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

N\bNA\bAM\bME\bE
     e\bex\bxp\bp, e\bex\bxp\bpm\bm1\b1, l\blo\bog\bg, l\blo\bog\bg1\b10\b0, l\blo\bog\bg1\b1p\bp, p\bpo\bow\bw - exponential, logarithm, power func-
     tions

S\bSY\bYN\bNO\bOP\bPS\bSI\bIS\bS
     #\b#i\bin\bnc\bcl\blu\bud\bde\be <\b<m\bma\bat\bth\bh.\b.h\bh>\b>

     _\bd_\bo_\bu_\bb_\bl_\be
     e\bex\bxp\bp(_\bd_\bo_\bu_\bb_\bl_\be _\bx);

     _\bd_\bo_\bu_\bb_\bl_\be
     e\bex\bxp\bpm\bm1\b1(_\bd_\bo_\bu_\bb_\bl_\be _\bx);

     _\bd_\bo_\bu_\bb_\bl_\be
     l\blo\bog\bg(_\bd_\bo_\bu_\bb_\bl_\be _\bx);

     _\bd_\bo_\bu_\bb_\bl_\be
     l\blo\bog\bg1\b10\b0(_\bd_\bo_\bu_\bb_\bl_\be _\bx);

     _\bd_\bo_\bu_\bb_\bl_\be
     l\blo\bog\bg1\b1p\bp(_\bd_\bo_\bu_\bb_\bl_\be _\bx);

     _\bd_\bo_\bu_\bb_\bl_\be
     p\bpo\bow\bw(_\bd_\bo_\bu_\bb_\bl_\be _\bx, _\bd_\bo_\bu_\bb_\bl_\be _\by);

D\bDE\bES\bSC\bCR\bRI\bIP\bPT\bTI\bIO\bON\bN
     The e\bex\bxp\bp() function computes the exponential value of the given argument
     _\bx.

     The e\bex\bxp\bpm\bm1\b1() function computes the value exp(x)-1 accurately even for tiny
     argument _\bx.

     The l\blo\bog\bg() function computes the value for the natural logarithm of the
     argument x.

     The l\blo\bog\bg1\b10\b0() function computes the value for the logarithm of argument _\bx
     to base 10.

     The l\blo\bog\bg1\b1p\bp() function computes the value of log(1+x) accurately even for
     tiny argument _\bx.

     The p\bpo\bow\bw() computes the value of _\bx to the exponent _\by.

E\bER\bRR\bRO\bOR\bR (\b(d\bdu\bue\be t\bto\bo R\bRo\bou\bun\bnd\bdo\bof\bff\bf e\bet\btc\bc.\b.)\b)
     exp(x), log(x), expm1(x) and log1p(x) are accurate to within an _\bu_\bp, and
     log10(x) to within about 2 _\bu_\bp_\bs; an _\bu_\bp is one _\bU_\bn_\bi_\bt in the _\bL_\ba_\bs_\bt _\bP_\bl_\ba_\bc_\be. The
     error in p\bpo\bow\bw(_\bx, _\by) is below about 2 _\bu_\bp_\bs when its magnitude is moderate,
     but increases as p\bpo\bow\bw(_\bx, _\by) 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 _\bu_\bp_\bs for VAX D, 300 _\bu_\bp_\bs for IEEE 754
     Double.  Moderate values of p\bpo\bow\bw() are accurate enough that p\bpo\bow\bw(_\bi_\bn_\bt_\be_\bg_\be_\br,
     _\bi_\bn_\bt_\be_\bg_\be_\br) is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE
     754.

R\bRE\bET\bTU\bUR\bRN\bN V\bVA\bAL\bLU\bUE\bES\bS
     These functions will return the approprate computation unless an error
     occurs or an argument is out of range.  The functions e\bex\bxp\bp(), e\bex\bxp\bpm\bm1\b1() and
     p\bpo\bow\bw() detect if the computed value will overflow, set the global variable
     _\be_\br_\br_\bn_\bo _\bt_\bo RANGE and cause a reserved operand fault on a VAX or Tahoe. The
     function p\bpo\bow\bw(_\bx, _\by) checks to see if _\bx < 0 and _\by is not an integer, in the
     event this is true, the global variable _\be_\br_\br_\bn_\bo is set to EDOM and on the
     VAX and Tahoe generate a reserved operand fault.  On a VAX and Tahoe,
     _\be_\br_\br_\bn_\bo is set to EDOM and the reserved operand is returned by log unless _\bx
     > 0, by l\blo\bog\bg1\b1p\bp() unless _\bx > -1.

N\bNO\bOT\bTE\bES\bS
     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 p\bpo\bow\bw(_\bx, _\b0) returns x**0 = 1 for all x including x = 0, Infin-
     ity (not found on a VAX), and _\bN_\ba_\bN (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
             _\bN_\ba_\bN) 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 _\ba_\bn_\by 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 _\bN_\ba_\bN**0 =
             1 too because x**0 = 1 for all finite and infinite x, i.e., inde-
             pendently of x.

S\bSE\bEE\bE A\bAL\bLS\bSO\bO
     math(3),  infnan(3)

H\bHI\bIS\bST\bTO\bOR\bRY\bY
     A e\bex\bxp\bp(), l\blo\bog\bg() and p\bpo\bow\bw() function appeared in Version 6 AT&T UNIX.  A
     l\blo\bog\bg1\b10\b0() function appeared in Version 7 AT&T UNIX.  The l\blo\bog\bg1\b1p\bp() and
     e\bex\bxp\bpm\bm1\b1() functions appeared in 4.3BSD.

4th Berkeley Distribution        June 4, 1993                                2