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EXP(3)                 FreeBSD Library Functions Manual                 EXP(3)

NAME
     exp, expf, exp10, exp10f, expm1, expm1f, log, logf, log10, log10f, log1p,
     log1pf, pow, powf - exponential, logarithm, power functions

LIBRARY
     Math Library (libm, -lm)

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() and the expf() functions compute the exponential value of the
     given argument x.

     The expm1() and the expm1f() functions compute the value exp(x)-1
     accurately even for tiny argument x.

     The log() and the logf() functions compute the value of the natural
     logarithm of argument x.

     The log10() and the log10f() functions compute the value of the logarithm
     of argument x to base 10.

     The log1p() and the log1pf() functions compute the value of log(1+x)
     accurately even for tiny argument x.

     The pow() and the powf() functions compute 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
     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.

RETURN VALUES
     These functions will return the appropriate computation unless an error
     occurs or an argument is out of range.  The functions exp(), expm1(),
     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 functions 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
     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
     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.

SEE ALSO
     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.

FreeBSD 11.0-PRERELEASE          July 31, 1991         FreeBSD 11.0-PRERELEASE

NAME | LIBRARY | SYNOPSIS | DESCRIPTION | ERROR (due to Roundoff etc.) | RETURN VALUES | NOTES | SEE ALSO | HISTORY

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