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NAME    [Toc]    [Back]

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

LIBRARY    [Toc]    [Back]

     Math Library (libm, -lm)

SYNOPSIS    [Toc]    [Back]

     #include <math.h>

     exp(double x);

     expf(float x);

     expm1(double x);

     expm1f(float x);

     log(double x);

     logf(float x);

     log10(double x);

     log10f(float x);

     log1p(double x);

     log1pf(float x);

     pow(double x, double y);

     powf(float x, float y);

DESCRIPTION    [Toc]    [Back]

     The exp() function computes the exponential value of the given argument

     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 argument

     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.

RETURN VALUES    [Toc]    [Back]

     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.  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 generate a reserved operand fault.  On a VAX, errno is set to EDOM
     and the reserved operand is returned by log unless x > 0, by log1p()
     unless x > -1.

ERRORS    [Toc]    [Back]

     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.

NOTES    [Toc]    [Back]

     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.

     The function 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

     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.

SEE ALSO    [Toc]    [Back]


STANDARDS    [Toc]    [Back]

     The exp(), log(), log10() and pow() functions conform to ANSI X3.159-1989
     (``ANSI C'').

HISTORY    [Toc]    [Back]

     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.

BSD                              July 31, 1991                             BSD
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