path: root/usr/src/man/man3lib/libm.3lib

'\" te
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.TH LIBM 3LIB "December 29, 2021"
.SH NAME
libm \- C math library
.SH SYNOPSIS
.nf
c99 [ \fIflag\fR... ] \fIfile\fR... \fB-lm\fR [ \fIlibrary\fR... ]
.fi

.SH DESCRIPTION
Functions in this library provide common elementary mathematical functions and
floating point environment routines defined by System V, ANSI C, POSIX, and so
on. See \fBstandards\fR(5). Additional functions in this library provide
extended support for handling floating point exceptions.
.SH INTERFACES
The shared object \fBlibm.so.2\fR provides the public interfaces defined below.
See \fBIntro\fR(3) for additional information on shared object interfaces.
.sp

.sp
.TS
tab(	);
lw(2.75i) lw(2.75i)
lw(2.75i) lw(2.75i)
.
\fBacos\fR	\fBacosf\fR
\fBacosh\fR	\fBacoshf\fR
\fBacoshl\fR	\fBacosl\fR
\fBasin\fR	\fBasinf\fR
\fBasinh\fR	\fBasinhf\fR
\fBasinhl\fR	\fBasinl\fR
\fBatan\fR	\fBatan2\fR
\fBatan2f\fR	\fBatan2l\fR
\fBatanf\fR	\fBatanh\fR
\fBatanhf\fR	\fBatanhl\fR
\fBatanl\fR	\fBcabs\fR
\fBcabsf\fR	\fBcabsl\fR
\fBcacos\fR	\fBcacosf\fR
\fBcacosh\fR	\fBcacoshf\fR
\fBcacoshl\fR	\fBcacosl\fR
\fBcarg\fR	\fBcargf\fR
\fBcargl\fR	\fBcasin\fR
\fBcasinf\fR	\fBcasinh\fR
\fBcasinhf\fR	\fBcasinhl\fR
\fBcasinl\fR	\fBcatan\fR
\fBcatanf\fR	\fBcatanh\fR
\fBcatanhf\fR	\fBcatanhl\fR
\fBcatanl\fR	\fBcbrt\fR
\fBcbrtf\fR	\fBcbrtl\fR
\fBccos\fR	\fBccosf\fR
\fBccosh\fR	\fBccoshf\fR
\fBccoshl\fR	\fBccosl\fR
\fBceil\fR	\fBceilf\fR
\fBceill\fR	\fBcexp\fR
\fBcexpf\fR	\fBcexpl\fR
\fBcimag\fR	\fBcimagf\fR
\fBcimagl\fR	\fBclog\fR
\fBclogf\fR	\fBclogl\fR
\fBconj\fR	\fBconjf\fR
\fBconjl\fR	\fBcopysign\fR
\fBcopysignf\fR	\fBcopysignl\fR
\fBcos\fR	\fBcosf\fR
\fBcosh\fR	\fBcoshf\fR
\fBcoshl\fR	\fBcosl\fR
\fBcpow\fR	\fBcpowf\fR
\fBcpowl\fR	\fBcproj\fR
\fBcprojf\fR	\fBcprojl\fR
\fBcreal\fR	\fBcrealf\fR
\fBcreall\fR	\fBcsin\fR
\fBcsinf\fR	\fBcsinh\fR
\fBcsinhf\fR	\fBcsinhl\fR
\fBcsinl\fR	\fBcsqrt\fR
\fBcsqrtf\fR	\fBcsqrtl\fR
\fBctan\fR	\fBctanf\fR
\fBctanh\fR	\fBctanhf\fR
\fBctanhl\fR	\fBctanl\fR
\fBerf\fR	\fBerfc\fR
\fBerfcf\fR	\fBerfcl\fR
\fBerff\fR	\fBerfl\fR
\fBexp\fR	\fBexp2\fR
\fBexp2f\fR	\fBexp2l\fR
\fBexpf\fR	\fBexpl\fR
\fBexpm1\fR	\fBexpm1f\fR
\fBexpm1l\fR	\fBfabs\fR
\fBfabsf\fR	\fBfabsl\fR
\fBfdim\fR	\fBfdimf\fR
\fBfdiml\fR	\fBfeclearexcept\fR
\fBfegetenv\fR	\fBfegetexceptflag\fR
\fBfegetround\fR	\fBfeholdexcept\fR
\fBferaiseexcept\fR	\fBfesetenv\fR
\fBfesetexceptflag\fR	\fBfesetround\fR
\fBfetestexcept\fR	\fBfeupdateenv\fR
\fBfex_get_handling\fR	\fBfex_get_log\fR
\fBfex_get_log_depth\fR	\fBfex_getexcepthandler\fR
\fBfex_log_entry\fR	\fBfex_merge_flags\fR
\fBfex_set_handling\fR	\fBfex_set_log\fR
\fBfex_set_log_depth\fR	\fBfex_setexcepthandler\fR
\fBfloor\fR	\fBfloorf\fR
\fBfloorl\fR	\fBfma\fR
\fBfmaf\fR	\fBfmal\fR
\fBfmax\fR	\fBfmaxf\fR
\fBfmaxl\fR	\fBfmin\fR
\fBfminf\fR	\fBfminl\fR
\fBfmod\fR	\fBfmodf\fR
\fBfmodl\fR	\fBfrexp\fR
\fBfrexpf\fR	\fBfrexpl\fR
\fBgamma\fR	\fBgamma_r\fR
\fBgammaf\fR	\fBgammaf_r\fR
\fBgammal\fR	\fBgammal_r\fR
\fBhypot\fR	\fBhypotf\fR
\fBhypotl\fR	\fBilogb\fR
\fBilogbf\fR	\fBilogbl\fR
\fBisnan\fR	\fBj0\fR
\fBj0f\fR	\fBj0l\fR
\fBj1\fR	\fBj1f\fR
\fBj1l\fR	\fBjn\fR
\fBjnf\fR	\fBjnl\fR
\fBldexp\fR	\fBldexpf\fR
\fBldexpl\fR	\fBlgamma\fR
\fBlgamma_r\fR	\fBlgammaf\fR
\fBlgammaf_r\fR	\fBlgammal\fR
\fBlgammal_r\fR	\fBllrint\fR
\fBllrintf\fR	\fBllrintl\fR
\fBllround\fR	\fBllroundf\fR
\fBllroundl\fR	\fBlog\fR
\fBlog10\fR	\fBlog10f\fR
\fBlog10l\fR	\fBlog1p\fR
\fBlog1pf\fR	\fBlog1pl\fR
\fBlog2\fR	\fBlog2f\fR
\fBlog2l\fR	\fBlogb\fR
\fBlogbf\fR	\fBlogbl\fR
\fBlogf\fR	\fBlogl\fR
\fBlrint\fR	\fBlrintf\fR
\fBlrintl\fR	\fBlround\fR
\fBlroundf\fR	\fBlroundl\fR
\fBmatherr\fR	\fBmodf\fR
\fBmodff\fR	\fBmodfl\fR
\fBnan\fR	\fBnanf\fR
\fBnanl\fR	\fBnearbyint\fR
\fBnearbyintf\fR	\fBnearbyintl\fR
\fBnextafter\fR	\fBnextafterf\fR
\fBnextafterl\fR	\fBnexttoward\fR
\fBnexttowardf\fR	\fBnexttowardl\fR
\fBpow\fR	\fBpowf\fR
\fBpowl\fR	\fBremainder\fR
\fBremainderf\fR	\fBremainderl\fR
\fBremquo\fR	\fBremquof\fR
\fBremquol\fR	\fBrint\fR
\fBrintf\fR	\fBrintl\fR
\fBround\fR	\fBroundf\fR
\fBroundl\fR	\fBscalb\fR
\fBscalbf\fR	\fBscalbl\fR
\fBscalbln\fR	\fBscalblnf\fR
\fBscalblnl\fR	\fBscalbn\fR
\fBscalbnf\fR	\fBscalbnl\fR
\fBsigngam\fR	\fBsigngamf\fR
\fBsigngaml\fR	\fBsignificand\fR
\fBsignificandf\fR	\fBsignificandl\fR
\fBsin\fR	\fBsincos\fR
\fBsincosf\fR	\fBsincosl\fR
\fBsinf\fR	\fBsinh\fR
\fBsinhf\fR	\fBsinhl\fR
\fBsinl\fR	\fBsqrt\fR
\fBsqrtf\fR	\fBsqrtl\fR
\fBtan\fR	\fBtanf\fR
\fBtanh\fR	\fBtanhf\fR
\fBtanhl\fR	\fBtanl\fR
\fBtgamma\fR	\fBtgammaf\fR
\fBtgammal\fR	\fBtrunc\fR
\fBtruncf\fR	\fBtruncl\fR
\fBy0\fR	\fBy0f\fR
\fBy0l\fR	\fBy1\fR
\fBy1f\fR	\fBy1l\fR
\fByn\fR	\fBynf\fR
\fBynl\fR	\fB\fR
.TE

.sp
.LP
The following interfaces are unique to the x86 and amd64 versions of this library:
.sp

.sp
.TS
tab(	);
lw(2.75i) lw(2.75i)
.
\fBfegetprec\fR	\fBfesetprec\fR
.TE

.SH ACCURACY
ISO/IEC 9899:1999, also known as C99, specifies the functions listed in the
following tables and states that the accuracy of these functions is
"implementation-defined". The information below characterizes the accuracy
of these functions as implemented in \fBlibm.so.2\fR. For each function,
the tables provide an upper bound on the largest error possible for any
argument and the largest error actually observed among a large sample of
arguments. Errors are expressed in "units in the last place", or ulps, relative
to the exact function value for each argument (regarding the argument as
exact). Ulps depend on the precision of the floating point format: if \fIy\fR
is the exact function value, \fIx\fR and \fIx\fR' are adjacent floating point
numbers such that \fIx\fR < \fIy\fR < \fIx\fR', and \fIx\fR'' is the computed
function value, then provided \fIx\fR, \fIx\fR', and \fIx\fR'' all lie in the
same binade, the error in \fIx\fR'' is |\fIy\fR - \fIx\fR''| / |\fIx\fR -
\fIx\fR'| ulps. In particular, when the error is less than one ulp, the computed
value is one of the two floating point numbers adjacent to the exact value.
.sp
.LP
The bounds and observed errors listed below apply only in the default floating
point modes. Specifically, on SPARC, these bounds assume the rounding direction
is round-to-nearest and non-standard mode is disabled. On x86, the bounds assume
the rounding direction is round-to-nearest and the rounding precision is
round-to-64-bits. Moreover, on x86, floating point function values are returned
in a floating point register in extended double precision format, but the bounds
below assume that the result value is then stored to memory in the format
corresponding to the function's type. On amd64, the bounds assume the rounding
direction in both the x87 floating point control word and the MXCSR is
round-to-nearest, the rounding precision in the x87 control word is
round-to-64-bits, and the FTZ and DAZ modes are disabled.
.sp
.LP
The error bounds listed below are believed to be correct, but smaller bounds
might be proved later. The observed errors are the largest ones currently known,
but larger errors might be discovered later. Numbers in the notes column refer
to the notes following the tables.
.SS "Real Functions"
.SS "Single precision real functions (SPARC, x86, and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBacosf\fR	1.0	< 1
\fBacoshf\fR	1.0	< 1
\fBasinf\fR	1.0	< 1
\fBasinhf\fR	1.0	< 1
\fBatanf\fR	1.0	< 1
\fBatan2f\fR	1.0	< 1
\fBatanhf\fR	1.0	< 1
\fBcbrtf\fR	1.0	< 1
\fBcosf\fR	1.0	< 1
\fBcoshf\fR	1.0	< 1
\fBerff\fR	1.0	< 1
\fBerfcf\fR	1.0	< 1
\fBexpf\fR	1.0	< 1
\fBexp2f\fR	1.0	< 1
\fBexpm1f\fR	1.0	< 1
\fBhypotf\fR	1.0	< 1
\fBlgammaf\fR	1.0	< 1
\fBlogf\fR	1.0	< 1
\fBlog10f\fR	1.0	< 1
\fBlog1pf\fR	1.0	< 1
\fBlog2f\fR	1.0	< 1
\fBpowf\fR	1.0	< 1
\fBsinf\fR	1.0	< 1
\fBsinhf\fR	1.0	< 1
\fBsqrtf\fR	0.5	0.500	[1]
\fBtanf\fR	1.0	< 1
\fBtanhf\fR	1.0	< 1
\fBtgammaf\fR	1.0	< 1
.TE

.SS "Double precision real functions (SPARC and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBacos\fR	1.0	< 1
\fBacosh\fR	4.0	1.878
\fBasin\fR	1.0	< 1
\fBasinh\fR	7.0	1.653
\fBatan\fR	1.0	<1
\fBatan2\fR	2.5	1.475
\fBatanh\fR	4.0	1.960
\fBcbrt\fR	1.0	< 1
\fBcos\fR	1.0	< 1
\fBcosh\fR	3.0	1.168
\fBerf\fR	4.0	0.959
\fBerfc\fR	6.0	2.816
\fBexp\fR	1.0	< 1
\fBexp2\fR	2.0	1.050
\fBexpm1\fR	1.0	< 1
\fBhypot\fR	1.0	< 1
\fBlgamma\fR	61.5	5.629	[2]
\fBlog\fR	1.0	< 1
\fBlog10\fR	3.5	1.592
\fBlog1p\fR	1.0	< 1
\fBlog2\fR	1.0	< 1
\fBpow\fR	1.0	< 1
\fBsin\fR	1.0	< 1
\fBsinh\fR	4.0	2.078
\fBsqrt\fR	0.5	0.500	[1]
\fBtan\fR	1.0	< 1
\fBtanh\fR	3.5	2.136
\fBtgamma\fR	1.0	< 1
.TE

.SS "Double precision real functions (x86)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBacos\fR	1.0	< 1
\fBacosh\fR	4.0	1.694
\fBasin\fR	1.0	< 1
\fBasinh\fR	7.0	1.493
\fBatan\fR	1.0	< 1
\fBatan2\fR	1.0	< 1
\fBatanh\fR	4.0	1.445
\fBcbrt\fR	1.0	< 1
\fBcos\fR	1.0	< 1
\fBcosh\fR	3.0	1.001
\fBerf\fR	4.0	0.932
\fBerfc\fR	6.0	2.728
\fBexp\fR	1.0	< 1
\fBexp2\fR	1.0	< 1
\fBexpm1\fR	1.0	< 1
\fBhypot\fR	1.0	< 1
\fBlgamma\fR	61.5	2.654	[2]
\fBlog\fR	1.0	< 1
\fBlog10\fR	1.0	< 1
\fBlog1p\fR	1.0	< 1
\fBlog2\fR	1.0	< 1
\fBpow\fR	1.0	< 1
\fBsin\fR	1.0	< 1
\fBsinh\fR	4.0	1.458
\fBsqrt\fR	0.5003	0.500	[1]
\fBtan\fR	1.0	< 1
\fBtanh\fR	3.5	1.592
\fBtgamma\fR	1.0	< 1
.TE

.SS "Quadruple precision real functions (SPARC)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBacosl\fR	3.5	1.771
\fBacoshl\fR	8.0	1.275
\fBasinl\fR	4.0	2.007
\fBasinhl\fR	9.0	1.823
\fBatanl\fR	1.0	< 1
\fBatan2l\fR	2.5	1.102
\fBatanhl\fR	4.0	1.970
\fBcbrtl\fR	1.0	< 1
\fBcosl\fR	1.0	< 1
\fBcoshl\fR	3.5	0.985
\fBerfl\fR	2.0	0.779
\fBerfcl\fR	68.5	13.923
\fBexpl\fR	1.0	< 1
\fBexp2l\fR	2.0	0.714
\fBexpm1l\fR	2.0	1.020
\fBhypotl\fR	1.0	< 1
\fBlgammal\fR	18.5	2.916	[2]
\fBlogl\fR	1.0	< 1
\fBlog10l\fR	3.5	1.156
\fBlog1pl\fR	2.0	1.216
\fBlog2l\fR	3.5	1.675
\fBpowl\fR	1.0	< 1
\fBsinl\fR	1.0	< 1
\fBsinhl\fR	4.5	1.589
\fBsqrtl\fR	0.5	0.500	[1]
\fBtanl\fR	4.5	2.380
\fBtanhl\fR	4.5	1.692
\fBtgammal\fR	1.0	< 1
.TE

.SS "Extended precision real functions (x86 and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBacosl\fR	3.0	1.868
\fBacoshl\fR	8.0	2.352
\fBasinl\fR	3.0	1.716
\fBasinhl\fR	9.0	2.346
\fBatanl\fR	1.0	< 1
\fBatan2l\fR	1.0	< 1
\fBatanhl\fR	4.0	2.438
\fBcbrtl\fR	1.0	< 1
\fBcosl\fR	1.0	< 1
\fBcoshl\fR	3.5	1.288
\fBerfl\fR	1.0	< 1
\fBerfcl\fR	78.5	13.407
\fBexpl\fR	3.5	1.291
\fBexp2l\fR	1.5	0.807
\fBexpm1l\fR	4.0	1.936
\fBhypotl\fR	3.5	2.087
\fBlgammal\fR	22.5	4.197	[2]
\fBlogl\fR	2.0	0.881
\fBlog10l\fR	2.0	1.284
\fBlog1pl\fR	5.0	2.370
\fBlog2l\fR	1.0	< 1
\fBpowl\fR	32770.0	4478.132
\fBsinl\fR	1.0	< 1
\fBsinhl\fR	4.5	2.356
\fBsqrtl\fR	0.5	0.500	[1]
\fBtanl\fR	4.5	2.366
\fBtanhl\fR	4.5	2.417
\fBtgammal\fR	1.0	< 1
.TE

.SS "Notes:"
.ne 2
.mk
.na
\fB[1]\fR
.ad
.RS 7n
.rt
On SPARC and amd64, \fBsqrtf\fR, \fBsqrt\fR, and \fBsqrtl\fR are correctly
rounded in accordance with IEEE 754. On x86, \fBsqrtl\fR is correctly rounded,
\fBsqrtf\fR is correctly rounded provided the result is narrowed to single
precision as discussed above, but \fBsqrt\fR might not be correctly rounded due
to "double rounding": when the intermediate value computed to extended precision
lies exactly halfway between two representable numbers in double precision, the
result of rounding the intermediate value to double precision is determined by
the round-ties-to-even rule. If this rule causes the second rounding to round in
the same direction as the first, the net rounding error can exceed 0.5 ulps.
(The error is bounded instead by 0.5*(1 + 2^-11) ulps.)
.RE

.sp
.ne 2
.mk
.na
\fB[2]\fR
.ad
.RS 7n
.rt
Error bounds for lgamma and lgammal apply only for positive arguments.
.RE

.SS "Complex functions"
The real-valued complex functions \fBcabsf\fR, \fBcabs\fR, \fBcabsl\fR,
\fBcargf\fR, \fBcarg\fR, and \fBcargl\fR are equivalent to the real functions
\fBhypotf\fR, \fBhypot\fR, \fBhypotl\fR, \fBatan2f\fR, \fBatan2\fR, and
\fBatan2l\fR, respectively. The error bounds and observed errors given above for
the latter functions also apply to the former.
.sp
.LP
The complex functions listed below are complex-valued. For each function, the
error bound shown applies separately to both the real and imaginary parts of the
result. (For example, both the real and imaginary parts of \fBcacosf\fR(\fIz\fR)
are accurate to within 1 ulp regardless of their magnitudes.) Similarly, the
largest observed error shown is the largest error found in either the real or
the imaginary part of the result.
.SS "Single precision complex functions (SPARC and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacosf\fR, \fBcacoshf\fR	1	< 1	[1]
\fBcasinf\fR, \fBcasinhf\fR	1	< 1
\fBcatanf\fR, \fBcatanhf\fR	6	< 1
\fBccosf\fR, \fBccoshf\fR	10	2.012
\fBcexpf\fR	3	2.239
\fBclogf\fR	3	< 1
\fBcpowf\fR	\(em	< 1	[2]
\fBcsinf\fR, \fBcsinhf\fR	10	2.009
\fBcsqrtf\fR	4	< 1
\fBctanf\fR, \fBctanhf\fR	13	6.987
.TE

.SS "Single precision complex functions (x86)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacosf\fR, \fBcacoshf\fR	1	< 1	[1]
\fBcasinf\fR, \fBcasinhf\fR	1	< 1
\fBcatanf\fR, \fBcatanhf\fR	6	< 1
\fBccosf\fR, \fBccoshf\fR	10	1.984
\fBcexpf\fR	3	1.984
\fBclogf\fR	3	< 1
\fBcpowf\fR	\(em	< 1	[2]
\fBcsinf\fR, \fBcsinhf\fR	10	1.973
\fBcsqrtf\fR	4	< 1
\fBctanf\fR, \fBctanhf\fR	13	4.657
.TE

.SS "Double precision complex functions (SPARC and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacos\fR, \fBcacosh\fR	9	3.831	[1]
\fBcasin\fR, \fBcasinh\fR	9	3.732
\fBcatan\fR, \fBcatanh\fR	6	4.179
\fBccos\fR, \fBccosh\fR	10	3.832
\fBcexp\fR	3	2.255
\fBclog\fR	3	2.870
\fBcpow\fR	-	-	[2]
\fBcsin\fR, \fBcsinh\fR	10	3.722
\fBcsqrt\fR	4	3.204
\fBctan\fR, \fBctanh\fR	13	7.143
.TE

.SS "Double precision complex functions (x86)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacos\fR, \fBcacosh\fR	9	3.624	[1]
\fBcasin\fR, \fBcasinh\fR	9	3.624
\fBcatan\fR, \fBcatanh\fR	6	2.500
\fBccos\fR, \fBccosh\fR	10	2.929
\fBcexp\fR	3	2.147
\fBclog\fR	3	1.927
\fBcpow\fR	-	-	[2]
\fBcsin\fR, \fBcsinh\fR	10	2.918
\fBcsqrt\fR	4	1.914
\fBctan\fR, \fBctanh\fR	13	4.630
.TE

.SS "Quadruple precision complex functions (SPARC)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacosl\fR, \fBcacoshl\fR	9	3	[1]
\fBcasinl\fR, \fBcasinhl\fR	9	3
\fBcatanl\fR, \fBcatanhl\fR	6	3
\fBccosl\fR, \fBccoshl\fR	10	3
\fBcexpl\fR	3	2
\fBclogl\fR	3	2
\fBcpowl\fR	-	-	[2]
\fBcsinl\fR, \fBcsinhl\fR	10	3
\fBcsqrtl\fR	4	3
\fBctanl\fR, \fBctanhl\fR	13	5
.TE

.SS "Extended precision complex functions (x86 and amd64)"

.TS
tab(	);
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
cw(1.38i) cw(1.38i) cw(1.38i) cw(1.38i)
.
	error bound	largest error
function	 (ulps)	observed (ulps)	notes
\fBcacosl\fR, \fBcacoshl\fR	9	2	[1]
\fBcasinl\fR, \fBcasinhl\fR	9	2
\fBcatanl\fR, \fBcatanhl\fR	6	2
\fBccosl\fR, \fBccoshl\fR	10	3
\fBcexpl\fR	3	2.699
\fBclogl\fR	3	1
\fBcpowl\fR	-	-	[2]
\fBcsinl\fR, \fBcsinhl\fR	10	3
\fBcsqrtl\fR	4	1.452
\fBctanl\fR, \fBctanhl\fR	13	5
.TE

.SS "Notes:"
.ne 2
.mk
.na
\fB[1]\fR
.ad
.RS 7n
.rt
The complex hyperbolic trigonometric functions are equivalent by symmetries to
their circular trigonometric counterparts. Because the implementations of these
functions exploit these symmetries, corresponding functions have the same error
bounds and observed errors.
.RE

.sp
.ne 2
.mk
.na
\fB[2]\fR
.ad
.RS 7n
.rt
For large arguments, the results computed by \fBcpowf\fR, \fBcpow\fR, and
\fBcpowl\fR can have unbounded relative error. It might be possible to give
error bounds for specific domains, but no such bounds are currently available.
The observed errors shown are for the domain {(\fIz\fR,\fIw\fR) :
\fBmax\fR(|\fBRe\fR \fIz\fR|, |\fBIm\fR \fIz\fR|, |\fBRe\fR \fIw\fR|, |\fBIm\fR
\fIw\fR|) <= 1}.
.RE

.SH FILES
.ne 2
.mk
.na
\fB\fB/lib/libm.so.2\fR\fR
.ad
.RS 21n
.rt
shared object
.RE

.sp
.ne 2
.mk
.na
\fB\fB/lib/64/libm.so.2\fR\fR
.ad
.RS 21n
.rt
64-bit shared object
.RE

.SH ATTRIBUTES
See \fBattributes\fR(5) for descriptions of the following attributes:
.sp

.sp
.TS
tab(	) box;
cw(2.75i) |cw(2.75i)
lw(2.75i) |lw(2.75i)
.
ATTRIBUTE TYPE	ATTRIBUTE VALUE
_
MT-Level	Safe with exceptions
.TE

.sp
.LP
As described on the \fBlgamma\fR(3M) manual page, \fBgamma()\fR and
\fBlgamma()\fR and their \fBfloat\fR and \fBlong double\fR counterparts are
Unsafe. All other functions in \fBlibm.so.2\fR are MT-Safe.
.SH SEE ALSO
\fBIntro\fR(3), \fBlgamma\fR(3M), \fBmath.h\fR(3HEAD), \fBattributes\fR(5),
\fBstandards\fR(5)