The
exp ();
and the
expf ();
functions compute the base
e
exponential value of the given argument
Fa x .
The
exp2 ();
and the
exp2f ();
functions compute the base 2 exponential of the given argument
Fa x .
The
expm1 ();
and the
expm1f ();
functions compute the value exp(x)-1 accurately even for tiny argument
Fa x .
The
log ();
and the
logf ();
functions compute the value of the natural logarithm of argument
Fa x .
The
log10 ();
and the
log10f ();
functions compute the value of the logarithm of argument
Fa x
to base 10.
The
log1p ();
and the
log1pf ();
functions compute
the value of log(1+x) accurately even for tiny argument
Fa x .
The
pow ();
and the
powf ();
functions compute the value
of
x
to the exponent
y
ERROR (due to Roundoff etc.)
The values of
exp (0 ,);
expm1 (0 ,);
exp2 (integer ,);
and
pow (integer integer);
are exact provided that they are representable.
Otherwise the error in these functions is generally below one
ulp
RETURN VALUES
These functions will return the appropriate computation unless an error
occurs or an argument is out of range.
The functions
pow (x y);
and
powf (x y);
raise an invalid exception and return an if
Fa x
< 0 and
Fa y
is not an integer.
An attempt to take the logarithm of 0 will result in
a divide-by-zero exception, and an infinity will be returned.
An attempt to take the logarithm of a negative number will
result in an invalid exception, and an will be generated.
NOTES
The functions exp(x)-1 and log(1+x) are called
expm1 and logp1 in
BASIC
on the Hewlett-Packard
HP -71B
and
APPLE
Macintosh,
EXP1
and
LN1
in Pascal, exp1 and log1 in C
on
APPLE
Macintoshes, where they have been provided to make
sure financial calculations of ((1+x)**n-1)/x, namely
expm1(n*log1p(x))/x, will be accurate when x is tiny.
They also provide accurate inverse hyperbolic functions.
The function
pow (x 0);
returns x**0 = 1 for all x including x = 0, , and .
Previous implementations of pow may
have defined x**0 to be undefined in some or all of these
cases.
Here are reasons for returning x**0 = 1 always:
Any program that already tests whether x is zero (or
infinite or ) before computing x**0 cannot care
whether 0**0 = 1 or not.
Any program that depends
upon 0**0 to be invalid is dubious anyway since that
expression's meaning and, if invalid, its consequences
vary from one computer system to another.
Some Algebra texts (e.g. Sigler's) define x**0 = 1 for
all x, including x = 0.
This is compatible with the convention that accepts a[0]
as the value of polynomial
Analysts will accept 0**0 = 1 despite that x**y can
approach anything or nothing as x and y approach 0
independently.
The reason for setting 0**0 = 1 anyway is this:
If x(z) and y(z) are
any
functions analytic (expandable
in power series) in z around z = 0, and if there
x(0) = y(0) = 0, then x(z)**y(z) -> 1 as z -> 0.
If 0**0 = 1, then
**0 = 1/0**0 = 1 too; and
then **0 = 1 too because x**0 = 1 for all finite
and infinite x, i.e., independently of x.
