[v10] Practical improvement to libgcc complex divide

Changes in this version from Version 9:

Replaced all uses of alloca with XALLOCAVEC in
c_cpp_builtins() in c-cppbuiltin.c
Did not replace alloca elsewhere in the same file.

Fixed type of name_len.
Fixed prototypes for abort & exit in new test programs.
Fixed spelling errors and omitted words in comments.
Changed XMTYPE to AMTYPE to avoid confusion with extended types.
Removed XCTYPE as unused. (A for additional)

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 - New test.
	* gcc.c-torture/execute/ieee/cdivchkld.c - New test.
---
 gcc/c-family/c-cppbuiltin.c                        |  58 +++++--
 .../gcc.c-torture/execute/ieee/cdivchkd.c          | 126 ++++++++++++++++
 .../gcc.c-torture/execute/ieee/cdivchkf.c          | 125 +++++++++++++++
 .../gcc.c-torture/execute/ieee/cdivchkld.c         | 168 +++++++++++++++++++++
 libgcc/config/rs6000/_divkc3.c                     | 102 ++++++++++++-
 libgcc/libgcc2.c                                   | 148 ++++++++++++++++--
 6 files changed, 697 insertions(+), 30 deletions(-)
 create mode 100644 gcc/testsuite/gcc.c-torture/execute/ieee/cdivchkd.c
 create mode 100644 gcc/testsuite/gcc.c-torture/execute/ieee/cdivchkf.c
 create mode 100644 gcc/testsuite/gcc.c-torture/execute/ieee/cdivchkld.c

Message ID	1617826875-12043-1-git-send-email-patrick.mcgehearty@oracle.com
State	New
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Series	[v10] Practical improvement to libgcc complex divide \| expand [v10] Practical improvement to libgcc complex divide

[v10] Practical improvement to libgcc complex divide

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