JPH0322025A - Exponential function arithmetic unit - Google Patents

Abstract

PURPOSE:To calculate an exponential function at a high speed with high accuracy by providing the table of the exponential function and respectively one executing the retrieval of the table, multiplication and addition. CONSTITUTION:A register 11 to divid an initial value (x) into a high-order digit H and a low-order digit L (x=H+L), a read only memory 14 to input the high-order digit H from the register 11 and to output the value of eH calcu lated in advance are provided. Then, a multiplier 17 to execute the multiplica tion concerning the output of the memory 14 and the output of the low-order digit L from the register 11 an adder 21 to add the output value of the multi plier 17 to the value of the memory 14 are provided. By the multiplier 17, the eH value of the memory 14 is multiplied with the low-order digit L and the output value of the multiplier 17 is added to the eH value of the memory 14 by the adder 21, then, an exponential function eX is calculated. Thus, arithmetic can be executed by one step and the exponential function eX can be calculated at the high speed with the high accuracy.

Description

[Detailed description of the invention] [Industrial application field] The present invention relates to an exponential function calculation device that can be pressed in a computer.

[Conventional technology]

In general, the exponential function e with the base e=Z71828...
x is one of the functions that must be provided in a computer that handles numerical calculations. When calculating this exponential function, it is not possible to apply the same algorithm to the entire range of arguments; instead, it is common to apply appropriate transformations to the arguments and perform calculations by limiting the range.

For example, when calculating e, i is an integer, 0≦x<1
Then t=i+X ・・・・・・・・・
・(1) et= e' X ex ・・
......(2). The range of i in which eX does not overflow or underflow is about -88≦i≦+88 in single precision according to the IEEE754 standard.It is also possible to prepare tables for D and e'. Therefore, it is sufficient to find e within the range of 0≦x<1.

Conventionally, the exponential function is obtained by the following Taylor polynomial approximation.

・・・・・・・・・{3} In this case, when the argument is necessary. The polynomial approximation described in this paper had the drawback of requiring a long computation time because calculations and additions were repeated.

On the other hand, in a microprogram brake calculator, the exponential function is expressed as 8 T L (Sequential Table
In some cases, it may be obtained by the Lockup method. This 8T
According to the l, modulus, the exponential function is the constant log(1+2
) can be obtained as follows.

■ Set Xo=x, yo=1.

■ For k=0.1, 2, ..., N, ■■t-Execute.

■ w=xk-1-log(1+2)...
・・・・・・{4}■ If W≧O, “k””
+ )'k = Yk-1 + Yk-I X 2'
・・・・・・・・・・・・(5) If w<0,
Xk = Xk-1, Yk = yi+-t ■ e
”=YN is obtained.

In the S '1' L method, in order to sum the N-digit exponential function,
It is necessary to perform/return the process described in ■■ above N times. 1, (5
) requires a shift of k digits, so a barrel shifter is required. Therefore, the STL method has the drawback of requiring a long calculation time, and is therefore not commonly used.

Another conventional example is an exponential function calculation device using a table l{OM. FIG. 4 is a block diagram showing the structure. The register 41 is a 20-bit register that stores the argument In the first step, when the value of X is input to the register 41, in the second step, }1, OM
Since the eX value is output from 42 to the register 43, Cx can be calculated for each step. However, in order to obtain a 2n-digit sheath degree, the RO of 2n bits}X2 #&
M is required. 49,152 if 2n=12 bits
A ROM of 402,653,184 bits is required, which is difficult to realize.Therefore, the conventional method is
It is only used when the number is 12 bits or less.

[Problem to be solved by the invention]

The conventional exponential curve calculation device described above has the following sets of points.

(1) Calculation time is long.

In the polynomial approximation method, to calculate equation (3), m operations and m additions are necessary. In order to solve the problem, m≧6
It must be. ') ishi, ride. It takes 6 calculations and 6 additions.

In addition, in the STL method, it takes k to extend the 24-bit Glaze solution.
=0. 1, . . . , 23, it is necessary to calculate equations (4) and (5), and it is necessary to perform a maximum of 24 shift operations and 48 additions and subtractions. If we prepare one set of barrels and two sets of adders/subtractors, (4),
Although the calculation of equation (5) is performed in one step, it takes the calculation time for 24 shifts, additions and subtractions.

(2) The hardware is large-scale.

In the polynomial approximation method, in order to achieve 2n-bit accuracy, it is necessary to perform multiplication D and addition with 20-bit accuracy. Therefore, a 20 bits x 2n bits multiplier and a 2n
A bit increaser and a small amount of constant R (JM are required. Since the multiplier is large-scale, one product increase is required when converting to LSL.

The STL method requires a sequencer as hardware in addition to a 20-hit barrel switch, a 2n-bit adder/subtractor, and a constant LOM. In the conventional example shown in Fig. 2, in order to obtain a 20-bit spread, 2n hits x 2 candy i-L (JM) are required. If 2n = 24 hits, 402,653,184 bits of ROM are required. Adashi,
LSt conversion is not possible.

The purpose of the present invention is to eliminate such drawbacks by having a table of ``e'' and performing one table search, one multiplication, and one addition, thereby achieving an exponential function at high speed and with a high degree of glaze. e
The object of the present invention is to provide an interexponential arithmetic device that can calculate X.

[Means to solve the problem]

The basic structure of the present invention is to use a computer to calculate the exponent number e! In the exponential function calculation device that calculates a read-only memory that outputs , a multiplier that multiplies the output of this memory and the output of the lower digit L of the register, and an adder that adds the output value of this multiplier to the value of the memory. The multiplier multiplies the C value in the memory by the lower digit L, and the adder adds the output value directly to e4 in the memory, thereby performing an exponent operation e. '' is taken as a t-% image.

〔Example〕

Next, the present invention will be explained in detail using the drawings.

FIG. 1 is a block diagram of an apparatus for calculating an exponential number eX according to an embodiment of the present invention. In the figure, register 11 is a 20-bit register that inputs variable Signal, R, OM14#ie 1st shift memorized 2
0 bit x 2” word ROM, signal l5 is 2 of ROM14
The n-bit output signal 16 is the output signal of the upper n bits of the ROM 14, the multiplier 17 is an nxn-bit multiplier that calculates the product of the signal 13 and the signal 16, and the signal 18 is the multiplier 1
7, the lower n bits of the output 20 bits, signal 19, the upper n bits of the output n bits of the rider 17, the higher n hit, signal 20
is No. 6 that outputs n-bit cello, adder 21 is a 20-hit adder that adds the value of signal 15 to the upper digit 20 and lower digit, and register 22 is a 20-bit adder that holds the output of adder 21. This is a register.

Next, the principle of exponent single number calculation, which is originally an excitation example, will be explained.

Here, x (Q≦X<l
), consider finding 2n-digit e”k. First,
Divide n digits into high-order digits H and low-order digits L.

x= I (xkX2-k)=H+Lk==1 However, xh=(1sO) 0≦x<1 ■= Old { xkX 2-' } 1. =1 L =,7. ．． (XkX 2-')...
・・・(6) ・・・・・・・・・(7) ・−・・・・・・・(8) ・・・・・・・・・(9) Here, the additive motion e” b== e”
......When the second term of the αυαυ equation is expanded by Taylor around L=O, the following equation is obtained.

Here, since 11<1 and L<2-', the term after L can be ignored. Therefore, the αυ formula can be approximated to the following formula.

e”=e”X (1+L)=e”+eHXL ・・・・・・・・・
...In the 0αυ formula, if a table ROM is prepared that inputs H and outputs the A value of e, e8 can be obtained by one multiplication and one addition. Since H<1 and L<2-n, the precision of the multiplication of the second term of equation 0 may be n digits. Therefore, e
To obtain 20 digits of precision for ``, we need 2n digits x 2n words.
x no table ROM14 and this 1{top n of OM14
A multiplier 17 that adds up the n digits of the digit and L, and an adder 1# unit 21 that adds the upper n digits of the multiplication result to the n digits of the output 2n.
Toka rL is good.

Next, using actual numerical values, the exponential function e is calculated according to this example.
The procedure for calculating X will be explained. Here, n=12, and the number is expressed in hexadecimal.

Since {aX input to the register 11 satisfies 0≦x<1, it is expressed as a 24-bit fixed-point number with the decimal point above the MOB. l output from the register 22
le'' is expressed as a fixed-point number with 2 bits of the integer part and 22 lower bits of the decimal point, such that 0≦e''<e.

First, write x=Q in register 11. When AIB345 is input, signal 12 is }i,=AIB, and signal 13 is L=
It becomes 345.

e"""=l.E16FID801-, but rounding is performed using the lower 22 bits of the decimal point, e,1.}
, 1 6 i" 2 0, and the output number 1 of ROM14
5 is 785 BCg. In the multiplier 17, the signal 13 and the signal 16 are multiplied.

345 x 786 = 785D51 Of this product, the lower 12 bits (signal 18) are not used, and the upper 12 bits (signal 19) are combined with signal 20 and output to adder 21. In adder 2l, signal 15 Toi No. 8 2
The addition of 0.19 is taken.

785B (.:8 + 000785 = 7
851) 51 This sum is entered in register 22, register 2
Since the direct representation of number 2 in 2 is a fixed point number with 2 bits in the integer part, this is equal to e''=l, E17544.

The true Ilk is e” = 1.E 1 7 5 4 3
9A...--te6 ruta, calculation accuracy is 2n=24
It's a bit.

As described above, from X on register 11 to register 22
The above eX is obtained. The time required for this calculation is approximately
This is the sum of the access time of M14, the calculation time of multiplier 18, and the calculation time of adder 21.

When calculating e”k using 24 bits f# & (n=12),
ROM14 has 24 hits x40961'd1 multiplier l8
The multiplier is 2 bits x 12 pins, and the adder 2l is 24
It is a bit, and is large enough to be integrated into an LSI.

FIG. 2 shows a pipeline/type calculation device for calculating an exponential function eX according to a second embodiment of the present invention. This arithmetic unit has the same multiplication method as that shown in FIG. 1, but registers are inserted at important points in the arithmetic unit to enable big-line operations.

In the figure, the difference from the first cause is that register 31 is RO
A 20-bit register that latches the output fa of M14,
Register 32 is a register that latches the value of signal 3, signal 33 is an output signal of the upper n hits of register 31, register 34 is a register that latches the lower n bits of the n bits output from multiplier 17, and register 35 is register 31 This is a register that latches the first shift.

Next, the operation shown in FIG. 2 will be explained.

FIG. 3 is a schematic diagram illustrating the operation of the exponential function calculation device shown in FIG. 2. The horizontal line 411 shows time, and shows how one shift of registers 11, 31, 32, 35, 34, and 22 changes at each step.

flmxl, x2, and x3 are input to the register 1l at steps 11, t2, and t3, respectively, and the intermediate results are input to the registers 31 and 32 at steps 12, 13, and 14 one step later. At steps 13, 14, and t5 two steps later, intermediate results are entered into registers 35 and 34. Step t4 after 3 steps,
15, At t6, the operation result ex1e is stored in the register 22.
, e is entered.

As described above, in this embodiment, Fi. e'' is calculated in 3 steps, but its $ calculation result is obtained at each step. The time for 1 step is 1 table search, 1 multiplication $ k
This is the maximum amount of time required for one call. Originally, the amount of node 9 in this embodiment was only increased by registers 31, 32, and 34.35 compared to the first embodiment, and the LS was sufficient.
It is large enough to be converted into an I.

When the D1 multiplier 17 is configured with a carry save adder and a carry propagation adder, substituting the carry propagation adder with the adder 21 shortens the calculation time and reduces the amount of hardware. . However, addition 521 requires a width of 3n bits.

〔Effect of the invention〕

As explained above, the exponential function calculation device according to the present invention has the following effects.

(1) Computation time is short.

The exponential function calculation device of the first embodiment can perform calculation in one step. The processing time for this l step is the sum of the table ROM read time, multiplication time, and addition time. Furthermore, in the second embodiment, the calculation is performed in three steps, and the processing time for one step is the maximum value of the table read time, multiplication time, and addition time. Since data can be input and output in a pipeline manner for each step, the average calculation time when performing a performance on a large amount of data is one step.

(2) The demonstration hardware is Kobuni 1, model.

The horizontal hardware is table ROM, i calculator, 7iT
It is a small collection, with only Ie Sanki.

In order to obtain a 20-bit performance f'1'4, a finger function calculation device using the conventional STL method requires a 2n-hit barre A7 and a 2n-bit adder. A sequencer was also required to control it. 1. In addition, in the conventional multi-η equation approximation, the 2n×2n board and 2
In addition, a sequencer was required for the n-bit processor.

The conventional arithmetic unit required a 20-bit x 2 word ROM, but the present invention's arithmetic unit requires a 20-bit x 2 word ROM.
``m's kLt)M, nXnk't's calculator, 2n
All you need is a bit accessory.

When converting to L81, the '@product of nXn bits' descendant is
Approximately 174 tales of a 2 n x 2 n bit multiplier.

tit, 2” #ROMH, 2 mOf-LO ▲
(approximately 1/90,000 when n=12)
That's fine.

[Brief explanation of drawings]

FIGS. 1 and 2 are block diagrams showing the first and second parts of the present invention. 11, 22, 31, 32, 34, 35, 41.43...
・Register, 14.42...approximately 0 to 1, 17...n power unit, 2l...assistor.

Claims (1)

[Claims] In an exponential function calculation device that calculates an exponential function e^x using a calculator, the initial value x is divided into upper digit H and lower digit L (x=H
+L), a read-only memory that inputs the upper digit H from this register and outputs the pre-calculated value of e^H, and multiplies the output of this memory and the output of the lower digit L of the register. and an adder that adds the output value of the multiplier to the value in the memory,
The multiplier multiplies the e^H value in the memory by the lower digit L, and the adder adds the multiplier output value to the e^H value in the memory to perform an exponent operation e^x. An exponential function calculation device characterized by: