# FreeBSD Manual Pages

MATH(3) FreeBSD Library Functions Manual MATH(3)NAMEmath - introduction to mathematical library functionsDESCRIPTIONThese functions constitute the C math library, The link editor searches this library under the -lm option. Declarations for these functions may be obtained from the include file <math.h>.LIST OF FUNCTIONSEach of the followingdoublefunctions has afloatcounterpart with an `f' appended to the name and alongdoublecounterpart with an `l' appended. As an example, thefloatandlongdoublecounterparts ofdoubleacos(doublex) arefloatacosf(floatx) andlongdoubleacosl(longdoublex), respectively.NameDescriptionErrorBound(ULPs)acos inverse trigonometric function 3 acosh inverse hyperbolic function 3 asin inverse trigonometric function 3 asinh inverse hyperbolic function 3 atan inverse trigonometric function 1 atanh inverse hyperbolic function 3 atan2 inverse trigonometric function 2 cbrt cube root 1 ceil integer no less than 0 copysign copy sign bit 0 cos trigonometric function 1 cosh hyperbolic function 3 erf error function ??? erfc complementary error function ??? exp exponential base e 1 expm1 exp(x)-1 1 fabs absolute value 0 floor integer no greater than 0 fmod remainder function ??? frexp extract mantissa and exponent 0 hypot Euclidean distance 1 ilogb exponent extraction 0 j0 bessel function ??? j1 bessel function ??? jn bessel function ??? ldexp multiply by power of 2 0 lgamma log gamma function ??? log natural logarithm 1 log10 logarithm to base 10 3 log1p log(1+x) 1 logb exponent extraction 0 modf extract fractional part ??? nextafter next representable value 0 pow exponential x**y 60-500 remainder remainder 0 rint round to nearest integer 0 round round to nearest integer 0 scalbln exponent adjustment 0 scalbn exponent adjustment 0 sin trigonometric function 1 sinh hyperbolic function 3 sqrt square root 1 tan trigonometric function 3 tanh hyperbolic function 3 tgamma gamma function ??? trunc round towards zero 0 y0 bessel function ??? y1 bessel function ??? yn bessel function ???NOTESVirtually all modern floating-point units attempt to support IEEE Stan- dard 754 for Binary Floating-Point Arithmetic. This standard does not cover particular routines in the math library except for the few docu- mented in ieee(3); it primarily defines representations of numbers and abstract properties of arithmetic operations relating to precision, rounding, and exceptional cases, as described below. The programs are accurate to within the numbers ofulps tabulated above; anulpis oneUnit in theLastPlace.IEEESTANDARD754Floating-PointArithmetic: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 Double-Precision numbers (they differ by 1ulp), 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 isGradual,rounding to the nearest integer multi- ple of 0.5**1074 = 4.9e-324. Zero is represented ambiguously as +0 or -0. Its sign transforms correctly through multiplication or divi- sion, 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 copy- sign(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 Infin- ity/Infinity are, like 0/0 and sqrt(-3), invalid operations that produceNaN. ... Reserved operands: there are 2**53-2 of them, all calledNaN(NotaNumber). Some, called SignalingNaNs, trap any floating-point opera- tion performed upon them; they are used to mark missing or uninitialized values, or nonexistent elements of arrays. The rest are QuietNaNs; they are the default results of Invalid Operations, and propagate through subsequent arithmetic oper- ations. If x != x then x isNaN; every other predicate (x > y, x = y, x < y, ...) is FALSE ifNaNis involved. NOTE: Trichotomy is violated byNaN. Besides being FALSE, predicates that entail ordered comparison, rather than mere (in)equality, signal Invalid Operation whenNaNis involved. Rounding: Every algebraic operation (+, -, *, /, sqrt) is rounded by default to within half anulp, and when the rounding error is exactly half anulpthen the rounded value's least signifi- cant 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 circumstance, so IEEE 754 provides rounding towards zero or towards +Infinity or towards -Infin- ity at the programmer's option. And the same kinds of round- ing 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 exceptions, listed below in declining order of probable importance. Exception Default Result Invalid OperationNaN, 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 exceptional is that no sin- gle 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 excep- tion 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 differ- ences cannot underflow in IEEE 754; if x != y then x-y is cor- rect to full precision and certainly nonzero regardless of how tiny it may be.) Products and quotients that underflow gradu- ally can lose accuracy gradually without vanishing, so compar- ing them with zero (as one might on a VAX) will not reveal the loss. Fortunately, if a gradually underflowed value is des- tined to be added to something bigger than the underflow threshold, as is almost always the case, digits lost to grad- ual underflow will not be missed because they would have been rounded off anyway. So gradual underflows are usuallyprov-ablyignorable. 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 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 offend- ing 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 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 programmer than the program. It classifies an exception in advance as a symptom of a program- mer'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 pro- grammer 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. 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 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 exceptions listed above, although the defini- tion of the function may be correlated intentionally with exception handling. The functions inlibmare 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 incabs,cbrt,hypot,log10andpowwhen it happens to be exact, thanks to fortuitous cancella- tion of errors. Otherwise, ... Invalid Operation is signaled only when any result butNaNwould 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 under- flow threshold. Inexact is signaled only when greater range or precision would be needed to represent the exact result.BUGSSeveral functions required by ISO/IEC 9899:1999 (``ISO C99'') are miss- ing, and many functions are not available in theirlongdoublevariants.SEE ALSOfenv(3), ieee(3) 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.HISTORYA math library with many of the present functions appeared in Version 7 AT&T UNIX. The library was substantially rewritten for 4.3BSD to provide better accuracy and speed on machines supporting either VAX or IEEE 754 floating-point. Most of this library was replaced with FDLIBM, developed at Sun Microsystems, in FreeBSD 1.1.5. FreeBSD 11.1 June 11, 2004 FreeBSD 11.1

NAME | DESCRIPTION | LIST OF FUNCTIONS | NOTES | BUGS | SEE ALSO | HISTORY

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