exp, expm1, log, log10, log1p, pow - exponential, logarithm, power
Exp returns the exponential function of x.
Exp10 returns the base 10 exponential 10x.
Exp2 returns the base 2 exponential 2x.
Expm1 returns exp(x)-1 accurately even for tiny x.
Log returns the natural logarithm of x.
Log10 returns the logarithm of x to base 10.
Log1p returns log(1+x) accurately even for tiny x.
Log2 returns the logarithm of x to base 2.
Pow(x,y) returns x**y.
ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within an
ulp, and log10(x) to within about 2 ulps; an ulp is one Unit in the
Last Place. The error in pow(x,y) is below about 2 ulps when its
magnitude is moderate, but increases as pow(x,y) approaches the
over/underflow thresholds until almost as many bits could be lost
as are occupied by the floating-point format’s exponent field; that
is 8 bits for VAX D and 11 bits for IEEE 754 Double. No such
drastic loss has been exposed by testing; the worst errors observed
have been below 20 ulps for VAX D, 300 ulps for IEEE 754 Double.
Moderate values of pow are accurate enough that
pow(integer,integer) is exact until it is bigger than 2**56 on a
VAX, 2**53 for IEEE 754.
Exp, exp10, exp2, expm1 and pow return +infinity on an overflow.
Pow(x,y) returns NaN when x < 0 and y is not an integer.
NaN is returned by log unless x > 0, by log1p unless x > -1.
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.
Pow(x,0) returns x**0 = 1 for
all x including x = 0, Infinity (not
found on a VAX), and NaN (the reserved operand on a VAX). 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) Any program that already
tests whether x is zero (or infinite
or NaN) 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.
(2) 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 as the value of polynomial
p(x) = a*x**0 + a*x**1 + a*x**2 +...+ a[n]*x**n
at x = 0 rather than reject a*0**0 as invalid.
(3) 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.
(4) If 0**0 = 1, then
infinity**0 = 1/0**0 = 1 too; and then NaN**0
= 1 too because x**0 = 1 for all finite and infinite x, i.e.,
independently of x.
Kwok-Choi Ng, W. Kahan