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  ANSI Common Lisp   12 Numbers   12.2 Dictionary of Numbers

12.2.31 exp, expt Function

exp number    result

expt base-number power-number    result

Arguments and Values:
number - a number.

base-number - a number.

power-number - a number.

result - a number.

exp and expt perform exponentiation.

exp returns e raised to the power number, where e is the base of the natural logarithms. exp has no branch cut.

expt returns base-number raised to the power power-number. If the base-number is a rational and power-number is an integer, the calculation is exact and the result will be of type rational; otherwise a floating-point approximation might result. For expt of a complex rational to an integer power, the calculation must be exact and the result is of type (or rational (complex rational)).

The result of expt can be a complex, even when neither argument is a complex, if base-number is negative and power-number is not an integer. The result is always the principal complex value. For example, (expt -8 1/3) is not permitted to return -2, even though -2 is one of the cube roots of -8. The principal cube root is a complex approximately equal to #C(1.0 1.73205), not -2.

expt is defined as bx = ex log b . This defines the principal values precisely. The range of expt is the entire complex plane. Regarded as a function of x, with b fixed, there is no branch cut. Regarded as a function of b, with x fixed, there is in general a branch cut along the negative real axis, continuous with quadrant II. The domain excludes the origin. By definition, 00=1. If b=0 and the real part of x is strictly positive, then bx=0. For all other values of x, 0x is an error.

When power-number is an integer 0, then the result is always the value one in the type of base-number, even if the base-number is zero (of any type). That is:

 (expt x 0) ==(coerce 1 (type-of x))
If power-number is a zero of any other type, then the result is also the value one, in the type of the arguments after the application of the contagion rules in Section Contagion in Numeric Operations, with one exception: the consequences are undefined if base-number is zero when power-number is zero and not of type integer.

 (exp 0)  1.0
 (exp 1)  2.718282
 (exp (log 5))  5.0 
 (expt 2 8)  256
 (expt 4 .5)  2.0
 (expt #c(0 1) 2)  -1
 (expt #c(2 2) 3)  #C(-16 16)
 (expt #c(2 2) 4)  -64 

See Also:
log, Section Rule of Float Substitutability

Implementations of expt are permitted to use different algorithms for the cases of a power-number of type rational and a power-number of type float.

Note that by the following logic, (sqrt (expt x 3)) is not equivalent to (expt x 3/2).

 (setq x (exp (/ (* 2 pi #c(0 1)) 3)))         ;exp(2.pi.i/3)
 (expt x 3)  1 ;except for round-off error
 (sqrt (expt x 3))  1 ;except for round-off error
 (expt x 3/2)  -1 ;except for round-off error

Allegro CL Implementation Details:

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