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Xnumbers Tutorial

Using extended numbers in Excel

If you try to enter a long number with more than 15 digits in a worksheet cell,

Excel automatically converts it in standard precision eliminating the extra

digits. The only way to preserve the accuracy is to convert the number in a

string. It can be done by prefixing it with the hyphen symbol ' .

This symbol is invisible in a cell but avoid the conversion.

Example: enter in a cell the number 1234567890123456789.

We have inserted the same

number with the hyphen in B2

and without the hyphen in B3.

Excel treats the first number as a

string and the second as a

numbers

Note also the different alignment

We have inserted a long

numbers with full precision as a

string in B2

If we try to multiply the cell B2 for

another number, example for 2,

Excel converts the string into

number before performing the

multiplication. In this way the

originally accuracy is destroyed

The only way to perform arithmetic operations

preserving the precision is to use the

multiprecision functions of the Xnumbers

library.

In that case we use the function xmult

Note from the alignment that the result is still

a string

You can also insert extended numbers directly in the function. Only remember that, for

preserving Excel to convert them, you must insert extended numbers like string, within quote

"...".

2469135780246913578 =xmult( "1234567890123456789" , 2 )

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Xnumbers Tutorial

Functions Handbook

Xnumbers includes an application for searching and pasting the Xnumbers functions,

cataloged by subject. This application can also submit the Xnumbers macros.

You can activate the Functions Handbook from the menu bar Help > Function manager.

Category: you can filter macros by category (Arithmetic, Statistical, Trigonometric, etc.)

Macro Type: filters by macro Functions, by macro Subroutines, or both

Paste Into: choose the cell you want to paste a function, default is the active cell

Search: searches macros by words or sub-words contained into the name or description. For

example, if you input "div" you list all macros that match words like (div, divisor, division,...)

You can associate more words in AND/OR. Separate words with comma "," for OR, with plus

"+", for AND. For example, if you type "+div +multi" you will get all the rows containing words

like (div, divisor, division,...) and words like (multi, multiprecision,...). On the contrary, if you

type "div, multi", you get all the rows that contain words like (div, divisor, division,...) or also the

words like (multi, multiprecision,...). Remember to choose also the Category and Macro Type.

Example, if you enter the word “hyperbolic”, setting the Category “complex”, you find the

hyperbolic functions restricted to the complex category.

Help: recalls the help-on-line for the selected function.

OK: insert the selected function into the worksheet ". This activates the standard Excel function

wizard panel. If the macro selected is a "sub", the OK button activates the macro.

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Xnumbers Tutorial

Precision

Most functions of this package have an optional parameter - Digit_Max - setting the maximum

number of significant digits for floating point computation, from 1 to 200 (default is 30). The

default can be changed from the menu X-Edit\Defaut Digits

This parameter also determines how the output is automatically formatted. If the result has

fewer integer digits than Digit_Max, then the output is in the plain decimal format

( 123.45, -0.0002364, 4000, etc.), otherwise, if the number of integer digits exceeds the

maximum number of digits allowed (significant or not), the output is automatically converted in

exponential format (1.23456789E+94).

The exponent can reach the extreme values of +/- 2,147,483,.647.

The output format is independent of the input format.

In synthesis, the Digit_Max parameter limits:

The significant digits of internal floating point computation

The maximum number output digits, significant or not.

The default of Digit_Max can be changed from the X-Edit menu . It affects all multiprecision

functions and macros.

Formatting Result

The user can not format an extended number with standard Excel number format tools,

because, it is a string for Excel. You can only change the alignment. To change it you can use

the usual standard Excel format tools.

It is possible to separate the digits of a x-numbers in groups, by the user function xFormat()

and xUnformat()

It work similar at the built-in function Format(x, "#,##0.00")

2,469,135,780,246,913,578 = xformat("2469135780246913578",3)

These functions were original developed by Ton Jeursen for the add-in XNUMBER95, the downgrade version of

XNUMBERS for Excel 5. Because they are very useful for examining long string of number, we have imported them in

this package

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Xnumbers Tutorial

Arithmetic Functions

Addition

xadd(a, b, [Digit_Max])

Performs the addition of two extended numbers: xadd(a, b) = a + b.

Subtraction

xsub(a, b, [Digit_Max])

Performs the subtraction of two extended numbers: xsub(a, b) = a − b.

NB. Do not use the operation xadd(a, -b) if “b” is an extended number. Excel converts “b” into

double, then changes its sign, and finally calls the xadd routine. By this time the original

precision of “b” is lost. If you want to change sign at an extended number and preserve its

precision use the function xneg()

Accuracy lack by subtraction

The subtraction is a critical operation from the point of view of numeric calculus. When the

operands are very near each others, this operation can cause a lack of accuracy. Of course

this can happen for addition when the operands are near and have opposit signs. Let’s see this

example. Assume one performs the following subtraction where the first operand has a

precision of 30 significant digits

(digits)

800000.008209750361424423316366

800000

0.008209750361424423316366

The subtraction is exact (no approximation has been entered). But the final result have 25 total

digits, of wich only 22 are significant. 8 significant digits are lost in this subtraction. We cannot

do anything

about this phenomenon, except to increase the precision of the operands, when

possible.

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Xnumbers Tutorial

Multiplication

xmult(a, b, [Digit_Max])

Performs the multiplication of two extended numbers: xmult(a, b) = a x b.

The product can often lead to long extended numbers. If the result has more integer digits than

the ones set by Digit_Max, then the function automatically converts the result into the

exponential format.

Division

xdiv(a, b, [Digit_Max])

Performs the division of two extended numbers: xdiv(a, b) = a / b.

If b = 0 the function returns “?”. The division can return long extended numbers even when the

operands are small. In the example below we see the well-known periodic division 122 / 7 =

17,428571

…. with 30 significant digits.

Inverse

xinv(x, [Digit_Max])

It returns the inverse of an extended number. If x = 0, the function returns “?”.

xinv(x) = 1 / x

Integer Division

xdivint(a, b)

Returns the quotient of the integer division for a>0 , b>0.

If b = 0 the function returns “?”.

xdivint(a, b)= q , where: a = b*q + r , with 0 ≤ r < b

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Xnumbers Tutorial

Integer Remainder

xdivrem(a, b)

Returns the remainder of the integer division for a>0 , b>0. If b = 0 the function returns “?”.

xdivrem(a, b)= r , where: a = b*q + r , with 0 ≤ r < b

This function is also called "a mod b "

How to test multiprecision functions ?

This test is the most important problem in developing multiprecision arithmetic. This activity,

absorbs almoust the 60% of the totally realization effort.

Apart the first immediate random tests, we can use many known formulas and algorithms. The

general selecting criterions are:

1. Formulas should be iterative

2. Formulas should have many arithmetic elementary operations

3. Final results should be easyly verified

4. Intermediate results should be easily verified

5. Algorithms should be stable

6. Efficency is not important

For example, a good arithmetic test is the Newton algoritm to compute the square root of a

number. The iterative formula:

x x

⋅

⋅ +

⋅

converges to 2

1/2

, starting from x

= 1.

We have rearranged the formula in order to increase the number of operations (remember: the

efficency is not important). In this way we can test mutliplication, division and addition.

1.5

1.41…

1.41421……

Here is a possible Excel arrangement. We have limited the number of the significant digits to

100 only for the picture dimensions, but there is no difficult to repeat it with the maximum digits.

For each iterate only the blu digits are exacts. We see the progressive convergence. By the

way, we note that this algorithm is also very efficent. The rate of convergence is quadratic. The

number of digits approximately doubles at each iteration (In fact this is just the algorithm used

by the xsqr multiprecision function)

But, as said, for testing,

the efficiency has no influence. It is important only that the algorithm

involves the most multiprecision functions as possible.

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Xnumbers Tutorial

Another algorithm quite suitable for testing multiprecision accuracy is the π approximatation by

continuous fraction

Initialize

X = 3

1/2

, Y = 1/2, T = 6

Iteration

X = (2 + X)

1/2

Y = Y/X

T = 2T

P = Y*T*(5Y

/112 + 3Y

/40 + Y

/6 + 1)

Accuracy: approximately 12 decimal digits every 5 iterations)

Below, step by step, a possible Excel arrangement:

The Digit_Max parameter is in the A1 cell. By this parameter we can modulate the arithmetic

accuracy. We have set 30 digits only for the picture dimensions. But you can try with 60, 100 or

more.

Note that, in order to have a more compact form, we have used the xeval function for

calculating the X and P formulas that are inserted into the cells B3 and E3 rispectively.

Selecting the last row (range A6:F6) and dragging it down, we get the following iteration table

The convergence to

is evident.

This version, studied by David Sloan (2003), full of many arithmetic operations, permitted us to detect a very hidden

bug in Xnumbers

Xnumbers Tutorial

Sum

xsum(v, [Digit_Max])

This is the extended version of the Excel built-in function SUM. It returns the sum of a vector of

numbers. The argument is a standard range of cells.

v v v v

...

= + + +

∑

Note that you can not use the standard function SUM, because it recognizes extended

numbers as strings and it excludes them from the calculus.

Product

xprod(v, [Digit_Max])

Returns the product of a vector of numbers.

i i

v v v v

...

= ⋅ ⋅ ⋅

∏

Note that the result is an extended number

even if all the factors are in standard

precision

Raise to power

xpow(x, n, [Digit_Max])

Returns the integer power od an extended number. xpow(x, n) = x

xpow("0.39155749636098981077147016011",90) = 1.9904508921478176508981155284E-7

xpow(5,81,60) = 5

= 413590306276513837435704346034981426782906055450439453125

xpow(122.5 ,1000) = 122.5

1000

= 1.36800819983562905361091390093E+2088

Note the exponent +2088 of the third result. Such kind of numbers can be managed only with

extended precision functions because they are out side of the standard limits for 32bit double

precision.

For not integer power see the exponential functions xexp and xexpa

Xnumbers Tutorial

Square Root

xsqr(x, [Digit_Max])

Returnts the square root of an extended number xsqr(x) =

The example below shows how to compute the

with 30 and 60 significant digits:

xsqr(2) = 1.41421356237309504880168872420969807

xsqr(2, 60) = 1.41421356237309504880168872420969807856967187537694807317667973799

- Root

xroot(x, n, [Digit_Max])

Returns the n

root of an extended number xroot(x, n)=

The root's index must be a positive integer.

The example below shows how to compute the

100

with 30 and 60 significant digits:

xroot(100,9) = 1.66810053720005875359979114908

xroot(100,9,60) = 1.66810053720005875359979114908865584747919268415239470704499

Absolute

xabs(x)

Returns the absolute value of an extended number xabs(x)= |x|

Do not use the built-in function "abs", as Excel converts x in double, then takes the absolute

value but, at that time, the original precision of x is lost.

Change sign

xneg(x)

Returns the opposite of an extended number: xneg(x) = −x

Do not use the operator “−“ (minus) for extended numbers. Otherwise Excel converts the

extended numbers into double and changes its sign but, at that time, the original precision is

lost. In the following example the cell B8 contains an extended number with18 digits. If you use

the “−“ as in the cell B9, you lose the last 3 digits. The function xneg(), as we can see in the cell

B10, preserves the original precision.

Xnumbers Tutorial

Integer part

xint(x)

Returns the integer part of an extended number, that is the greatest integer less than or equal

to x.

Examples:

xint(2.99) = 2

xint(2.14) = 2

xint(-2.14) = −3

xint(-2.99) = −3

xint(12345675.00000001) = 12345675

xint(−12345675.00000001) = −12345676

Decimal part

xdec(x)

Returns the decimal part of an extended number

Examples:

xdec(2.99) = 0.99

xdec(-2.14) = - 0.14

Truncating

xtrunc(x, [dec])

Returns the truncated number; the parameter "dec" sets the number of decimals to keep

(default 0). It works like Excel function TRUNC. “dec” can be negative; in that case x is cut to

the integer number, counting back from the decimal point. See the following examples.

Examples:

number

dec

number truncated

4074861.076055370173

-2

4074800

4074861.076055370173

-1

4074860

4074861.076055370173

4074861

4074861.076055370173

4074861

4074861.076055370173

4074861.07

4074861.076055370173

4074861.076

Rounding

=xround(x, [dec])

Rounds an extended number, the parameter "dec" sets the decimal number of is to keep

(default 0). It works like standard round function. “dec” can be negative, in that case x is

rounded to the integer number, starting to count back from decimal point. See the following

examples.

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