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log(3)

NAME

     exp, expf, expm1, expm1f, log, logf, log10, log10f, log1p, log1pf, pow,
     powf -- exponential, logarithm, power functions

LIBRARY

     Math Library (libm, -lm)

SYNOPSIS

     #include <math.h>

     double
     exp(double x);

     float
     expf(float x);

     double
     expm1(double x);

     float
     expm1f(float x);

     double
     log(double x);

     float
     logf(float x);

     double
     log10(double x);

     float
     log10f(float x);

     double
     log1p(double x);

     float
     log1pf(float x);

     double
     pow(double x, double y);

     float
     powf(float x, float y);

DESCRIPTION

     The exp() and the expf() functions compute the exponential value of the
     given argument x.

     The expm1() and the expm1f() functions compute the value exp(x)-1 accu-
     rately even for tiny argument x.

     The log() and the logf() functions compute the value of the natural loga-
     rithm of argument x.

     The log10() and the log10f() functions compute the value of the logarithm

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 moder-
     ate, but increases as pow(x, y) approaches the over/underflow thresholds
     until almost as many bits could be lost as are occupied by the float-
     ing-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.

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 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.

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 Pas-
     cal, exp1 and log1 in C on APPLE Macintoshes, where they have been pro-
     vided 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, infin-
     ity (not found on a VAX), and NaN (the reserved operand on a VAX).  Pre-
     vious 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 any-
	     way since that expression's meaning and, if invalid, its conse-
	     quences 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[0] as the value of polynomial

		   p(x) = a[0]*x**0 + a[1]*x**1 + a[2]*x**2 +...+ a[n]*x**n

	     at x = 0 rather than reject a[0]*0**0 as invalid.

     3.      Analysts will accept 0**0 = 1 despite that x**y can approach any-
	     thing 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.

     A exp(), log() and pow() functions appeared in Version 6 AT&T UNIX.  A
     log10() function appeared in Version 7 AT&T UNIX.	The log1p() and
     expm1() functions appeared in 4.3BSD.

FreeBSD 5.4		       January 14, 2005 		   FreeBSD 5.4

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