# 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

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