The exp and the expf functions compute the base
.Ms e exponential value of the given argument x.
The exp2 and the exp2f functions compute the base 2 exponential of the given argument x.
The expm1 and the expm1f functions compute the value exp(x)-1 accurately even for tiny argument x.
The log and the logf functions compute the value of the natural logarithm of argument x.
The log10 and the log10f functions compute the value of the logarithm of argument x to base 10.
The log1p and the log1pf functions compute the value of log(1+x) accurately even for tiny argument 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.
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 NaN if x < 0 and 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 NaN will be generated.
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, oo, and NaN . 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 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 expressions meaning and, if invalid, its consequences vary from one computer system to another.
- Some Algebra texts (e.g. Siglers) 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.
- 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
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 oo**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.