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EXP(3)			 BSD Library Functions Manual			EXP(3)

NAME
     exp, expf, exp2, exp2f, expm1, expm1f, log, logf, log2, log2f, log10,
     log10f, log1p, log1pf, pow, powf — exponential, logarithm, power func‐
     tions

LIBRARY
     Math Library (libm, -lm)

SYNOPSIS
     #include <math.h>

     double
     exp(double x);

     float
     expf(float x);

     double
     exp2(double x);

     float
     exp2f(float x);

     double
     expm1(double x);

     float
     expm1f(float x);

     double
     log(double x);

     float
     logf(float x);

     double
     log2(double x);

     float
     log2f(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 base e exponential value
     of the given argument x.

     The exp2(), and exp2f() functions compute the base 2 exponential of the
     given argument x.

     The expm1() and the expm1f() functions computes the value exp(x)-1 accu‐
     rately 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 log2() and the log2f() functions compute the value of the logarithm
     of argument x to base 2.

     The pow() and powf() functions compute the value of x to the exponent y.

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.  The func‐
     tion 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
     The values of exp(x), expm1(x), exp2(x), log(x), and log1p(x), are exact
     provided that they are representable.  Otherwise the error in these func‐
     tions is generally below one ulp.	The values of log10(x) are 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 expo‐
     nent 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.  Moder‐
     ate values of pow(x, y) 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
     The functions exp(x - 1) and log(1 + x) are called expm1(x) and logp1(x)
     in BASIC on the Hewlett-Packard HP-71B and APPLE Macintosh, EXP1 and LN1
     in Pascal, exp1(x) and log1(x) 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, Infin‐
     ity (not found on a VAX), and NaN (the reserved operand on a VAX).	 Pre‐
     vious 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 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,
	     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., inde‐
	     pendently of x.

SEE ALSO
     math(3)

STANDARDS
     The exp(), log(), log10() and pow() functions conform to ANSI X3.159-1989
     (“ANSI C89”).  The exp2(), exp2f(), expf(), expm1(), expm1f(), log1p(),
     log1pf(), log2(), log2f(), log10f(), logf(), and powf() functions conform
     to ISO/IEC 9899:1999 (“ISO C99”).

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.

BSD				  May 3, 2010				   BSD
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