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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
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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
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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
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libc.h is only for weak_alias so include it directly where it is used
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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.
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copy the fix from i386: return -1 instead of exp2l(x)-1 when x <= -65
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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
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only fma used these macros and the explicit union is clearer
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* use new ldshape union consistently
* add ld128 support to frexpl
* simplify sqrtl comment (ld64 is not just arm)
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remove STRICT_ASSIGN (c99 semantics is assumed) and use the conventional
union to prepare the scaling factor (so libm.h is no longer needed)
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in lgammal don't handle 1 and 2 specially, in fma use the new ldshape
union instead of ld80 one.
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* 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)
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* 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)
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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.
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* added ld128 support from freebsd fdlibm (untested)
* using new ldshape union instead of IEEEl2bits
* inexact status flag is not supported
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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
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* 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
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* 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)
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use -1/(x*x) instead of -1/(x+0) to return -inf, -0+0 is -0 in
downward rounding mode
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* 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)
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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
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apparently this label change was not carried over when adapting the
changes from the i386 version.
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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)
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* remove volatile hacks
* don't care about inexact flag for now (removed all the +-tiny)
* fix atanl to raise underflow properly
* remove signed int arithmetics
* use pi/2 instead of pi_o_2 (gcc generates the same code, which is not
correct, but it does not matter: we mainly care about nearest rounding)
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underflow is raised by an inexact subnormal float store,
since subnormal operations are slow, check the underflow
flag and skip the store if it's already raised
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for these functions f(x)=x for small inputs, because f(0)=0 and
f'(0)=1, but for subnormal values they should raise the underflow
flag (required by annex F), if they are approximated by a polynomial
around 0 then spurious underflow should be avoided (not required by
annex F)
all these functions should raise inexact flag for small x if x!=0,
but it's not required by the standard and it does not seem a worthy
goal, so support for it is removed in some cases.
raising underflow:
- x*x may not raise underflow for subnormal x if FLT_EVAL_METHOD!=0
- x*x may raise spurious underflow for normal x if FLT_EVAL_METHOD==0
- in case of double subnormal x, store x as float
- in case of float subnormal x, store x*x as float
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The underflow exception is not raised correctly in some
cornercases (see previous fma commit), added comments
with examples for fmaf, fmal and non-x86 fma.
In fmaf store the result before returning so it has the
correct precision when FLT_EVAL_METHOD!=0
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1) in downward rounding fma(1,1,-1) should be -0 but it was 0 with
gcc, the code was correct but gcc does not support FENV_ACCESS ON
so it used common subexpression elimination where it shouldn't have.
now volatile memory access is used as a barrier after fesetround.
2) in directed rounding modes there is no double rounding issue
so the complicated adjustments done for nearest rounding mode are
not needed. the only exception to this rule is raising the underflow
flag: assume "small" is an exactly representible subnormal value in
double precision and "verysmall" is a much smaller value so that
(long double)(small plus verysmall) == small
then
(double)(small plus verysmall)
raises underflow because the result is an inexact subnormal, but
(double)(long double)(small plus verysmall)
does not because small is not a subnormal in long double precision
and it is exact in double precision.
now this problem is fixed by checking inexact using fenv when the
result is subnormal
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* use unsigned arithmetics
* use unsigned to store arg reduction quotient (so n&3 is understood)
* remove z=0.0 variables, use literal 0
* raise underflow and inexact exceptions properly when x is small
* fix spurious underflow in tanl
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* use unsigned arithmetics on the representation
* store arg reduction quotient in unsigned (so n%2 would work like n&1)
* use different convention to pass the arg reduction bit to __tan
(this argument used to be 1 for even and -1 for odd reduction
which meant obscure bithacks, the new n&1 is cleaner)
* raise inexact and underflow flags correctly for small x
(tanl(x) may still raise spurious underflow for small but normal x)
(this exception raising code increases codesize a bit, similar fixes
are needed in many other places, it may worth investigating at some
point if the inexact and underflow flags are worth raising correctly
as this is not strictly required by the standard)
* tanf manual reduction optimization is kept for now
* tanl code path is cleaned up to follow similar logic to tan and tanf
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When FLT_EVAL_METHOD!=0 (only i386 with x87 fp) the excess
precision of an expression must be removed in an assignment.
(gcc needs -fexcess-precision=standard or -std=c99 for this)
This is done by extra load/store instructions which adds code
bloat when lot of temporaries are used and it makes the result
less precise in many cases.
Using double_t and float_t avoids these issues on i386 and
it makes no difference on other archs.
For now only a few functions are modified where the excess
precision is clearly beneficial (mostly polynomial evaluations
with temporaries).
object size differences on i386, gcc-4.8:
old new
__cosdf.o 123 95
__cos.o 199 169
__sindf.o 131 95
__sin.o 225 203
__tandf.o 207 151
__tan.o 605 499
erff.o 1470 1416
erf.o 1703 1649
j0f.o 1779 1745
j0.o 2308 2274
j1f.o 1602 1568
j1.o 2286 2252
tgamma.o 1431 1424
math/*.o 64164 63635
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common part of erf and erfc was put in a separate function which
saved some space and the new code is using unsigned arithmetics
erfcf had a bug: for some inputs in [7.95,8] the result had
more than 60ulp error: in expf(-z*z - 0.5625f) the argument
must be exact but not enough lowbits of z were zeroed,
-SET_FLOAT_WORD(z, ix&0xfffff000);
+SET_FLOAT_WORD(z, ix&0xffffe000);
fixed the issue
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both jn and yn functions had integer overflow issues for large
and small n
to handle these issues nm1 (== |n|-1) is used instead of n and -n
in the code and some loops are changed to make sure the iteration
counter does not overflow
(another solution could be to use larger integer type or even double
but that has more size and runtime cost, on x87 loading int64_t or
even uint32_t into an fpu register is more than two times slower than
loading int32_t, and using double for n slows down iteration logic)
yn(-1,0) now returns inf
posix2008 specifies that on overflow and at +-0 all y0,y1,yn functions
return -inf, this is not consistent with math when n<0 odd integer in yn
(eg. when x->0, yn(-1,x)->inf, but historically yn(-1,0) seems to be
special cased and returned -inf)
some threshold values in jnf and ynf were fixed that seems to be
incorrectly copy-pasted from the double version
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a common code path in j1 and y1 was factored out so the resulting
object code is a bit smaller
unsigned int arithmetics is used for bit manipulation
j1(-inf) now returns 0 instead of -0
an incorrect threshold in the common code of j1f and y1f got fixed
(this caused spurious overflow and underflow exceptions)
the else branch in pone and pzero functions are fixed
(so code analyzers dont warn about uninitialized values)
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a common code path in j0 and y0 was factored out so the resulting
object code is smaller
unsigned int arithmetics is used for bit manipulation
the logic of j0 got a bit simplified (x < 1 case was handled
separately with a bit higher precision than now, but there are large
errors in other domains anyway so that branch has been removed)
some threshold values were adjusted in j0f and y0f
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previously 0x1p-1000 and 0x1p1000 was used for raising inexact
exception like x+tiny (when x is big) or x+huge (when x is small)
the rational is that these float consts are large enough
(0x1p-120 + 1 raises inexact even on ld128 which has 113 mant bits)
and float consts maybe smaller or easier to load on some platforms
(on i386 this reduced the object file size by 4bytes in some cases)
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this is not a full rewrite just fixes to the special case logic:
+-0 and non-integer x<INT_MIN inputs incorrectly raised invalid
exception and for +-0 the return value was wrong
so integer test and odd/even test for negative inputs are changed
and a useless overflow test was removed
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comments are kept in the double version of the function
compared to fdlibm/freebsd we partition the domain into one
more part and select different threshold points:
now the [log(5/3)/2,log(3)/2] and [log(3)/2,inf] domains
should have <1.5ulp error
(so only the last bit may be wrong, assuming good exp, expm1)
(note that log(3)/2 and log(5/3)/2 are the points where tanh
changes resolution: tanh(log(3)/2)=0.5, tanh(log(5/3)/2)=0.25)
for some x < log(5/3)/2 (~=0.2554) the error can be >1.5ulp
but it should be <2ulp
(the freebsd code had some >2ulp errors in [0.255,1])
even with the extra logic the new code produces smaller
object files
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comments are kept in the double version of the function
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changed the algorithm: large input is not special cased
(when exp(-x) is small compared to exp(x))
and the threshold values are reevaluated
(fdlibm code had a log(2)/2 cutoff for which i could not find
justification, log(2) seems to be a better threshold and this
was verified empirically)
the new code is simpler, makes smaller binaries and should be
faster for common cases
the old comments were removed as they are no longer true for the
new algorithm and the fdlibm copyright was dropped as well
because there is no common code or idea with the original anymore
except for trivial ones.
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