exp, expm1, log, log10, log1p, pow - exponential, logarithm, power
Exp returns the exponential function of x.
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.
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, expm1 and pow return the reserved operand on a VAX when the correct value would overflow, and they set errno to ERANGE. Pow(x,y)
returns the reserved operand on a VAX and sets errno to EDOM when x < 0 and y is not an integer.
On a VAX, errno is set to EDOM and the reserved operand 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
always:(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., indepen-
dently of x.
Kwok-Choi Ng, W. Kahan
4th Berkeley Distribution May 27, 1986 EXP(3M)