<html><head><title>libm</title></head><body>
<h2>libm</h2>
-<p>This page is about designing libm for the
-<a href="http://www.etalabs.net/musl/">musl</a> libc.
-<p>Writing the math code from scratch is a huge work
-so already existing code is used.
+<p>This page is about libm for the
+<a href="http://www.musl-libc.org/">musl</a> libc.
<ul>
-<li><a href="#prototype">Prototype</a>
<li><a href="#sources">Sources</a>
-<li><a href="#issues">Design issues</a>
+<li><a href="#rules">General rules</a>
<li><a href="#representation">Representation</a>
<li><a href="#ugly">Ugly</a>
<li><a href="#implementations">libm implementations</a>
<li><a href="#tests">libm tests</a>
</ul>
-<h3><a name="prototype" href="#prototype">Prototype</a></h3>
-<ul>
-<li><a href="/git/?p=libm">browse</a> with gitweb
-<li>over git (recommended)
-<pre>
-git clone git://nsz.repo.hu:45100/repo/libm
-</pre>
-<li>over http (slower)
-<pre>
-git clone http://nsz.repo.hu/repo/libm
-</pre>
-<li>over ssh (passwd is anon)
-<pre>
-git clone ssh://anon@nsz.repo.hu:45022/repo/libm
-</pre>
-</ul>
-
<h3><a name="sources" href="#sources">Sources</a></h3>
-<p>The math code is mostly from freebsd which in turn
-is based on <a href="http://www.netlib.org/fdlibm/">fdlibm</a>.
+<p>Writing math code from scratch is a huge work so already existing code is
+used. Several math functions are taken from the
+<a href="http://freebsd.org/">freebsd</a> libm and a few from the
+<a href="http://openbsd.org/">openbsd</a> libm implementations.
+Both of them are based on <a href="http://www.netlib.org/fdlibm/">fdlibm</a>.
+The freebsd libm seems to be the most well maintained and most correct version
+of fdlibm.
<p>sources:
<ul>
<li>freebsd /lib/msun (<a href="http://svnweb.FreeBSD.org/base/head/lib/msun/">browse</a>)
</pre>
</ul>
-<h3><a name="issues" href="#issues">Design issues</a></h3>
-
+<h3><a name="rules" href="#rules">General rules</a></h3>
<ul>
-<li>Code organization
-<p>The long double code uses ifdefs and macro hacks to allow
-ld64, ld80 and ld128 implementation in the same file.
-<p>Pending questions:
+<li>Assumptions about floating-point representation and arithmetics
+(see <a href="http://port70.net/~nsz/c/c99/n1256.html#F.2">c99 annex F.2</a>):
<ul>
-<li>What are the correct long double ifdefs?
-<li>Check fpu precision setting at runtime in long double code?
-<li>support c99 tgmath.h?
-<li>TODO: complex
-</ul>
-
-<li>Workarounds
-<p>The bsd libm code has many workarounds for various
-compiler issues. It's not clear what's the best way
-to handle the uglyness.
-<p>Pending questions:
-<ul>
-<li>Use STDC pragmas (eventhough gcc does not support them)?
-<li>Where should we use volatile (to avoid evaluation precision and const folding issues)?
-<li>What tricks to use for proper exception raising? (huge*huge, etc)
-<li>Should signaling nans be considered?
-<li>Which compiler flags should be used (-frounding-math, -ffloat-store)?
-<li>TODO: inline/macro optimization for small functions (isnan, isinf, signbit, creal, cimag,..)
-</ul>
-
-<li>Code cleanups
-<p>The fdlibm code has code style and correctness issues
-which might worth addressing.
-<p>Pending questions:
-<ul>
-<li>Should 0x1p0 float format be preferred over decimal format?
-<li>How much c99 is allowed? (inline, compound literals, 0x1p0,..)
-<li>Which one should be used 0f, 0l, 0.0, static const double zero = 0,.. ?
-<li>Can we use a polyeval(coeffs, x) function instead of writing out the poly?
-<li>TODO: use uint32_t instead of int32_t (to avoid signed int overflow and right shifts)
-<li>TODO: prefer isnan, signbit,.. over inplace bithacks
+<li>float is ieee binary32
+<li>double is ieee binary64
+<li>long double is either ieee binary64 or little-endian 80bit extended precision (x87 fpu)
</ul>
+(other long double representations may be supported in the future, until then
+long double math functions will be missing on non-supported platforms)
+<li>On soft-float architectures fenv should not be expected to work according to
+the c and ieee standards (ie. rounding modes and exceptions are not supported,
+the fenv functions are dummy ones)
+<li>Floating-point exception flags should be properly set in math functions
+according to c99 annex F, but without using fenv.h
+(eg. overflow flag can be raised by 0x1p900*0x1p900, because this works even
+without fenv support)
+<li>Most functions need to be precise only in nearest rounding mode.
+<li>Returned floating-point values should be correctly rounded in most cases,
+but the last bit may be wrong:
+<pre>
+ |error| < 1.5 ulp
+</pre>
+should hold for most functions.
+(error is the difference between the exact result and the calculated
+floating-point value)
+(in theory correct rounding can be achieved but with big implementation cost,
+see <a href="http://lipforge.ens-lyon.fr/www/crlibm/">crlibm</a>)
+<li>At least the following functions must be correctly rounded:
+ceil, copysign, fabs, fdim, floor, fma, fmax, fmin, fmod, frexp, ldexp, logb,
+modf, nearbyint, nextafter, nexttoward, rint, remainder, remquo, round, scalbln,
+scalbn, sqrt, trunc.
+<li>Mathematical properties of functions should be as expected
+(monotonicity, range, symmetries).
+<li>If the FPU precision is altered then nothing is guaranteed to work.
+(ie. when long double does not have full 80bit precision on i386 then things
+may break, this also means that compiler must spill fpu registers at 80bits
+precision)
+<li>Signaling NaN is not supported
+<li>Quiet NaN is supported but all NaNs are treated equally without special
+attention to the internal representation of a NaN
+(eg. the sign of NaN may not be preserved).
+<li>Most gcc bug workarounds should be removed from the code
+(STRICT_ASSIGN macro is used when excessive precision is harmful and
+FORCE_EVAL when expressions must be evaluated for their side-effect, other
+usage of volatile is not justified, hacks around long double constants are
+not justified eventhough gcc can miscompile those with non-default FPU setting)
+<li>When excessive precision is not harmful, temporary variables
+should be float_t or double_t (so on i386 no superfluous store is
+generated)
+<li>Whenever fenv is accessed the FENV_ACCESS pragma of c99 should be used
+(eventhough gcc does not yet support it), and all usage of optional FE_
+macros should be protected by #ifdef
+<li>For bit manipulation of floating-point values an union should be used
+(eg. union {float f; uint32_t i;})
+<li>uint32_t and uint64_t should be used for bit manipulations.
+(eg signed int must not be used in bit shifts etc when it might invoke
+undefined or implementation defined behaviour).
+<li>POSIX namespace rules must be respected.
+<li>c99 hexfloat syntax (0x1.0p0) should be used when it makes the code
+clearer, but not in public header files
+(those should be c++ and ansi c compatible)
+<li>The 'f' suffix should be used for single precision values (0.1f) when the
+value cannot be exactly represented or the type of the arithmetics is important
+((float)0.1 is not ok, that style may lead to double rounding issues, but eg.
+1.0 or 0.5 may be used instead of 1.0f or 0.5f in some cases)
+<li>Prefer classification macros (eg. isnan) over inplace bit hacks.
+<li>For every math function there should be a c implementation.
+(a notable exception now is sqrtl, since most fpu has instruction for it
+and on soft-float architectures long double == double)
+<li>The c implementation of a long double function should use ifdefs with the
+LDBL_MANT_DIG etc constants from float.h for architecture specific
+implementations.
+<li>In the musl source tree math.h functions go to src/math, complex.h functions
+to src/complex and fenv.h functions to src/fenv. And files are named after the
+functions they implement.
</ul>
<h3><a name="representation" href="#representation">Representation</a></h3>
<p>
Binary representation of floating point numbers matter
-because lots of bithacks are needed in the math code.
+because bit hacks are often needed in the math code.
+(in particular bit hacks are used instead of relational operations for nan
+and sign checks becuase relational operators raise invalid fp exception on nan
+and they treat -0.0 and +0.0 equally and more often than not these are not desired)
<p>
-float and double bit manipulation can be handled
-in a portable way in c:
+float and double bit manipulation can be handled in a portable way in c using
+union types:
<ul>
-<li>float is ieee binary32
-<li>double is ieee binary64
+<li>union {float f; uint32_t i;};
+<li>union {double f; uint64_t i;};
</ul>
-(<a href="http://repo.or.cz/w/c-standard.git/blob_plain/HEAD:/n1256.html#F.2">C99 annex F</a>
-makes these mandatory)
-<p>
-The endianness may still vary, but that can be worked
-around by using a union with a single large enough
-unsigned int. (The only exception is probably arm/oabi fpa
-where the word order of a double is not consistent with the
-endianness [<a href="http://wiki.debian.org/ArmEabiPort#ARM_floating_points">debian wiki on arm</a>],
-but we probably won't support that)
-<p>
-long double bit manipulation is harder as there are
-various representations:
+(assuming the bits in the object representation of 32bit and 64bit unsigned ints
+map to the floating-point representation according to ieee-754, this is not
+always the case, eg. old
+<a href="http://wiki.debian.org/ArmEabiPort#ARM_floating_points">arm floating-point accelerator</a>
+(FPA) used mixed endian double representation, but musl does not support the old
+arm ABI)
+<p>
+long double bit manipulation is harder as there are various representations
+and some of them don't map to any unsigned integer type:
<ul>
<li>ld64: long double is the same as double (ieee binary64)
<li>ld80: 80bit extended precision format [<a href="https://en.wikipedia.org/wiki/Extended_precision">wikipedia</a>]
</ul>
(and other non-standard formats are not supported)
<p>
-ld64 is easy to handle: all long double functions
-are just wrappers around the corresponding double ones
-(aliasing is non-conformant)
+In case of ld64 the bit manipulation is the same as with double
+and all long double math functions can be just wrappers around the
+corresponding double ones.
+(using symbol aliasing on the linker level is non-conformant
+since functions would not have unique address then)
<p>
-ld80 is the most common (i386, x86_64), it means
-64bit significand with explicit msb (inconsistent with other ieee formats),
-15bit exp, 1 sign bit.
+ld80 is the most common long double on linux (i386 and x86_64 abi),
+it means 64bit significand with explicit msb
+(inconsistent with other ieee formats), 15bit exp, 1 sign bit.
+The m68k (and m88k) architecture uses the same format, but different endianness:
+<ul>
+<li>union {long double f; struct{uint64_t m; uint16_t se; uint16_t pad;} i;}; // x86
+<li>union {long double f; struct{uint16_t se; uint16_t pad; uint64_t m;} i;}; // m68k
+</ul>
+where m is the significand and se is the sign and exponent.
<p>
-ld128 is rare (sparc64 with software emulation), it means
-113bit significand with implicit msb, 15bit exp, 1 sign bit.
+ld128 is rare (eg. sparc64 with software emulation), it means
+113bit significand with implicit msb, 15bit exp, 1 sign bit:
+<ul>
+<li>union {long double f; struct{uint16_t se; uint16_t hi; uint32_t mid; uint64_t lo;} i;};
+</ul>
<p>
-Endianness can vary (although on the supported i386 and x86_64 it is the same)
-and there is no large enough unsigned int to handle it.
-(In case of ld80 internal padding can vary as well, eg
-m68k and m88k cpus use different ld80 than the intel ones.)
-So there can be further variations in the binary representation
-than just ld80 and ld128.
-
+There are other non-conformant long double types: eg. the old SVR4 abi for ppc
+uses 128 bit long doubles, but it's software emulated and traditionally
+implemented using
+<a href="https://en.wikipedia.org/wiki/Quadruple_precision#Double-double_arithmetic">two doubles</a>
+(also called ibm long double as this is what ibm aix used on ppc).
+The ibm s390 supports the ieee 754-2008 compliant binary128 floating-point
+format, but previous ibm machines (S/370, S/360) used slightly different
+representation.
+<p>
+This variation shows the difficulty to consistently handle
+long double: the solution is to use ifdefs based on float.h and
+on the endianness and write different code for different architectures.
<h3><a name="ugly" href="#ugly">Ugly</a></h3>
<p>The ugly parts of libm hacking.
<p>Some notes are from:
<a href="http://www.vinc17.org/research/extended.en.html">http://www.vinc17.org/research/extended.en.html</a>
+<p>Useful info about floating-point in gcc:
+<a href="http://gcc.gnu.org/wiki/FloatingPointMath">http://gcc.gnu.org/wiki/FloatingPointMath</a>
<ul>
<li>Double rounding:
and then round to 64bit when storing it, this can
give different result than a single 64bit rounding.
(on x86-linux the default fpu setting is to round the
-results in extended precision, this only affects x87 instructions, not see2 etc)
+results in extended precision, this only affects x87 instructions, not sse2 etc)
(freebsd and openbsd use double precision by default)
<p>
So x = a+b may give different results depending on
<p>
<a href="http://repo.or.cz/w/c-standard.git/blob_plain/HEAD:/n1256.html#F.7.3">C99 annex F</a>
prohibits double rounding, but that's non-normative.
+<p>
+Note that the value of the result can only be ruined by
+double rounding in nearest rounding mode, but the double
+rounding issue haunts directed rounding modes as well:
+raising the underflow flag might be omitted.
+On x86 with downward rounding
+<pre>
+(double)(0x1p-1070 + 0x1p-2000L)
+</pre>
+does not raise underflow (only inexact) eventhough the
+final result is an inexact subnormal.
+
<li>Wider exponent range (x87 issue):
<p>
may be false when the two sides
are kept in different precision.
(This is not an x87 specific problem, it matters whenever there
-is a higher precision fp type than the currently used one.
+is a higher precision fp type than the currently used one and
+FLT_EVAL_METHOD!=0.
It goes away if the highest precision (long double) is used
everywhere, but that can have a huge penalty).
+(clang uses sse by default on i386 with FLT_EVAL_METHOD==0,
+while gcc uses the 80bit x87 fp registers and FLT_EVAL_METHOD==2)
<p>
C99 has a way to control this (see
<a href="http://repo.or.cz/w/c-standard.git/blob_plain/HEAD:/n1256.html#5.1.2.3">5.1.2.3 example 4</a>,
gcc 4.5 fixed it with '-fexcess-precision=standard'
(it is enabled by '-std=c99', but the default is
'-fexcess-precision=fast')
+(An alternative solution would be if gcc spilled the
+registers with temporary results without rounding,
+storing the 80 bit registers entirely in memory
+which would make the behaviour under FLT_EVAL_METHOD==2
+mode more predictable)
<p>
The workaround for older gcc is to force the
compiler to store the intermediate results:
what we want and don't depend on the implicit
excess precision).
+<li>Float literals
+<p>
+The standard allows 1 ulp errors in the conversion
+of decimal floating-point literals into floating-point
+values (it only requires the same result for the same
+literal thus <tt>1.1 - 1.1</tt> is always 0,
+but <tt>1.1 - 11e-1</tt> maybe +-0x1p-52 or 0).
+<p>
+A reasonable compiler always use correctly rounded
+conversion according to the default (nearest) rounding
+mode, but there are exceptions:
+the x87 has builtin constants which are faster to load
+from hardware than from memory
+(and the hw has sticky bit set correctly for rounding).
+gcc can recognize these constants so an operation on
+<pre>
+3.141592653589793238462643383L
+</pre>
+can turn into code that uses the <tt>fldpi</tt>
+instruction instead of memory loads.
+The only issue is that fldpi depends on
+the current rounding mode at runtime
+so the result can indeed be 1 ulp off compared
+to the compile-time rounded value.
+<p>
+According to the freebsd libm code gcc truncates long double
+const literals on i386.
+I assume this happens because freebsd uses 64bit long doubles by default
+(double precision) and gcc incorrectly uses the precision setting of the
+host platform instead of the target one, but i did not observe this on linux.
+(as a workaround sometimes double-double arithmetics was used
+to initialize long doubles on i386, but most of these should be
+fixed in musl's math code now)
+
<li>Compiler optimizations:
<p>
Runtime and compile time semantics may be different
C99 actually allows most of these optimizations
but they can be turned off with STDC pragmas (see
<a href="http://repo.or.cz/w/c-standard.git/blob_plain/HEAD:/n1256.html#6.10.6">6.10.6</a>).
-Unfortunately <a href="http://gcc.gnu.org/c99status.html">gcc does not support these pragmas</a>.
+Unfortunately <a href="http://gcc.gnu.org/c99status.html">gcc does not support these pragmas</a>
+nor clang (<a href="http://llvm.org/bugs/show_bug.cgi?id=8100">clang bug 8100</a>).
<p>
FENV_ACCESS ON tells the compiler that the code wants
to access the floating point environment (eg. set different rounding mode)
static const volatile two52 = 0x1p52;
</pre>
and using the '-frounding-math' gcc flag.
-<p>
-(According the freebsd libm code gcc truncates
-long double const literals on i386.
-I haven't yet verified if this still the case,
-but as a workaround double-double arithmetics is used:
-initializing the long double constant from two doubles)
<li>ld80 vs ld128
<p>
then simple arithmetics can be be used just for their
exception raising side effect
(eg. 1/0.0 to raise divbyzero), however beaware
-of compiler optimizations (dead code elimination,..).
+of compiler optimizations (constant folding and dead code elimination,..).
<p>
Unfortunately gcc does not always take fp exceptions into
account: a simple x = 1e300*1e300; may not raise overflow
exception at runtime, but get optimized into x = +inf.
see compiler optimizations above.
<p>
-Another x87 gcc bug related to fp exceptions is that
+Another x87 gcc bug related to fp exceptions is that in some cases
comparision operators (==, <, etc) don't raise invalid
when an operand is nan
(eventhough this is required by ieee + c99 annex F).
(see <a href="http://gcc.gnu.org/bugzilla/show_bug.cgi?id=52451">gcc bug52451</a>).
<p>
-The ieee standard defines signaling and quite nan
+The ieee standard defines signaling and quiet nan
floating-point numbers as well.
The c99 standard only considers quiet nan, but it allows
signaling nans to be supported as well.
Without signaling nans x * 1 is equivalent to x,
but if signaling nan is supported then the former
raises an invalid exception.
-This may complicates things further if one wants to write
+This may complicate things further if one wants to write
portable fp math code.
<p>
A further libm design issue is the math_errhandling macro:
are implemented as a single asm instruction (eg sqrt),
the only way to set errno is to query the fp exception flags
and then set the errno variable based on that.
-So eventhough errno may be convenient in libm it is
+So eventhough errno may be convenient, in libm it is
not the right thing to do.
+<p>
+For soft-float targets however errno seems to be the only option
+(which means annex K cannot be fully supported, as it requires
+the support of exception flags).
+The problem is that at context switches the fpu status should
+be saved and restored which is done by the kernel on hard-fp
+architectures when the state is in an fpu status word.
+In case of soft-fp emulation this must be done by the c runtime:
+context switches between threads can be supported with thread local
+storage of the exception state, but signal handlers may do floating-point
+arithmetics which should not alter the fenv state.
+Wrapping signal handlers is not possible/difficult for various
+reasons and the compiler cannot know which functions will be used
+as signal handlers, so the c runtime has no way to guarantee that
+signal handlers do not alter the fenv.
<li>Complex arithmetics
<p>
<p>
The freebsd libm code has many inconsistencies
(naming conventions, 0x1p0 notation vs decimal notation,..),
-one of them is the integer type used for bitmanipulations:
+one of them is the integer type used for bit manipulations:
The bits of a double are unpacked into one of
-int32_t, uint32_t and u_int32_t
+int, int32_t, uint32_t and u_int32_t
integer types.
<p>
-int32_t is used most often which is wrong because of
-implementation defined signed int representation.
+int32_t is used the most often which is not wrong in itself
+but it is used incorrectly in many places.
+<p>
+int is a bit worse because unlike int32_t it is not guaranteed
+to be 32bit two's complement representation. (but of course in
+practice they are the same)
+<p>
+The issues found so far are left shift of negative integers
+(undefined behaviour), right shift of negative integers
+(implementation defined behaviour), signed overflow
+(implementation defined behaviour), unsigned to signed conversion
+(implementation defined behaviour).
<p>
-In general signed int is not handled carefully
-in the libm code: scalbn even depends on signed int overflow.
+It is easy to avoid these issues without performance impact,
+but a bit of care should be taken around bit manipulations.
</ul>
<h3><a name="implementations" href="#implementations">libm implementations</a></h3>
projects / www / blobdiff