Power
Chapter I  integer exponent
PROBLEM
A frequently occuring operation in numbertheoretic computations is raising a number to a power another number, also known as exponentiation. We would like an efficient way to compute X**Z (X to the Z power), where X is a real number and Z is a nonnegative integer. Usually we use the ** operator for calculation of the power function,
raising a real number to a integer exponent. Another way to this calculation is
the following algorithm POWER stated by alKashi about 1414 C.E.
IMPLEMENTATION
Unit: internal function, external function without procedure statement
Parameters: real X, nonnegative integer Z
Returns: X**Z
POWER: procedure
parse arg X, Z; Pwr = 1
do forever
if Z // 2 then Pwr = Pwr * X
Z = Z % 2
if Z = 0 then return Pwr
X = X * X
end

EXAMPLE
[Regina] The following program ends with Error 26 ... line 3: Invalid whole number
/* BERNOULLI computes e */
numeric digits 20; Z = 1E+9
X = 1 + 1 / Z
say "e =" FORMAT(X**Z,,8)

But program
/* BERNOULLI computes e */
numeric digits 20; Z = 1E+9
X = 1 + 1 / Z
say "e =" FORMAT(POWER(X,Z),,8)
exit
POWER: procedure
...

displays e = 2.71828183
Chapter II  real exponent
The following function REPOWER computes the result by raising the
real number X to the given real exponent Z.
IMPLEMENTATION
Unit: internal function
Parameters: real numbers X and Z, a positive integer P  number of significant digits of result, default is 9
Interface: the functions EXP and LN
Returns: X**Z or writes error message REAL_POWER: Result is undefined
REPOWER: procedure
parse arg X, Z, P
if P = "" then P = 9; P = P + 4; numeric digits P
if DATATYPE(Z, "Whole_number")
then
if X = 0 & Z <= 0
then call ERROR "REPOWER: Result is undefined"
else return X ** Z
else
select
when X > 0 then return EXP(Z * LN(X, P), P)
when X = 0 & Z > 0 then return 0
otherwise
call ERROR "REPOWER: Result is undefined"
end
ERROR: say ARG(1); exit

CONNECTIONS Exponential function
Natural logarithms
Select computation of S = 0.5 * N**(2/3)
Technique: Bit array encoded as decimal number
Literature
Knuth D. E. Seminumerical Algorithms, vol. 2 of The Art of Computer Programming AddisonWesley, 1973
