4.4BSD/usr/share/man/cat3/expm1.0

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

NNAAMMEE
     eexxpp, eexxppmm11, lloogg, lloogg1100, lloogg11pp, ppooww - exponential, logarithm, power func-
     tions

SSYYNNOOPPSSIISS
     ##iinncclluuddee <<mmaatthh..hh>>

     _d_o_u_b_l_e
     eexxpp(_d_o_u_b_l_e _x);

     _d_o_u_b_l_e
     eexxppmm11(_d_o_u_b_l_e _x);

     _d_o_u_b_l_e
     lloogg(_d_o_u_b_l_e _x);

     _d_o_u_b_l_e
     lloogg1100(_d_o_u_b_l_e _x);

     _d_o_u_b_l_e
     lloogg11pp(_d_o_u_b_l_e _x);

     _d_o_u_b_l_e
     ppooww(_d_o_u_b_l_e _x, _d_o_u_b_l_e _y);

DDEESSCCRRIIPPTTIIOONN
     The eexxpp() function computes the exponential value of the given argument
     _x.

     The eexxppmm11() function computes the value exp(x)-1 accurately even for tiny
     argument _x.

     The lloogg() function computes the value for the natural logarithm of the
     argument x.

     The lloogg1100() function computes the value for the logarithm of argument _x
     to base 10.

     The lloogg11pp() function computes the value of log(1+x) accurately even for
     tiny argument _x.

     The ppooww() computes the value of _x to the exponent _y.

EERRRROORR ((dduuee ttoo RRoouunnddooffff eettcc..))
     exp(x), log(x), expm1(x) and log1p(x) are accurate to within an _u_p, and
     log10(x) to within about 2 _u_p_s; an _u_p is one _U_n_i_t in the _L_a_s_t _P_l_a_c_e. The
     error in ppooww(_x, _y) is below about 2 _u_p_s when its magnitude is moderate,
     but increases as ppooww(_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 _u_p_s for VAX D, 300 _u_p_s for IEEE 754
     Double.  Moderate values of ppooww() are accurate enough that ppooww(_i_n_t_e_g_e_r,
     _i_n_t_e_g_e_r) is exact until it is bigger than 2**56 on a VAX, 2**53 for IEEE
     754.

RREETTUURRNN VVAALLUUEESS
     These functions will return the approprate computation unless an error
     occurs or an argument is out of range.  The functions eexxpp(), eexxppmm11() and
     ppooww() detect if the computed value will overflow, set the global variable
     _e_r_r_n_o _t_o RANGE and cause a reserved operand fault on a VAX or Tahoe. The
     function ppooww(_x, _y) checks to see if _x < 0 and _y is not an integer, in the
     event this is true, the global variable _e_r_r_n_o is set to EDOM and on the
     VAX and Tahoe generate a reserved operand fault.  On a VAX and Tahoe,
     _e_r_r_n_o is set to EDOM and the reserved operand is returned by log unless _x
     > 0, by lloogg11pp() unless _x > -1.

NNOOTTEESS
     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 ppooww(_x, _0) returns x**0 = 1 for all x including x = 0, Infin-
     ity (not found on a VAX), and _N_a_N (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
             _N_a_N) 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 _a_n_y 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 _N_a_N**0 =
             1 too because x**0 = 1 for all finite and infinite x, i.e., inde-
             pendently of x.

SSEEEE AALLSSOO
     math(3),  infnan(3)

HHIISSTTOORRYY
     A eexxpp(), lloogg() and ppooww() function appeared in Version 6 AT&T UNIX.  A
     lloogg1100() function appeared in Version 7 AT&T UNIX.  The lloogg11pp() and
     eexxppmm11() functions appeared in 4.3BSD.

4th Berkeley Distribution        June 4, 1993                                2