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gcc/libgcc/libgcc2.c
Patrick McGehearty 54f0224d55 Practical improvement to libgcc complex divide 
Correctness and performance test programs used during development of
this project may be found in the attachment to:
https://www.mail-archive.com/gcc-patches@gcc.gnu.org/msg254210.html

Summary of Purpose

This patch to libgcc/libgcc2.c __divdc3 provides an
opportunity to gain important improvements to the quality of answers
for the default complex divide routine (half, float, double, extended,
long double precisions) when dealing with very large or very small exponents.

The current code correctly implements Smith's method (1962) [2]
further modified by c99's requirements for dealing with NaN (not a
number) results. When working with input values where the exponents
are greater than *_MAX_EXP/2 or less than -(*_MAX_EXP)/2, results are
substantially different from the answers provided by quad precision
more than 1% of the time. This error rate may be unacceptable for many
applications that cannot a priori restrict their computations to the
safe range. The proposed method reduces the frequency of
"substantially different" answers by more than 99% for double
precision at a modest cost of performance.

Differences between current gcc methods and the new method will be
described. Then accuracy and performance differences will be discussed.

Background

This project started with an investigation related to
https://gcc.gnu.org/bugzilla/show_bug.cgi?id=59714.  Study of Beebe[1]
provided an overview of past and recent practice for computing complex
divide. The current glibc implementation is based on Robert Smith's
algorithm [2] from 1962.  A google search found the paper by Baudin
and Smith [3] (same Robert Smith) published in 2012. Elen Kalda's
proposed patch [4] is based on that paper.

I developed two sets of test data by randomly distributing values over
a restricted range and the full range of input values. The current
complex divide handled the restricted range well enough, but failed on
the full range more than 1% of the time. Baudin and Smith's primary
test for "ratio" equals zero reduced the cases with 16 or more error
bits by a factor of 5, but still left too many flawed answers. Adding
debug print out to cases with substantial errors allowed me to see the
intermediate calculations for test values that failed. I noted that
for many of the failures, "ratio" was a subnormal. Changing the
"ratio" test from check for zero to check for subnormal reduced the 16
bit error rate by another factor of 12. This single modified test
provides the greatest benefit for the least cost, but the percentage
of cases with greater than 16 bit errors (double precision data) is
still greater than 0.027% (2.7 in 10,000).

Continued examination of remaining errors and their intermediate
computations led to the various tests of input value tests and scaling
to avoid under/overflow. The current patch does not handle some of the
rare and most extreme combinations of input values, but the random
test data is only showing 1 case in 10 million that has an error of
greater than 12 bits. That case has 18 bits of error and is due to
subtraction cancellation. These results are significantly better
than the results reported by Baudin and Smith.

Support for half, float, double, extended, and long double precision
is included as all are handled with suitable preprocessor symbols in a
single source routine. Since half precision is computed with float
precision as per current libgcc practice, the enhanced algorithm
provides no benefit for half precision and would cost performance.
Further investigation showed changing the half precision algorithm
to use the simple formula (real=a*c+b*d imag=b*c-a*d) caused no
loss of precision and modest improvement in performance.

The existing constants for each precision:
float: FLT_MAX, FLT_MIN;
double: DBL_MAX, DBL_MIN;
extended and/or long double: LDBL_MAX, LDBL_MIN
are used for avoiding the more common overflow/underflow cases.  This
use is made generic by defining appropriate __LIBGCC2_* macros in
c-cppbuiltin.c.

Tests are added for when both parts of the denominator have exponents
small enough to allow shifting any subnormal values to normal values
all input values could be scaled up without risking overflow. That
gained a clear improvement in accuracy. Similarly, when either
numerator was subnormal and the other numerator and both denominator
values were not too large, scaling could be used to reduce risk of
computing with subnormals.  The test and scaling values used all fit
within the allowed exponent range for each precision required by the C
standard.

Float precision has more difficulty with getting correct answers than
double precision. When hardware for double precision floating point
operations is available, float precision is now handled in double
precision intermediate calculations with the simple algorithm the same
as the half-precision method of using float precision for intermediate
calculations. Using the higher precision yields exact results for all
tested input values (64-bit double, 32-bit float) with the only
performance cost being the requirement to convert the four input
values from float to double. If double precision hardware is not
available, then float complex divide will use the same improved
algorithm as the other precisions with similar change in performance.

Further Improvement

The most common remaining substantial errors are due to accuracy loss
when subtracting nearly equal values. This patch makes no attempt to
improve that situation.

NOTATION

For all of the following, the notation is:
Input complex values:
  a+bi  (a= real part, b= imaginary part)
  c+di
Output complex value:
  e+fi = (a+bi)/(c+di)

For the result tables:
current = current method (SMITH)
b1div = method proposed by Elen Kalda
b2div = alternate method considered by Elen Kalda
new = new method proposed by this patch

DESCRIPTIONS of different complex divide methods:

NAIVE COMPUTATION (-fcx-limited-range):
  e = (a*c + b*d)/(c*c + d*d)
  f = (b*c - a*d)/(c*c + d*d)

Note that c*c and d*d will overflow or underflow if either
c or d is outside the range 2^-538 to 2^512.

This method is available in gcc when the switch -fcx-limited-range is
used. That switch is also enabled by -ffast-math. Only one who has a
clear understanding of the maximum range of all intermediate values
generated by an application should consider using this switch.

SMITH's METHOD (current libgcc):
  if(fabs(c)<fabs(d) {
    r = c/d;
    denom = (c*r) + d;
    e = (a*r + b) / denom;
    f = (b*r - a) / denom;
  } else {
    r = d/c;
    denom = c + (d*r);
    e = (a + b*r) / denom;
    f = (b - a*r) / denom;
  }

Smith's method is the current default method available with __divdc3.

Elen Kalda's METHOD

Elen Kalda proposed a patch about a year ago, also based on Baudin and
Smith, but not including tests for subnormals:
https://gcc.gnu.org/legacy-ml/gcc-patches/2019-08/msg01629.html [4]
It is compared here for accuracy with this patch.

This method applies the most significant part of the algorithm
proposed by Baudin&Smith (2012) in the paper "A Robust Complex
Division in Scilab" [3]. Elen's method also replaces two divides by
one divide and two multiplies due to the high cost of divide on
aarch64. In the comparison sections, this method will be labeled
b1div. A variation discussed in that patch which does not replace the
two divides will be labeled b2div.

  inline void improved_internal (MTYPE a, MTYPE b, MTYPE c, MTYPE d)
  {
    r = d/c;
    t = 1.0 / (c + (d * r));
    if (r != 0) {
        x = (a + (b * r)) * t;
        y = (b - (a * r)) * t;
    }  else {
    /* Changing the order of operations avoids the underflow of r impacting
     the result. */
        x = (a + (d * (b / c))) * t;
        y = (b - (d * (a / c))) * t;
    }
  }

  if (FABS (d) < FABS (c)) {
      improved_internal (a, b, c, d);
  } else {
      improved_internal (b, a, d, c);
      y = -y;
  }

NEW METHOD (proposed by patch) to replace the current default method:

The proposed method starts with an algorithm proposed by Baudin&Smith
(2012) in the paper "A Robust Complex Division in Scilab" [3]. The
patch makes additional modifications to that method for further
reductions in the error rate. The following code shows the #define
values for double precision. See the patch for #define values used
for other precisions.

  #define RBIG ((DBL_MAX)/2.0)
  #define RMIN (DBL_MIN)
  #define RMIN2 (0x1.0p-53)
  #define RMINSCAL (0x1.0p+51)
  #define RMAX2  ((RBIG)*(RMIN2))

  if (FABS(c) < FABS(d)) {
  /* prevent overflow when arguments are near max representable */
  if ((FABS (d) > RBIG) || (FABS (a) > RBIG) || (FABS (b) > RBIG) ) {
      a = a * 0.5;
      b = b * 0.5;
      c = c * 0.5;
      d = d * 0.5;
  }
  /* minimize overflow/underflow issues when c and d are small */
  else if (FABS (d) < RMIN2) {
      a = a * RMINSCAL;
      b = b * RMINSCAL;
      c = c * RMINSCAL;
      d = d * RMINSCAL;
  }
  else {
    if(((FABS (a) < RMIN) && (FABS (b) < RMAX2) && (FABS (d) < RMAX2)) ||
       ((FABS (b) < RMIN) && (FABS (a) < RMAX2) && (FABS (d) < RMAX2))) {
        a = a * RMINSCAL;
        b = b * RMINSCAL;
        c = c * RMINSCAL;
        d = d * RMINSCAL;
    }
  }
  r = c/d; denom = (c*r) + d;
  if( r > RMIN ) {
      e = (a*r + b) / denom   ;
      f = (b*r - a) / denom
  } else {
      e = (c * (a/d) + b) / denom;
      f = (c * (b/d) - a) / denom;
  }
  }
[ only presenting the fabs(c) < fabs(d) case here, full code in patch. ]

Before any computation of the answer, the code checks for any input
values near maximum to allow down scaling to avoid overflow.  These
scalings almost never harm the accuracy since they are by 2. Values that
are over RBIG are relatively rare but it is easy to test for them and
allow aviodance of overflows.

Testing for RMIN2 reveals when both c and d are less than [FLT|DBL]_EPSILON.
By scaling all values by 1/EPSILON, the code converts subnormals to normals,
avoids loss of accuracy and underflows in intermediate computations
that otherwise might occur. If scaling a and b by 1/EPSILON causes either
to overflow, then the computation will overflow whatever method is used.

Finally, we test for either a or b being subnormal (RMIN) and if so,
for the other three values being small enough to allow scaling.  We
only need to test a single denominator value since we have already
determined which of c and d is larger.

Next, r (the ratio of c to d) is checked for being near zero. Baudin
and Smith checked r for zero. This code improves that approach by
checking for values less than DBL_MIN (subnormal) covers roughly 12
times as many cases and substantially improves overall accuracy. If r
is too small, then when it is used in a multiplication, there is a
high chance that the result will underflow to zero, losing significant
accuracy. That underflow is avoided by reordering the computation.
When r is subnormal, the code replaces a*r (= a*(c/d)) with ((a/d)*c)
which is mathematically the same but avoids the unnecessary underflow.

TEST Data

Two sets of data are presented to test these methods. Both sets
contain 10 million pairs of complex values.  The exponents and
mantissas are generated using multiple calls to random() and then
combining the results. Only values which give results to complex
divide that are representable in the appropriate precision after
being computed in quad precision are used.

The first data set is labeled "moderate exponents".
The exponent range is limited to -DBL_MAX_EXP/2 to DBL_MAX_EXP/2
for Double Precision (use FLT_MAX_EXP or LDBL_MAX_EXP for the
appropriate precisions.
The second data set is labeled "full exponents".
The exponent range for these cases is the full exponent range
including subnormals for a given precision.

ACCURACY Test results:

Note: The following accuracy tests are based on IEEE-754 arithmetic.

Note: All results reporteed are based on use of fused multiply-add. If
fused multiply-add is not used, the error rate increases, giving more
1 and 2 bit errors for both current and new complex divide.
Differences between using fused multiply and not using it that are
greater than 2 bits are less than 1 in a million.

The complex divide methods are evaluated by determining the percentage
of values that exceed differences in low order bits.  If a "2 bit"
test results show 1%, that would mean that 1% of 10,000,000 values
(100,000) have either a real or imaginary part that differs from the
quad precision result by more than the last 2 bits.

Results are reported for differences greater than or equal to 1 bit, 2
bits, 8 bits, 16 bits, 24 bits, and 52 bits for double precision.  Even
when the patch avoids overflows and underflows, some input values are
expected to have errors due to the potential for catastrophic roundoff
from floating point subtraction. For example, when b*c and a*d are
nearly equal, the result of subtraction may lose several places of
accuracy. This patch does not attempt to detect or minimize this type
of error, but neither does it increase them.

I only show the results for Elen Kalda's method (with both 1 and
2 divides) and the new method for only 1 divide in the double
precision table.

In the following charts, lower values are better.

current - current complex divide in libgcc
b1div - Elen Kalda's method from Baudin & Smith with one divide
b2div - Elen Kalda's method from Baudin & Smith with two divides
new   - This patch which uses 2 divides

===================================================
Errors   Moderate Dataset
gtr eq     current    b1div      b2div        new
======    ========   ========   ========   ========
 1 bit    0.24707%   0.92986%   0.24707%   0.24707%
 2 bits   0.01762%   0.01770%   0.01762%   0.01762%
 8 bits   0.00026%   0.00026%   0.00026%   0.00026%
16 bits   0.00000%   0.00000%   0.00000%   0.00000%
24 bits         0%         0%         0%         0%
52 bits         0%         0%         0%         0%
===================================================
Table 1: Errors with Moderate Dataset (Double Precision)

Note in Table 1 that both the old and new methods give identical error
rates for data with moderate exponents. Errors exceeding 16 bits are
exceedingly rare. There are substantial increases in the 1 bit error
rates for b1div (the 1 divide/2 multiplys method) as compared to b2div
(the 2 divides method). These differences are minimal for 2 bits and
larger error measurements.

===================================================
Errors   Full Dataset
gtr eq     current    b1div      b2div        new
======    ========   ========   ========   ========
 1 bit      2.05%   1.23842%    0.67130%   0.16664%
 2 bits     1.88%   0.51615%    0.50354%   0.00900%
 8 bits     1.77%   0.42856%    0.42168%   0.00011%
16 bits     1.63%   0.33840%    0.32879%   0.00001%
24 bits     1.51%   0.25583%    0.24405%   0.00000%
52 bits     1.13%   0.01886%    0.00350%   0.00000%
===================================================
Table 2: Errors with Full Dataset (Double Precision)

Table 2 shows significant differences in error rates. First, the
difference between b1div and b2div show a significantly higher error
rate for the b1div method both for single bit errros and well
beyond. Even for 52 bits, we see the b1div method gets completely
wrong answers more than 5 times as often as b2div. To retain
comparable accuracy with current complex divide results for small
exponents and due to the increase in errors for large exponents, I
choose to use the more accurate method of two divides.

The current method has more 1.6% of cases where it is getting results
where the low 24 bits of the mantissa differ from the correct
answer. More than 1.1% of cases where the answer is completely wrong.
The new method shows less than one case in 10,000 with greater than
two bits of error and only one case in 10 million with greater than
16 bits of errors. The new patch reduces 8 bit errors by
a factor of 16,000 and virtually eliminates completely wrong
answers.

As noted above, for architectures with double precision
hardware, the new method uses that hardware for the
intermediate calculations before returning the
result in float precision. Testing of the new patch
has shown zero errors found as seen in Tables 3 and 4.

Correctness for float
=============================
Errors   Moderate Dataset
gtr eq     current     new
======    ========   ========
 1 bit   28.68070%         0%
 2 bits   0.64386%         0%
 8 bits   0.00401%         0%
16 bits   0.00001%         0%
24 bits         0%         0%
=============================
Table 3: Errors with Moderate Dataset (float)

=============================
Errors   Full Dataset
gtr eq     current     new
======    ========   ========
 1 bit     19.98%         0%
 2 bits     3.20%         0%
 8 bits     1.97%         0%
16 bits     1.08%         0%
24 bits     0.55%         0%
=============================
Table 4: Errors with Full Dataset (float)

As before, the current method shows an troubling rate of extreme
errors.

There very minor changes in accuracy for half-precision since the code
changes from Smith's method to the simple method. 5 out of 1 million
test cases show correct answers instead of 1 or 2 bit errors.
libgcc computes half-precision functions in float precision
allowing the existing methods to avoid overflow/underflow issues
for the allowed range of exponents for half-precision.

Extended precision (using x87 80-bit format on x86) and Long double
(using IEEE-754 128-bit on x86 and aarch64) both have 15-bit exponents
as compared to 11-bit exponents in double precision. We note that the
C standard also allows Long Double to be implemented in the equivalent
range of Double. The RMIN2 and RMINSCAL constants are selected to work
within the Double range as well as with extended and 128-bit ranges.
We will limit our performance and accurancy discussions to the 80-bit
and 128-bit formats as seen on x86 here.

The extended and long double precision investigations were more
limited. Aarch64 does not support extended precision but does support
the software implementation of 128-bit long double precision. For x86,
long double defaults to the 80-bit precision but using the
-mlong-double-128 flag switches to using the software implementation
of 128-bit precision. Both 80-bit and 128-bit precisions have the same
exponent range, with the 128-bit precision has extended mantissas.
Since this change is only aimed at avoiding underflow/overflow for
extreme exponents, I studied the extended precision results on x86 for
100,000 values. The limited exponent dataset showed no differences.
For the dataset with full exponent range, the current and new values
showed major differences (greater than 32 bits) in 567 cases out of
100,000 (0.56%). In every one of these cases, the ratio of c/d or d/c
(as appropriate) was zero or subnormal, indicating the advantage of
the new method and its continued correctness where needed.

PERFORMANCE Test results

In order for a library change to be practical, it is necessary to show
the slowdown is tolerable. The slowdowns observed are much less than
would be seen by (for example) switching from hardware double precison
to a software quad precision, which on the tested machines causes a
slowdown of around 100x).

The actual slowdown depends on the machine architecture. It also
depends on the nature of the input data. If underflow/overflow is
rare, then implementations that have strong branch prediction will
only slowdown by a few cycles. If underflow/overflow is common, then
the branch predictors will be less accurate and the cost will be
higher.

Results from two machines are presented as examples of the overhead
for the new method. The one labeled x86 is a 5 year old Intel x86
processor and the one labeled aarch64 is a 3 year old arm64 processor.

In the following chart, the times are averaged over a one million
value data set. All values are scaled to set the time of the current
method to be 1.0. Lower values are better. A value of less than 1.0
would be faster than the current method and a value greater than 1.0
would be slower than the current method.

================================================
               Moderate set          full set
               x86  aarch64        x86  aarch64
========     ===============     ===============
float         0.59    0.79        0.45    0.81
double        1.04    1.24        1.38    1.56
long double   1.13    1.24        1.29    1.25
================================================
Table 5: Performance Comparisons (ratio new/current)

The above tables omit the timing for the 1 divide and 2 multiply
comparison with the 2 divide approach.

The float results show clear performance improvement due to using the
simple method with double precision for intermediate calculations.

The double results with the newer method show less overhead for the
moderate dataset than for the full dataset. That's because the moderate
dataset does not ever take the new branches which protect from
under/overflow. The better the branch predictor, the lower the cost
for these untaken branches. Both platforms are somewhat dated, with
the x86 having a better branch predictor which reduces the cost of the
additional branches in the new code. Of course, the relative slowdown
may be greater for some architectures, especially those with limited
branch prediction combined with a high cost of misprediction.

The long double results are fairly consistent in showing the moderate
additional cost of the extra branches and calculations for all cases.

The observed cost for all precisions is claimed to be tolerable on the
grounds that:

(a) the cost is worthwhile considering the accuracy improvement shown.
(b) most applications will only spend a small fraction of their time
    calculating complex divide.
(c) it is much less than the cost of extended precision
(d) users are not forced to use it (as described below)

Those users who find this degree of slowdown unsatisfactory may use
the gcc switch -fcx-fortran-rules which does not use the library
routine, instead inlining Smith's method without the C99 requirement
for dealing with NaN results. The proposed patch for libgcc complex
divide does not affect the code generated by -fcx-fortran-rules.

SUMMARY

When input data to complex divide has exponents whose absolute value
is less than half of *_MAX_EXP, this patch makes no changes in
accuracy and has only a modest effect on performance.  When input data
contains values outside those ranges, the patch eliminates more than
99.9% of major errors with a tolerable cost in performance.

In comparison to Elen Kalda's method, this patch introduces more
performance overhead but reduces major errors by a factor of
greater than 4000.

REFERENCES

[1] Nelson H.F. Beebe, "The Mathematical-Function Computation Handbook.
Springer International Publishing AG, 2017.

[2] Robert L. Smith. Algorithm 116: Complex division.  Commun. ACM,
 5(8):435, 1962.

[3] Michael Baudin and Robert L. Smith. "A robust complex division in
Scilab," October 2012, available at http://arxiv.org/abs/1210.4539.

[4] Elen Kalda: Complex division improvements in libgcc
https://gcc.gnu.org/legacy-ml/gcc-patches/2019-08/msg01629.html

2020-12-08  Patrick McGehearty  <patrick.mcgehearty@oracle.com>

gcc/c-family/
	* c-cppbuiltin.c (c_cpp_builtins): Add supporting macros for new
	complex divide
libgcc/
	* libgcc2.c (XMTYPE, XCTYPE, RBIG, RMIN, RMIN2, RMINSCAL, RMAX2):
	Define.
	(__divsc3, __divdc3, __divxc3, __divtc3): Improve complex divide.
	* config/rs6000/_divkc3.c (RBIG, RMIN, RMIN2, RMINSCAL, RMAX2):
	Define.
	(__divkc3): Improve complex divide.
gcc/testsuite/
	* gcc.c-torture/execute/ieee/cdivchkd.c: New test.
	* gcc.c-torture/execute/ieee/cdivchkf.c: Likewise.
	* gcc.c-torture/execute/ieee/cdivchkld.c: Likewise.
2021-04-28 21:54:44 +02:00

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/* More subroutines needed by GCC output code on some machines. */
/* Compile this one with gcc. */
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This file is part of GCC.
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<http://www.gnu.org/licenses/>. */
#include "tconfig.h"
#include "tsystem.h"
#include "coretypes.h"
#include "tm.h"
#include "libgcc_tm.h"
#ifdef HAVE_GAS_HIDDEN
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# define LIBGCC2_MAX_UNITS_PER_WORD 8
#elif (MIN_UNITS_PER_WORD > 2 \
|| (MIN_UNITS_PER_WORD > 1 && __SIZEOF_LONG_LONG__ > 4))
# define LIBGCC2_MAX_UNITS_PER_WORD 4
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# define LIBGCC2_MAX_UNITS_PER_WORD MIN_UNITS_PER_WORD
#endif
/* Work out what word size we are using for this compilation.
The value can be set on the command line. */
#ifndef LIBGCC2_UNITS_PER_WORD
#define LIBGCC2_UNITS_PER_WORD LIBGCC2_MAX_UNITS_PER_WORD
#endif
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#include "libgcc2.h"
#ifdef DECLARE_LIBRARY_RENAMES
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#if defined (L_negdi2)
DWtype
__negdi2 (DWtype u)
{
const DWunion uu = {.ll = u};
const DWunion w = { {.low = -uu.s.low,
.high = -uu.s.high - ((UWtype) -uu.s.low > 0) } };
return w.ll;
}
#endif
#ifdef L_addvsi3
Wtype
__addvSI3 (Wtype a, Wtype b)
{
Wtype w;
if (__builtin_add_overflow (a, b, &w))
abort ();
return w;
}
#ifdef COMPAT_SIMODE_TRAPPING_ARITHMETIC
SItype
__addvsi3 (SItype a, SItype b)
{
SItype w;
if (__builtin_add_overflow (a, b, &w))
abort ();
return w;
}
#endif /* COMPAT_SIMODE_TRAPPING_ARITHMETIC */
#endif
#ifdef L_addvdi3
DWtype
__addvDI3 (DWtype a, DWtype b)
{
DWtype w;
if (__builtin_add_overflow (a, b, &w))
abort ();
return w;
}
#endif
#ifdef L_subvsi3
Wtype
__subvSI3 (Wtype a, Wtype b)
{
Wtype w;
if (__builtin_sub_overflow (a, b, &w))
abort ();
return w;
}
#ifdef COMPAT_SIMODE_TRAPPING_ARITHMETIC
SItype
__subvsi3 (SItype a, SItype b)
{
SItype w;
if (__builtin_sub_overflow (a, b, &w))
abort ();
return w;
}
#endif /* COMPAT_SIMODE_TRAPPING_ARITHMETIC */
#endif
#ifdef L_subvdi3
DWtype
__subvDI3 (DWtype a, DWtype b)
{
DWtype w;
if (__builtin_sub_overflow (a, b, &w))
abort ();
return w;
}
#endif
#ifdef L_mulvsi3
Wtype
__mulvSI3 (Wtype a, Wtype b)
{
Wtype w;
if (__builtin_mul_overflow (a, b, &w))
abort ();
return w;
}
#ifdef COMPAT_SIMODE_TRAPPING_ARITHMETIC
SItype
__mulvsi3 (SItype a, SItype b)
{
SItype w;
if (__builtin_mul_overflow (a, b, &w))
abort ();
return w;
}
#endif /* COMPAT_SIMODE_TRAPPING_ARITHMETIC */
#endif
#ifdef L_negvsi2
Wtype
__negvSI2 (Wtype a)
{
Wtype w;
if (__builtin_sub_overflow (0, a, &w))
abort ();
return w;
}
#ifdef COMPAT_SIMODE_TRAPPING_ARITHMETIC
SItype
__negvsi2 (SItype a)
{
SItype w;
if (__builtin_sub_overflow (0, a, &w))
abort ();
return w;
}
#endif /* COMPAT_SIMODE_TRAPPING_ARITHMETIC */
#endif
#ifdef L_negvdi2
DWtype
__negvDI2 (DWtype a)
{
DWtype w;
if (__builtin_sub_overflow (0, a, &w))
abort ();
return w;
}
#endif
#ifdef L_absvsi2
Wtype
__absvSI2 (Wtype a)
{
const Wtype v = 0 - (a < 0);
Wtype w;
if (__builtin_add_overflow (a, v, &w))
abort ();
return v ^ w;
}
#ifdef COMPAT_SIMODE_TRAPPING_ARITHMETIC
SItype
__absvsi2 (SItype a)
{
const SItype v = 0 - (a < 0);
SItype w;
if (__builtin_add_overflow (a, v, &w))
abort ();
return v ^ w;
}
#endif /* COMPAT_SIMODE_TRAPPING_ARITHMETIC */
#endif
#ifdef L_absvdi2
DWtype
__absvDI2 (DWtype a)
{
const DWtype v = 0 - (a < 0);
DWtype w;
if (__builtin_add_overflow (a, v, &w))
abort ();
return v ^ w;
}
#endif
#ifdef L_mulvdi3
DWtype
__mulvDI3 (DWtype u, DWtype v)
{
/* The unchecked multiplication needs 3 Wtype x Wtype multiplications,
but the checked multiplication needs only two. */
const DWunion uu = {.ll = u};
const DWunion vv = {.ll = v};
if (__builtin_expect (uu.s.high == uu.s.low >> (W_TYPE_SIZE - 1), 1))
{
/* u fits in a single Wtype. */
if (__builtin_expect (vv.s.high == vv.s.low >> (W_TYPE_SIZE - 1), 1))
{
/* v fits in a single Wtype as well. */
/* A single multiplication. No overflow risk. */
return (DWtype) uu.s.low * (DWtype) vv.s.low;
}
else
{
/* Two multiplications. */
DWunion w0 = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low};
DWunion w1 = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.high};
if (vv.s.high < 0)
w1.s.high -= uu.s.low;
if (uu.s.low < 0)
w1.ll -= vv.ll;
w1.ll += (UWtype) w0.s.high;
if (__builtin_expect (w1.s.high == w1.s.low >> (W_TYPE_SIZE - 1), 1))
{
w0.s.high = w1.s.low;
return w0.ll;
}
}
}
else
{
if (__builtin_expect (vv.s.high == vv.s.low >> (W_TYPE_SIZE - 1), 1))
{
/* v fits into a single Wtype. */
/* Two multiplications. */
DWunion w0 = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low};
DWunion w1 = {.ll = (UDWtype) (UWtype) uu.s.high
* (UDWtype) (UWtype) vv.s.low};
if (uu.s.high < 0)
w1.s.high -= vv.s.low;
if (vv.s.low < 0)
w1.ll -= uu.ll;
w1.ll += (UWtype) w0.s.high;
if (__builtin_expect (w1.s.high == w1.s.low >> (W_TYPE_SIZE - 1), 1))
{
w0.s.high = w1.s.low;
return w0.ll;
}
}
else
{
/* A few sign checks and a single multiplication. */
if (uu.s.high >= 0)
{
if (vv.s.high >= 0)
{
if (uu.s.high == 0 && vv.s.high == 0)
{
const DWtype w = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low;
if (__builtin_expect (w >= 0, 1))
return w;
}
}
else
{
if (uu.s.high == 0 && vv.s.high == (Wtype) -1)
{
DWunion ww = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low};
ww.s.high -= uu.s.low;
if (__builtin_expect (ww.s.high < 0, 1))
return ww.ll;
}
}
}
else
{
if (vv.s.high >= 0)
{
if (uu.s.high == (Wtype) -1 && vv.s.high == 0)
{
DWunion ww = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low};
ww.s.high -= vv.s.low;
if (__builtin_expect (ww.s.high < 0, 1))
return ww.ll;
}
}
else
{
if ((uu.s.high & vv.s.high) == (Wtype) -1
&& (uu.s.low | vv.s.low) != 0)
{
DWunion ww = {.ll = (UDWtype) (UWtype) uu.s.low
* (UDWtype) (UWtype) vv.s.low};
ww.s.high -= uu.s.low;
ww.s.high -= vv.s.low;
if (__builtin_expect (ww.s.high >= 0, 1))
return ww.ll;
}
}
}
}
}
/* Overflow. */
abort ();
}
#endif
/* Unless shift functions are defined with full ANSI prototypes,
parameter b will be promoted to int if shift_count_type is smaller than an int. */
#ifdef L_lshrdi3
DWtype
__lshrdi3 (DWtype u, shift_count_type b)
{
if (b == 0)
return u;
const DWunion uu = {.ll = u};
const shift_count_type bm = W_TYPE_SIZE - b;
DWunion w;
if (bm <= 0)
{
w.s.high = 0;
w.s.low = (UWtype) uu.s.high >> -bm;
}
else
{
const UWtype carries = (UWtype) uu.s.high << bm;
w.s.high = (UWtype) uu.s.high >> b;
w.s.low = ((UWtype) uu.s.low >> b) | carries;
}
return w.ll;
}
#endif
#ifdef L_ashldi3
DWtype
__ashldi3 (DWtype u, shift_count_type b)
{
if (b == 0)
return u;
const DWunion uu = {.ll = u};
const shift_count_type bm = W_TYPE_SIZE - b;
DWunion w;
if (bm <= 0)
{
w.s.low = 0;
w.s.high = (UWtype) uu.s.low << -bm;
}
else
{
const UWtype carries = (UWtype) uu.s.low >> bm;
w.s.low = (UWtype) uu.s.low << b;
w.s.high = ((UWtype) uu.s.high << b) | carries;
}
return w.ll;
}
#endif
#ifdef L_ashrdi3
DWtype
__ashrdi3 (DWtype u, shift_count_type b)
{
if (b == 0)
return u;
const DWunion uu = {.ll = u};
const shift_count_type bm = W_TYPE_SIZE - b;
DWunion w;
if (bm <= 0)
{
/* w.s.high = 1..1 or 0..0 */
w.s.high = uu.s.high >> (W_TYPE_SIZE - 1);
w.s.low = uu.s.high >> -bm;
}
else
{
const UWtype carries = (UWtype) uu.s.high << bm;
w.s.high = uu.s.high >> b;
w.s.low = ((UWtype) uu.s.low >> b) | carries;
}
return w.ll;
}
#endif
#ifdef L_bswapsi2
SItype
__bswapsi2 (SItype u)
{
return ((((u) & 0xff000000u) >> 24)
| (((u) & 0x00ff0000u) >> 8)
| (((u) & 0x0000ff00u) << 8)
| (((u) & 0x000000ffu) << 24));
}
#endif
#ifdef L_bswapdi2
DItype
__bswapdi2 (DItype u)
{
return ((((u) & 0xff00000000000000ull) >> 56)
| (((u) & 0x00ff000000000000ull) >> 40)
| (((u) & 0x0000ff0000000000ull) >> 24)
| (((u) & 0x000000ff00000000ull) >> 8)
| (((u) & 0x00000000ff000000ull) << 8)
| (((u) & 0x0000000000ff0000ull) << 24)
| (((u) & 0x000000000000ff00ull) << 40)
| (((u) & 0x00000000000000ffull) << 56));
}
#endif
#ifdef L_ffssi2
#undef int
int
__ffsSI2 (UWtype u)
{
UWtype count;
if (u == 0)
return 0;
count_trailing_zeros (count, u);
return count + 1;
}
#endif
#ifdef L_ffsdi2
#undef int
int
__ffsDI2 (DWtype u)
{
const DWunion uu = {.ll = u};
UWtype word, count, add;
if (uu.s.low != 0)
word = uu.s.low, add = 0;
else if (uu.s.high != 0)
word = uu.s.high, add = W_TYPE_SIZE;
else
return 0;
count_trailing_zeros (count, word);
return count + add + 1;
}
#endif
#ifdef L_muldi3
DWtype
__muldi3 (DWtype u, DWtype v)
{
const DWunion uu = {.ll = u};
const DWunion vv = {.ll = v};
DWunion w = {.ll = __umulsidi3 (uu.s.low, vv.s.low)};
w.s.high += ((UWtype) uu.s.low * (UWtype) vv.s.high
+ (UWtype) uu.s.high * (UWtype) vv.s.low);
return w.ll;
}
#endif
#if (defined (L_udivdi3) || defined (L_divdi3) || \
defined (L_umoddi3) || defined (L_moddi3))
#if defined (sdiv_qrnnd)
#define L_udiv_w_sdiv
#endif
#endif
#ifdef L_udiv_w_sdiv
#if defined (sdiv_qrnnd)
#if (defined (L_udivdi3) || defined (L_divdi3) || \
defined (L_umoddi3) || defined (L_moddi3))
static inline __attribute__ ((__always_inline__))
#endif
UWtype
__udiv_w_sdiv (UWtype *rp, UWtype a1, UWtype a0, UWtype d)
{
UWtype q, r;
UWtype c0, c1, b1;
if ((Wtype) d >= 0)
{
if (a1 < d - a1 - (a0 >> (W_TYPE_SIZE - 1)))
{
/* Dividend, divisor, and quotient are nonnegative. */
sdiv_qrnnd (q, r, a1, a0, d);
}
else
{
/* Compute c1*2^32 + c0 = a1*2^32 + a0 - 2^31*d. */
sub_ddmmss (c1, c0, a1, a0, d >> 1, d << (W_TYPE_SIZE - 1));
/* Divide (c1*2^32 + c0) by d. */
sdiv_qrnnd (q, r, c1, c0, d);
/* Add 2^31 to quotient. */
q += (UWtype) 1 << (W_TYPE_SIZE - 1);
}
}
else
{
b1 = d >> 1; /* d/2, between 2^30 and 2^31 - 1 */
c1 = a1 >> 1; /* A/2 */
c0 = (a1 << (W_TYPE_SIZE - 1)) + (a0 >> 1);
if (a1 < b1) /* A < 2^32*b1, so A/2 < 2^31*b1 */
{
sdiv_qrnnd (q, r, c1, c0, b1); /* (A/2) / (d/2) */
r = 2*r + (a0 & 1); /* Remainder from A/(2*b1) */
if ((d & 1) != 0)
{
if (r >= q)
r = r - q;
else if (q - r <= d)
{
r = r - q + d;
q--;
}
else
{
r = r - q + 2*d;
q -= 2;
}
}
}
else if (c1 < b1) /* So 2^31 <= (A/2)/b1 < 2^32 */
{
c1 = (b1 - 1) - c1;
c0 = ~c0; /* logical NOT */
sdiv_qrnnd (q, r, c1, c0, b1); /* (A/2) / (d/2) */
q = ~q; /* (A/2)/b1 */
r = (b1 - 1) - r;
r = 2*r + (a0 & 1); /* A/(2*b1) */
if ((d & 1) != 0)
{
if (r >= q)
r = r - q;
else if (q - r <= d)
{
r = r - q + d;
q--;
}
else
{
r = r - q + 2*d;
q -= 2;
}
}
}
else /* Implies c1 = b1 */
{ /* Hence a1 = d - 1 = 2*b1 - 1 */
if (a0 >= -d)
{
q = -1;
r = a0 + d;
}
else
{
q = -2;
r = a0 + 2*d;
}
}
}
*rp = r;
return q;
}
#else
/* If sdiv_qrnnd doesn't exist, define dummy __udiv_w_sdiv. */
UWtype
__udiv_w_sdiv (UWtype *rp __attribute__ ((__unused__)),
UWtype a1 __attribute__ ((__unused__)),
UWtype a0 __attribute__ ((__unused__)),
UWtype d __attribute__ ((__unused__)))
{
return 0;
}
#endif
#endif
#if (defined (L_udivdi3) || defined (L_divdi3) || \
defined (L_umoddi3) || defined (L_moddi3) || \
defined (L_divmoddi4))
#define L_udivmoddi4
#endif
#ifdef L_clz
const UQItype __clz_tab[256] =
{
0,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,
8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8
};
#endif
#ifdef L_clzsi2
#undef int
int
__clzSI2 (UWtype x)
{
Wtype ret;
count_leading_zeros (ret, x);
return ret;
}
#endif
#ifdef L_clzdi2
#undef int
int
__clzDI2 (UDWtype x)
{
const DWunion uu = {.ll = x};
UWtype word;
Wtype ret, add;
if (uu.s.high)
word = uu.s.high, add = 0;
else
word = uu.s.low, add = W_TYPE_SIZE;
count_leading_zeros (ret, word);
return ret + add;
}
#endif
#ifdef L_ctzsi2
#undef int
int
__ctzSI2 (UWtype x)
{
Wtype ret;
count_trailing_zeros (ret, x);
return ret;
}
#endif
#ifdef L_ctzdi2
#undef int
int
__ctzDI2 (UDWtype x)
{
const DWunion uu = {.ll = x};
UWtype word;
Wtype ret, add;
if (uu.s.low)
word = uu.s.low, add = 0;
else
word = uu.s.high, add = W_TYPE_SIZE;
count_trailing_zeros (ret, word);
return ret + add;
}
#endif
#ifdef L_clrsbsi2
#undef int
int
__clrsbSI2 (Wtype x)
{
Wtype ret;
if (x < 0)
x = ~x;
if (x == 0)
return W_TYPE_SIZE - 1;
count_leading_zeros (ret, x);
return ret - 1;
}
#endif
#ifdef L_clrsbdi2
#undef int
int
__clrsbDI2 (DWtype x)
{
const DWunion uu = {.ll = x};
UWtype word;
Wtype ret, add;
if (uu.s.high == 0)
word = uu.s.low, add = W_TYPE_SIZE;
else if (uu.s.high == -1)
word = ~uu.s.low, add = W_TYPE_SIZE;
else if (uu.s.high >= 0)
word = uu.s.high, add = 0;
else
word = ~uu.s.high, add = 0;
if (word == 0)
ret = W_TYPE_SIZE;
else
count_leading_zeros (ret, word);
return ret + add - 1;
}
#endif
#ifdef L_popcount_tab
const UQItype __popcount_tab[256] =
{
0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,
1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,
1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,
2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,
1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,
2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,
2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,
3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8
};
#endif
#if defined(L_popcountsi2) || defined(L_popcountdi2)
#define POPCOUNTCST2(x) (((UWtype) x << __CHAR_BIT__) | x)
#define POPCOUNTCST4(x) (((UWtype) x << (2 * __CHAR_BIT__)) | x)
#define POPCOUNTCST8(x) (((UWtype) x << (4 * __CHAR_BIT__)) | x)
#if W_TYPE_SIZE == __CHAR_BIT__
#define POPCOUNTCST(x) x
#elif W_TYPE_SIZE == 2 * __CHAR_BIT__
#define POPCOUNTCST(x) POPCOUNTCST2 (x)
#elif W_TYPE_SIZE == 4 * __CHAR_BIT__
#define POPCOUNTCST(x) POPCOUNTCST4 (POPCOUNTCST2 (x))
#elif W_TYPE_SIZE == 8 * __CHAR_BIT__
#define POPCOUNTCST(x) POPCOUNTCST8 (POPCOUNTCST4 (POPCOUNTCST2 (x)))
#endif
#endif
#ifdef L_popcountsi2
#undef int
int
__popcountSI2 (UWtype x)
{
/* Force table lookup on targets like AVR and RL78 which only
pretend they have LIBGCC2_UNITS_PER_WORD 4, but actually
have 1, and other small word targets. */
#if __SIZEOF_INT__ > 2 && defined (POPCOUNTCST) && __CHAR_BIT__ == 8
x = x - ((x >> 1) & POPCOUNTCST (0x55));
x = (x & POPCOUNTCST (0x33)) + ((x >> 2) & POPCOUNTCST (0x33));
x = (x + (x >> 4)) & POPCOUNTCST (0x0F);
return (x * POPCOUNTCST (0x01)) >> (W_TYPE_SIZE - __CHAR_BIT__);
#else
int i, ret = 0;
for (i = 0; i < W_TYPE_SIZE; i += 8)
ret += __popcount_tab[(x >> i) & 0xff];
return ret;
#endif
}
#endif
#ifdef L_popcountdi2
#undef int
int
__popcountDI2 (UDWtype x)
{
/* Force table lookup on targets like AVR and RL78 which only
pretend they have LIBGCC2_UNITS_PER_WORD 4, but actually
have 1, and other small word targets. */
#if __SIZEOF_INT__ > 2 && defined (POPCOUNTCST) && __CHAR_BIT__ == 8
const DWunion uu = {.ll = x};
UWtype x1 = uu.s.low, x2 = uu.s.high;
x1 = x1 - ((x1 >> 1) & POPCOUNTCST (0x55));
x2 = x2 - ((x2 >> 1) & POPCOUNTCST (0x55));
x1 = (x1 & POPCOUNTCST (0x33)) + ((x1 >> 2) & POPCOUNTCST (0x33));
x2 = (x2 & POPCOUNTCST (0x33)) + ((x2 >> 2) & POPCOUNTCST (0x33));
x1 = (x1 + (x1 >> 4)) & POPCOUNTCST (0x0F);
x2 = (x2 + (x2 >> 4)) & POPCOUNTCST (0x0F);
x1 += x2;
return (x1 * POPCOUNTCST (0x01)) >> (W_TYPE_SIZE - __CHAR_BIT__);
#else
int i, ret = 0;
for (i = 0; i < 2*W_TYPE_SIZE; i += 8)
ret += __popcount_tab[(x >> i) & 0xff];
return ret;
#endif
}
#endif
#ifdef L_paritysi2
#undef int
int
__paritySI2 (UWtype x)
{
#if W_TYPE_SIZE > 64
# error "fill out the table"
#endif
#if W_TYPE_SIZE > 32
x ^= x >> 32;
#endif
#if W_TYPE_SIZE > 16
x ^= x >> 16;
#endif
x ^= x >> 8;
x ^= x >> 4;
x &= 0xf;
return (0x6996 >> x) & 1;
}
#endif
#ifdef L_paritydi2
#undef int
int
__parityDI2 (UDWtype x)
{
const DWunion uu = {.ll = x};
UWtype nx = uu.s.low ^ uu.s.high;
#if W_TYPE_SIZE > 64
# error "fill out the table"
#endif
#if W_TYPE_SIZE > 32
nx ^= nx >> 32;
#endif
#if W_TYPE_SIZE > 16
nx ^= nx >> 16;
#endif
nx ^= nx >> 8;
nx ^= nx >> 4;
nx &= 0xf;
return (0x6996 >> nx) & 1;
}
#endif
#ifdef L_udivmoddi4
#ifdef TARGET_HAS_NO_HW_DIVIDE
#if (defined (L_udivdi3) || defined (L_divdi3) || \
defined (L_umoddi3) || defined (L_moddi3) || \
defined (L_divmoddi4))
static inline __attribute__ ((__always_inline__))
#endif
UDWtype
__udivmoddi4 (UDWtype n, UDWtype d, UDWtype *rp)
{
UDWtype q = 0, r = n, y = d;
UWtype lz1, lz2, i, k;
/* Implements align divisor shift dividend method. This algorithm
aligns the divisor under the dividend and then perform number of
test-subtract iterations which shift the dividend left. Number of
iterations is k + 1 where k is the number of bit positions the
divisor must be shifted left to align it under the dividend.
quotient bits can be saved in the rightmost positions of the dividend
as it shifts left on each test-subtract iteration. */
if (y <= r)
{
lz1 = __builtin_clzll (d);
lz2 = __builtin_clzll (n);
k = lz1 - lz2;
y = (y << k);
/* Dividend can exceed 2 ^ (width - 1) - 1 but still be less than the
aligned divisor. Normal iteration can drops the high order bit
of the dividend. Therefore, first test-subtract iteration is a
special case, saving its quotient bit in a separate location and
not shifting the dividend. */
if (r >= y)
{
r = r - y;
q = (1ULL << k);
}
if (k > 0)
{
y = y >> 1;
/* k additional iterations where k regular test subtract shift
dividend iterations are done. */
i = k;
do
{
if (r >= y)
r = ((r - y) << 1) + 1;
else
r = (r << 1);
i = i - 1;
} while (i != 0);
/* First quotient bit is combined with the quotient bits resulting
from the k regular iterations. */
q = q + r;
r = r >> k;
q = q - (r << k);
}
}
if (rp)
*rp = r;
return q;
}
#else
#if (defined (L_udivdi3) || defined (L_divdi3) || \
defined (L_umoddi3) || defined (L_moddi3) || \
defined (L_divmoddi4))
static inline __attribute__ ((__always_inline__))
#endif
UDWtype
__udivmoddi4 (UDWtype n, UDWtype d, UDWtype *rp)
{
const DWunion nn = {.ll = n};
const DWunion dd = {.ll = d};
DWunion rr;
UWtype d0, d1, n0, n1, n2;
UWtype q0, q1;
UWtype b, bm;
d0 = dd.s.low;
d1 = dd.s.high;
n0 = nn.s.low;
n1 = nn.s.high;
#if !UDIV_NEEDS_NORMALIZATION
if (d1 == 0)
{
if (d0 > n1)
{
/* 0q = nn / 0D */
udiv_qrnnd (q0, n0, n1, n0, d0);
q1 = 0;
/* Remainder in n0. */
}
else
{
/* qq = NN / 0d */
if (d0 == 0)
d0 = 1 / d0; /* Divide intentionally by zero. */
udiv_qrnnd (q1, n1, 0, n1, d0);
udiv_qrnnd (q0, n0, n1, n0, d0);
/* Remainder in n0. */
}
if (rp != 0)
{
rr.s.low = n0;
rr.s.high = 0;
*rp = rr.ll;
}
}
#else /* UDIV_NEEDS_NORMALIZATION */
if (d1 == 0)
{
if (d0 > n1)
{
/* 0q = nn / 0D */
count_leading_zeros (bm, d0);
if (bm != 0)
{
/* Normalize, i.e. make the most significant bit of the
denominator set. */
d0 = d0 << bm;
n1 = (n1 << bm) | (n0 >> (W_TYPE_SIZE - bm));
n0 = n0 << bm;
}
udiv_qrnnd (q0, n0, n1, n0, d0);
q1 = 0;
/* Remainder in n0 >> bm. */
}
else
{
/* qq = NN / 0d */
if (d0 == 0)
d0 = 1 / d0; /* Divide intentionally by zero. */
count_leading_zeros (bm, d0);
if (bm == 0)
{
/* From (n1 >= d0) /\ (the most significant bit of d0 is set),
conclude (the most significant bit of n1 is set) /\ (the
leading quotient digit q1 = 1).
This special case is necessary, not an optimization.
(Shifts counts of W_TYPE_SIZE are undefined.) */
n1 -= d0;
q1 = 1;
}
else
{
/* Normalize. */
b = W_TYPE_SIZE - bm;
d0 = d0 << bm;
n2 = n1 >> b;
n1 = (n1 << bm) | (n0 >> b);
n0 = n0 << bm;
udiv_qrnnd (q1, n1, n2, n1, d0);
}
/* n1 != d0... */
udiv_qrnnd (q0, n0, n1, n0, d0);
/* Remainder in n0 >> bm. */
}
if (rp != 0)
{
rr.s.low = n0 >> bm;
rr.s.high = 0;
*rp = rr.ll;
}
}
#endif /* UDIV_NEEDS_NORMALIZATION */
else
{
if (d1 > n1)
{
/* 00 = nn / DD */
q0 = 0;
q1 = 0;
/* Remainder in n1n0. */
if (rp != 0)
{
rr.s.low = n0;
rr.s.high = n1;
*rp = rr.ll;
}
}
else
{
/* 0q = NN / dd */
count_leading_zeros (bm, d1);
if (bm == 0)
{
/* From (n1 >= d1) /\ (the most significant bit of d1 is set),
conclude (the most significant bit of n1 is set) /\ (the
quotient digit q0 = 0 or 1).
This special case is necessary, not an optimization. */
/* The condition on the next line takes advantage of that
n1 >= d1 (true due to program flow). */
if (n1 > d1 || n0 >= d0)
{
q0 = 1;
sub_ddmmss (n1, n0, n1, n0, d1, d0);
}
else
q0 = 0;
q1 = 0;
if (rp != 0)
{
rr.s.low = n0;
rr.s.high = n1;
*rp = rr.ll;
}
}
else
{
UWtype m1, m0;
/* Normalize. */
b = W_TYPE_SIZE - bm;
d1 = (d1 << bm) | (d0 >> b);
d0 = d0 << bm;
n2 = n1 >> b;
n1 = (n1 << bm) | (n0 >> b);
n0 = n0 << bm;
udiv_qrnnd (q0, n1, n2, n1, d1);
umul_ppmm (m1, m0, q0, d0);
if (m1 > n1 || (m1 == n1 && m0 > n0))
{
q0--;
sub_ddmmss (m1, m0, m1, m0, d1, d0);
}
q1 = 0;
/* Remainder in (n1n0 - m1m0) >> bm. */
if (rp != 0)
{
sub_ddmmss (n1, n0, n1, n0, m1, m0);
rr.s.low = (n1 << b) | (n0 >> bm);
rr.s.high = n1 >> bm;
*rp = rr.ll;
}
}
}
}
const DWunion ww = {{.low = q0, .high = q1}};
return ww.ll;
}
#endif
#endif
#ifdef L_divdi3
DWtype
__divdi3 (DWtype u, DWtype v)
{
Wtype c = 0;
DWunion uu = {.ll = u};
DWunion vv = {.ll = v};
DWtype w;
if (uu.s.high < 0)
c = ~c,
uu.ll = -uu.ll;
if (vv.s.high < 0)
c = ~c,
vv.ll = -vv.ll;
w = __udivmoddi4 (uu.ll, vv.ll, (UDWtype *) 0);
if (c)
w = -w;
return w;
}
#endif
#ifdef L_moddi3
DWtype
__moddi3 (DWtype u, DWtype v)
{
Wtype c = 0;
DWunion uu = {.ll = u};
DWunion vv = {.ll = v};
DWtype w;
if (uu.s.high < 0)
c = ~c,
uu.ll = -uu.ll;
if (vv.s.high < 0)
vv.ll = -vv.ll;
(void) __udivmoddi4 (uu.ll, vv.ll, (UDWtype*)&w);
if (c)
w = -w;
return w;
}
#endif
#ifdef L_divmoddi4
DWtype
__divmoddi4 (DWtype u, DWtype v, DWtype *rp)
{
Wtype c1 = 0, c2 = 0;
DWunion uu = {.ll = u};
DWunion vv = {.ll = v};
DWtype w;
DWtype r;
if (uu.s.high < 0)
c1 = ~c1, c2 = ~c2,
uu.ll = -uu.ll;
if (vv.s.high < 0)
c1 = ~c1,
vv.ll = -vv.ll;
w = __udivmoddi4 (uu.ll, vv.ll, (UDWtype*)&r);
if (c1)
w = -w;
if (c2)
r = -r;
*rp = r;
return w;
}
#endif
#ifdef L_umoddi3
UDWtype
__umoddi3 (UDWtype u, UDWtype v)
{
UDWtype w;
(void) __udivmoddi4 (u, v, &w);
return w;
}
#endif
#ifdef L_udivdi3
UDWtype
__udivdi3 (UDWtype n, UDWtype d)
{
return __udivmoddi4 (n, d, (UDWtype *) 0);
}
#endif
#ifdef L_cmpdi2
cmp_return_type
__cmpdi2 (DWtype a, DWtype b)
{
return (a > b) - (a < b) + 1;
}
#endif
#ifdef L_ucmpdi2
cmp_return_type
__ucmpdi2 (UDWtype a, UDWtype b)
{
return (a > b) - (a < b) + 1;
}
#endif
#if defined(L_fixunstfdi) && LIBGCC2_HAS_TF_MODE
UDWtype
__fixunstfDI (TFtype a)
{
if (a < 0)
return 0;
/* Compute high word of result, as a flonum. */
const TFtype b = (a / Wtype_MAXp1_F);
/* Convert that to fixed (but not to DWtype!),
and shift it into the high word. */
UDWtype v = (UWtype) b;
v <<= W_TYPE_SIZE;
/* Remove high part from the TFtype, leaving the low part as flonum. */
a -= (TFtype)v;
/* Convert that to fixed (but not to DWtype!) and add it in.
Sometimes A comes out negative. This is significant, since
A has more bits than a long int does. */
if (a < 0)
v -= (UWtype) (- a);
else
v += (UWtype) a;
return v;
}
#endif
#if defined(L_fixtfdi) && LIBGCC2_HAS_TF_MODE
DWtype
__fixtfdi (TFtype a)
{
if (a < 0)
return - __fixunstfDI (-a);
return __fixunstfDI (a);
}
#endif
#if defined(L_fixunsxfdi) && LIBGCC2_HAS_XF_MODE
UDWtype
__fixunsxfDI (XFtype a)
{
if (a < 0)
return 0;
/* Compute high word of result, as a flonum. */
const XFtype b = (a / Wtype_MAXp1_F);
/* Convert that to fixed (but not to DWtype!),
and shift it into the high word. */
UDWtype v = (UWtype) b;
v <<= W_TYPE_SIZE;
/* Remove high part from the XFtype, leaving the low part as flonum. */
a -= (XFtype)v;
/* Convert that to fixed (but not to DWtype!) and add it in.
Sometimes A comes out negative. This is significant, since
A has more bits than a long int does. */
if (a < 0)
v -= (UWtype) (- a);
else
v += (UWtype) a;
return v;
}
#endif
#if defined(L_fixxfdi) && LIBGCC2_HAS_XF_MODE
DWtype
__fixxfdi (XFtype a)
{
if (a < 0)
return - __fixunsxfDI (-a);
return __fixunsxfDI (a);
}
#endif
#if defined(L_fixunsdfdi) && LIBGCC2_HAS_DF_MODE
UDWtype
__fixunsdfDI (DFtype a)
{
/* Get high part of result. The division here will just moves the radix
point and will not cause any rounding. Then the conversion to integral
type chops result as desired. */
const UWtype hi = a / Wtype_MAXp1_F;
/* Get low part of result. Convert `hi' to floating type and scale it back,
then subtract this from the number being converted. This leaves the low
part. Convert that to integral type. */
const UWtype lo = a - (DFtype) hi * Wtype_MAXp1_F;
/* Assemble result from the two parts. */
return ((UDWtype) hi << W_TYPE_SIZE) | lo;
}
#endif
#if defined(L_fixdfdi) && LIBGCC2_HAS_DF_MODE
DWtype
__fixdfdi (DFtype a)
{
if (a < 0)
return - __fixunsdfDI (-a);
return __fixunsdfDI (a);
}
#endif
#if defined(L_fixunssfdi) && LIBGCC2_HAS_SF_MODE
UDWtype
__fixunssfDI (SFtype a)
{
#if LIBGCC2_HAS_DF_MODE
/* Convert the SFtype to a DFtype, because that is surely not going
to lose any bits. Some day someone else can write a faster version
that avoids converting to DFtype, and verify it really works right. */
const DFtype dfa = a;
/* Get high part of result. The division here will just moves the radix
point and will not cause any rounding. Then the conversion to integral
type chops result as desired. */
const UWtype hi = dfa / Wtype_MAXp1_F;
/* Get low part of result. Convert `hi' to floating type and scale it back,
then subtract this from the number being converted. This leaves the low
part. Convert that to integral type. */
const UWtype lo = dfa - (DFtype) hi * Wtype_MAXp1_F;
/* Assemble result from the two parts. */
return ((UDWtype) hi << W_TYPE_SIZE) | lo;
#elif FLT_MANT_DIG < W_TYPE_SIZE
if (a < 1)
return 0;
if (a < Wtype_MAXp1_F)
return (UWtype)a;
if (a < Wtype_MAXp1_F * Wtype_MAXp1_F)
{
/* Since we know that there are fewer significant bits in the SFmode
quantity than in a word, we know that we can convert out all the
significant bits in one step, and thus avoid losing bits. */
/* ??? This following loop essentially performs frexpf. If we could
use the real libm function, or poke at the actual bits of the fp
format, it would be significantly faster. */
UWtype shift = 0, counter;
SFtype msb;
a /= Wtype_MAXp1_F;
for (counter = W_TYPE_SIZE / 2; counter != 0; counter >>= 1)
{
SFtype counterf = (UWtype)1 << counter;
if (a >= counterf)
{
shift |= counter;
a /= counterf;
}
}
/* Rescale into the range of one word, extract the bits of that
one word, and shift the result into position. */
a *= Wtype_MAXp1_F;
counter = a;
return (DWtype)counter << shift;
}
return -1;
#else
# error
#endif
}
#endif
#if defined(L_fixsfdi) && LIBGCC2_HAS_SF_MODE
DWtype
__fixsfdi (SFtype a)
{
if (a < 0)
return - __fixunssfDI (-a);
return __fixunssfDI (a);
}
#endif
#if defined(L_floatdixf) && LIBGCC2_HAS_XF_MODE
XFtype
__floatdixf (DWtype u)
{
#if W_TYPE_SIZE > __LIBGCC_XF_MANT_DIG__
# error
#endif
XFtype d = (Wtype) (u >> W_TYPE_SIZE);
d *= Wtype_MAXp1_F;
d += (UWtype)u;
return d;
}
#endif
#if defined(L_floatundixf) && LIBGCC2_HAS_XF_MODE
XFtype
__floatundixf (UDWtype u)
{
#if W_TYPE_SIZE > __LIBGCC_XF_MANT_DIG__
# error
#endif
XFtype d = (UWtype) (u >> W_TYPE_SIZE);
d *= Wtype_MAXp1_F;
d += (UWtype)u;
return d;
}
#endif
#if defined(L_floatditf) && LIBGCC2_HAS_TF_MODE
TFtype
__floatditf (DWtype u)
{
#if W_TYPE_SIZE > __LIBGCC_TF_MANT_DIG__
# error
#endif
TFtype d = (Wtype) (u >> W_TYPE_SIZE);
d *= Wtype_MAXp1_F;
d += (UWtype)u;
return d;
}
#endif
#if defined(L_floatunditf) && LIBGCC2_HAS_TF_MODE
TFtype
__floatunditf (UDWtype u)
{
#if W_TYPE_SIZE > __LIBGCC_TF_MANT_DIG__
# error
#endif
TFtype d = (UWtype) (u >> W_TYPE_SIZE);
d *= Wtype_MAXp1_F;
d += (UWtype)u;
return d;
}
#endif
#if (defined(L_floatdisf) && LIBGCC2_HAS_SF_MODE) \
|| (defined(L_floatdidf) && LIBGCC2_HAS_DF_MODE)
#define DI_SIZE (W_TYPE_SIZE * 2)
#define F_MODE_OK(SIZE) \
(SIZE < DI_SIZE \
&& SIZE > (DI_SIZE - SIZE + FSSIZE) \
&& !AVOID_FP_TYPE_CONVERSION(SIZE))
#if defined(L_floatdisf)
#define FUNC __floatdisf
#define FSTYPE SFtype
#define FSSIZE __LIBGCC_SF_MANT_DIG__
#else
#define FUNC __floatdidf
#define FSTYPE DFtype
#define FSSIZE __LIBGCC_DF_MANT_DIG__
#endif
FSTYPE
FUNC (DWtype u)
{
#if FSSIZE >= W_TYPE_SIZE
/* When the word size is small, we never get any rounding error. */
FSTYPE f = (Wtype) (u >> W_TYPE_SIZE);
f *= Wtype_MAXp1_F;
f += (UWtype)u;
return f;
#elif (LIBGCC2_HAS_DF_MODE && F_MODE_OK (__LIBGCC_DF_MANT_DIG__)) \
|| (LIBGCC2_HAS_XF_MODE && F_MODE_OK (__LIBGCC_XF_MANT_DIG__)) \
|| (LIBGCC2_HAS_TF_MODE && F_MODE_OK (__LIBGCC_TF_MANT_DIG__))
#if (LIBGCC2_HAS_DF_MODE && F_MODE_OK (__LIBGCC_DF_MANT_DIG__))
# define FSIZE __LIBGCC_DF_MANT_DIG__
# define FTYPE DFtype
#elif (LIBGCC2_HAS_XF_MODE && F_MODE_OK (__LIBGCC_XF_MANT_DIG__))
# define FSIZE __LIBGCC_XF_MANT_DIG__
# define FTYPE XFtype
#elif (LIBGCC2_HAS_TF_MODE && F_MODE_OK (__LIBGCC_TF_MANT_DIG__))
# define FSIZE __LIBGCC_TF_MANT_DIG__
# define FTYPE TFtype
#else
# error
#endif
#define REP_BIT ((UDWtype) 1 << (DI_SIZE - FSIZE))
/* Protect against double-rounding error.
Represent any low-order bits, that might be truncated by a bit that
won't be lost. The bit can go in anywhere below the rounding position
of the FSTYPE. A fixed mask and bit position handles all usual
configurations. */
if (! (- ((DWtype) 1 << FSIZE) < u
&& u < ((DWtype) 1 << FSIZE)))
{
if ((UDWtype) u & (REP_BIT - 1))
{
u &= ~ (REP_BIT - 1);
u |= REP_BIT;
}
}
/* Do the calculation in a wider type so that we don't lose any of
the precision of the high word while multiplying it. */
FTYPE f = (Wtype) (u >> W_TYPE_SIZE);
f *= Wtype_MAXp1_F;
f += (UWtype)u;
return (FSTYPE) f;
#else
#if FSSIZE >= W_TYPE_SIZE - 2
# error
#endif
/* Finally, the word size is larger than the number of bits in the
required FSTYPE, and we've got no suitable wider type. The only
way to avoid double rounding is to special case the
extraction. */
/* If there are no high bits set, fall back to one conversion. */
if ((Wtype)u == u)
return (FSTYPE)(Wtype)u;
/* Otherwise, find the power of two. */
Wtype hi = u >> W_TYPE_SIZE;
if (hi < 0)
hi = -(UWtype) hi;
UWtype count, shift;
#if !defined (COUNT_LEADING_ZEROS_0) || COUNT_LEADING_ZEROS_0 != W_TYPE_SIZE
if (hi == 0)
count = W_TYPE_SIZE;
else
#endif
count_leading_zeros (count, hi);
/* No leading bits means u == minimum. */
if (count == 0)
return Wtype_MAXp1_F * (FSTYPE) (hi | ((UWtype) u != 0));
shift = 1 + W_TYPE_SIZE - count;
/* Shift down the most significant bits. */
hi = u >> shift;
/* If we lost any nonzero bits, set the lsb to ensure correct rounding. */
if ((UWtype)u << (W_TYPE_SIZE - shift))
hi |= 1;
/* Convert the one word of data, and rescale. */
FSTYPE f = hi, e;
if (shift == W_TYPE_SIZE)
e = Wtype_MAXp1_F;
/* The following two cases could be merged if we knew that the target
supported a native unsigned->float conversion. More often, we only
have a signed conversion, and have to add extra fixup code. */
else if (shift == W_TYPE_SIZE - 1)
e = Wtype_MAXp1_F / 2;
else
e = (Wtype)1 << shift;
return f * e;
#endif
}
#endif
#if (defined(L_floatundisf) && LIBGCC2_HAS_SF_MODE) \
|| (defined(L_floatundidf) && LIBGCC2_HAS_DF_MODE)
#define DI_SIZE (W_TYPE_SIZE * 2)
#define F_MODE_OK(SIZE) \
(SIZE < DI_SIZE \
&& SIZE > (DI_SIZE - SIZE + FSSIZE) \
&& !AVOID_FP_TYPE_CONVERSION(SIZE))
#if defined(L_floatundisf)
#define FUNC __floatundisf
#define FSTYPE SFtype
#define FSSIZE __LIBGCC_SF_MANT_DIG__
#else
#define FUNC __floatundidf
#define FSTYPE DFtype
#define FSSIZE __LIBGCC_DF_MANT_DIG__
#endif
FSTYPE
FUNC (UDWtype u)
{
#if FSSIZE >= W_TYPE_SIZE
/* When the word size is small, we never get any rounding error. */
FSTYPE f = (UWtype) (u >> W_TYPE_SIZE);
f *= Wtype_MAXp1_F;
f += (UWtype)u;
return f;
#elif (LIBGCC2_HAS_DF_MODE && F_MODE_OK (__LIBGCC_DF_MANT_DIG__)) \
|| (LIBGCC2_HAS_XF_MODE && F_MODE_OK (__LIBGCC_XF_MANT_DIG__)) \
|| (LIBGCC2_HAS_TF_MODE && F_MODE_OK (__LIBGCC_TF_MANT_DIG__))
#if (LIBGCC2_HAS_DF_MODE && F_MODE_OK (__LIBGCC_DF_MANT_DIG__))
# define FSIZE __LIBGCC_DF_MANT_DIG__
# define FTYPE DFtype
#elif (LIBGCC2_HAS_XF_MODE && F_MODE_OK (__LIBGCC_XF_MANT_DIG__))
# define FSIZE __LIBGCC_XF_MANT_DIG__
# define FTYPE XFtype
#elif (LIBGCC2_HAS_TF_MODE && F_MODE_OK (__LIBGCC_TF_MANT_DIG__))
# define FSIZE __LIBGCC_TF_MANT_DIG__
# define FTYPE TFtype
#else
# error
#endif
#define REP_BIT ((UDWtype) 1 << (DI_SIZE - FSIZE))
/* Protect against double-rounding error.
Represent any low-order bits, that might be truncated by a bit that
won't be lost. The bit can go in anywhere below the rounding position
of the FSTYPE. A fixed mask and bit position handles all usual
configurations. */
if (u >= ((UDWtype) 1 << FSIZE))
{
if ((UDWtype) u & (REP_BIT - 1))
{
u &= ~ (REP_BIT - 1);
u |= REP_BIT;
}
}
/* Do the calculation in a wider type so that we don't lose any of
the precision of the high word while multiplying it. */
FTYPE f = (UWtype) (u >> W_TYPE_SIZE);
f *= Wtype_MAXp1_F;
f += (UWtype)u;
return (FSTYPE) f;
#else
#if FSSIZE == W_TYPE_SIZE - 1
# error
#endif
/* Finally, the word size is larger than the number of bits in the
required FSTYPE, and we've got no suitable wider type. The only
way to avoid double rounding is to special case the
extraction. */
/* If there are no high bits set, fall back to one conversion. */
if ((UWtype)u == u)
return (FSTYPE)(UWtype)u;
/* Otherwise, find the power of two. */
UWtype hi = u >> W_TYPE_SIZE;
UWtype count, shift;
count_leading_zeros (count, hi);
shift = W_TYPE_SIZE - count;
/* Shift down the most significant bits. */
hi = u >> shift;
/* If we lost any nonzero bits, set the lsb to ensure correct rounding. */
if ((UWtype)u << (W_TYPE_SIZE - shift))
hi |= 1;
/* Convert the one word of data, and rescale. */
FSTYPE f = hi, e;
if (shift == W_TYPE_SIZE)
e = Wtype_MAXp1_F;
/* The following two cases could be merged if we knew that the target
supported a native unsigned->float conversion. More often, we only
have a signed conversion, and have to add extra fixup code. */
else if (shift == W_TYPE_SIZE - 1)
e = Wtype_MAXp1_F / 2;
else
e = (Wtype)1 << shift;
return f * e;
#endif
}
#endif
#if defined(L_fixunsxfsi) && LIBGCC2_HAS_XF_MODE
UWtype
__fixunsxfSI (XFtype a)
{
if (a >= - (DFtype) Wtype_MIN)
return (Wtype) (a + Wtype_MIN) - Wtype_MIN;
return (Wtype) a;
}
#endif
#if defined(L_fixunsdfsi) && LIBGCC2_HAS_DF_MODE
UWtype
__fixunsdfSI (DFtype a)
{
if (a >= - (DFtype) Wtype_MIN)
return (Wtype) (a + Wtype_MIN) - Wtype_MIN;
return (Wtype) a;
}
#endif
#if defined(L_fixunssfsi) && LIBGCC2_HAS_SF_MODE
UWtype
__fixunssfSI (SFtype a)
{
if (a >= - (SFtype) Wtype_MIN)
return (Wtype) (a + Wtype_MIN) - Wtype_MIN;
return (Wtype) a;
}
#endif
/* Integer power helper used from __builtin_powi for non-constant
exponents. */
#if (defined(L_powisf2) && LIBGCC2_HAS_SF_MODE) \
|| (defined(L_powidf2) && LIBGCC2_HAS_DF_MODE) \
|| (defined(L_powixf2) && LIBGCC2_HAS_XF_MODE) \
|| (defined(L_powitf2) && LIBGCC2_HAS_TF_MODE)
# if defined(L_powisf2)
# define TYPE SFtype
# define NAME __powisf2
# elif defined(L_powidf2)
# define TYPE DFtype
# define NAME __powidf2
# elif defined(L_powixf2)
# define TYPE XFtype
# define NAME __powixf2
# elif defined(L_powitf2)
# define TYPE TFtype
# define NAME __powitf2
# endif
#undef int
#undef unsigned
TYPE
NAME (TYPE x, int m)
{
unsigned int n = m < 0 ? -(unsigned int) m : (unsigned int) m;
TYPE y = n % 2 ? x : 1;
while (n >>= 1)
{
x = x * x;
if (n % 2)
y = y * x;
}
return m < 0 ? 1/y : y;
}
#endif
#if((defined(L_mulhc3) || defined(L_divhc3)) && LIBGCC2_HAS_HF_MODE) \
|| ((defined(L_mulsc3) || defined(L_divsc3)) && LIBGCC2_HAS_SF_MODE) \
|| ((defined(L_muldc3) || defined(L_divdc3)) && LIBGCC2_HAS_DF_MODE) \
|| ((defined(L_mulxc3) || defined(L_divxc3)) && LIBGCC2_HAS_XF_MODE) \
|| ((defined(L_multc3) || defined(L_divtc3)) && LIBGCC2_HAS_TF_MODE)
#undef float
#undef double
#undef long
#if defined(L_mulhc3) || defined(L_divhc3)
# define MTYPE HFtype
# define CTYPE HCtype
# define AMTYPE SFtype
# define MODE hc
# define CEXT __LIBGCC_HF_FUNC_EXT__
# define NOTRUNC (!__LIBGCC_HF_EXCESS_PRECISION__)
#elif defined(L_mulsc3) || defined(L_divsc3)
# define MTYPE SFtype
# define CTYPE SCtype
# define AMTYPE DFtype
# define MODE sc
# define CEXT __LIBGCC_SF_FUNC_EXT__
# define NOTRUNC (!__LIBGCC_SF_EXCESS_PRECISION__)
# define RBIG (__LIBGCC_SF_MAX__ / 2)
# define RMIN (__LIBGCC_SF_MIN__)
# define RMIN2 (__LIBGCC_SF_EPSILON__)
# define RMINSCAL (1 / __LIBGCC_SF_EPSILON__)
# define RMAX2 (RBIG * RMIN2)
#elif defined(L_muldc3) || defined(L_divdc3)
# define MTYPE DFtype
# define CTYPE DCtype
# define MODE dc
# define CEXT __LIBGCC_DF_FUNC_EXT__
# define NOTRUNC (!__LIBGCC_DF_EXCESS_PRECISION__)
# define RBIG (__LIBGCC_DF_MAX__ / 2)
# define RMIN (__LIBGCC_DF_MIN__)
# define RMIN2 (__LIBGCC_DF_EPSILON__)
# define RMINSCAL (1 / __LIBGCC_DF_EPSILON__)
# define RMAX2 (RBIG * RMIN2)
#elif defined(L_mulxc3) || defined(L_divxc3)
# define MTYPE XFtype
# define CTYPE XCtype
# define MODE xc
# define CEXT __LIBGCC_XF_FUNC_EXT__
# define NOTRUNC (!__LIBGCC_XF_EXCESS_PRECISION__)
# define RBIG (__LIBGCC_XF_MAX__ / 2)
# define RMIN (__LIBGCC_XF_MIN__)
# define RMIN2 (__LIBGCC_XF_EPSILON__)
# define RMINSCAL (1 / __LIBGCC_XF_EPSILON__)
# define RMAX2 (RBIG * RMIN2)
#elif defined(L_multc3) || defined(L_divtc3)
# define MTYPE TFtype
# define CTYPE TCtype
# define MODE tc
# define CEXT __LIBGCC_TF_FUNC_EXT__
# define NOTRUNC (!__LIBGCC_TF_EXCESS_PRECISION__)
# define RBIG (__LIBGCC_TF_MAX__ / 2)
# define RMIN (__LIBGCC_TF_MIN__)
# define RMIN2 (__LIBGCC_TF_EPSILON__)
# define RMINSCAL (1 / __LIBGCC_TF_EPSILON__)
# define RMAX2 (RBIG * RMIN2)
#else
# error
#endif
#define CONCAT3(A,B,C) _CONCAT3(A,B,C)
#define _CONCAT3(A,B,C) A##B##C
#define CONCAT2(A,B) _CONCAT2(A,B)
#define _CONCAT2(A,B) A##B
#define isnan(x) __builtin_isnan (x)
#define isfinite(x) __builtin_isfinite (x)
#define isinf(x) __builtin_isinf (x)
#define INFINITY CONCAT2(__builtin_huge_val, CEXT) ()
#define I 1i
/* Helpers to make the following code slightly less gross. */
#define COPYSIGN CONCAT2(__builtin_copysign, CEXT)
#define FABS CONCAT2(__builtin_fabs, CEXT)
/* Verify that MTYPE matches up with CEXT. */
extern void *compile_type_assert[sizeof(INFINITY) == sizeof(MTYPE) ? 1 : -1];
/* Ensure that we've lost any extra precision. */
#if NOTRUNC
# define TRUNC(x)
#else
# define TRUNC(x) __asm__ ("" : "=m"(x) : "m"(x))
#endif
#if defined(L_mulhc3) || defined(L_mulsc3) || defined(L_muldc3) \
|| defined(L_mulxc3) || defined(L_multc3)
CTYPE
CONCAT3(__mul,MODE,3) (MTYPE a, MTYPE b, MTYPE c, MTYPE d)
{
MTYPE ac, bd, ad, bc, x, y;
CTYPE res;
ac = a * c;
bd = b * d;
ad = a * d;
bc = b * c;
TRUNC (ac);
TRUNC (bd);
TRUNC (ad);
TRUNC (bc);
x = ac - bd;
y = ad + bc;
if (isnan (x) && isnan (y))
{
/* Recover infinities that computed as NaN + iNaN. */
_Bool recalc = 0;
if (isinf (a) || isinf (b))
{
/* z is infinite. "Box" the infinity and change NaNs in
the other factor to 0. */
a = COPYSIGN (isinf (a) ? 1 : 0, a);
b = COPYSIGN (isinf (b) ? 1 : 0, b);
if (isnan (c)) c = COPYSIGN (0, c);
if (isnan (d)) d = COPYSIGN (0, d);
recalc = 1;
}
if (isinf (c) || isinf (d))
{
/* w is infinite. "Box" the infinity and change NaNs in
the other factor to 0. */
c = COPYSIGN (isinf (c) ? 1 : 0, c);
d = COPYSIGN (isinf (d) ? 1 : 0, d);
if (isnan (a)) a = COPYSIGN (0, a);
if (isnan (b)) b = COPYSIGN (0, b);
recalc = 1;
}
if (!recalc
&& (isinf (ac) || isinf (bd)
|| isinf (ad) || isinf (bc)))
{
/* Recover infinities from overflow by changing NaNs to 0. */
if (isnan (a)) a = COPYSIGN (0, a);
if (isnan (b)) b = COPYSIGN (0, b);
if (isnan (c)) c = COPYSIGN (0, c);
if (isnan (d)) d = COPYSIGN (0, d);
recalc = 1;
}
if (recalc)
{
x = INFINITY * (a * c - b * d);
y = INFINITY * (a * d + b * c);
}
}
__real__ res = x;
__imag__ res = y;
return res;
}
#endif /* complex multiply */
#if defined(L_divhc3) || defined(L_divsc3) || defined(L_divdc3) \
|| defined(L_divxc3) || defined(L_divtc3)
CTYPE
CONCAT3(__div,MODE,3) (MTYPE a, MTYPE b, MTYPE c, MTYPE d)
{
#if defined(L_divhc3) \
|| (defined(L_divsc3) && defined(__LIBGCC_HAVE_HWDBL__) )
/* Half precision is handled with float precision.
float is handled with double precision when double precision
hardware is available.
Due to the additional precision, the simple complex divide
method (without Smith's method) is sufficient to get accurate
answers and runs slightly faster than Smith's method. */
AMTYPE aa, bb, cc, dd;
AMTYPE denom;
MTYPE x, y;
CTYPE res;
aa = a;
bb = b;
cc = c;
dd = d;
denom = (cc * cc) + (dd * dd);
x = ((aa * cc) + (bb * dd)) / denom;
y = ((bb * cc) - (aa * dd)) / denom;
#else
MTYPE denom, ratio, x, y;
CTYPE res;
/* double, extended, long double have significant potential
underflow/overflow errors that can be greatly reduced with
a limited number of tests and adjustments. float is handled
the same way when no HW double is available.
*/
/* Scale by max(c,d) to reduce chances of denominator overflowing. */
if (FABS (c) < FABS (d))
{
/* Prevent underflow when denominator is near max representable. */
if (FABS (d) >= RBIG)
{
a = a / 2;
b = b / 2;
c = c / 2;
d = d / 2;
}
/* Avoid overflow/underflow issues when c and d are small.
Scaling up helps avoid some underflows.
No new overflow possible since c&d < RMIN2. */
if (FABS (d) < RMIN2)
{
a = a * RMINSCAL;
b = b * RMINSCAL;
c = c * RMINSCAL;
d = d * RMINSCAL;
}
else
{
if (((FABS (a) < RMIN) && (FABS (b) < RMAX2) && (FABS (d) < RMAX2))
|| ((FABS (b) < RMIN) && (FABS (a) < RMAX2)
&& (FABS (d) < RMAX2)))
{
a = a * RMINSCAL;
b = b * RMINSCAL;
c = c * RMINSCAL;
d = d * RMINSCAL;
}
}
ratio = c / d;
denom = (c * ratio) + d;
/* Choose alternate order of computation if ratio is subnormal. */
if (FABS (ratio) > RMIN)
{
x = ((a * ratio) + b) / denom;
y = ((b * ratio) - a) / denom;
}
else
{
x = ((c * (a / d)) + b) / denom;
y = ((c * (b / d)) - a) / denom;
}
}
else
{
/* Prevent underflow when denominator is near max representable. */
if (FABS (c) >= RBIG)
{
a = a / 2;
b = b / 2;
c = c / 2;
d = d / 2;
}
/* Avoid overflow/underflow issues when both c and d are small.
Scaling up helps avoid some underflows.
No new overflow possible since both c&d are less than RMIN2. */
if (FABS (c) < RMIN2)
{
a = a * RMINSCAL;
b = b * RMINSCAL;
c = c * RMINSCAL;
d = d * RMINSCAL;
}
else
{
if (((FABS (a) < RMIN) && (FABS (b) < RMAX2) && (FABS (c) < RMAX2))
|| ((FABS (b) < RMIN) && (FABS (a) < RMAX2)
&& (FABS (c) < RMAX2)))
{
a = a * RMINSCAL;
b = b * RMINSCAL;
c = c * RMINSCAL;
d = d * RMINSCAL;
}
}
ratio = d / c;
denom = (d * ratio) + c;
/* Choose alternate order of computation if ratio is subnormal. */
if (FABS (ratio) > RMIN)
{
x = ((b * ratio) + a) / denom;
y = (b - (a * ratio)) / denom;
}
else
{
x = (a + (d * (b / c))) / denom;
y = (b - (d * (a / c))) / denom;
}
}
#endif
/* Recover infinities and zeros that computed as NaN+iNaN; the only
cases are nonzero/zero, infinite/finite, and finite/infinite. */
if (isnan (x) && isnan (y))
{
if (c == 0.0 && d == 0.0 && (!isnan (a) || !isnan (b)))
{
x = COPYSIGN (INFINITY, c) * a;
y = COPYSIGN (INFINITY, c) * b;
}
else if ((isinf (a) || isinf (b)) && isfinite (c) && isfinite (d))
{
a = COPYSIGN (isinf (a) ? 1 : 0, a);
b = COPYSIGN (isinf (b) ? 1 : 0, b);
x = INFINITY * (a * c + b * d);
y = INFINITY * (b * c - a * d);
}
else if ((isinf (c) || isinf (d)) && isfinite (a) && isfinite (b))
{
c = COPYSIGN (isinf (c) ? 1 : 0, c);
d = COPYSIGN (isinf (d) ? 1 : 0, d);
x = 0.0 * (a * c + b * d);
y = 0.0 * (b * c - a * d);
}
}
__real__ res = x;
__imag__ res = y;
return res;
}
#endif /* complex divide */
#endif /* all complex float routines */
/* From here on down, the routines use normal data types. */
#define SItype bogus_type
#define USItype bogus_type
#define DItype bogus_type
#define UDItype bogus_type
#define SFtype bogus_type
#define DFtype bogus_type
#undef Wtype
#undef UWtype
#undef HWtype
#undef UHWtype
#undef DWtype
#undef UDWtype
#undef char
#undef short
#undef int
#undef long
#undef unsigned
#undef float
#undef double
#ifdef L__gcc_bcmp
/* Like bcmp except the sign is meaningful.
Result is negative if S1 is less than S2,
positive if S1 is greater, 0 if S1 and S2 are equal. */
int
__gcc_bcmp (const unsigned char *s1, const unsigned char *s2, size_t size)
{
while (size > 0)
{
const unsigned char c1 = *s1++, c2 = *s2++;
if (c1 != c2)
return c1 - c2;
size--;
}
return 0;
}
#endif
/* __eprintf used to be used by GCC's private version of <assert.h>.
We no longer provide that header, but this routine remains in libgcc.a
for binary backward compatibility. Note that it is not included in
the shared version of libgcc. */
#ifdef L_eprintf
#ifndef inhibit_libc
#undef NULL /* Avoid errors if stdio.h and our stddef.h mismatch. */
#include <stdio.h>
void
__eprintf (const char *string, const char *expression,
unsigned int line, const char *filename)
{
fprintf (stderr, string, expression, line, filename);
fflush (stderr);
abort ();
}
#endif
#endif
#ifdef L_clear_cache
/* Clear part of an instruction cache. */
void
__clear_cache (void *beg __attribute__((__unused__)),
void *end __attribute__((__unused__)))
{
#ifdef CLEAR_INSN_CACHE
/* Cast the void* pointers to char* as some implementations
of the macro assume the pointers can be subtracted from
one another. */
CLEAR_INSN_CACHE ((char *) beg, (char *) end);
#endif /* CLEAR_INSN_CACHE */
}
#endif /* L_clear_cache */
#ifdef L_trampoline
/* Jump to a trampoline, loading the static chain address. */
#if defined(WINNT) && ! defined(__CYGWIN__)
#include <windows.h>
int getpagesize (void);
int mprotect (char *,int, int);
int
getpagesize (void)
{
#ifdef _ALPHA_
return 8192;
#else
return 4096;
#endif
}
int
mprotect (char *addr, int len, int prot)
{
DWORD np, op;
if (prot == 7)
np = 0x40;
else if (prot == 5)
np = 0x20;
else if (prot == 4)
np = 0x10;
else if (prot == 3)
np = 0x04;
else if (prot == 1)
np = 0x02;
else if (prot == 0)
np = 0x01;
else
return -1;
if (VirtualProtect (addr, len, np, &op))
return 0;
else
return -1;
}
#endif /* WINNT && ! __CYGWIN__ */
#ifdef TRANSFER_FROM_TRAMPOLINE
TRANSFER_FROM_TRAMPOLINE
#endif
#endif /* L_trampoline */
#ifndef __CYGWIN__
#ifdef L__main
#include "gbl-ctors.h"
/* Some systems use __main in a way incompatible with its use in gcc, in these
cases use the macros NAME__MAIN to give a quoted symbol and SYMBOL__MAIN to
give the same symbol without quotes for an alternative entry point. You
must define both, or neither. */
#ifndef NAME__MAIN
#define NAME__MAIN "__main"
#define SYMBOL__MAIN __main
#endif
#if defined (__LIBGCC_INIT_SECTION_ASM_OP__) \
|| defined (__LIBGCC_INIT_ARRAY_SECTION_ASM_OP__)
#undef HAS_INIT_SECTION
#define HAS_INIT_SECTION
#endif
#if !defined (HAS_INIT_SECTION) || !defined (OBJECT_FORMAT_ELF)
/* Some ELF crosses use crtstuff.c to provide __CTOR_LIST__, but use this
code to run constructors. In that case, we need to handle EH here, too.
But MINGW32 is special because it handles CRTSTUFF and EH on its own. */
#ifdef __MINGW32__
#undef __LIBGCC_EH_FRAME_SECTION_NAME__
#endif
#ifdef __LIBGCC_EH_FRAME_SECTION_NAME__
#include "unwind-dw2-fde.h"
extern unsigned char __EH_FRAME_BEGIN__[];
#endif
/* Run all the global destructors on exit from the program. */
void
__do_global_dtors (void)
{
#ifdef DO_GLOBAL_DTORS_BODY
DO_GLOBAL_DTORS_BODY;
#else
static func_ptr *p = __DTOR_LIST__ + 1;
while (*p)
{
p++;
(*(p-1)) ();
}
#endif
#if defined (__LIBGCC_EH_FRAME_SECTION_NAME__) && !defined (HAS_INIT_SECTION)
{
static int completed = 0;
if (! completed)
{
completed = 1;
__deregister_frame_info (__EH_FRAME_BEGIN__);
}
}
#endif
}
#endif
#ifndef HAS_INIT_SECTION
/* Run all the global constructors on entry to the program. */
void
__do_global_ctors (void)
{
#ifdef __LIBGCC_EH_FRAME_SECTION_NAME__
{
static struct object object;
__register_frame_info (__EH_FRAME_BEGIN__, &object);
}
#endif
DO_GLOBAL_CTORS_BODY;
atexit (__do_global_dtors);
}
#endif /* no HAS_INIT_SECTION */
#if !defined (HAS_INIT_SECTION) || defined (INVOKE__main)
/* Subroutine called automatically by `main'.
Compiling a global function named `main'
produces an automatic call to this function at the beginning.
For many systems, this routine calls __do_global_ctors.
For systems which support a .init section we use the .init section
to run __do_global_ctors, so we need not do anything here. */
extern void SYMBOL__MAIN (void);
void
SYMBOL__MAIN (void)
{
/* Support recursive calls to `main': run initializers just once. */
static int initialized;
if (! initialized)
{
initialized = 1;
__do_global_ctors ();
}
}
#endif /* no HAS_INIT_SECTION or INVOKE__main */
#endif /* L__main */
#endif /* __CYGWIN__ */
#ifdef L_ctors
#include "gbl-ctors.h"
/* Provide default definitions for the lists of constructors and
destructors, so that we don't get linker errors. These symbols are
intentionally bss symbols, so that gld and/or collect will provide
the right values. */
/* We declare the lists here with two elements each,
so that they are valid empty lists if no other definition is loaded.
If we are using the old "set" extensions to have the gnu linker
collect ctors and dtors, then we __CTOR_LIST__ and __DTOR_LIST__
must be in the bss/common section.
Long term no port should use those extensions. But many still do. */
#if !defined(__LIBGCC_INIT_SECTION_ASM_OP__)
#if defined (TARGET_ASM_CONSTRUCTOR) || defined (USE_COLLECT2)
func_ptr __CTOR_LIST__[2] = {0, 0};
func_ptr __DTOR_LIST__[2] = {0, 0};
#else
func_ptr __CTOR_LIST__[2];
func_ptr __DTOR_LIST__[2];
#endif
#endif /* no __LIBGCC_INIT_SECTION_ASM_OP__ */
#endif /* L_ctors */
#endif /* LIBGCC2_UNITS_PER_WORD <= MIN_UNITS_PER_WORD */
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