Calculate exponential function with Taylor series in SQL Server

Introduction

In mathematics, the exponential function is the function e^x, where e is the number (approximately 2.718281828) such that the function e^x is its own derivative. The exponential function can be characterized in many ways, one of the most common characterizations is with the infinite Taylor series.

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18^th century.

I implement the exponential function in two different ways, one with recursion and the other without, I used Common Table Expression; for more details about using and implementing CTE check this nice article: CTE in SQL Servers.

Implement with CTE 

To find out what I do in my code, as the picture shows I need to calculate the factorial of digits and power of them, I create a table with an identity row by cte:

As default number of recursion in cte is 100, I limit the identity number to 100, so the series continues 100 times.

;WITH CTE AS
(SELECT cast(1.0 as float) AS rowNumber
 UNION ALL
 SELECT cast(cast(rowNumber as float)+1 as float)
 from CTE
 WHERE rowNumber < 100)
 select *
 FROM CTE

And the result is numbers 1 to 100:

Then for each number I need its factorial. For this I add an additional column, and it is the result of multiplication with the last rows for each row:

;WITH CTE AS
(SELECT cast(1.0 as float) AS i, 
        cast(1.0 as float) AS factorial
 UNION ALL
 SELECT cast(cast(i as float)+1 as float),
        cast(cast(factorial as float)*(cast(i as float)+1)as float)
 FROM CTE
 WHERE  i < 100) 
select * from CTE

And the result is as below:

At last for each row I want the power of rowNumber for x and I use the power function.

As first the row which is created is the result of @x ^1. In the second row I start the power with rowNumber +1.

    declare @x  float = 3
;WITH CTE AS
(SELECT cast(1.0 as float) AS rowNumber, 
        cast(1.0 as float) AS factorial,
    cast(@x as float)as [power]

 UNION ALL
 SELECT cast(cast(rowNumber as float)+1 as float),
        cast(cast(factorial as float)*(cast(rowNumber as float)+1)as float),
    cast(power(cast(@x as float),cast(rowNumber+1 as float)) as float)
 FROM CTE
 WHERE  rowNumber < 100) 
select * from CTE

And the result is as below:

At last I added the division of power to the factorial in each row with others by dynamic concatenation:

create FUNCTION dbo.EpowerX(@x int)
RETURNS FLOAT
AS
BEGIN
declare @Result float
;WITH N AS
(SELECT cast(1.0 as float) AS i, 
        cast(1.0 as float) AS f,
        cast(@x as float)as g
 UNION ALL
 SELECT cast(cast(i as float)+1 as float),
        cast(cast(f as float)*(cast(i as float)+1)as float),
        cast(power(cast(@x as float),cast(i+1 as float)) as float)
 FROM N
 WHERE  i < 100) 
select @Result=isnull(@Result,cast(@x as float))+ cast(g as float)/cast(f as float) 
from N
where i !=1
OPTION (MAXRECURSION 0);
RETURN @Result + 1
END 

select cast(cast(dbo.EpowerX(30) as float) as decimal(38,3)) as myex
select cast(cast(EXP(30)as float) as decimal(38,3)) sqlexp

And the result is :