2.11BSD#431 - /usr/man/cat3/log.0

EXP(3M)             UNIX Programmer's Manual		  EXP(3M)


NAME
     exp, expm1, log, log10, log1p, pow - exponential, logarithm,
     power

SYNOPSIS
     #include <math.h>

     double exp(x)
     double x;

     double expm1(x)
     double x;

     double log(x)
     double x;

     double log10(x)
     double x;

     double log1p(x)
     double x;

     double pow(x,y)
     double x,y;

DESCRIPTION
     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.


Printed 11/26/99	  May 27, 1986				1


EXP(3M)             UNIX Programmer's Manual		  EXP(3M)


DIAGNOSTICS
     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.

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.

     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).  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[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 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.


Printed 11/26/99	  May 27, 1986				2


EXP(3M)             UNIX Programmer's Manual		  EXP(3M)


SEE ALSO
     math(3M), infnan(3M)

AUTHOR
     Kwok-Choi Ng, W. Kahan


Printed 11/26/99	  May 27, 1986				3