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, 11 bits for
          IEEE 754 Double.  No such drastic loss has been exposed by
          testing; the worst errors observed have been below 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**53 for IEEE 754.




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     EXP(3M)                                                   EXP(3M)



     DIAGNOSTICS
          Exp, expm1 and pow return infinity when the correct value
          would overflow.

     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.

          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(3M)

     AUTHOR
          Kwok-Choi Ng, W. Kahan




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     EXP(3M)                                                   EXP(3M)



     BUGS
          log(x) returns log(1 x 1) if x<0.

     ORIGIN
          MIPS Computer Systems


















































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