EXP(3)                    OpenBSD Programmer's Manual                   EXP(3)

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



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() function computes the exponential value of the given argument
     x.

     The expm1() function computes the value exp(x)-1 accurately even for tiny
     argument x.

     The log() function computes the value of the natural logarithm of argu-
     ment x.

     The log10() function computes the value of the logarithm of argument x to
     base 10.

     The log1p() function computes the value of log(1+x) accurately even for
     tiny argument x.

     The pow() computes the value of x to the exponent 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 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 exp(), expm1() and
     pow() detect if the computed value will overflow, set the global variable
     errno to ERANGE and cause a reserved operand fault on a VAX or Tahoe. The
     function pow(x, y) checks to see if x < 0 and y is not an integer, in the
     event this is true, the global variable errno is set to EDOM and on the
     VAX and Tahoe generate a reserved operand fault.  On a VAX and Tahoe,
     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 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).
     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 al-
     ways:

     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, in-
             cluding x = 0.  This is compatible with the convention that ac-
             cepts 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., inde-
             pendently of x.

SEE ALSO
     infnan(3),  math(3)

HISTORY
     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.

OpenBSD 2.6                      July 31, 1991                               3