highly efficient general purpose generators of exponential and logarithmic functions are described. Prior art general purpose logarithmic operations, when com¬ pared with additions and subtractions and multiplications, have been in¬ efficient and expensive — so much so that the inefficiency and expense are widely regarded as inevitable penalties on the logarithmic operation it¬ self. For example, a commercial conventional numerical processor might execute logarithmic operations ten to twenty times slower than even its comparatively slow and expensive multiplication operation. The present invention targets problems which have been thought unsolvable. problems which have not been recognized as prob¬ lems, and problems which are major obstacles to practical implementation of generators of generalized functions. Major problems not recognized as such in the prior art include areas of analysis, notation, and terminology associated with such commonly used functions as the elementary expo¬ nential and logarithmic functions. These areas need to be extended in a number of distinctly fundamental directions. Description of the Prior Art Function Generation by Polynomial Approximation Polynomial approximation forms, and particularly minimax polynomials, have been used most widely for generating fixed-point func¬ tions or floating-point functions of restricted range. However, such poly¬ nomial function generators usually require many multiplication operations and. therefore, cannot execute as rapidly as a basic arithmetic operation. Two exemplary references are "Near-Minimax Polynomial Approximations and Partitioning of Intervals. " by W. Fraser and J. F. Hart, in the Com¬ munications of the A CM. August. 1964. 486-489 and "Floating Point Algo¬ rithm Design. " in Computer Design. June. 1987. 107-114 by R. L. Cassola. Function Generation by Direct Table Look-up Direct table look-up of pre-computed function values in a random access memory is one technique for rapidly generating logical or mathematical functions. This technique has generally required the least possible execution time when generating fixed-point numerical functions. However, direct table look-up can require massive tables even for compar¬ atively minor levels of precision. A completely general radix-r 77 -digit-wide logical or math¬ ematical function of an n-digit-wide radix-7' argument requires r n table entries of m digits per entry for a total table size of rar" stored digits. For m = n this is nr n which grows to astronomical proportions long be¬ fore reaching the levels of precision needed for general purpose compu¬ tation. This well-known extreme growth rate of direct-look-up tables as a function of the n digits of precision is clearly shown for r = 2 by the following bit-widtJt:byte-size pairs: 8-bits:256-bytes: 16-bits:131 ,072-bytes; 24-bits:44,040,192-bytes: 32-bits:17.179,869.184-bytes. The steep growth exhibited here is but an absolute upper limit which is always applicable because absolutely all possible combina¬ tions of such inputs and outputs for every such particular function can be specified completely in a table of such size, even when no special properties of the function have been identified. Thus there can be no n-digit function of an n-digit argument demanding -more than the indicated direct-look-up table size. A table look-up process is known to be functionally equiv¬ alent to a rule-like process implemented with logic gates. An exemplary reference is ''Minimal Sets of Distinct Literals for a Logically Passive Func¬ tion. " authored by R. C. De Vries in the Journal of the Association for Computing Machinery. 431-443. July. 1971. For example, when a major portion of a look-up table contains the same value, then that portion of the table may be replaced by comparatively simple logical operations. Hence the possibility remains that for particular functions the table requirements may well be but a tiny fraction of the upper limits given above. Therefore, when the tables for direct table look-up are too large and expensive, other methods are needed to obtain the desired re¬ sults. In particular, indirect table look-up techniques may be used, i.e., employing any kind of operation which can be construed to be usefully available together with some form of abbreviated look-up. (Such an op¬ eration may possibly include yet other look-ups. ) In particular, interpo¬ lation between, or extrapolation about, comparatively sparse table entries permits a drastic reduction in the number of required table entries when generation of a fixed-point approximation of a differentiable function is desired. Attempts have been made to develop table look-up schemes applicable to generating mathematical functions which are not limited by the large function table requirements of direct table look-up techniques. One such technique that is limited to generating particular trigonometric functions is described in the U. S. Patent No. 4.077.063 to Lind. L τ . S. Patent No. 4.078.250 to Windsor et al. purports to describe a technique applicable to the generation of logarithms and exponentials. The disclosure thereof however is, when analyzed, incorrect and inoperative. While the Singer Company, assignee of the 4.078.250 patent, may well have attempted to make an operative form of such a system, no report thereof was found in a search of the technical literature by the present inventor. Other function generators are known. Exemplary refer¬ ences of logarithmic and/or exponential function generation are as follows: U. S. Patent No. 3.036.774 to Brinkerhoff. U. S. Patent No. 3.099,742 to Bvrne. U. S. Patent No. 3.194.951 to Schaefer. U. S. Patent No. 3.402.285 to Wang, U. S. Patent No. 3,436,533 to Moore et al., U. S. Patent No. 4,046,999 to Katsuoka et al., U. S. Patent No. 4,062,014 to Rothgordt et al., and U. S. Patent No. 4,158,889 to Monden. The generation of an exponential function employing vec¬ torized floating-point multiplications in the interpolation between table look-up values is known. An exemplary reference is "Cyber 205: A Fast Vectorized Exponential Function," in Supercomputer 13, May, 1986. Generation of other functions is known, e.g.. an exemplary reference of arctangent generation is U. S. Patent No. 4,164,022 to Rat- tlingourd et al. During the period from about January to March 1979 the inventor herein contracted with the AiResearch Manufacturing Company of the Garrett Corporation (now the AiResearch Los Angeles Division of the Allied-Signal Aerospace Company) of Torrance. California to design and develop software for microprocessor-based controllers of aircraft cabin pressure. In the course of that work, the inventor herein developed and built a system which used mathematical functions which were precursors of the present invention. In particular, the double-byte fraction functions log 2 ( l + x) and 2 X — 1 as functions of a double-byte fraction argument x were generated indirectly from tables. These tables allowed comparatively direct look-up of all 256 double-byte values of each of these double-byte functions as functions of a general single-byte fraction argument. However, straightforward logarithmic calculation of single- byte interpolating corrections for both functions was used in that system in order to produce the double-byte values of both functions that were not in the tables. In particular, when the least significant byte of the argu¬ ment was nonzero, the calculation included logarithmizing that nonzero byte value. The standard iterative technique was employed for normal¬ izing this byte value as 2 m ( 1 + g) in terms of an iterative!}' decremented integer value rn and a nonnegative fraction byte g derived from iterative left shifting. The byte value g was then used as a look-up index into the table of fractional logarithmic table values to complete the logarithmiza- tion. The multiplications and divisions of the calculation were mapped into additions and subtractions, respectively, of logarithms. The resulting single-byte mantissa was used as a look-up index into the table of fractional exponential values. After adding the value one to the looked-up fraction, the antilogarithm (i.e.. exponential) was then completed by straightfor¬ ward iterative denormalization to yield the single-byte correction value. The operations for normalization and denormalization re¬ quired not only the 8L+8R = 16 bit places of comparatively economical left-right "fixed " or "constant " shifting, but also up to a maximum of 8L+9R=17 bit places of comparatively slow left-right "Variable " iterative shifting which has been determined to be comparatively inefficient in retro¬ spect. At that time the calculation represented the best available solution to the difficult problem of real-time calculation of logarithms and exponen¬ tials with both precision and performance adequate for use in high perfor¬ mance sampled-data control systems based on a microprocessor contain¬ ing no instruction for direct multiplication. Thus no improvements λvere suggested or even suspected of being possible. The present invention rep¬ resents an advance over the inventor ' s prior unpublished but commercially exploited work. No work is known which suggests the present invention. Summary of the Invention According to the invention a generalized apparatus and method for generation of mathematical functions in connection with a computer processor enabling extremely rapid computation of such functions as the logarithmic, exponential, trigonometric and hyperbolic functions, as well as numerous special purpose engineering functions. The invention em- ploys compact look-up tables and. in order to determine values not found in the tables, uses value interpolation which, according to the invention, is oriented toward reducing the required range of quantities used in the value interpolation. In particular, the evaluation of logarithmic quantities of minimal range requires zero or minimal left-right digit-place shifting, which provides economy and greatly speeds up the computation of a value interpolation that uses logarithmic quantities. In specific embodiments, specific techniques are employed for value interpolation to attain results at desired levels of speed, precision and economy. Applications include alternatives to conventional hardware multipliers and numerical processors. Individual embodiments may be re¬ alized either directly in hardware or in software. Multiple embodiments may be realized as software or hardware function libraries. Coupled multi¬ ple embodiments may be realized as general purpose hardware or software numerical processors. The highest speeds of computation are achieved when the invention is implemented directly in dedicated hardware for general pur¬ pose computation as well as for more specialized applications such as signal and image processing requiring particularly high performance. The hard¬ ware and its associated operations can, however, be implemented partic¬ ularly simply. This simplicit3 r permits economical firmware and software implementations of the invention to provide new levels of performance which approach that of existing hardware function generators. Such soft¬ ware and firmware implementations provide qualitatively new levels of combined performance, function, economy, and ease-of-use even on the simpler microprocessors possessing no instruction for direct multiplication. This invention can be used for: numerical computation, mathematical special functions, mathematical function libraries, special¬ ized application functions, logarithmic converters and normalizers. expo- nential converters and denormalizers. and generally in applications re¬ lated to fixed-point computation, floating-point computation, real-time sampled-data control, process control, signal processing, signal condition¬ ing, telemetry, engine control, motor control, guidance, navigation, statis¬ tical analysis, data reduction, avionics, bionics, nucleonics, radar, sonar, microprocessors, reduced instruction set computers (RISCs), pipe-lined processors, parallel processors, and stand-alone processors. Brief Description of the Drawings Figure 1 is a block diagram of a general purpose function generator employing a compact look-up table and exponential/logarithmic (E/L) interpolation computation requiring minimal left-right digit-place shifting according to the invention. Figure 2 is a block diagram of the general 2n-bit generator of the binary logarithm of ( 1 + the 2n-bit-fraction argument), i.e., the BLD2 function. Figure 3 is a block diagram of the general 2n-bit generator of the binary (exponential) function ( — l + 2 **(2n-bit-fraction argument ) ). i.e.. the BXD2 function. Figure 4 is a block diagram of the general 3n-bit generator of the binary logarithm of ( 1 + the 2n-bit-fraction argument), i.e.. the BLD3 function. Figure 5 is a block diagram of the general 3n-bit generator of the binary (exponential) function ( — l + 2 * *(2n-bit-fraction argument ) ), i.e.. the BXD3 function. Figure 6 is a schematic diagram that shows the low logic gate count of a high speed barrel shifter for dynamically shifting k + 1, k, or " — 1 bit places, wherein the arbitrary fixed k bit places of shift represents a hard- wired displacement of the output relative to the input. Description of the Specific Embodiments In order to assist in understanding the invention, the follow¬ ing definitions and terminology are used. Conventions, Notation, and Terminology General Terminology. The terms computer, data pro¬ cessing system, number processing system, and number processor are used interchangeably in the following descriptions. Similarly, interpolation and extrapolation are also used mostly interchangeably. The term "object" refers to any basic system construct such as a look-up table, function gen¬ erator, look-up-table value correction calculator, memory page or segment, or pipe-line stage. A memory is referred to as a random access memory in order to distinguish it from a slower serial access memory without regard to any read-write or read-only capability. Whenever digit , bit. byte, or word positions are described using numerical indices, position zero is the position immediately to the left of the radix point with positive positions further to the left. Negative positions are to the right of the radix point. Thus a shift of +k bit places corresponds to a fc-bit-place left shift or a right shift of —k bit places. The basic objects used in the invention usually perform unsealed (e.g. , fixed-point) numerical operations. The set of such objects includes signal detectors and modulators configured as logical gates, data registers, adders, subtracters, shifters, random access memories, function look-up tables, function-value interpolators, exponential function gener¬ ators, logarithmic function generators, and generators for scientific li¬ brary functions, particularly as based upon the product-of-exponentials and "continued-exponential " series expansions as introduced for the first time hereinafter and in the related U.S . patent applications mentioned above. However, the comparatively expensive component of a conven¬ tional processor, i.e., the multiplier, usually is not required. Notation for Logarithms and Exponentials. The fol¬ lowing definitions are used herein: axpu ≡ a u and algit ≡ log α u (1) for 0 < a. Other letters or symbols can be substituted for a. except that usually bxp u ≡ 2 U and blgw ≡ log 9 w, (2) (i.e.. b ≡ 2 wherein the "b" is a mnemonic for "binary"), dxpw ≡ 10" and dig t ≡ log 10 w, (3) (i.e., d ≡ 10 wherein the "d" is a mnemonic for "decimal") and expu ≡ e υ and elgu ≡ log e u ≡ I u. (4) There are also the constants defined as follows: q a ≡ alge = 1/lnα (5) Q„ ≡ slgq a — (ln(l/lnα))/lnα (6) wherein e is the base of the natural logarithm. When a = 2 ≡ b. the subscript is often omitted, i.e., q ≡ blge = l/ln2 (7) Q ≡ bigg = (In(l/ln2))/ln2. (8) Furthermore, the newly named "diff-exponentials " are de¬ fined as follows: axd' ≡ — 1 + axpt/ ≡ a 11 — 1 (9) hxάti ≡ -1 + bxpα ≡ 2 V - 1 (10) exάu ≡ — 1 + exp?/ ≡ e" — 1. etc. (11) Th e respective inverse "diff-logarithms" are defined as fol¬ lows: ald ≡ alg(l + u) (12) bld w = blg(l + u) (13) eld w ≡ elg(l + u) ≡ ln(l + w) (14) and so on. Thus for a given base the diff-logarithm and diff-exponential are inverse functions. There is the further convention that the upper-case forms AXP. AXD. ALG. and ALD refer to functions whose arguments are re¬ stricted to particular domains chosen by the designer, e.g., for imple¬ mentation in hardware. However, the arguments for the upper-case BLD and BXD functions are specifically restricted to be nonnegative fractions, thereby also restricting the BLD and BXD values themselves to this same fraction range. Generalized Function Generation Using a Look-up-Table and E/L Interpolation Computation Introduction. Ultrafast function generation by direct look¬ up from tables is known to be much too expensive except for very low pre¬ cision. Hence such table-based generation has been regarded as virtually useless. Nevertheless, according to the invention, this ultra rapid generation of the low-precision logarithmic and exponential functions is employed to effect a comparatively low-precision exponential/logarithmic (E/L) computation that requires minimal iterative left-right digit-place shifting to interpolate values within a compact look-up table for a net gain in precision while retaining high levels of execution speed and economy. Furthermore, extending the interpolation computation to higher orders without using direct multiplication operations suggests the use of logarithmic and exponential functions only if these transformations can be generated particularly rapidly rnd economically, a circumstance usually considered to be distinctly uncharacteristic of these functions. Thus, the following description and analysis shows that, according to the invention, drastic improvements in efficiency and economy are achieved through the elimination of nearly all iterative left-right digit-place shift¬ ing from the interpolation computation. These improvements affect the generation of exponential and logarithmic functions, the generation of gen¬ eralized functions, and thence conventional and novel applications of these efficient function generators. First and Second-Order Interpolation. This technique is useful where the desired function is to be characterized as f(x) . Gener¬ ation of a radix-?" fixed-point function f(x) as a function of a radix-?' fixed- point argument x may proceed according to the partitioning x = u* + hv* wherein a dominant value u * and an adjunct value r* are. respectively, the values of the most significant and next most significant portions of the digits of x. A constant integer bias ξ may be introduced such that u * + hv * = {u * - hξ) + h{ξ + v * ) = u + hv. u = u * - hξ. v = ξ + v* for integer constant ξ wherein 0 < \ξ\ < r — 1. \h\ = r ~n < 1 for positive integer n, 0 < v * < 1. and ξ < v < ξ + 1. The generator employs an interpolation computation that includes first-order and second-order differential effects to interpolate val¬ ues between comparatively sparse look-up table values. According to the invention. E/L computational techniques that are oriented toward requir¬ ing minimal iterative lefi-rigld digit-place shifting provide this interpola¬ tion. Using the notation: rig a" ≡ log,, α * and rxp ≡ r 1 . f ≡ f u ) = f( u* — hξ ) ≡ f * ( tι* ). and indicating the derivative with respect to u by a prime, the following approximation is used: f(u + hv) = X + h ψr ~χ ξ + hwr ~x v * + hs w rxp W . (15) X ≡ X(u * ) = / + ( 1 - α) wr- λ , (16) i = j * + /ι C * . W * = V + Y, (17) C * = 5 w s d rxp(Q 7 . + V - Y + _ ), (18) s w = s v 5 y , sr f = sign /", (19) 5,. = sign υ, s y = sign(/' — wr ~λ ) , (20) V = rlg |υ|. V = rlg |/' - rr- λ | , (21 ) £> = £(« * ) = rig | i/"| . (22) wherein α- is an arbitrary constant, ' and λ are signed integer constants of small magnitude chosen according to the invention in conjunction with the choice of / such that the quantity f*'{u* ) ≡ \f — ψr ~λ \ is of restricted dynamic range. That is, /*' is of the form r m g wherein 1 < g < r, and the range of values of the signed integer m is substantially independent of n and comparatively restricted. The scaling multiplication ?- ~λ x v* may be implemented by minimal shifting. Similarly, the scaling multiplication v x (7 ,-λ r* ) may be efficiently realized with only a very few additions or subtractions. These parameters permit a useful degree of adjustment to the allowed range of ' without reducing the digit width of v* and thus sacrificing precision. The values of Λ " . Y . and D are functions only of u * . Hence, even at worst, these values may be generated efficiently, either individually or in various combinations, with or without Q r . from look-up tables of but modest size. More importantly, when the relevant interval of /' includes values near zero, particular choices of values for these parameters can greath" reduce both the dynamic range and the associated quantity of required digit places of slow and expensive left-right "variable " shifting associated with the normalization-denormalization of the E/L calculation. However, the fixed "constant" shifting relative to particular data regis¬ ter boundaries is comparatively economical, e.g., an 8-bit shift in a byte- oriented processor is economically effected directly as a data transfer rather than a slower iterative process. Thus, instead of a slower executing large amount of left- right shifting, there is but the minor cost of adding or subtracting a small multiple of an only slightly shifted v * . Hence, second-order interpolation (i.e.. with a double-width correction) can be generated provided only that the rig and rxp functions are made available with an absolute error whose magnitude is less than or on the order of h '2 . High performance forms of such double-width generators for the rig and rxp functions may be con¬ structed similarly, but with first-order (i.e., single- width) rig and rxp tables for the single-width interpolation computation. First-order interpolation corresponds to omitting the C * term which depends upon /". Third- Order and Higher- Order Interpolation. The foregoing second-order interpolation computation can be extended to third and higher orders by following the same general principles which extend the first-order computation to second order. Generalized Hardware Function Generator. Figure 1 shows the combinatorial flow-through organization of a generalized 2n- digit function generator 201. The primary function map 206 includes at least one random access memory and may include other components such as adders, subtracters or other combinatorial logic, particularly when gen¬ erating certain functions for which v * 205 is conveniently also connected as an input to 206. The secondary function map 207 indicates that a secondary function signal that depends on u * 204 must be routed to a port of adder 212. In particular, when the secondary function 207 of this map is comparatively simple, such as when it equals an unbiased or biased u* for example. 207 may be implemented more economically as an adder or other combinatorial logic without including a more expensive random access memory, as hereinafter shown. Similarly, because the primary func¬ tion signal 209 is a function of u*, a most significant portion of 209 may be economically included as an integral part of the secondary function map 207 when generating certain functions, as shown hereinafter. The function generator 201 employs an E/L computation of an (unsigned) correction value that is used for interpolating values not directly extracted from a primary map 206 that includes a look-up table. Pipe-lining is particularly easy for such a combinatorial (i.e.. no feed-back loops) circuit. The circuit can be clocked as a single unit, or else synchro¬ nizing delays can be distributed throughout the circuit in order to reduce the execution time of the pipe-line stages, thereby allowing a corresponding increase in the clocking rate. The generalized function generator 201 comprises a 2n- digit input first data bus 202 coupled to a partitioning means 203. The partitioning means 203 partitions the digits of the input first data bus 202 into two n-digit portions, which are a most significant (MS) (dominant) second data bus 204 and a next most significant (NS) (adjunct) third data bus 205 corresponding to values u* and v*, respectively. The second data bus 204 is coupled to the input of the /* (primary) look-up table means 206. The 2n-digit primary function signal on the fourth data bus 209 of the /* look-up table means 206 is a value of the function f * { u * ) ≡ f{ u* — hξ) ≡ f( u ) wherein ξ is an arbitrary (positive) bias value that is usually related to a maximum magnitude of v * . The second data bus 204 is also coupled to a secondary function generating means 207 for generating a secondary function signal log fl \f * '{u * ) \ ≡ log α \f'{ u * - hξ ) \ ≡ log 0 |/'( u) |. wherein 0 < a. as the output value on the fifth data bus 210. The fifth data bus 210 is coupled to a first data input of an n-digit first adder 212. The adjunct third data bus 205 which possesses a value ι>*, is coupled to an adjunct function generating means 208 for generating an adjunct function signal 211 comprising an approximation of the modified logarithmic function value log * ( *) ≡ log α |ξ + f * | ≡ log α \v\ . The arbitrary bias value ξ is chosen to reduce the dynamic range of the quantity \v\ ≡ \ξ + v* \, thereby reducing the amount of left shifting required in evaluating log α \v\ ≡ log α |ξ + v* \ and in turn reducing the amount of right shifting required in evaluating a subsequent antilogarithm. The output of the adjunct second function generating means 208 as the sixth data bus 211 is coupled to the second data input of the n-digit first adder 212. The output from the first adder 212 is coupled to a seventh data bus 213. The seventh data bus 213 is coupled to the input of a third function generating means 214 providing the base-α antilogarithm function value as the output coupled to an eighth data bus 215. The data value output of the antilogarithm generating means 214 is an (unsigned) interpolation correction value. The 2n-digit fourth data bus 209 is coupled to the LS n-digit portion of a first data input of a 2?ϊ-digit second adder 218. The eighth data bus 215 is coupled to the second data input of the 2n-digit second adder 218. The output of the second adder 218 is coupled to the ninth data bus 216 for generating the output of the typical function generator 201. Signed Interpolation Correction Values. Permitting the bias value, the derivative value, and the adjunct portion value to be signed quantities allows further latitude in the foregoing interpolation. In particular, a signed interpolation correction value is provided in an elementary manner by: 1 ) coupling a sign providing means ( not shown ) to said partitioning means 203 for extracting a sign corresponding to the product of the sign of f*'{u*) times the sign of the quantity r ________ ξ + v* as output and 2) coupling the output of said sign providing means (not shown) to the output of said antilogarithm generating means for attaching the provided sign to the output of said antilogarithm generating means to produce a signed interpolation correction value. General Logarithmic Function Generation Using a Look-up- Table and E/L Interpolation Computation Introduction. This technique is useful where the desired function is to be characterized as log,. f(x). Such functions are particularly useful in a generalized base-?' E/L computational system which is described by the present inventor for the first time in a copending U.S. patent ap¬ plication entitled "Method and Apparatus for Exponential/Logarithmic Computation." In a generalized base-?" E/L computational system, general posit e exponential quantities g and £ are represented by fixed-point log¬ arithmic quantities G and Z. respectively, wherein g = r G and £ = r z . Evaluating g(z ) as a general E/L function of £ corresponds to generating G(r z ) = τlg g(r z ) as a fixed-point function of the fixed-point argument Z. First and Second- Order Interpolation. The E/L func¬ tion /(rlg £ ) ≡ /(rlg r ) ≡ f{ Z) is effected by generating rlg /( ) as a radix-?' fixed-point function of the radix-?- fixed-point argument Z . The digits of Z can be partitioned such that Z = u * + hv * wherein h = r ~n < 1 with 0 < v* < 1. Lipon biasing v * by an integer ξ of small magnitude, there is rlg /(Z) = rlg /(n * + hv * ) = ήg f(C - hξ + h(ξ + v * ) ) = ή g f(u + hv) wherein υ = C — hξ and v = ξ -f v * . A generator for the function rlg/(ιz + hv) uses values in¬ terpolated between look-up table values and may employ an interpolation computation that includes first and second- rder differential effects. Ac¬ cording to the invention, this interpolation employs E/L computational techniques oriented toward requiring minimal left-right digit-place shift¬ ing. Indicating the derivative with respect to u by a prime, the following approximation is used: ήgf{u + hv) ≥. X + hs w τxpW. (23) W = W * + hC * {W * .u). (24) T < r * = rlg vq r r V + Qr-X + Y (25) f C * {W * ,u) = s w s d τxp{W * + D) (26) f ≡f(u), X≡X(u) = ήgf, (27) y = rl H, Y = (28) q r ≡ήge. Q r = ήgq r . (29) wherein each scaling by h can be effected as a right shift of n digit places. The value for ξ is chosen to reduce the dynamic range (and expensive left- right iterative shifting) of the calculation. The values of Λ " . Y , and D are functions only of it. Thus, even at worst, these values may be generated efficiently, either individually or in various combinations, without or with Q r . from look-up tables of but modest size. Although exponential and logarithmic transformations are required at execution time, given the D values (possibly from a look-up table), direct fixed-point multiplication operations (other than by signs or by h) are not needed. The corresponding relations for first-order interpolation are derived simply by omitting C* which is equivalent to assigning W = W*. Third and Higher-Order Interpolation. The foregoing second-order interpolation method can be extended to third and higher orders by following the same general principles which extend the first- order method to second order. General ALG and AXP Function Generators The Basic Equations. The foregoing methods of interpo¬ lation can be applied to the generation of logarithms and exponentials of general base a using radix-?- numbers. The generalized base-α logarithmic and exponential func¬ tions {x) and E{x) are defined as follows: L ) = f{a.A,C.x) = A + alg{C + x) (33) E(a') = g(α.F.x) — — + axp . (34) Defining x = u * + hv* = u + hv wherein = it* — hd. i' = d + v* . and // = C": the continued-exponential series expansion for L(x) can take the form of: {u + hv) = L(u) + h{F + E{R-h{F + E{R' -h(F + E(R" + -..)))))) (35) wherein = L(f') + Q α - L{u). v' = v - C. Q α = algg α , q α = alge. ' = i? + alg(l/2). and R" = B + alg(5/12). The product-of-exponentials series expansion for E(x) can take the form of an exponential of a sum as follows: E{u + hυ) ' = E{u) + h{F + E{L{υ') - Q' a + u * + h±v * oo + ∑ (-l) i+1 h 2 ' (F + E(K ά + j2L(v'))))), 3 = wherein B 2j is the 2 -th Bernoulli number. The first few of the Bernoulli numbers are: B 0 = 1. B λ = - 1/2, B 2 = 1/6. £ 3 = 0, B = -1/30, B b = 0, B 6 = 1/42, B 7 = 0 and B 8 = -1/30. The halving of v* is easily handled in most fixed-point num¬ ber systems in popular use wherein the radix is a multiple of two. (This division can be performed with the L and E functions if necessary.) The term j2L(v') is efficiently implemented as a sequence of additions of L(v') or 2L(v') . The remaining multiplications by h are efficiently realized by shifting as all quantities are taken to be radix-?- values. The extent of each table must be sufficient to accommodate at least the full ranges of all indices accessing that table. First-Order Interpolation for L2(u + hv). Retrieve the look-up table value L12(u) = (u) with an absolute error on the order of h 2 . Indicating by LI . El and Q\ a . respectively, the approximations of the L and E operations (as derived from direct table look-up. say) and the constant value Q a with errors less than or on the order of h. evaluate: B = Ll( t/) + Ql a — Ll ( u ). wherein v' = v — C = d -\- v * — C. and L2( ιι -\- Jι,v) = ~ V2(ιι ) + 1ι(F -- El (B) ), which corresponds to an L operation with an absolute error on the order of h 2 . First-Order Interpolation for E2(ιι + hv) . Retrieve the look-up table value E12( u ) = E( t/ ) with an absolute error on the order of h '2 . Indicating by LI . El . and Ql' a . respectively, the approximations of the L and E operations and the constant value Q' a with errors on the order of h, evaluate: B = Ll (ι ) — QV a + C wherein υ' = v — C. and E2(u + hv) = E{u) + h(F + El(B)) , which corresponds to an E operation possessing an absolute error on the order of h 2 . Second- Order Interpolation for L3(n -j- hv). Retrieve the look-up table value Ll23(u) = L(w) with an absolute error on the order of /r 3 . With LI . El . L2, E2 and L12 as described above and Ql2 a as an approximation of the constant value Q a with an error on the order of h 2 . evaluate: B = L2(ι )+Ql2 fl -L12(u) wherein v' = v-C. B' = i?+alg( l/2), T = i? - /ι( + El(i?') ) , and finally L3{ u + hv) = L123(w) + h{F + E2{T)), which corresponds to an L operation possessing an absolute error on the order of h Α . Second-Order Interpolation for Ε3(ιι + hv) . Retrieve the look-up table value El23(ιt) = Ε(u) with absolute error on the order of 3 . With LI, El, L2. and E2 as described above and Q12' α as an approximation of the constant value Q' a with an error on the order of h 2 . evaluate: B = L2( ι-') - Q12' α + it * + h±v * wherein v' = v - C ' . R! = i? + alg( l/2) . T = i? - ι(i :ι + El(i?')), and finally E3(n + hv) = E123( u) + h(F - -E2{T) ) . which corresponds to an E operation possessing an absolute error on the order of //A Generators Based on Third and Higher-Order Inter¬ polation. Third and higher-order interpolations involve using more terms from the approximating series expansions and table values of successively higher precision together with the generators of lower orders. Each re¬ quired fixed-point multiplication is degenerate, in that it can always be implemented either as a single (or at most a double) addition or as a shift. When a = r. the foregoing methods for generating exponen- 99- tial and logarithmic functions are particularly simple and efficient because scaling by a can then be realized efficiently as a shift. This permits the look-up tables to be much smaller than in the general case. In any such case, as for a = r = 2. the above description requires little modification in order to account for the fact that the base-?' exponential of an integer argument is most efficiently realized as a shift. The values of the various parameters are chosen at the convenience of the system designer with the required quantity of table entries and the necessary precision of such en¬ tries depending on those choices. Such tradeoffs are analogous to those which are required for functional approximation in floating-point systems. BLD and BXD Function Generators Definitions. Regarding notation, bid ≡ blg(l + ) ≡ log 2 (l + 2-) (37) bxd a: ≡≡ -1 + bxp rr ≡ 2 X - 1 (38) q ≡ blg e = l/ hι 2 (39) Q ≡ big g = (In( l/ ln 2))/ ln 2. (40) Also. x\ and {x} represent, respectively, the signed integer and unsigned fraction portions of x. F2 and F3. respectively, refer to 2n-bit and 3n-bit unsigned fractions. The bits of the (n + ?n)-bit unsigned fraction value F are partitioned into a MS ?ι-bit fraction value u and a LS m-bit fraction value v such that F = u + hv = u' + h( l + v), 0 < u < 1, 0 < v < 1, υ! = it - h and h = 2 → < 1. However, both BLD ≡≡ bld F and XΩ F ≡ bxά F are defined only for 0 < F < 1. Thus. 0 < B O F < 1 and 0 < BXD < 1 with BLD(BXD ) ≡ BXD(BLD ) ≡ F. High- performance implementations of these fraction BLD and BXD functions are particularly desirable in a binary E/L computational system that is described by the present inventor for the first time in a cop ending U.S. application entitled "Method and Apparatus for Exponential/Logarithmic Computation." The Basic Equations. According to the invention, the interpolations of the invention are based on the general relations of the combination of equations (41) through (46) as follows implemented in a computer system. The basic interpolation relations are the following: bxd( u + hv) = bxd u + (bxp v!) bxd h{ 1 + v) (41) blά{υ + hv) = blάu' + b\ά{hq- 1 2 R ) (42) i? = bldυ + Q-bldι . (43) Thus, B = BLD v + Q- BLD υ! when u φ 0 and B = BLD v+Q+1- BLD(1 -2h) when u = 0. The above general product-of-exponentials series expansion for the binary function bxd/ϊ(l + v) may be written in "binary notation" as follows: bxd/i(l + f) = hbxp(BLOv-Q + + v) wherein B 2j is the 2 -th Bernoulli number. For the binary function blά(lιq ~1 2 R ). the above general continued-exponential series expansion can be extended and written in "binary notation " as follows: blά(hq- l 2 R ) = hbxp(B-h 2 bxp(B-lι bxp(B -h§ Q -bxp(B - h^ 0 b p(B -ΛfHilbxp(i? +...)))))) (45) = h bxp{B - h bxp(B' - h bxv{B" -hbxp(B'" + ...)))). B' = B + blg(l/2), B" = B + blg(5/12), iT = H-fblg(47/120). (46) The most significant n-bit and 2π-bit portions of Q may be identified as Q\ and QV2. respectively. Look-up tables BLDT. BXDT. BLDT * . and BXDT* are constructed to provide at least (n + ?rz)-bit values of bid g. bxάg. {bld(g — h)}. and {bxά(g — /?)}. respectively, for any n-bit fraction g. The most significant n-bit.2n-bit. and 3n-bit portions of the entries of the four tables may be identified, respectively, as BLDTl(p), BLDT12(o). BLDT123(ø): BXDTl(o). BXDT12(p). BXDT123(#): BLDTl * (α). BLDT12 * ((/). BLDT123 * (o); and BXDTU( ). BXDT12 * (o). BXDT123 * (#). The quantities BLDTl( ). BLDT2(ø). and BLDT3(o). re¬ spectively, are fractional and comprise most significant (MS), next most significant (NS). and least significant (LS) n-bit portions of BLDT123(g). Corresponding quantities are defined similarly for the MS. NS. and LS n- bit portions of the BXDT123(ø). BLDT123 * (o). and BXDT123 * (p). These quantities have fractional values such that BLDT123tø) = BLDTl(o) + /ϊBLDT2( 5 ) + /rBLDT3(o) (47) BXDT123(</) = BXDTl(#) + /iBXDT2(α) + /7 BXDT3( ) (48) BLDT123 * ( . g) = BLDTl * (.g) + h BLDT2 * (o) + / BLDT3 * (o) (49) - Or BXDT123 * ( ) - BXDTl * ( ) + hBXOT2 * {g) + h 2 BXDT3 * (#). (50) First-Order Interpolation for BLD2(F2). For F2 = u + hv (i.e.,. m = n), there is bid it' + h\ά(hq ~l 2 R ) = bid u' + hbxp B. Thus BLD2(u - hv), as an approximation of BLD(n + hv) to about 2n-bits of precision, can be generated as follows: If u φ 0, then: 1) Look up BLDT12(w'); 2) Look up BLDTl(τ); 3) Sum R =BLDTl(υ)+Ql-BLDTl(n'): 4) Partition the dig¬ its of R into m.f , wherein m = R is the signed integer portion of R, ( — 1 < m < 1), and / = {R} is the unsigned fraction portion of R: 5) Look up BXDT1(/); and 6) Form the sum BLO2{u + hv) =BLDT12(n) + /ι2 m (l+BXDTl(/)) as the output: Or else, if u = 0, then: 1) Look up BLDTl(r): 2) Form the sum R =BLDTl(ι>)+Ql + 1-BLDT1(1 - 2h) 3) Partition the digits of R into m.f, wherein 0 < m = [R\ < 1 and / = {R},' 4) Look up BXDT1(/); and 5) Form the sum BLD2(n + hv) = -1+BLDT12(1 - 2h) + h2 m {l+BXDTl{f)) as the output. First-Order Interpolation for BXD2(F2). For F2 = it + hυ (i.e.. m = n). there is bxdu' + (bxp u')bxάh{\ + v) = bxd it' + h bxp(BLD v — Q + u'). Thus BXD2( u + hv). as an approximation of BXD(ι/ + hv) to about 2n bits of precision, can be generated as follows: If u φ 0. then: 1) Look up BXDT12(n'); 2) Look up BLDTl(r): 3) Form the sum R =BLDTl(r) - Ql + u': 4) Partition the digits of R into m.f. wherein —1 < m = [R\ < 1 and / = {R}: 5) Look up BXDT1( ): and 6) Form the sum BXD2(υ + /u-)=BXDT12(u') + /ι2 m (l+BXDTl(/)) as the output. Or else, if u = 0. then: 1) Look up BLDTl(r): 2) Form the sum R =BLDTl(ι>) - Ql - h: 3) Partition the digits of R into m.f, wherein -1 < m = [R\ < 0 and / = {i?}: 4) Look up BXDT1(/ ' ); and. 5) With D12 as a 2n-bit approximation of the constant D = 1 + bxd(-/) = ± + ±BXD(l-/z), form the sum BXD2(u + = -1 + £>12 + fc2 m (l+BXDTl(/)) as the output. Second-Order Interpolation for BLD3(F3). For F3 = u + hv (i.e., m = 2n), there is = bldu' + hbxp{R - h^bxpR). Thus BLD3(w + hv), as an approximation of BLD(w + hv) to about 3n bits of precision, can be generated as follows: If u φ 0, then: 1) Look up BLDT123(n'): 2) Generate BLD2(r): 3) Sum R =BLD2(ι>)+Q12-BLDT12(u'): 4) Partition the digits of R into m.f . wherein — 1 < m = [R < 1 and / = {R}: 5) Look up BXDT1(/); 6) Subtract the correction: R' = R - h ±2 m (l+BXDTl(/)); 7) Partition the digits of R' into m'.f. wherein —l < m' = [R' < 1 and /' = {R'}; 8) Generate BXD2(/'); and 9) Form the sum BLD3(w + hv)=B DT123{u') + h 2 m '{l+BXO2{f')) as the output: Or else, if υ = 0. then: 1) Generate BLD2(r): 2) Form the sum R =BLD2(r) + Q12 + 1 - BLDT12(1 - 2h) 3) Partition the digits of R into m.f . wherein —1 < m = [R\ < 1 and / = {R}: 4) Look up BXDT1(/): 5) Subtract the correction: R' = R - h 2 m (l+BXDTl( )); 6) Partition the digits of R' into m'.f, wherein — 1 < m' = [i?'J < 1 and / ' ' = {R'}; 7) Generate BXD2(/'): and 8) Form the sum BLD3(u + hv) = -1+BLDT123(1 - 2/ + /i2 m '(l+BXD2(/')) as the output. Second-Order Interpolation for BXD3(F3). For F3 = it + hv (i.e.. n/ = 2n). there is bxd*/' + (bxp ιι')bxάh{l + v) = bxd u! + h bxp(BLD7' — Q + u' + + v)). Thus BXD3(n + hv). as an approxi¬ mation of BXD(a + hv) to about 3n bits of precision, can be generated as follows: If u φ 0. then: 1 ) Look up BLDT123(n') : 2 ) Generate BLD2(TJ ) ; 3) Form the sum R =BLO2{v ) - Q12 + υ' + h±{ l + v) 4) Partition the digits of R into m.f, wherein — 1 < m = \_R\ < 1 and / = {R}; 5) Generate BXD2(J); and 6) Form the sum BXD3(υ + + h 2 m { l+BXO2{f)) as the output. Or else, if u = 0, then: 1) Generate BLD2(v) ; 2) Form the sum R ) - Q12 - h + + v); 3) Partition the digits of R into m.f . wherein — 1 < m — \ R\ < 0 and / = {R} 4) Generate BXD 2(/); and 5 ) With D123 as a 3n-bit approximation of the constant D = 1 + bxcl(-/?) = + ± BXD(1 - h), form the sum BXD3( u + hv) = -1 + £>123 + h 2 m ( l+BXD2(/)) as the output. Third and Higher-Order Interpolation. The third- order and higher-order interpolations merely involve employing succes¬ sively more terms from the series expansions and successively more accu¬ rate table values together with the generators of lower orders. Hardware BLD2 Generator. Figure 2 shows the com¬ binatorial flow-through organization of the general 2n-bit BLD2 function generator 1. Pipe-lining is particularly easy for such a combinatorial (i.e., no feed-back loops) circuit. The circuit can be clocked as a single unit, or else synchronizing delays can be distributed throughout the circuit in order to reduce the execution time of the pipe-line stages, thereby allowing a corresponding increase in the clocking rate. The BLD2 function generator 1 comprises a 2n-bit input first data bus 2 coupled to a partitioning means 3. The partitioning means 3 partitions the MS 2n bits of the input data bus 2 into two n-bit data busses, which are a most significant (MS ) second bus 6 and a least signif¬ icant (LS ) third bus 10. The second bus line 6 is coupled to the inputs of the first and second look-up tables 7 (BLDT1 * ) and 8 (BLDT2 * ). respec- tively. The n-bit third bus 10 is coupled to the input of the third look-up table 11 (BLDTl ) . The n-bit output of the BLDTl look-up table 11 is coupled to the first input of an n-bit subtracter 12. The n-bit output of the first look-up table 7 (BLDTl * ) is coupled to the second input of the subtracter 12. The input bits of a first port of an n-bit first adder 13 are determined by a fixed value input from a first element 21 , which is set to the n-bit constant fraction Ql (an n-bit approximation of the constant fraction Q — big big e = .875D3 . . . Q ) . The n-bit output of the subtracter 12 is coupled to the second input of the first adder 13. The n-bit output of the first adder 13 is coupled to the input of the fourth look-up table 14 (BXDT1 ) . The n-bit second bus 6 is coupled to the inputs of a nonzero- detector n-input OR gate 18. The output of the OR gate 18 is coupled to a first input of a 2-input AND gate 19. The carry-out borrow 20 of the subtracter 12 is coupled to the second input of the 2-input AND gate 19. The output 4 of the AND gate 19 is coupled to the ( — ) control input of the shifter 16. The carry-out 17 of the first adder 13 is coupled to the ( + ) control input of the shifter 16. The n-bit output of the fourth look-up table 14 (BXDT1 ) is coupled to the LS n-bit input of the 2n-bit shifter 16 (Figure 6 configured with a 0-bit-place hard-wired displacement of the output relative to the input ) . The MS n-bit input of the shifter 16 is determined by a fixed value input from a second element 15. which clears the MS n — 1 bits all to 0s and sets the LS bit to 1. The ( + ) and ( — ) control inputs of the shifter 16 determine the 1. 0. or — 1 bit-places of (left ) shifting of the 2n-bit output of the shifter 16 which is coupled to a first input of the 2n-bit second adder 9. The n-bit outputs of the first and second look-up tables 7 and 8. respectively, are coupled to the second port of the 2n-bit second adder 9 wherein they, respectively, constitute the MS and LS n-bits of a 2n-bit approximation of the unsigned fraction value {bld(ι/ — h)} wherein u is the n-bit fraction value on the n-bit second data bus 6. The 2n-bit output of the 2n-bit second adder 9 constitutes the 2n-bit output of the BLD2 function generator 1. A further implementation alternative is to have a single random access memory perform the function of the biasing adder 13. the BXDT1 function block 14 and the shifter 16, although this requires at least triple the number of entries that is required for the BXDT1 block alone. Hardware BXD2 Generator. Figure 3 shows the com¬ binatorial flow-through organization of the general 2n-bit BXD2 function generator 42 comprising a 2n-bit input first data bus 23 coupled to a partitioning means 24. The partitioning means 24 partitions the MS 2n bits of the input data bus 23 into two n-bit data busses, which are a MS second bus 25 and a LS third bus 29. The n-bit second bus line 25 is coupled to the inputs of the first and second look-up tables 26 (BXDT1*) and 27 (BXDT2* ). respectively. The second bus line 25 is coupled to the first input of an 77- bit subtracter 31. The third bus line 29 is coupled to the input of the third look-up table 30 (BLDTl ). The n-bit output of the third look-up table 30 (BLDTl ) is coupled to a first input of an n-bit first adder 32. The input bits of the second port of the n-bit subtracter 31 are determined by a fixed value input from a first element 40. which is set to the value Ql + h (an n-bit approximation of the constant fraction Q = big big e = .875D3 . . . 16 incremented by 1 . The n-bit output of the subtracter 31 is coupled to the second input of the first adder 32. The output of the first adder 32 is coupled to the input of a fourth look-up table 33 (BXDT1 ). Th e n-bit second bus 25 is coupled to the inputs of a zero- detector n-input NOR gate 34. The output of the NOR gate 34 is coupled to a first input of a 2-input AND gate 35. The carry- out borrow 41 of the subtracter 31 is coupled to the second input of the AND gate 35. The output 37 of the AND gate 35 is coupled to the ( — ) control input of a 2n-bit shifter 36. The carry-out 38 of the n-bit first adder 32 is coupled to the (+) control input of the shifter 36. The n-bit output of the fourth look-up table 33 (BXDT1) is coupled to the LS n-bit input of the 2n-bit shifter 36 (Figure 6 configured with a O-bit-place hard-wired displacement of the output relative to the input ) . The MS n-bit input of the shifter 36 is determined by a fixed value input from a second element 39. which clears the MS n — 1 bits all to Os and sets the LS bit to 1. The (+ ) and ( — ) control inputs of the shifter 36 determine the 1, 0. or —1 bit-places of (left) shifting of the 2n-bit output of the shifter 36 which is coupled to a first input of a 2n-bit second adder 28. The n-bit outputs of the first and second look-up tables 26 and 27, respectively, are coupled to the second input of the 2n-bit second adder 28 wherein they, respectively, constitute the MS and LS n bits of a 2n-bit approximation of the fraction value {bxd(ιt — h)} wherein u is the n-bit fraction value of the n-bit second bus 25. The 2n-bit output 22 of the 2n- bit adder 28 constitutes the 2n-bit output of the BLD2 function generator 42. Hardware BLD3 Generator. Figure 4 shows the com¬ binatorial flow-through organization of the general 3n-bit BLD3 function generator 70 comprising a 3n-bit input first data bus 43 coupled to a par¬ titioning means 44. The partitioning means 44 partitions the MS 3n bits of the input first data bus 43 into three n-bit data busses, which are a MS second bus 45. a NS third bus 46 and a LS fourth bus 47. The second bus line 45 is coupled to the inputs of the first, second and third look-up tables 48 (BLDTl * ) . 49 (BLDT2 * ) and 66 (BLDT3 * ) . respectively. The third and fourth busses 46 and 47, respectively, are coupled to the MS and LS n-bit inputs, respectively, of a 2n-bit BLD2 function generator 52 (Figure 2). The 2n-bit output of the BLD2 function generator 52 is coupled to a first input of a 2π-bit first adder 53. The input bits of the first port of a 2n-bit first subtracter 50 are determined by a fixed value input of from a first element 68, which is set to the value Q12 ( a 2n-bit approximation of the constant fraction Q = big big e = .875D3BA705 . . -ic ) . The n-bit outputs of the first and second look-up tables 48 and 49. respectively, are coupled to the MS and LS n-bit inputs, respectively, of the second port of the first subtracter 50. The 2n-bit output of the first subtracter 50 is coupled to the second input of the 2n- bit first adder 53. The 2n-bit output of the first adder 53 is coupled to a first input of a 2n-bit second subtracter 60. The MS n bits of the 2n-bit output of the first adder 53 is coupled to the input of a fourth look-up table 55 ( BXDT1 ). The n-bit output of the fourth look-up table 55 (BXDT1 ) is coupled to the LS n- bit data input of a 2n-bit first shifter 58 (Figure 6 configured with the output hard- wired — 1 bit-place displacement to the left. i.e.. +1 bit-place to the right, relative to the input. The MS n-bit input of the 2n-bit first shifter 58 is determined by a fixed value input of a second element 56, which clears the MS n — 1 bits all to Os and sets the LS bit to 1. The ( + ) and ( — ) control inputs of the first shifter 58 determine the 0, — 1 , or —2 bit places of (left ) shifting of the 2n-bit output of the first shifter 58. The output of the first shifter 58 is coupled to the second input of the 2n-bit second subtracter 60. The 2n-bit output of the second subtracter 60 is coupled to the 2n-bit input of a 27/-bit BXD2 function generator 63 (Figure 3 ) . The n-bit second bus 45 is coupled to the inputs of a nonzero-detector n-input first OR gate 57. The output of the first OR gate 57 is coupled to a first input of a 2-input AND gate 59. The carry- out 54 of the first adder 53 is coupled to the ( + ) control input of the 2n-bit first shifter 58. The carry-out borrow 51 of the first subtracter 50 is coupled to the second input of the AND gate 59. The output of the AND gate 59 is coupled to the ( — ) control input of the first shifter 58. The output of the AND gate 59 is coupled to a first input of a 2-input second OR gate 62. The carry-out borrow 61 of the second subtracter 60 is coupled to the second input of the second OR gate 62. The output of the second OR gate 62 is coupled to the ( — ) control input of a 3n-bit second shifter 65. The carry-out 54 of the first adder 53 is coupled to the ( + ) control input of the 3n-bit second shifter 65. The 2n-bit output of the second subtracter 60 is coupled to the input of a 2?i-bit BXD2 function generator 63 (Figure 3 ). The 2n-bit output of the BXD2 function generator 63 is coupled to the LS 2n-bit data input of the 3n-bit second shifter 65 (Figure 6 configured with a 0-bit-place hard-wired displacement of the output relative to the input). The MS n-bit input of the second shifter 65 is determined by a fixed value input of a third element 64, which clears the MS n — 1 bits all to Os and sets the LS bit to 1. The ( + ) and ( — ) control inputs of the second shifter 65 determine the 1. 0. or —1 bit places of (left) shifting of the 3n-bit output of the second shifter 65. The output of the second shifter 65 is coupled to a first input of a 3n-bit second adder 67. The n-bit outputs of the first, second, and third look-up tables 48. 49. and 66. respectively, are coupled to a second input of the 3n-bit second adder 67 wherein they constitute, respectively, the MS. NS and LS 77-bit portions of a 3n-bit approximation of the unsigned fraction value {bld(υ — h)} wherein u is the n-bit fraction value of the n-bit second bus 45. The 377-bit output 69 of the second adder 67 constitutes the output of the BLD3 function generator 70. Hardware BXD3 Generator. Figure 5 shows the com¬ binatorial flow-through organization of the general 3 n-bit BXD3 function generator 71 comprising a 3n-bit input first data bus 72 ccupled to a par¬ titioning means 73. The partitioning means 73 partitions the MS 3n bits of the input first data bus 72 into three n-bit data busses, which are a most significant (MS) second bus 75, a next most significant (NS) third bus 76. and a least significant (LS) fourth bus 77. The second bus line 75 is coupled to the inputs of the first, second, and third look-up tables 78 (BXDT1 * ) . 79 (BXDT2*), and 91 (BXDT3 * ). respectively. The third and fourth busses 76 and 77. respectively, are coupled to the MS and LS n-bit inputs of a 2n-bit BLD2 function generator 83 (Figure 2). The 2n-bit output of the BLD2 function generator 83 is coupled to a first input of a 2n-bit first adder 84. The MS n — 1 bits of the n-bit NS third bus 76 are coupled to the input of a first element 80. The n-bit output of the first element 80 is simply a MS fixed constant 1-bit concatenated with the (n — l)-bit input of the first element 80. The output of the first element 80 is coupled to the LS n-bit input of the first port of a 2n-bit subtracter 81. The second bus 75 is coupled to the MS n-bit input of the first port of the 2/7-bit subtracter 81. The input bits of the second port of the subtracter 81 are determined by a fixed value input from a second element 93. which is set to the 2n-bit constant fraction Ql2 + /i (a 2n-bit approximation of the constant fraction Q = big big e = .875D3BA705 . . . \ Q incremented by ή ). The 2n-bit output of the subtracter 81 is coupled to the second input of the 2n-bit first adder 84. The 2?v,-bit output of the first adder 84 is coupled to the input of a 27i-bit BXD2 function generator 88 (Figure 3 ) . The n-bit second bus 75 is coupled to the inputs of a nonzero-detector //-input NOR gate 86. The output of the NOR gate 86 is coupled to a first input of a 2-input OR gate 87. The carry-out borrow 82 of the subtracter 81 is coupled to the second input of the OR gate 87. The output of the OR gate 87 is coupled to the ( — ) control input of a 3n-bit shifter 90. The carry-out 85 from the first adder 84 is coupled to the ( + ) control input of the shifter 90. The 2?7-bit output of the BXD2 function generator 88 is coupled to the LS 2n-bit input of the 3n-bit shifter 90. The MS n-bit input of the shifter 90 is determined by a fixed value input from a third element 89. which clears the MS n — 1 bits all to 0s and sets the LS bit to 1. The ( + ) and ( — ) control inputs of the shifter 90 determine the 1, 0, or —1 bit places of shifting of the 3?i-bit output of the shifter 90. The output of the shifter 90 is coupled to a first input of a 3/7-bit second adder 92. The n-bit outputs of the first, second, and third look-up tables 78, 79. and 91. respectively are coupled to the MS. NS. and LS n-bit inputs, respectively, of the second port of the 3n-bit second adder 92 wherein they constitute a 3n-bit approximation of the unsigned fraction value {bxd( u — h)} wherein u is the n-bit fraction value of the n-bit second bus 75. The 3n-bit output of the second adder 92 constitutes the output of the 3n-bit BXD3 function generator 71. A further implementation alternative is to have a single random access memory perform the function of the biasing subtracter 31. the BXDT1 function block 33 and the shifter 36. although this at least triples the number of entries required for the BXDT1 block alone. Reduction and Elimination of Tables. Referring to Figure 1. the functional blocks 206. 207. 208 and 214. respectively la¬ beled '7*. " "log / * '. " "log *" and "anti-log. " may each be implemented using random access memory look-up tables. However, the last three of these blocks and a most significant portion of the first block constitute generators for functions that have derivatives of restricted magnitude and thus each can be implemented using a type of generator, such as the gen- erator 201 diagrammed in Figure 1. that is suitable for generating such functions. This approach, together with pipe-lining, can substantially re¬ duce the size of total memory required while retaining substantially all of the throughput speed, although latency will likely be extended. Furthermore, some useful functions may be generated us¬ ing a generator implementation that completely replaces memory in the log / * ' block by lower cost combinatorial logic such as adders, subtracters or multiplexers, etc. A first example is the generator of a logarithmic function that is diagrammed in Figure 2 with the BLDTl * block and the BLDT2 * block together constituting the /* block of Figure 1. The most signifi¬ cant BLDTl* block also constitutes a major portion of the log *' block in Figure 1. The generator diagrammed in Figure 2 generates a binary loga¬ rithmic function, but the BLDTl* block, the BLDT2* block and the bias value 21 can be changed to provide generation of logarithmic functions of other base values. A second example is the generator of an exponential func¬ tion that is diagrammed in Figure 3 with the BXDT1* block and the BXDT2* block together constituting the /* block of Figure 1. The most significant (dominant) portion u 25 of the input 23 also constitutes a major portion of the log /*' block in Figure 1. The generator of Figure 3 generates a binary exponential function, but the BXDT1 * block, the BXDT2 * block and the bias value 40 can be changed to provide generation of exponential functions of other base values. Further examples of tables performing multiple duties can be found in the generation of "rnurlus " functions as next described. Generation of Murlus Functions. The pair of relations comprising 1 ) the ordinary differential equation f a [x) — a ~ f" { ~x ) and 2) the functional equation f a {x) = x + f a { — ) bare solution functions that are parametrized by the parameter a. These solution functions f a {x) = aml + ( ) and f a (x) = aml_ ( ') are herein called, respectively, base-α murlus functions of the first kind and the second kind. (Regarding notation, ami x ≡ aml± x when both kinds of functions are included.) The base-α murlus function of the first kind of an argument x is aml + x ≡ amr x ≡ alg(α 2' + 1 ) which is herein called a base-α mur function of x. This mur function converts an input logarithm of a number into a logarithm of a sum of that number and one. The base-α murlus function of the second kind of an argument x is aml_ x ≡ als x ≡ alg \a x - 11 which is herein called a base-α lus function of x. This lus function converts an input logarithm of a number into a logarithm of the absolute value of a difference between that number and one. The murlus functions are useful for performing arithmetic operations on numbers when only logarithms of those numbers are available for processing. The properties 0 < amr (a:) < 1 for x < 0 and — 1 < als( ' ) < 0 for x < — 1 ensure that the table values can be strictly limited in mag¬ nitude except close to the logarithmic singularity of als(α') at x = 0. The generalized generator design diagrammed in Figure 1 provides an approximation to a designated function f( u * + hv* ) according to f(ιι * + hv * ) ≥ f(C - hξ) + Λ rxp(rlg(ξ + v * ) + rlg(/'(n * - hξ) ) (51 ) which can be applied to generating a murlus function using look-up ta¬ ble values from the primary function block 206 to produce the secondary function signal 210 produced by the secondary function block 207. Fur¬ thermore, the murlus functional equation permits generation of a murlus function of a signed argument to be performed efficiently and economi¬ cal!)- using a primary look-up table that contains values only of a murlus function of a negative argument, thereby halving the required quantity of entries in the look-up table of the primary map 206. This ma}' be shown as follows: For 0 < x = u * + hv* and 0 < u * + hξ. the approximating equation (51) when applied to generating a murlus function of a positive argument may be written as aml(a;) = aml(w* + hv * ) = u + aml(— u* + hξ) - hξ--hτxp( ή g(ξ + v*) -aml(-κ * + /z rlg(α)). For 0 < x = u* + hv* and 0 < u* + hξ, the approximating equation (51) when applied to generating a murlus function of a negative value —x (i.e.. of a negative of an actually positive argument x) may be written as aml(- ") = aml(-u * - hv * ) = aml(-u * + Jiξ) - hξ - hv* + hτxp{ήg{ξ + v*) - aml(-n* + hξ)ήg{a)). Thus, the primary map 206 need include only look-up val¬ ues aml( — u* + hξ) — hξ in either of the foregoing cases. A MS portion of such values can then also be used to form the argument of the rxp function in equation (51). particularly when / • = a and rlg = 1. thereby reducing the quantity of executed additions as well as reducing total table size. Fi¬ nally, selectively adding u* or — hv* in the final sum as required may be economically implemented using a multiplexer and an adder.

Claims: What is claimed is: 1. Apparatus for generating an output signal λvhich ap¬ proximates a designated differentiable function of an input argument sig¬ nal, comprising: means coupled to receive said input argument signal for partitioning between a most significant dominant portion and a next most significant adjunct portion; means coupled to receive said adjunct portion for mapping into an adjunct function signal in accordance with an adjunct map com¬ prising a first logarithmic function of a sum comprising said adjunct por¬ tion and a predetermined first bias that is related to a maximum magnitude of said adjunct portion; means comprising random access memory coupled to re¬ ceive said dominant portion for mapping into a primary function signal in accordance with a primary map that is related to said designated function of a difference between said dominant portion and said first bias: means coupled to receive said dominant portion for map¬ ping into a secondary function signal in accordance with a secondary map that is related to said first logarithmic function of a first derivative of said designated function of said difference between said dominant portion and said first bias; means comprising combinatorial logic coupled to receive said secondary function signal and said adjunct function signal for com¬ bining into an interim signal: means coupled to receive said interim signal for mapping into a correction signal in accordance with an exponential map that is re¬ lated to a first exponential function, corresponding to said first logarithmic function, of said interim signal: and means coupled to receive said primary function signal and said correction signal for adding to produce said output signal. 2. The apparatus as set forth in claim 1 wherein: said designated function comprises a second exponential function: said primary function signal comprises said second expo¬ nential function of a difference between said dominant portion and said first bias: said secondary function signal comprises said dominant por¬ tion: and said correction signal comprises said first exponential func¬ tion of a sum comprising said interim signal and a second bias that is related to a base of said second exponential function. 3. The apparatus as set forth in claim 2 wherein a base of said first logarithmic function, said first exponential function and said second exponential function is two. 4. The apparatus as set forth in claim 1 wherein: said designated function comprises a second logarithmic- function: said primary function signal comprises said second logarith¬ mic function of a difference between said dominant portion and said first bias: said secondary function signal approximates a negative of a most significant portion of said primary function signal; and said correction signal comprises said first exponential func¬ tion of a sum comprising said interim signal and a second bias that is related to a base of said second logarithmic function. 5. The apparatus as set forth in claim 4 wherein a base of said first logarithmic function, said second logarithmic function and said first exponential function is two. 6. The apparatus as set forth in claim 1 wherein: said input argument signal is positive: said designated function comprises a murlus function of said positive input argument signal; said primary map comprises a difference between said murlus function of a difference between said first bias and said dominant portion and said first bias: said secondary function signal comprises an approximation of a most significant portion of a negative of a product comprising said primary function signal scaled by a factor that is related to a base of said murlus function and a base of said logarithmic function and said exponential function: and said adding means is further coupled to receive said domi¬ nant portion. 7. The apparatus as set forth in claim 6 wherein said base of said murlus function and said base of said logarithmic function and said exponential function is two. 8. The apparatus as set forth in claim 1 wherein: said input argument signal is positive: said designated function comprises a murlus function of a negative of said positive input argument signal; said primary map comprises a difference between said murlus function of a difference between said first bias and said dominant portion and said first bias: said secondary function signal comprises an approximation of a most significant portion of a negative of a product comprising said primary function signal scaled by a factor that is related to a base of said murlus function and said base of said logarithmic function and said exponential function; and said adding means is further coupled to receive a negative adjunct signal; further comprising: means coupled to recede said adjunct portion for negating to produce said negative adjunct signal. 9. The apparatus as set forth in claim 8 wherein said base of said murlus function and said base of said logarithmic function and said exponential function is two. 10. The apparatus as set forth in claim 9 wherein a radix of said input argument signal, said base of said murlus function and said base of said logarithmic function and said exponential function is two. 11. A method for generating output data which approxi¬ mates a designated diff ' erentiable function of input argument data, com¬ prising the steps of: partitioning said input argument data between a most sig¬ nificant dominant portion and a next most significant adjunct portion; mapping said adjunct portion into adjunct function data in accordance with an adjunct map comprising a first logarithmic function of a sum comprising said adjunct portion and a predetermined first bias that is related to a maximum magnitude of said adjunct portion; mapping said dominant portion into primary function data in accordance with a primary map that uses random access memory data that are related to said designated function of a difference between said dominant portion and said first bias: mapping said dominant portion into secondary function data in accordance with a secondary map that is related to said first log¬ arithmic function of a first derivative of said designated functicn of said difference between said dominant portion and said first bias; combining said secondary function data and said adjunct function data into interim data; mapping said interim data into correction data in accor¬ dance with an exponential map that is related to a first exponential func¬ tion, corresponding to said first logarithmic function, of said interim data: and adding said primary function data and said correction data to produce said output data. 12. The method as set forth in claim 11 wherein: said designated function comprises a second exponential function: said primary function data comprises said second exponen¬ tial function of a difference between said dominant portion and said first bias: said secondary function data comprises said dominant por¬ tion: and said correction data comprises said first exponential func¬ tion of a sum comprising said interim data and a second bias that is related to a base of said second exponential function. 13. The method as set forth in claim 12 wherein a base of said first logarithmic function, said first exponential function and said second exponential function is two. 14. The method as set forth in claim 11 wherein: said designated function comprises a second logarithmic function: said primary function data comprises said second logarith¬ mic function of a difference between said dominant portion and said first bias; said secondary function data approximates a negative of a most significant portion of said primary function data: and said correction data comprises said first exponential func¬ tion of a sum comprising said interim data and a second bias that is related to a base of said second logarithmic function. 15. The method as set forth in claim 14 wherein a base of said first logarithmic function, said second logarithmic function and said first exponential function is two. 16. The method as set forth in claim 11 wherein: said input argument data is positive: said designated function comprises a murlus function of said positive input argument data; said primary map comprises a difference between said murlus function of a difference between said first bias and said dominant portion and said first bias: said secondary function data comprises an approximation of a most significant portion of a negative of a product comprising said primary function data scaled by a factor that is related to a base of said murlus function and a base of said logarithmic function and said exponen¬ tial function: and said adding means is further coupled to receive said domi¬ nant jjortion. 17. The method as set forth in claim 16 wherein said base of said murlus function and said base of said logarithmic function and said exponential function is two. 18. The method as set forth in claim 11 wherein: said input, argument data is positive; said designated function comprises a murlus function of a negative of said positive input argument data; said primary map comprises a difference between said murlus function of a difference between said first bias and said dominant portion and said first bias; said secondary function data comprises an approximation of a most significant portion of a negative of a product comprising said primary function data scaled by a factor that is related to a base of said murlus function and a base of said logarithmic function and said exponen¬ tial function; and said adding means is further coupled to receive negative adjunct data; further comprising: means coupled to receive said adjunct portion for negating to produce said negative adjunct data. 19. The method as set forth in claim 18 wherein said base of said murlus function and said base of said logarithmic function and said exponential function is two. 20. The method as set forth in claim 19 wherein a radix of said input argument data, said base of said murlus function and said base of said logarithmic function and said exponential function is two.

Inventors: PICKETT LESTER CARYL, PICKETT, LESTER, CARYL
Applicants: PICKETT LESTER CARYL, PICKETT, LESTER, CARYL
Application year: 1993
Main classification: G06F15
IPC codes: G06F1, G06F7, G06F17