Some early x86_64 cpus (released before 2006) did not support sahf/lahf
instructions so they should be avoided (intel manual says they are only
supported if CPUID.80000001H:ECX.LAHF-SAHF[bit 0] = 1).
The workaround simplifies exp2l and expm1l because fucomip can be
used instead of the fucomp;fnstsw;sahf sequence copied from i386.
In fmodl and remainderl sahf is replaced by a simple bit test.
the idiomatic rounding of x is
n = x + toint - toint;
where toint is either 1/EPSILON (x is non-negative) or 1.5/EPSILON
(x may be negative and nearest rounding mode is assumed) and EPSILON is
according to the evaluation precision (the type of toint is not very
important, because single precision float can represent the 1/EPSILON of
ieee binary128).
in case of FLT_EVAL_METHOD!=0 this avoids a useless store to double or
float precision, and the long double code became cleaner with
1/LDBL_EPSILON instead of ifdefs for toint.
__rem_pio2f and __rem_pio2 functions slightly changed semantics:
on i386 a double-rounding is avoided so close to half-way cases may
get evaluated differently eg. as sin(pi/4-eps) instead of cos(pi/4+eps)
The old code used the rounding idiom incorrectly:
y = (double)(x + 0x1p52) - 0x1p52;
the cast is useless if FLT_EVAL_METHOD==0 and causes a second rounding
if FLT_EVAL_METHOD==2 which can give incorrect result in nearest rounding
mode, so the correct idiom is to add/sub a power-of-2 according to the
characteristics of double_t.
This did not cause actual bug because only i386 is affected where rint
is implemented in asm.
Other rounding functions use a similar idiom, but they give correct
results because they only rely on getting a neighboring integer result
and the rounding direction is fixed up separately independently of the
current rounding mode. However they should be fixed to use the idiom
correctly too.
previously the external definitions of these functions were omitted on
archs where long double is the same as double, since the code paths in
the math.h macros which would call them are unreachable. however, even
if they are unreachable, the definitions are still mandatory. omitting
them is invalid C, and in the case of a non-optimizing compiler, will
result in a link error.
__polevll, __p1evll and exp10l were provided on archs when long double
is the same as double. The first two were completely unused and exp10l
can be a wrapper around exp10.
modfl and sincosl were passing long double* instead of double*
to the wrapped double precision functions (on archs where long
double and double have the same size).
This is fixed now by using temporaries (this is not optimized
to a single branch so the generated code is a bit bigger).
Found by Morten Welinder.
- remove the HAVE_EFFICIENT_IRINT case: fn is an exact integer, so
it can be converted to int32_t a bit more efficiently than with a
cast (the rounding mode change can be avoided), but musl does not
support this case on any arch.
- __rem_pio2: use double_t where possible
- __rem_pio2f: use less assignments to avoid stores on i386
- use unsigned int bit manipulation (and union instead of macros)
- use hexfloat literals instead of named constants
* simplify sin_pi(x) (don't care about inexact here, the result is
inexact anyway, and x is not so small to underflow)
* in lgammal add the previously removed special case for x==1 and
x==2 (to fix the sign of zero in downward rounding mode)
* only define lgammal on supported long double platforms
* change tgamma so the generated code is a bit smaller
The log, log2 and log10 functions share a lot of code and to a lesser
extent log1p too. A small part of the code was kept separately in
__log1p.h, but since it did not capture much of the common code and
it was inlined anyway, it did not solve the issue properly. Now the
log functions have significant code duplication, which may be resolved
later, until then they need to be modified together.
logl, log10l, log2l, log1pl:
* Fix the sign when the return value should be -inf.
* Remove the volatile hack from log10l (seems unnecessary)
log1p, log1pf:
* Change the handling of small inputs: only |x|<2^-53 is special
(then it is enough to return x with the usual subnormal handling)
this fixes the sign of log1p(0) in downward rounding.
* Do not handle the k==0 case specially (other than skipping the
elaborate argument reduction)
* Do not handle 1+x close to power-of-two specially (this code was
used rarely, did not give much speed up and the precision wasn't
better than the general)
* Fix the correction term formula (c=1-(u-x) was used incorrectly
when x<1 but (double)(x+1)==2, this was not a critical issue)
* Use the exact same method for calculating log(1+f) as in log
(except in log1p the c correction term is added to the result).
log, logf, log10, log10f, log2, log2f:
* Use double_t and float_t consistently.
* Now the first part of log10 and log2 is identical to log (until the
return statement, hopefully this makes maintainence easier).
* Most special case formulas were removed (close to power-of-two and
k==0 cases), they increase the code size without providing precision
or performance benefits (and obfuscate the code).
Only x==1 is handled specially so in downward rounding mode the
sign of zero is correct (the general formula happens to give -0).
* For x==0 instead of -1/0.0 or -two54/0.0, return -1/(x*x) to force
raising the exception at runtime.
* Arg reduction code is changed (slightly simplified)
* The thresholds for arg reduction to [sqrt(2)/2,sqrt(2)] are now
consistently the [0x3fe6a09e00000000,0x3ff6a09dffffffff] and the
[0x3f3504f3,0x3fb504f2] intervals for double and float reductions
respectively (the exact threshold values are not critical)
* Remove the obsolete comment for the FLT_EVAL_METHOD!=0 case in log2f
(The same code is used for all eval methods now, on i386 slightly
simpler code could be used, but we have asm there anyway)
all:
* Fix signed int arithmetics (using unsigned for bitmanipulation)
* Fix various comments
the underlying problem was not incorrect sign extension (fixed in the
previous commit to this file by nsz) but that code that treats "long"
as 32-bit was copied blindly from i386 to x86_64.
now lrintl is identical to llrintl on x86_64, as it should be.
gcc did not always drop excess precision according to c99 at assignments
before version 4.5 even if -std=c99 was requested which caused badly
broken mathematical functions on i386 when FLT_EVAL_METHOD!=0
but STRICT_ASSIGN was not used consistently and it is worked around for
old compilers with -ffloat-store so it is no longer needed
the new convention is to get the compiler respect c99 semantics and when
excess precision is not harmful use float_t or double_t or to specialize
code using FLT_EVAL_METHOD
apparently gnulib requires invalid long double representations
to be handled correctly in printf so we classify them according
to how the fpu treats them: bad inf is nan, bad nan is nan,
bad normal is nan and bad subnormal/zero is minimal normal
in atanh exception handling was left to the called log functions,
but the argument to those functions could underflow or overflow.
use double_t and float_t to avoid some useless stores on x86
acosh(x) is invalid for x<1, acoshf tried to be clever using
signed comparisions to handle all x<2 the same way, but the
formula was wrong on large negative values.
there were two problems:
* omitted underflow on subnormal results: exp2l(-16383.5) was calculated
as sqrt(2)*2^-16384, the last bits of sqrt(2) are zero so the down scaling
does not underflow eventhough the result is in subnormal range
* spurious underflow for subnormal inputs: exp2l(0x1p-16400) was evaluated
as f2xm1(x)+1 and f2xm1 raised underflow (because inexact subnormal result)
the first issue is fixed by raising underflow manually if x is in
(-32768,-16382] and not integer (x-0x1p63+0x1p63 != x)
the second issue is fixed by treating x in (-0x1p64,0x1p64) specially
for these fixes the special case handling was completely rewritten
* use float_t and double_t
* cleanup subnormal handling
* bithacks according to the new convention (ldshape for long double
and explicit unions for float and double)
* don't care about inexact flag
* use double_t and float_t (faster, smaller, more precise on x86)
* exp: underflow when result is zero or subnormal and not -inf
* exp2: underflow when result is zero or subnormal and not exact
* expm1: underflow when result is zero or subnormal
* expl: don't underflow on -inf
* exp2: fix incorrect comment
* expm1: simplify special case handling and overflow properly
* expm1: cleanup final scaling and fix negative left shift ub (twopk)
ld128 support was added to internal kernel functions (__cosl, __sinl,
__tanl, __rem_pio2l) from freebsd (not tested, but should be a good
start for when ld128 arch arrives)
__rem_pio2l had some code cleanup, the freebsd ld128 code seems to
gather the results of a large reduction with precision loss (fixed
the bug but a todo comment was added for later investigation)
the old copyright was removed from the non-kernel wrapper functions
(cosl, sinl, sincosl, tanl) since these are trivial and the interesting
parts and comments had been already rewritten.
method: if there is a large difference between the scale of x and y
then the larger magnitude dominates, otherwise reduce x,y so the
argument of sqrt (x*x+y*y) does not overflow or underflow and calculate
the argument precisely using exact multiplication. If the argument
has less error than 1/sqrt(2) ~ 0.7 ulp, then the result has less error
than 1 ulp in nearest rounding mode.
the original fdlibm method was the same, except it used bit hacks
instead of dekker-veltkamp algorithm, which is problematic for long
double where different representations are supported. (the new hypot
and hypotl code should be smaller and faster on 32bit cpu archs with
fast fpu), the new code behaves differently in non-nearest rounding,
but the error should be still less than 2ulps.
ld80 and ld128 are supported
* results are exact
* modfl follows truncl (raises inexact flag spuriously now)
* modf and modff only had cosmetic cleanup
* remainder is just a wrapper around remquo now
* using iterative shift+subtract for remquo and fmod
* ld80 and ld128 are supported as well
* faster, smaller, cleaner implementation than the bit hacks of fdlibm
* use arithmetics like y=(double)(x+0x1p52)-0x1p52, which is an integer
neighbor of x in all rounding modes (0<=x<0x1p52) and only use bithacks
when that's faster and smaller (for float it usually is)
* the code assumes standard excess precision handling for casts
* long double code supports both ld80 and ld128
* nearbyint is not changed (it is a wrapper around rint)
* consistent code style
* explicit union instead of typedef for double and float bit access
* turn FENV_ACCESS ON to make 0/0.0f raise invalid flag
* (untested) ld128 version of ilogbl (used by logbl which has ld128 support)
new ldshape union, ld128 support is kept, code that used the old
ldshape union was rewritten (IEEEl2bits union of freebsd libm is
not touched yet)
ld80 __fpclassifyl no longer tries to handle invalid representation
if FLT_EVAL_METHOD!=0 check if (double)(1/x) is subnormal and not a
power of 2 (if 1/x is power of 2 then either it is exact or the
long double to double rounding already raised inexact and underflow)
Szabolcs Nagy