MirOS Manual: math(3)

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

NAME

     math - introduction to mathematical library functions

LIBRARY

     libm

SYNOPSIS

     #include <math.h>

DESCRIPTION

     These functions constitute the C libm. Declarations for these functions
     may be obtained from the include file #include <math.h>

List of Functions


     Name         Man page     Description
     acos         acos(3)      inverse trigonometric function
     acosh        acosh(3)     inverse hyperbolic function
     asin         asin(3)      inverse trigonometric function
     asinh        asinh(3)     inverse hyperbolic function
     atan         atan(3)      inverse trigonometric function
     atanh        atanh(3)     inverse hyperbolic function
     atan2        atan2(3)     inverse trigonometric function
     cbrt         sqrt(3)      cube root
     ceil         ceil(3)      integer no less
     copysign     copysign(3)  copy sign bit
     cos          cos(3)       trigonometric function
     cosh         cosh(3)      hyperbolic function
     erf          erf(3)       error function
     erfc         erf(3)       complementary error function
     exp          exp(3)       exponential                    1
     expm1        exp(3)       exp(x)-1                       1
     fabs         fabs(3)      absolute value
     finite       finite(3)    test for finity
     floor        floor(3)     integer no greater
     fmod         fmod(3)      remainder                      ???
     hypot        hypot(3)     Euclidean distance
     ilogb        ilogb(3)     exponent extraction
     isinf        isinf(3)     test for infinity
     isnan        isnan(3)     test for not-a-number
     j0           j0(3)        Bessel function
     j1           j0(3)        Bessel function
     jn           j0(3)        Bessel function
     lgamma       lgamma(3)    log gamma function
     log          log(3)       natural logarithm
     log10        log(3)       logarithm to base
     log1p        log(3)       log(1+x)                       1
     nan          nan(3)       return quiet NaN
     nextafter    nextafter(3) next representable number
     pow          pow(3)       exponential x**y
     remainder    remainder(3) remainder                      0
     rint         rint(3)      round to nearest
     scalbn       scalbn(3)    exponent adjustment
     sin          sin(3)       trigonometric function
     sinh         sinh(3)      hyperbolic function
     sqrt         sqrt(3)      square root
     tan          tan(3)       trigonometric function
     tanh         tanh(3)      hyperbolic function
     trunc        trunc(3)     nearest integral value
     y0           j0(3)        Bessel function
     y1           j0(3)        Bessel function
     yn           j0(3)        Bessel function

List of Defined Values

     Name            Value                       Description
     M_E             2.7182818284590452354       e
     M_LOG2E         1.4426950408889634074       log 2e
     M_LOG10E        0.43429448190325182765      log 10e
     M_LN2           0.69314718055994530942      log e2
     M_LN10          2.30258509299404568402      log e10
     M_PI            3.14159265358979323846      pi
     M_PI_2          1.57079632679489661923      pi/2
     M_PI_4          0.78539816339744830962      pi/4
     M_1_PI          0.31830988618379067154      1/pi
     M_2_PI          0.63661977236758134308      2/pi
     M_2_SQRTPI      1.12837916709551257390      2/sqrt(pi)
     M_SQRT2         1.41421356237309504880      sqrt(2)
     M_SQRT1_2       0.70710678118654752440      1/sqrt(2)

NOTES

     In 4.3 BSD, distributed from the University of California in late 1985,
     most of the foregoing functions come in two versions, one for the
     double-precision "D" format in the DEC VAX-11 family of computers, anoth-
     er for double-precision arithmetic conforming to the IEEE Standard 754
     for Binary Floating-Point Arithmetic. The two versions behave very simi-
     larly, 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 in which incidents like the following had been re-
     ported:

           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 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 significant bits, roughly like 17 significant de-
                   cimals. 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 cus-
                   tomarily flushed quietly to zero. CAUTION: It is possible
                   to have x / y and yet x-y = 0 because of underflow. Simi-
                   larly x > y > 0 cannot prevent either x*y = 0 or y/x = 0
                   from happening without warning.

           Zero is represented ambiguously: Although 2**55 different represen-
                   tations of zero are accepted by the hardware, only the ob-
                   vious 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 in-
                   valid operations that customarily stop computation 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 round-
                   ing is away from 0.

     Except for its narrow range, D_floating-point is one of the better com-
     puter 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 arith-
     metic operation that over/underflowed would behave. Similarly, expres-
     sions 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 computa-
     tion! 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 foreseeable more flexible but still uniform
     scheme being developed to handle all floating-point arithmetic exceptions
     neatly.

     How do the functions in 4.3 BSD's new math library 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 puri-
     tanical 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.

     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 in-
     tended 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 oth-
     er design for computer arithmetic. VLSI chips that conform to some ver-
     sion of that standard have been produced by a host of manufacturers,
     among them ...

     Intel i8087, i80287      National Semiconductor 32081
     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 han-
     dling rounding and exceptions like over/underflow. The DEC VAX
     G_floating-point format is very similar to the IEEE 754 Double format, so
     similar that the C programs for the IEEE versions of most of the elemen-
     tary functions listed above could easily be converted to run on a Micro-
     VAX, though nobody has volunteered to do that yet.

     The codes in 4.3 BSD's libm for machines that conform to IEEE 754 are in-
     tended primarily for the National Semiconductor 32081 and WTL 1164/65. To
     use these codes with the Intel or Zilog chips, or with the Apple Macin-
     tosh 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(), 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 significant 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 ack-
     nowledged, and provided 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 anything worse than the new
     libm.

     Properties of IEEE 754 Double-Precision:

           Wordsize: 64 bits, 8 bytes.

           Radix: Binary.

           Precision: 53 significant bits, roughly like 16 significant de-
                   cimals. 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 multiplication 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 re-
                   veal 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 In-
                   finity = 1/(x-y) / -1/(y-x) = - Infinity .

           Infinity is signed: it persists when added to itself or to any fin-
                   ite number. Its sign transforms correctly through multipli-
                   cation and division, and Infinity (finite)/±  = ±0
                   (nonzero)/0 = ± Infinity. But oo-oo, oo*0 and oo/oo 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 subsequent 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 in-
                   volved.

           Rounding: Every algebraic operation (+, -, *, /, /) 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 sig-
                   nificant 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 ... despite that both the quo-
                   tients 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 pro-
                   vides rounding towards zero or towards +Infinity or towards
                   -Infinity at the programmer's option. And the same kinds of
                   rounding are specified 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 excep-
                   tions, listed below in declining order of probable impor-
                   tance.

                   Exception             Default Result
                   Invalid Operation     NaN, or FALSE
                   Overflow              ±oo
                   Divide by Zero        ±oo
                   Underflow             Gradual Underflow
                   Inexact               Rounded value

                   NOTE: An Exception is not an Error unless handled badly.
                   What makes a class of exceptions exceptional is that no
                   single default response can be satisfactory in every in-
                   stance. On the other hand, if a default response will serve
                   most instances satisfactorily, the unsatisfactory instances
                   cannot justify aborting computation every time the excep-
                   tion 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 pro-
          gram 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 quo-
          tients 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 certainly
          nonzero regardless of how tiny it may be.) Products and quotients
          that underflow gradually can lose accuracy gradually without vanish-
          ing, so comparing 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 something bigger than the underflow thres-
          hold, 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 underflows are usually provably ignorable. The same can-
          not 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 in-
          cident to be handled by means traditionally 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 statement, an ex-
              ception 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 environ-
          ment, is more for the programmer than the program. It classifies an
          exception in advance as a symptom of a programmer's error; the ex-
          ception suspends execution as near as it can to the offending opera-
          tion so that the programmer can look around to see how it happened.
          Quite often the first several exceptions turn out to be quite unex-
          ceptionable, 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 document.

     The crucial problem for exception handling is the problem of Scope, and
     the problem's solution is understood, but not enough manpower was avail-
     able to implement it fully in time to be distributed in 4.3 BSD's libm.
     Ideally, each elementary function should act as if it were indivisible,
     or atomic, in the sense that ...

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

     2.   Any exception signaled should be identified with that function rath-
          er than with one of its subroutines.

     3.   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 exceptions listed above, although the definition of
          the function may be correlated intentionally with exception han-
          dling.

     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 sig-
     nal no inappropriate exception except possibly ...

           Over/Underflow
           when a result, if properly computed, might have lain barely within
           range, and

           Inexact in cbrt(),hypot(),
           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.

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 Ra-
     dix- and Word-length-independent Standard for Floating-point Arithmetic"
     by W. J. Cody et al. The manuals for Pascal, C and BASIC on the Apple Ma-
     cintosh 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 per-
     tain to superseded drafts of the standard.

BUGS

     When signals are appropriate, they are emitted by certain operations
     within the codes, so a subroutine-trace may be 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 otherwise get
     correct results despite division by zero.

MirOS BSD #10-current         February 23, 2007                              6

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