# 2.11BSD#431 - /usr/man/cat3/math.0

    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

    NAME
         math - introduction to mathematical library functions

    DESCRIPTION
         These functions constitute the C math library, libm. The
         link editor searches this library under the "-lm" option.
         Declarations for these functions may be obtained from the
         include file <math.h>.  The Fortran math library is
         described in ``man 3f intro''.

    LIST OF FUNCTIONS
         Name      Appears on Page	  Description		    Error Bound (ULPs)
         acos	 sin.3m       inverse trigonometric function	 3
         acosh	 asinh.3m     inverse hyperbolic function	 3
         asin	 sin.3m       inverse trigonometric function	 3
         asinh	 asinh.3m     inverse hyperbolic function	 3
         atan	 sin.3m       inverse trigonometric function	 1
         atanh	 asinh.3m     inverse hyperbolic function	 3
         atan2	 sin.3m       inverse trigonometric function	 2
         cabs	 hypot.3m     complex absolute value		 1
         cbrt	 sqrt.3m      cube root                          1
         ceil	 floor.3m     integer no less than		 0
         copysign	 ieee.3m      copy sign bit			 0
         cos	 sin.3m       trigonometric function		 1
         cosh	 sinh.3m      hyperbolic function		 3
         drem	 ieee.3m      remainder                          0
         erf	 erf.3m       error function			???
         erfc	 erf.3m       complementary error function	???
         exp	 exp.3m       exponential			 1
         expm1	 exp.3m       exp(x)-1				 1
         fabs	 floor.3m     absolute value			 0
         floor	 floor.3m     integer no greater than		 0
         hypot	 hypot.3m     Euclidean distance		 1
         infnan	 infnan.3m    signals exceptions
         j0          j0.3m	      bessel function			???
         j1          j0.3m	      bessel function			???
         jn          j0.3m	      bessel function			???
         lgamma	 lgamma.3m    log gamma function; (formerly gamma.3m)
         log	 exp.3m       natural logarithm                  1
         logb	 ieee.3m      exponent extraction		 0
         log10	 exp.3m       logarithm to base 10		 3
         log1p	 exp.3m       log(1+x)				 1
         pow	 exp.3m       exponential x**y		       60-500
         rint	 floor.3m     round to nearest integer		 0
         scalb	 ieee.3m      exponent adjustment		 0
         sin	 sin.3m       trigonometric function		 1
         sinh	 sinh.3m      hyperbolic function		 3
         sqrt	 sqrt.3m      square root			 1
         tan	 sin.3m       trigonometric function		 3
         tanh	 sinh.3m      hyperbolic function		 3
         y0          j0.3m	      bessel function			???
         y1          j0.3m	      bessel function			???

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

         yn          j0.3m	      bessel function			???

    NOTES
         In 4.3 BSD, distributed from the University of California in
         late 1985, most of the foregoing functions come in two ver-
         sions, one for the double-precision "D" format in the DEC
         VAX-11 family of computers, another for double-precision
         arithmetic conforming to the IEEE Standard 754 for Binary
         Floating-Point Arithmetic.  The two versions behave very
         similarly, as should be expected from programs more accurate
         and robust than was the norm when UNIX was born.  For
         instance, the programs are accurate to within the numbers of
         ulps tabulated above; an ulp is one Unit in the Last Place.
         And the programs have been cured of anomalies that afflicted
         the older math library libm in which incidents like the fol-
         lowing had been reported:
    	  sqrt(-1.0) = 0.0 and log(-1.0) = -1.7e38.
    	  cos(1.0e-11) > cos(0.0) > 1.0.
    	  pow(x,1.0) != x when x = 2.0, 3.0, 4.0, ..., 9.0.
    	  pow(-1.0,1.0e10) trapped on Integer Overflow.
    	  sqrt(1.0e30) and sqrt(1.0e-30) were very slow.
         However the two versions do differ in ways that have to be
         explained, to which end the following notes are provided.

         DEC VAX-11 D_floating-point:

         This is the format for which the original math library libm
         was developed, and to which this manual is still principally
         dedicated.  It is the double-precision format for the PDP-11
         and the earlier VAX-11 machines; VAX-11s after 1983 were
         provided with an optional "G" format closer to the IEEE
         double-precision format.  The earlier DEC MicroVAXs have no
         D format, only G double-precision. (Why?  Why not?)

         Properties of D_floating-point:
    	  Wordsize: 64 bits, 8 bytes.  Radix: Binary.
    	  Precision: 56 sig.  bits, roughly like 17 sig.
    	  decimals.
    	       If x and x' are consecutive positive
    	       D_floating-point numbers (they differ by 1 ulp),
    	       then
    	       1.3e-17 < 0.5**56 < (x'-x)/x < 0.5**55 < 2.8e-17.
    	  Range: Overflow threshold  = 2.0**127 = 1.7e38.
    		 Underflow threshold = 0.5**128 = 2.9e-39.
    		 NOTE:	THIS RANGE IS COMPARATIVELY NARROW.
    	       Overflow customarily stops computation.
    	       Underflow is customarily flushed quietly to zero.
    	       CAUTION:
    		    It is possible to have x != y and yet x-y = 0
    		    because of underflow.  Similarly x > y > 0
    		    cannot prevent either x*y = 0 or  y/x = 0
    		    from happening without warning.

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

    	  Zero is represented ambiguously.
    	       Although 2**55 different representations of zero
    	       are accepted by the hardware, only the obvious
    	       representation is ever produced.  There is no -0
    	       on a VAX.
    	  Infinity is not part of the VAX architecture.
    	  Reserved operands:
    	       of the 2**55 that the hardware recognizes, only
    	       one of them is ever produced.  Any floating-point
    	       operation upon a reserved operand, even a MOVF or
    	       MOVD, customarily stops computation, so they are
    	       not much used.
    	  Exceptions:
    	       Divisions by zero and operations that overflow are
    	       invalid operations that customarily stop computa-
    	       tion or, in earlier machines, produce reserved
    	       operands that will stop computation.
    	  Rounding:
    	       Every rational operation  (+, -, *, /) on a VAX
    	       (but not necessarily on a PDP-11), if not an
    	       over/underflow nor division by zero, is rounded to
    	       within half an ulp, and when the rounding error is
    	       exactly half an ulp then rounding is away from 0.

         Except for its narrow range, D_floating-point is one of the
         better computer arithmetics designed in the 1960's.  Its
         properties are reflected fairly faithfully in the elementary
         functions for a VAX distributed in 4.3 BSD.  They
         over/underflow only if their results have to lie out of
         range or very nearly so, and then they behave much as any
         rational arithmetic operation that over/underflowed would
         behave.  Similarly, expressions like log(0) and atanh(1)
         behave like 1/0; and sqrt(-3) and acos(3) behave like 0/0;
         they all produce reserved operands and/or stop computation!
         The situation is described in more detail in manual pages.
    	  This response seems excessively punitive, so it is
    	  destined to be replaced at some time in the fore-
    	  seeable future by a more flexible but still uni-
    	  form scheme being developed to handle all
    	  floating-point arithmetic exceptions neatly.	See
    	  infnan(3M) for the present state of affairs.

         How do the functions in 4.3 BSD's new libm for UNIX compare
         with their counterparts in DEC's VAX/VMS library?	Some of
         the VMS functions are a little faster, some are a little
         more accurate, some are more puritanical about exceptions
         (like pow(0.0,0.0) and atan2(0.0,0.0)), and most occupy much
         more memory than their counterparts in libm.  The VMS codes
         interpolate in large table to achieve speed and accuracy;
         the libm codes use tricky formulas compact enough that all
         of them may some day fit into a ROM.

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

         More important, DEC regards the VMS codes as proprietary and
         guards them zealously against unauthorized use.  But the
         libm codes in 4.3 BSD are intended for the public domain;
         they may be copied freely provided their provenance is
         always acknowledged, and provided users assist the authors
         in their researches by reporting experience with the codes.
         Therefore no user of UNIX on a machine whose arithmetic
         resembles VAX D_floating-point need use anything worse than
         the new libm.

         IEEE STANDARD 754 Floating-Point Arithmetic:

         This standard is on its way to becoming more widely adopted
         than any other design for computer arithmetic.  VLSI chips
         that conform to some version of that standard have been pro-
         duced by a host of manufacturers, among them ...
    	  Intel i8087, i80287	   National Semiconductor  32081
    	  Motorola 68881	   Weitek WTL-1032, ... , -1165
    	  Zilog Z8070		   Western Electric (AT&T) WE32106.
         Other implementations range from software, done thoroughly
         in the Apple Macintosh, through VLSI in the Hewlett-Packard
         9000 series, to the ELXSI 6400 running ECL at 3 Megaflops.
         Several other companies have adopted the formats of IEEE 754
         without, alas, adhering to the standard's way of handling
         rounding and exceptions like over/underflow.  The DEC VAX
         G_floating-point format is very similar to the IEEE 754 Dou-
         ble format, so similar that the C programs for the IEEE ver-
         sions of most of the elementary functions listed above could
         easily be converted to run on a MicroVAX, though nobody has
         volunteered to do that yet.

         The codes in 4.3 BSD's libm for machines that conform to
         IEEE 754 are intended primarily for the National Semi. 32081
         and WTL 1164/65.  To use these codes with the Intel or Zilog
         chips, or with the Apple Macintosh or ELXSI 6400, is to
         forego the use of better codes provided (perhaps freely) by
         those companies and designed by some of the authors of the
         codes above.  Except for atan, cabs, cbrt, erf, erfc, hypot,
         j0-jn, lgamma, pow and y0-yn, the Motorola 68881 has all the
         functions in libm on chip, and faster and more accurate; it,
         Apple, the i8087, Z8070 and WE32106 all use 64 sig.  bits.
         The main virtue of 4.3 BSD's libm codes is that they are
         intended for the public domain; they may be copied freely
         provided their provenance is always acknowledged, and pro-
         vided users assist the authors in their researches by
         reporting experience with the codes.  Therefore no user of
         UNIX on a machine that conforms to IEEE 754 need use any-
         thing worse than the new libm.

         Properties of IEEE 754 Double-Precision:
    	  Wordsize: 64 bits, 8 bytes.  Radix: Binary.
    	  Precision: 53 sig.  bits, roughly like 16 sig.

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

    	  decimals.
    	       If x and x' are consecutive positive
    	       Double-Precision numbers (they differ by 1 ulp),
    	       then
    	       1.1e-16 < 0.5**53 < (x'-x)/x < 0.5**52 < 2.3e-16.
    	  Range: Overflow threshold  = 2.0**1024 = 1.8e308
    		 Underflow threshold = 0.5**1022 = 2.2e-308
    	       Overflow goes by default to a signed Infinity.
    	       Underflow is Gradual, rounding to the nearest
    	       integer multiple of 0.5**1074 = 4.9e-324.
    	  Zero is represented ambiguously as +0 or -0.
    	       Its sign transforms correctly through multiplica-
    	       tion or division, and is preserved by addition of
    	       zeros with like signs; but x-x yields +0 for every
    	       finite x.  The only operations that reveal zero's
    	       sign are division by zero and copysign(x,+0).  In
    	       particular, comparison (x > y, x > y, etc.) cannot
    	       be affected by the sign of zero; but if finite x =
    	       y then Infinity = 1/(x-y) != -1/(y-x) = -Infinity.
    	  Infinity is signed.
    	       it persists when added to itself or to any finite
    	       number.	Its sign transforms correctly through
    	       multiplication and division, and
    	       (finite)/+Infinity = +0 (nonzero)/0 = +Infinity.
    	       But Infinity-Infinity, Infinity*0 and
    	       Infinity/Infinity are, like 0/0 and sqrt(-3),
    	       invalid operations that produce NaN. ...
    	  Reserved operands:
    	       there are 2**53-2 of them, all called NaN (Not a
    	       Number).  Some, called Signaling NaNs, trap any
    	       floating-point operation performed upon them; they
    	       are used to mark missing or uninitialized values,
    	       or nonexistent elements of arrays.  The rest are
    	       Quiet NaNs; they are the default results of
    	       Invalid Operations, and propagate through subse-
    	       quent arithmetic operations.  If x != x then x is
    	       NaN; every other predicate (x > y, x = y, x < y,
    	       ...) is FALSE if NaN is involved.
    	       NOTE: Trichotomy is violated by NaN.
    		    Besides being FALSE, predicates that entail
    		    ordered comparison, rather than mere
    		    (in)equality, signal Invalid Operation when
    		    NaN is involved.
    	  Rounding:
    	       Every algebraic operation (+, -, *, /, sqrt) is
    	       rounded by default to within half an ulp, and when
    	       the rounding error is exactly half an ulp then the
    	       rounded value's least significant bit is zero.
    	       This kind of rounding is usually the best kind,
    	       sometimes provably so; for instance, for every x =
    	       1.0, 2.0, 3.0, 4.0, ..., 2.0**52, we find
    	       (x/3.0)*3.0 == x and (x/10.0)*10.0 == x and ...

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

    	       despite that both the quotients and the products
    	       have been rounded.  Only rounding like IEEE 754
    	       can do that.  But no single kind of rounding can
    	       be proved best for every circumstance, so IEEE 754
    	       provides rounding towards zero or towards +Infin-
    	       ity or towards -Infinity at the programmer's
    	       option.	And the same kinds of rounding are speci-
    	       fied for Binary-Decimal Conversions, at least for
    	       magnitudes between roughly 1.0e-10 and 1.0e37.
    	  Exceptions:
    	       IEEE 754 recognizes five kinds of floating-point
    	       exceptions, listed below in declining order of
    	       probable importance.
    		    Exception		   Default Result
    		    __________________________________________
    		    Invalid Operation	   NaN, or FALSE
    		    Overflow		   +Infinity
    		    Divide by Zero	   +Infinity
    		    Underflow		   Gradual Underflow
    		    Inexact		   Rounded value
    	       NOTE:  An Exception is not an Error unless handled
    	       badly.  What makes a class of exceptions excep-
    	       tional is that no single default response can be
    	       satisfactory in every instance.	On the other
    	       hand, if a default response will serve most
    	       instances satisfactorily, the unsatisfactory
    	       instances cannot justify aborting computation
    	       every time the exception occurs.

    	  For each kind of floating-point exception, IEEE 754
    	  provides a Flag that is raised each time its exception
    	  is signaled, and stays raised until the program resets
    	  it.  Programs may also test, save and restore a flag.
    	  Thus, IEEE 754 provides three ways by which programs
    	  may cope with exceptions for which the default result
    	  might be unsatisfactory:

    	  1)  Test for a condition that might cause an exception
    	      later, and branch to avoid the exception.

    	  2)  Test a flag to see whether an exception has
    	      occurred since the program last reset its flag.

    	  3)  Test a result to see whether it is a value that
    	      only an exception could have produced.
    	      CAUTION: The only reliable ways to discover whether
    	      Underflow has occurred are to test whether products
    	      or quotients lie closer to zero than the underflow
    	      threshold, or to test the Underflow flag.  (Sums
    	      and differences cannot underflow in IEEE 754; if x
    	      != y then x-y is correct to full precision and cer-
    	      tainly nonzero regardless of how tiny it may be.)

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

    	      Products and quotients that underflow gradually can
    	      lose accuracy gradually without vanishing, so com-
    	      paring them with zero (as one might on a VAX) will
    	      not reveal the loss.  Fortunately, if a gradually
    	      underflowed value is destined to be added to some-
    	      thing bigger than the underflow threshold, as is
    	      almost always the case, digits lost to gradual
    	      underflow will not be missed because they would
    	      have been rounded off anyway.  So gradual under-
    	      flows are usually provably ignorable.  The same
    	      cannot be said of underflows flushed to 0.

    	  At the option of an implementor conforming to IEEE 754,
    	  other ways to cope with exceptions may be provided:

    	  4)  ABORT.  This mechanism classifies an exception in
    	      advance as an incident to be handled by means trad-
    	      itionally associated with error-handling statements
    	      like "ON ERROR GO TO ...".  Different languages
    	      offer different forms of this statement, but most
    	      share the following characteristics:

    	  -   No means is provided to substitute a value for the
    	      offending operation's result and resume computation
    	      from what may be the middle of an expression.  An
    	      exceptional result is abandoned.

    	  -   In a subprogram that lacks an error-handling state-
    	      ment, an exception causes the subprogram to abort
    	      within whatever program called it, and so on back
    	      up the chain of calling subprograms until an
    	      error-handling statement is encountered or the
    	      whole task is aborted and memory is dumped.

    	  5)  STOP.  This mechanism, requiring an interactive
    	      debugging environment, is more for the programmer
    	      than the program.  It classifies an exception in
    	      advance as a symptom of a programmer's error; the
    	      exception suspends execution as near as it can to
    	      the offending operation so that the programmer can
    	      look around to see how it happened.  Quite often
    	      the first several exceptions turn out to be quite
    	      unexceptionable, so the programmer ought ideally to
    	      be able to resume execution after each one as if
    	      execution had not been stopped.

    	  6)  ... Other ways lie beyond the scope of this docu-
    	      ment.

         The crucial problem for exception handling is the problem of
         Scope, and the problem's solution is understood, but not
         enough manpower was available to implement it fully in time

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

         to be distributed in 4.3 BSD's libm.  Ideally, each elemen-
         tary function should act as if it were indivisible, or
         atomic, in the sense that ...

         i)    No exception should be signaled that is not deserved
    	   by the data supplied to that function.

         ii)   Any exception signaled should be identified with that
    	   function rather than with one of its subroutines.

         iii)  The internal behavior of an atomic function should not
    	   be disrupted when a calling program changes from one
    	   to another of the five or so ways of handling excep-
    	   tions listed above, although the definition of the
    	   function may be correlated intentionally with excep-
    	   tion handling.

         Ideally, every programmer should be able conveniently to
         turn a debugged subprogram into one that appears atomic to
         its users.  But simulating all three characteristics of an
         atomic function is still a tedious affair, entailing hosts
         of tests and saves-restores; work is under way to ameliorate
         the inconvenience.

         Meanwhile, the functions in libm are only approximately
         atomic.  They signal no inappropriate exception except pos-
         sibly ...
    	  Over/Underflow
    	       when a result, if properly computed, might have
    	       lain barely within range, and
    	  Inexact in cabs, cbrt, hypot, log10 and pow
    	       when it happens to be exact, thanks to fortuitous
    	       cancellation of errors.
         Otherwise, ...
    	  Invalid Operation is signaled only when
    	       any result but NaN would probably be misleading.
    	  Overflow is signaled only when
    	       the exact result would be finite but beyond the
    	       overflow threshold.
    	  Divide-by-Zero is signaled only when
    	       a function takes exactly infinite values at finite
    	       operands.
    	  Underflow is signaled only when
    	       the exact result would be nonzero but tinier than
    	       the underflow threshold.
    	  Inexact is signaled only when
    	       greater range or precision would be needed to
    	       represent the exact result.

    BUGS
         When signals are appropriate, they are emitted by certain
         operations within the codes, so a subroutine-trace may be

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    MATH(3M)	    UNIX Programmer's Manual		 MATH(3M)

         needed to identify the function with its signal in case
         method 5) above is in use.  And the codes all take the IEEE
         754 defaults for granted; this means that a decision to trap
         all divisions by zero could disrupt a code that would other-
         wise get correct results despite division by zero.

    SEE ALSO
         An explanation of IEEE 754 and its proposed extension p854
         was published in the IEEE magazine MICRO in August 1984
         under the title "A Proposed Radix- and
         Word-length-independent Standard for Floating-point Arith-
         metic" by W. J. Cody et al.  The manuals for Pascal, C and
         BASIC on the Apple Macintosh document the features of IEEE
         754 pretty well.  Articles in the IEEE magazine COMPUTER
         vol. 14 no. 3 (Mar.  1981), and in the ACM SIGNUM Newsletter
         Special Issue of Oct. 1979, may be helpful although they
         pertain to superseded drafts of the standard.

    AUTHOR
         W. Kahan, with the help of Z-S. Alex Liu, Stuart I.
         McDonald, Dr. Kwok-Choi Ng, Peter Tang.

    Printed 11/26/99	  May 27, 1986				9

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