MATH(3) BSD Programmer's Manual MATH(3)
math - introduction to mathematical library functions
These functions constitute the C libm. Declarations for these functions may be obtained from the include file #include <math.h>
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
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)
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.
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.
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.
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.
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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