4.3BSD - /usr/man/cat3/log.0

EXP(3M)                                                                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.

DIAGNOSTICS
       Exp,  expm1  and pow return the reserved operand on a VAX when the cor‐
       rect 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 pro‐
       vide 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[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.

       (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.

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

AUTHOR
       Kwok-Choi Ng, W. Kahan


4th Berkeley Distribution        May 27, 1986                          EXP(3M)