+
+
+
+MATH(3) BSD Programmer's Manual MATH(3)
+
+
+N\bNA\bAM\bME\bE
+ math - introduction to mathematical library functions
+
+D\bDE\bES\bSC\bCR\bRI\bIP\bPT\bTI\bIO\bON\bN
+ These functions constitute the C math library, _\bl_\bi_\bb_\bm_\b. The
+ link editor searches this library under the "-lm" option.
+ Declarations for these functions may be obtained from the
+ include file <_\bm_\ba_\bt_\bh_\b._\bh>. The Fortran math library is
+ described in ``man 3f intro''.
+
+L\bLI\bIS\bST\bT O\bOF\bF F\bFU\bUN\bNC\bCT\bTI\bIO\bON\bNS\bS
+ _\bN_\ba_\bm_\be _\bA_\bp_\bp_\be_\ba_\br_\bs _\bo_\bn _\bP_\ba_\bg_\be _\bD_\be_\bs_\bc_\br_\bi_\bp_\bt_\bi_\bo_\bn _\bE_\br_\br_\bo_\br _\bB_\bo_\bu_\bn_\bd _\b(_\bU_\bL_\bP_\bs_\b)
+ 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 ???
+
+
+
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+ y1 j0.3m bessel function ???
+ yn j0.3m bessel function ???
+
+N\bNO\bOT\bTE\bES\bS
+ 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, another for dou-
+ ble-precision arithmetic conforming to the IEEE Standard
+ 754 for Binary Floating-Point Arithmetic. The two ver-
+ sions 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 _\bu_\bl_\bps tabulated above; an _\bu_\bl_\bp is one
+ _\bUnit in the _\bLast _\bPlace. And the programs have been cured
+ of anomalies that afflicted the older math library _\bl_\bi_\bb_\bm in
+ which incidents like the following 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.
+
+ D\bDE\bEC\bC V\bVA\bAX\bX-\b-1\b11\b1 D\bD_\b_f\bfl\blo\boa\bat\bti\bin\bng\bg-\b-p\bpo\boi\bin\bnt\bt:\b:
+
+ This is the format for which the original math library
+ _\bl_\bi_\bb_\bm was developed, and to which this manual is still
+ principally dedicated. It is _\bt_\bh_\be 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
+ _\bu_\bl_\bp), 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.
+
+
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+ 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 hap-
+ pening without warning.
+ 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 over-
+ flow are invalid 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 _\bu_\bl_\bp, and when
+ the rounding error is exactly half an _\bu_\bl_\bp
+ 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 ele-
+ mentary 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 computa-
+ tion! The situation is described in more detail in manual
+ pages.
+ _\bT_\bh_\bi_\bs _\br_\be_\bs_\bp_\bo_\bn_\bs_\be _\bs_\be_\be_\bm_\bs _\be_\bx_\bc_\be_\bs_\bs_\bi_\bv_\be_\bl_\by _\bp_\bu_\bn_\bi_\bt_\bi_\bv_\be_\b, _\bs_\bo
+ _\bi_\bt _\bi_\bs _\bd_\be_\bs_\bt_\bi_\bn_\be_\bd _\bt_\bo _\bb_\be _\br_\be_\bp_\bl_\ba_\bc_\be_\bd _\ba_\bt _\bs_\bo_\bm_\be _\bt_\bi_\bm_\be _\bi_\bn
+ _\bt_\bh_\be _\bf_\bo_\br_\be_\bs_\be_\be_\ba_\bb_\bl_\be _\bf_\bu_\bt_\bu_\br_\be _\bb_\by _\ba _\bm_\bo_\br_\be _\bf_\bl_\be_\bx_\bi_\bb_\bl_\be _\bb_\bu_\bt
+ _\bs_\bt_\bi_\bl_\bl _\bu_\bn_\bi_\bf_\bo_\br_\bm _\bs_\bc_\bh_\be_\bm_\be _\bb_\be_\bi_\bn_\bg _\bd_\be_\bv_\be_\bl_\bo_\bp_\be_\bd _\bt_\bo _\bh_\ba_\bn_\bd_\bl_\be
+ _\ba_\bl_\bl _\bf_\bl_\bo_\ba_\bt_\bi_\bn_\bg_\b-_\bp_\bo_\bi_\bn_\bt _\ba_\br_\bi_\bt_\bh_\bm_\be_\bt_\bi_\bc _\be_\bx_\bc_\be_\bp_\bt_\bi_\bo_\bn_\bs
+ _\bn_\be_\ba_\bt_\bl_\by_\b. _\bS_\be_\be _\bi_\bn_\bf_\bn_\ba_\bn_\b(_\b3_\bM_\b) _\bf_\bo_\br _\bt_\bh_\be _\bp_\br_\be_\bs_\be_\bn_\bt _\bs_\bt_\ba_\bt_\be
+
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+ _\bo_\bf _\ba_\bf_\bf_\ba_\bi_\br_\bs_\b.
+
+ How do the functions in 4.3 BSD's new _\bl_\bi_\bb_\bm for UNIX com-
+ pare 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
+ _\bl_\bi_\bb_\bm. The VMS codes interpolate in large table to achieve
+ speed and accuracy; the _\bl_\bi_\bb_\bm 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 _\bl_\bi_\bb_\bm codes in 4.3 BSD are intended for the public
+ domain; they may be copied freely provided their prove-
+ nance 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 _\bl_\bi_\bb_\bm.
+
+ I\bIE\bEE\bEE\bE S\bST\bTA\bAN\bND\bDA\bAR\bRD\bD 7\b75\b54\b4 F\bFl\blo\boa\bat\bti\bin\bng\bg-\b-P\bPo\boi\bin\bnt\bt A\bAr\bri\bit\bth\bhm\bme\bet\bti\bic\bc:\b:
+
+ 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 produced 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 stan-
+ dard's way of handling 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
+ elementary functions listed above could easily be con-
+ verted to run on a MicroVAX, though nobody has volunteered
+ to do that yet.
+
+ The codes in 4.3 BSD's _\bl_\bi_\bb_\bm 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
+
+
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+ freely) by those companies and designed by some of the
+ authors of the codes above. Except for _\ba_\bt_\ba_\bn, _\bc_\ba_\bb_\bs, _\bc_\bb_\br_\bt,
+ _\be_\br_\bf, _\be_\br_\bf_\bc, _\bh_\by_\bp_\bo_\bt, _\bj_\b0_\b-_\bj_\bn, _\bl_\bg_\ba_\bm_\bm_\ba, _\bp_\bo_\bw and _\by_\b0_\b-_\by_\bn, the
+ Motorola 68881 has all the functions in _\bl_\bi_\bb_\bm 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 _\bl_\bi_\bb_\bm codes is that they are intended for the public
+ domain; they may be copied freely provided their prove-
+ nance 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
+ that conforms to IEEE 754 need use anything worse than the
+ new _\bl_\bi_\bb_\bm.
+
+ Properties of IEEE 754 Double-Precision:
+ Wordsize: 64 bits, 8 bytes. Radix: Binary.
+ Precision: 53 sig. bits, roughly like 16 sig.
+ decimals.
+ If x and x' are consecutive positive Dou-
+ ble-Precision numbers (they differ by 1
+ _\bu_\bl_\bp), 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 Infin-
+ ity.
+ Underflow is _\bG_\br_\ba_\bd_\bu_\ba_\bl_\b, rounding to the near-
+ est integer multiple of 0.5**1074 =
+ 4.9e-324.
+ Zero is represented ambiguously as +0 or -0.
+ Its sign transforms correctly through multi-
+ plication 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 divi-
+ sion by zero and copysign(x,+-0). In par-
+ ticular, 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 cor-
+ rectly through multiplication and division,
+ and (finite)/+-Infinity = +-0 (nonzero)/0 =
+ +-Infinity. But Infinity-Infinity, Infin-
+ ity*0 and Infinity/Infinity are, like 0/0
+ and sqrt(-3), invalid operations that pro-
+ duce _\bN_\ba_\bN. ...
+ Reserved operands:
+
+
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+ there are 2**53-2 of them, all called _\bN_\ba_\bN
+ (_\bNot _\ba _\bNumber). Some, called Signaling
+ _\bN_\ba_\bNs, trap any floating-point operation per-
+ formed upon them; they are used to mark
+ missing or uninitialized values, or nonexis-
+ tent elements of arrays. The rest are Quiet
+ _\bN_\ba_\bNs; they are the default results of
+ Invalid Operations, and propagate through
+ subsequent arithmetic operations. If x != x
+ then x is _\bN_\ba_\bN; every other predicate (x > y,
+ x = y, x < y, ...) is FALSE if _\bN_\ba_\bN is
+ involved.
+ NOTE: Trichotomy is violated by _\bN_\ba_\bN.
+ Besides being FALSE, predicates that
+ entail ordered comparison, rather
+ than mere (in)equality, signal
+ Invalid Operation when _\bN_\ba_\bN is
+ involved.
+ Rounding:
+ Every algebraic operation (+, -, *, /, sqrt)
+ is rounded by default to within half an _\bu_\bl_\bp,
+ and when the rounding error is exactly half
+ an _\bu_\bl_\bp 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 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 cir-
+ cumstance, so IEEE 754 provides 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 float-
+ ing-point exceptions, listed below in
+ declining order of probable importance.
+ Exception Default Result
+ __________________________________________
+ Invalid Operation _\bN_\ba_\bN, or FALSE
+ Overflow +-Infinity
+ Divide by Zero +-Infinity
+ Underflow Gradual Underflow
+ Inexact Rounded value
+ NOTE: An Exception is not an Error unless
+
+
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+ handled badly. What makes a class of excep-
+ tions exceptional is that no single default
+ response can be satisfactory in every
+ instance. On the other hand, if a default
+ response will serve most instances satisfac-
+ torily, 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 excep-
+ tion is signaled, and stays raised until the pro-
+ gram 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 excep-
+ tion 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 can-
+ not 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 vanishing,
+ 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
+ threshold, as is almost always the case, digits
+ lost to gradual underflow will not be missed
+ because they would have been rounded off any-
+ way. So gradual underflows are usually _\bp_\br_\bo_\bv_\b-
+ _\ba_\bb_\bl_\by 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 pro-
+ vided:
+
+ 4) ABORT. This mechanism classifies an exception
+
+
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+ in advance as an incident 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 aban-
+ doned.
+
+ -- In a subprogram that lacks an error-handling
+ statement, 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 encoun-
+ tered or the whole task is aborted and memory
+ is dumped.
+
+ 5) STOP. This mechanism, requiring an interactive
+ debugging environment, is more for the program-
+ mer than the program. It classifies an excep-
+ tion 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 excep-
+ tions 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
+ document.
+
+ 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 to be distributed in 4.3 BSD's _\bl_\bi_\bb_\bm. Ideally, each
+ elementary 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 subrou-
+ tines.
+
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+ 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
+ exceptions listed above, although the definition of
+ the function may be correlated intentionally with
+ exception handling.
+
+ Ideally, every programmer should be able _\bc_\bo_\bn_\bv_\be_\bn_\bi_\be_\bn_\bt_\bl_\by 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 amelio-
+ rate the inconvenience.
+
+ Meanwhile, the functions in _\bl_\bi_\bb_\bm are only approximately
+ atomic. They signal no inappropriate exception except
+ possibly ...
+ Over/Underflow
+ when a result, if properly computed, might
+ have lain barely within range, and
+ Inexact in _\bc_\ba_\bb_\bs, _\bc_\bb_\br_\bt, _\bh_\by_\bp_\bo_\bt, _\bl_\bo_\bg_\b1_\b0 and _\bp_\bo_\bw
+ when it happens to be exact, thanks to for-
+ tuitous cancellation of errors.
+ Otherwise, ...
+ Invalid Operation is signaled only when
+ any result but _\bN_\ba_\bN would probably be mis-
+ leading.
+ 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.
+
+B\bBU\bUG\bGS\bS
+ 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.
+
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+ 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 Newslet-
+ ter Special Issue of Oct. 1979, may be helpful although
+ they pertain to superseded drafts of the standard.
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projects / unix-history / commitdiff