Implement x^(e) in Extended Pascal, without using exponentiation operators - pascal

Question

Part of a Pascal ISO 10206 program I am building, requires me to implement a function to exponentiate a real number (x) to Eulers number (e), without using any exponentiation functions already included in Pascal(**,pow,exp...).
I have been trying different algorithms for hours but I cant figure out how to do it without using the already included exponentiation functions.
Any help would be appreciated. Any mathematical algorithm of some sort etc... Thanks in advance.

Answer 1

As others have said, it doesn't make sense not to use Exp() or a function based upon it. But if you really must use something else, and don't want to get too technical/mathematical, then the following should work (the real algorithm is far more complicated and requires much more knowledge of math).
The program uses a combination of the first N terms of Taylor series for the fraction of X and binary exponentiation for the integer part of X. It is probably not very fast, but pretty accurate, even for larger exponents. For comparison, I also display Exp(X). If your Pascal has a Double or Extended type, use those instead of Real.
program SimplePower;
{ Required for Delphi, you can omit it in other Pascals: }
{$APPTYPE CONSOLE}
{ Returns approximate value of e^X using sum of first N terms of Taylor series.
Works fine with X values between 0 and 1.0 and N ~ 30. }
function Exponential(N: Integer; X: Real): Real;
var
I: Integer;
begin
Result := 1.0;
for I := N - 1 downto 1 do
Result := 1.0 + X * Result / I;
end;
{ Binary exponentiation of Base by integer Exponent. }
function IntegerExp(Base: Real; Exponent: Integer): Real;
begin
Result := 1.0;
while Exponent > 0 do
begin
if Odd(Exponent) then
Result := Result * Base;
Base := Base * Base;
Exponent := Exponent shr 1;
end;
end;
{ Combines IntegerExp function for integral part with
Exponential function for fractional part. }
function MyExp(N: Integer; X: Real): Real;
const
E = 2.7182818284590452353602874713527; { from Google: "e euler" }
var
Factor: Real;
Fraction: Real;
begin
Fraction := Exponential(N, Frac(X));
Factor := IntegerExp(E, Trunc(X));
Result := Factor * Fraction;
end;
{ Simple demo: }
const
N = 30;
X = 73.4567890242421234;
begin
Writeln('MyExp(', N, ', ', X:22:18, ') = ', MyExp(N, X):22:18);
Writeln('Exp(', X:22:18, ') = ', Exp(X):22:18);
end.
Ref:
Taylor series for exponentiation
Binary exponentiation
I did not do anything for negative exponents, but just know that Exp(-x) = 1/Exp(x). You could amend MyExpwith that knowledge.

Answer 2

I used the solution pointed out by #RudyVelthuis in some other post, but modified it a bit. It is based upon that x^0.5 = sqrt(x), which we can use to our advantage. Ill leave the Pascal ISO 10206 code I used attached. Thank you all for your help, specially Rudy.
function MyPow(base,power,precision:real):real;
begin
if (power<0) then MyPow:=1/MyPow(base,-power,precision)
else if (power>=10) then MyPow:=sqr(MyPow(base,power/2,precision/2))
else if (power>=1) then MyPow:=base*MyPow(base,power-1,precision)
else if (precision>=1) then MyPow:=sqrt(base)
else MyPow:=sqrt(MyPow(base,power*2,precision*2));
end;