[1/4,v2] Optimized generic expf and exp2f with wrappers

Message ID 59B90C2E.9050806@arm.com
State New
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  • Optimized generic expf and exp2f
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Commit Message

Szabolcs Nagy Sept. 13, 2017, 10:45 a.m.
Based on new expf and exp2f code from
https://github.com/ARM-software/optimized-routines/

with the new expf benchmark
https://sourceware.org/ml/libc-alpha/2017-08/msg01126.html
aarch64 expf reciprocal-throughput: 3.3x faster
aarch64 expf latency: 1.7x faster
aarch64 libm.so size: -8byte data, -2140byte text
expf/exp2f worst case nearest rounding ulp error: 0.502

Error checks are inline and errno setting is in separate
tail called functions, but the wrappers are kept in this
patch to handle the _LIB_VERSION==_SVID_ case.  (So e.g.
errno is set twice for expf calls and once __expf_finite
calls now on targets where the new code is used.)

Double precision arithmetics is used which is expected
to be faster on most targets (including soft-float) than
using single precision and it is easier to get good
precision result with it.

Const data is kept in a separate translation unit which
complicates maintenance a bit, but is expected to give
good code for literal loads on most targets and allows
sharing data across expf, exp2f and powf. (This data is
disabled on i386 and ia64 which have their own expf, exp2f
and powf code.)

Some configuration is in a new math_config.h the settings
may need further discussion.

Some details may need target specific tweaks:
- best convert and round to int operation in the arg
reduction may be different across targets.
- code was optimized on fma target, optimal polynomial
eval may be different without fma.
- gcc does not always generate good code for fp bit
representation access via unions or it may be inherently
slow on some target.

The libm-test-ulps will need adjustment because..
- The argument reduction ideally uses nearest rounded rint,
but that is not efficient on most targets, so the polynomial
can get evaluated on a wider interval in non-nearest
rounding mode making 1 ulp errors common in that case.
- The polynomial is evaluated such that it has 1 ulp error
on negative tiny inputs with upward rounding, but in
exchange the evaluation is better pipelined.


v2:
- keep the compat wrappers for now.
- cosmetic changes to math_config.h.
- disable math_errf.c and e_exp2f_data.c on i386 and ia64.
- remove unused t_exp2f.h

2017-09-13  Szabolcs Nagy  <szabolcs.nagy@arm.com>

	* math/Makefile (type-float-routines): Add math_errf and e_exp2f_data.
	* sysdeps/aarch64/fpu/math_private.h (TOINT_INTRINSICS): Define.
	(roundtoint, converttoint): Likewise.
	* sysdeps/i386/fpu/e_exp2f_data.c: New file.
	* sysdeps/i386/fpu/math_errf.c: New file.
	* sysdeps/ia64/fpu/e_exp2f_data.c: New file.
	* sysdeps/ia64/fpu/math_errf.c: New file.
	* sysdeps/ieee754/flt-32/e_expf.c: New implementation.
	* sysdeps/ieee754/flt-32/e_exp2f.c: New implementation.
	* sysdeps/ieee754/flt-32/e_exp2f_data.c: New file.
	* sysdeps/ieee754/flt-32/math_config.h: New file.
	* sysdeps/ieee754/flt-32/math_errf.c: New file.
	* sysdeps/ieee754/flt-32/t_exp2f.h: Remove.

Comments

Joseph Myers Sept. 13, 2017, 12:41 p.m. | #1
On Wed, 13 Sep 2017, Szabolcs Nagy wrote:

> with the new expf benchmark
> https://sourceware.org/ml/libc-alpha/2017-08/msg01126.html

We need to get this reviewed.  Is it being pinged?  Are the benchtests 
maintainers looking at it?
Joseph Myers Sept. 13, 2017, 8:13 p.m. | #2
On Wed, 13 Sep 2017, Szabolcs Nagy wrote:

> - disable math_errf.c and e_exp2f_data.c on i386 and ia64.

sysdeps/m68k/m680x0/fpu has its own e_expf.c and e_exp2f.c, I would expect 
those new files to be disabled there as well.
Szabolcs Nagy Sept. 14, 2017, 9:39 a.m. | #3
On 13/09/17 21:13, Joseph Myers wrote:
> On Wed, 13 Sep 2017, Szabolcs Nagy wrote:
> 
>> - disable math_errf.c and e_exp2f_data.c on i386 and ia64.
> 
> sysdeps/m68k/m680x0/fpu has its own e_expf.c and e_exp2f.c, I would expect 
> those new files to be disabled there as well.
> 

yes i made a number of mistakes:
- m68k/m680x0/fpu does not need the math_errf/e_exp2f_data code
- powerpc64/fpu/multiarch/e_expf-ppc64.c needs to suppress
libm_hidden_proto too now that it overrides __expf
(otherwise there is a hidden proto of __ieee754_expf_ppc64)
- sysdeps/*/w_expf.c does not work to include srcdir/math/w_expf.c
because the generated builddir/math/w_expf.c gets included.
(i'm not yet sure what the right fix for this is, i can just
copy the math/w_expf.c code around instead of including it
or use a file name that does not collide with the generated files)
- the new versioned symbols are missing on ia64 too because
it uses its own w_expf wrappers.

i'm trying to fix these.
Joseph Myers Sept. 14, 2017, 11:48 a.m. | #4
On Thu, 14 Sep 2017, Szabolcs Nagy wrote:

> - sysdeps/*/w_expf.c does not work to include srcdir/math/w_expf.c
> because the generated builddir/math/w_expf.c gets included.
> (i'm not yet sure what the right fix for this is, i can just
> copy the math/w_expf.c code around instead of including it
> or use a file name that does not collide with the generated files)

I'd suggest not having a srcdir/math/w_expf.c.  When you need a new 
w_expf.c, put it in sysdeps/ieee754/flt-32 instead.
Szabolcs Nagy Sept. 14, 2017, 12:04 p.m. | #5
On 14/09/17 10:39, Szabolcs Nagy wrote:
> On 13/09/17 21:13, Joseph Myers wrote:
>> On Wed, 13 Sep 2017, Szabolcs Nagy wrote:
>>
>>> - disable math_errf.c and e_exp2f_data.c on i386 and ia64.
>>
>> sysdeps/m68k/m680x0/fpu has its own e_expf.c and e_exp2f.c, I would expect 
>> those new files to be disabled there as well.
>>
> 
> yes i made a number of mistakes:
> - m68k/m680x0/fpu does not need the math_errf/e_exp2f_data code
> - powerpc64/fpu/multiarch/e_expf-ppc64.c needs to suppress
> libm_hidden_proto too now that it overrides __expf
> (otherwise there is a hidden proto of __ieee754_expf_ppc64)
> - sysdeps/*/w_expf.c does not work to include srcdir/math/w_expf.c
> because the generated builddir/math/w_expf.c gets included.
> (i'm not yet sure what the right fix for this is, i can just
> copy the math/w_expf.c code around instead of including it
> or use a file name that does not collide with the generated files)

i guess i can just do
#include <sysdeps/../math/w_expf.c>
to make it clear that i want the file from the srcdir

> - the new versioned symbols are missing on ia64 too because
> it uses its own w_expf wrappers.

i will try to add .symver directives to the ia64
asm files in some ways.

but overall the symbol version fixes need a lot of
changes.. do we really need to carry asm math code
for barely supported targets? do ppl really care
if m68k/ia64/i386 get suboptimal math code?
Joseph Myers Sept. 14, 2017, 12:45 p.m. | #6
On Thu, 14 Sep 2017, Szabolcs Nagy wrote:

> but overall the symbol version fixes need a lot of
> changes.. do we really need to carry asm math code
> for barely supported targets? do ppl really care
> if m68k/ia64/i386 get suboptimal math code?

The first question would be whether it's suboptimal anyway.  That needs 
benchmarking on hardware, which presumably architecture maintainers could 
do for m68k/ia64.  For i386 there are both generic and SSE2 versions of 
expf to benchmark.

It's far from obvious that just because the x87 f2xm1 instruction exists 
it's efficient to use (especially since it needs to produce a result to 
the precision of long double, and expf only needs to produce a result to 
the precision of float).
Szabolcs Nagy Sept. 14, 2017, 3:08 p.m. | #7
On 14/09/17 12:48, Joseph Myers wrote:
> On Thu, 14 Sep 2017, Szabolcs Nagy wrote:
> 
>> - sysdeps/*/w_expf.c does not work to include srcdir/math/w_expf.c
>> because the generated builddir/math/w_expf.c gets included.
>> (i'm not yet sure what the right fix for this is, i can just
>> copy the math/w_expf.c code around instead of including it
>> or use a file name that does not collide with the generated files)
> 
> I'd suggest not having a srcdir/math/w_expf.c.  When you need a new 
> w_expf.c, put it in sysdeps/ieee754/flt-32 instead.
> 

well eventually i need the w_expf.c at many places
except in sysdeps/ieee754/flt-32 where the generic
code needs no wrapping.

so i have to put it somewhere else such that the
target specific w_expf.c can include it.

Patch

From 9feaa15d988bcd48a399b8616eb234de805e7b5e Mon Sep 17 00:00:00 2001
From: Szabolcs Nagy <szabolcs.nagy@arm.com>
Date: Wed, 6 Sep 2017 17:42:00 +0100
Subject: [PATCH 1/4] expf

---
 math/Makefile                         |   2 +-
 sysdeps/aarch64/fpu/math_private.h    |  20 ++
 sysdeps/i386/fpu/e_exp2f_data.c       |   1 +
 sysdeps/i386/fpu/math_errf.c          |   1 +
 sysdeps/ia64/fpu/e_exp2f_data.c       |   1 +
 sysdeps/ia64/fpu/math_errf.c          |   1 +
 sysdeps/ieee754/flt-32/e_exp2f.c      | 170 ++++++----------
 sysdeps/ieee754/flt-32/e_exp2f_data.c |  44 +++++
 sysdeps/ieee754/flt-32/e_expf.c       | 185 ++++++++----------
 sysdeps/ieee754/flt-32/math_config.h  | 114 +++++++++++
 sysdeps/ieee754/flt-32/math_errf.c    |  76 ++++++++
 sysdeps/ieee754/flt-32/t_exp2f.h      | 351 ----------------------------------
 12 files changed, 399 insertions(+), 567 deletions(-)
 create mode 100644 sysdeps/i386/fpu/e_exp2f_data.c
 create mode 100644 sysdeps/i386/fpu/math_errf.c
 create mode 100644 sysdeps/ia64/fpu/e_exp2f_data.c
 create mode 100644 sysdeps/ia64/fpu/math_errf.c
 create mode 100644 sysdeps/ieee754/flt-32/e_exp2f_data.c
 create mode 100644 sysdeps/ieee754/flt-32/math_config.h
 create mode 100644 sysdeps/ieee754/flt-32/math_errf.c
 delete mode 100644 sysdeps/ieee754/flt-32/t_exp2f.h

diff --git a/math/Makefile b/math/Makefile
index 0601f3ac43..04586156f8 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -115,7 +115,7 @@  type-double-routines := branred doasin dosincos halfulp mpa mpatan2	\
 
 # float support
 type-float-suffix := f
-type-float-routines := k_rem_pio2f
+type-float-routines := k_rem_pio2f math_errf e_exp2f_data
 
 # _Float128 support
 type-float128-suffix := f128
diff --git a/sysdeps/aarch64/fpu/math_private.h b/sysdeps/aarch64/fpu/math_private.h
index 807111ea5a..775237ffb2 100644
--- a/sysdeps/aarch64/fpu/math_private.h
+++ b/sysdeps/aarch64/fpu/math_private.h
@@ -319,6 +319,26 @@  libc_feresetround_noex_aarch64_ctx (struct rm_ctx *ctx)
 #define libc_feresetround_noexf_ctx	libc_feresetround_noex_aarch64_ctx
 #define libc_feresetround_noexl_ctx	libc_feresetround_noex_aarch64_ctx
 
+/* Hack: only include the large arm_neon.h when needed.  */
+#ifdef _MATH_CONFIG_H
+#include <arm_neon.h>
+
+/* ACLE intrinsics for frintn and fcvtns instructions.  */
+#define TOINT_INTRINSICS 1
+
+static inline double_t
+roundtoint (double_t x)
+{
+  return vget_lane_f64 (vrndn_f64 (vld1_f64 (&x)), 0);
+}
+
+static inline uint64_t
+converttoint (double_t x)
+{
+  return vcvtnd_s64_f64 (x);
+}
+#endif
+
 #include_next <math_private.h>
 
 #endif
diff --git a/sysdeps/i386/fpu/e_exp2f_data.c b/sysdeps/i386/fpu/e_exp2f_data.c
new file mode 100644
index 0000000000..1cc8931700
--- /dev/null
+++ b/sysdeps/i386/fpu/e_exp2f_data.c
@@ -0,0 +1 @@ 
+/* Not needed.  */
diff --git a/sysdeps/i386/fpu/math_errf.c b/sysdeps/i386/fpu/math_errf.c
new file mode 100644
index 0000000000..1cc8931700
--- /dev/null
+++ b/sysdeps/i386/fpu/math_errf.c
@@ -0,0 +1 @@ 
+/* Not needed.  */
diff --git a/sysdeps/ia64/fpu/e_exp2f_data.c b/sysdeps/ia64/fpu/e_exp2f_data.c
new file mode 100644
index 0000000000..1cc8931700
--- /dev/null
+++ b/sysdeps/ia64/fpu/e_exp2f_data.c
@@ -0,0 +1 @@ 
+/* Not needed.  */
diff --git a/sysdeps/ia64/fpu/math_errf.c b/sysdeps/ia64/fpu/math_errf.c
new file mode 100644
index 0000000000..1cc8931700
--- /dev/null
+++ b/sysdeps/ia64/fpu/math_errf.c
@@ -0,0 +1 @@ 
+/* Not needed.  */
diff --git a/sysdeps/ieee754/flt-32/e_exp2f.c b/sysdeps/ieee754/flt-32/e_exp2f.c
index 567d3ff6d0..be53e5c699 100644
--- a/sysdeps/ieee754/flt-32/e_exp2f.c
+++ b/sysdeps/ieee754/flt-32/e_exp2f.c
@@ -1,7 +1,6 @@ 
-/* Single-precision floating point 2^x.
-   Copyright (C) 1997-2017 Free Software Foundation, Inc.
+/* Single-precision 2^x function.
+   Copyright (C) 2017 Free Software Foundation, Inc.
    This file is part of the GNU C Library.
-   Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
 
    The GNU C Library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
@@ -17,116 +16,73 @@ 
    License along with the GNU C Library; if not, see
    <http://www.gnu.org/licenses/>.  */
 
-/* The basic design here is from
-   Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical
-   Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft.,
-   17 (1), March 1991, pp. 26-45.
-   It has been slightly modified to compute 2^x instead of e^x, and for
-   single-precision.
-   */
-#ifndef _GNU_SOURCE
-# define _GNU_SOURCE
-#endif
-#include <stdlib.h>
-#include <float.h>
-#include <ieee754.h>
 #include <math.h>
-#include <fenv.h>
-#include <inttypes.h>
-#include <math_private.h>
-
-#include "t_exp2f.h"
-
-static const float TWOM100 = 7.88860905e-31;
-static const float TWO127 = 1.7014118346e+38;
+#include <stdint.h>
+#include "math_config.h"
+
+/*
+EXP2F_TABLE_BITS = 5
+EXP2F_POLY_ORDER = 3
+
+ULP error: 0.502 (nearest rounding.)
+Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.)
+Wrong count: 168353 (all nearest rounding wrong results with fma.)
+Non-nearest ULP error: 1 (rounded ULP error)
+*/
+
+#define N (1 << EXP2F_TABLE_BITS)
+#define T __exp2f_data.tab
+#define C __exp2f_data.poly
+#define SHIFT __exp2f_data.shift_scaled
+
+static inline uint32_t
+top12 (float x)
+{
+  return asuint (x) >> 20;
+}
 
 float
 __ieee754_exp2f (float x)
 {
-  static const float himark = (float) FLT_MAX_EXP;
-  static const float lomark = (float) (FLT_MIN_EXP - FLT_MANT_DIG - 1);
-
-  /* Check for usual case.  */
-  if (isless (x, himark) && isgreaterequal (x, lomark))
+  uint32_t abstop;
+  uint64_t ki, t;
+  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
+  double_t kd, xd, z, r, r2, y, s;
+
+  xd = (double_t) x;
+  abstop = top12 (x) & 0x7ff;
+  if (__glibc_unlikely (abstop >= top12 (128.0f)))
     {
-      static const float THREEp14 = 49152.0;
-      int tval, unsafe;
-      float rx, x22, result;
-      union ieee754_float ex2_u, scale_u;
-
-      if (fabsf (x) < FLT_EPSILON / 4.0f)
-	return 1.0f + x;
-
-      {
-	SET_RESTORE_ROUND_NOEXF (FE_TONEAREST);
-
-	/* 1. Argument reduction.
-	   Choose integers ex, -128 <= t < 128, and some real
-	   -1/512 <= x1 <= 1/512 so that
-	   x = ex + t/512 + x1.
-
-	   First, calculate rx = ex + t/256.  */
-	rx = x + THREEp14;
-	rx -= THREEp14;
-	x -= rx;  /* Compute x=x1. */
-	/* Compute tval = (ex*256 + t)+128.
-	   Now, t = (tval mod 256)-128 and ex=tval/256  [that's mod, NOT %;
-	   and /-round-to-nearest not the usual c integer /].  */
-	tval = (int) (rx * 256.0f + 128.0f);
-
-	/* 2. Adjust for accurate table entry.
-	   Find e so that
-	   x = ex + t/256 + e + x2
-	   where -7e-4 < e < 7e-4, and
-	   (float)(2^(t/256+e))
-	   is accurate to one part in 2^-64.  */
-
-	/* 'tval & 255' is the same as 'tval%256' except that it's always
-	   positive.
-	   Compute x = x2.  */
-	x -= __exp2f_deltatable[tval & 255];
-
-	/* 3. Compute ex2 = 2^(t/255+e+ex).  */
-	ex2_u.f = __exp2f_atable[tval & 255];
-	tval >>= 8;
-	/* x2 is an integer multiple of 2^-30; avoid intermediate
-	   underflow from the calculation of x22 * x.  */
-	unsafe = abs(tval) >= -FLT_MIN_EXP - 32;
-	ex2_u.ieee.exponent += tval >> unsafe;
-	scale_u.f = 1.0;
-	scale_u.ieee.exponent += tval - (tval >> unsafe);
-
-	/* 4. Approximate 2^x2 - 1, using a second-degree polynomial,
-	   with maximum error in [-2^-9 - 2^-14, 2^-9 + 2^-14]
-	   less than 1.3e-10.  */
-
-	x22 = (.24022656679f * x + .69314736128f) * ex2_u.f;
-      }
-
-      /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex).  */
-      result = x22 * x + ex2_u.f;
-
-      if (!unsafe)
-	return result;
-      else
-	{
-	  result *= scale_u.f;
-	  math_check_force_underflow_nonneg (result);
-	  return result;
-	}
-    }
-  /* Exceptional cases:  */
-  else if (isless (x, himark))
-    {
-      if (isinf (x))
-	/* e^-inf == 0, with no error.  */
-	return 0;
-      else
-	/* Underflow */
-	return TWOM100 * TWOM100;
+      /* |x| >= 128 or x is nan.  */
+      if (asuint (x) == asuint (-INFINITY))
+	return 0.0f;
+      if (abstop >= top12 (INFINITY))
+	return x + x;
+      if (x > 0.0f)
+	return __math_oflowf (0);
+      if (x <= -150.0f)
+	return __math_uflowf (0);
+#if WANT_ERRNO_UFLOW
+      if (x < -149.0f)
+	return __math_may_uflowf (0);
+#endif
     }
-  else
-    /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
-    return TWO127*x;
+
+  /* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k.  */
+  kd = (double) (xd + SHIFT); /* Rounding to double precision is required.  */
+  ki = asuint64 (kd);
+  kd -= SHIFT; /* k/N for int k.  */
+  r = xd - kd;
+
+  /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
+  t = T[ki % N];
+  t += ki << (52 - EXP2F_TABLE_BITS);
+  s = asdouble (t);
+  z = C[0] * r + C[1];
+  r2 = r * r;
+  y = C[2] * r + 1;
+  y = z * r2 + y;
+  y = y * s;
+  return (float) y;
 }
 strong_alias (__ieee754_exp2f, __exp2f_finite)
diff --git a/sysdeps/ieee754/flt-32/e_exp2f_data.c b/sysdeps/ieee754/flt-32/e_exp2f_data.c
new file mode 100644
index 0000000000..390dcae333
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/e_exp2f_data.c
@@ -0,0 +1,44 @@ 
+/* Shared data between expf, exp2f and powf.
+   Copyright (C) 2017 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include "math_config.h"
+
+#define N (1 << EXP2F_TABLE_BITS)
+
+const struct exp2f_data __exp2f_data = {
+  /* tab[i] = uint(2^(i/N)) - (i << 52-BITS)
+     used for computing 2^(k/N) for an int |k| < 150 N as
+     double(tab[k%N] + (k << 52-BITS)) */
+  .tab = {
+0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51,
+0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1,
+0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d,
+0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585,
+0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13,
+0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d,
+0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069,
+0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540,
+  },
+  .shift_scaled = 0x1.8p+52 / N,
+  .poly = { 0x1.c6af84b912394p-5, 0x1.ebfce50fac4f3p-3, 0x1.62e42ff0c52d6p-1 },
+  .shift = 0x1.8p+52,
+  .invln2_scaled = 0x1.71547652b82fep+0 * N,
+  .poly_scaled = {
+0x1.c6af84b912394p-5/N/N/N, 0x1.ebfce50fac4f3p-3/N/N, 0x1.62e42ff0c52d6p-1/N
+  },
+};
diff --git a/sysdeps/ieee754/flt-32/e_expf.c b/sysdeps/ieee754/flt-32/e_expf.c
index 782072f213..b8244a82cf 100644
--- a/sysdeps/ieee754/flt-32/e_expf.c
+++ b/sysdeps/ieee754/flt-32/e_expf.c
@@ -1,7 +1,6 @@ 
-/* Single-precision floating point e^x.
-   Copyright (C) 1997-2017 Free Software Foundation, Inc.
+/* Single-precision e^x function.
+   Copyright (C) 2017 Free Software Foundation, Inc.
    This file is part of the GNU C Library.
-   Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
 
    The GNU C Library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
@@ -17,117 +16,87 @@ 
    License along with the GNU C Library; if not, see
    <http://www.gnu.org/licenses/>.  */
 
-/* How this works:
-
-   The input value, x, is written as
-
-   x = n * ln(2) + t/512 + delta[t] + x;
-
-   where:
-   - n is an integer, 127 >= n >= -150;
-   - t is an integer, 177 >= t >= -177
-   - delta is based on a table entry, delta[t] < 2^-28
-   - x is whatever is left, |x| < 2^-10
-
-   Then e^x is approximated as
-
-   e^x = 2^n ( e^(t/512 + delta[t])
-	       + ( e^(t/512 + delta[t])
-		   * ( p(x + delta[t] + n * ln(2)) - delta ) ) )
-
-   where
-   - p(x) is a polynomial approximating e(x)-1;
-   - e^(t/512 + delta[t]) is obtained from a table.
-
-   The table used is the same one as for the double precision version;
-   since we have the table, we might as well use it.
-
-   It turns out to be faster to do calculations in double precision than
-   to perform an 'accurate table method' expf, because of the range reduction
-   overhead (compare exp2f).
-   */
-#include <float.h>
-#include <ieee754.h>
 #include <math.h>
-#include <fenv.h>
-#include <inttypes.h>
-#include <math_private.h>
-
-extern const float __exp_deltatable[178];
-extern const double __exp_atable[355] /* __attribute__((mode(DF))) */;
-
-static const float TWOM100 = 7.88860905e-31;
-static const float TWO127 = 1.7014118346e+38;
+#include <stdint.h>
+#include "math_config.h"
+
+/*
+EXP2F_TABLE_BITS = 5
+EXP2F_POLY_ORDER = 3
+
+ULP error: 0.502 (nearest rounding.)
+Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.)
+Wrong count: 170635 (all nearest rounding wrong results with fma.)
+Non-nearest ULP error: 1 (rounded ULP error)
+*/
+
+#define N (1 << EXP2F_TABLE_BITS)
+#define InvLn2N __exp2f_data.invln2_scaled
+#define T __exp2f_data.tab
+#define C __exp2f_data.poly_scaled
+
+static inline uint32_t
+top12 (float x)
+{
+  return asuint (x) >> 20;
+}
 
 float
 __ieee754_expf (float x)
 {
-  static const float himark = 88.72283935546875;
-  static const float lomark = -103.972084045410;
-  /* Check for usual case.  */
-  if (isless (x, himark) && isgreater (x, lomark))
+  uint32_t abstop;
+  uint64_t ki, t;
+  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
+  double_t kd, xd, z, r, r2, y, s;
+
+  xd = (double_t) x;
+  abstop = top12 (x) & 0x7ff;
+  if (__glibc_unlikely (abstop >= top12 (88.0f)))
     {
-      static const float THREEp42 = 13194139533312.0;
-      static const float THREEp22 = 12582912.0;
-      /* 1/ln(2).  */
-#undef M_1_LN2
-      static const float M_1_LN2 = 1.44269502163f;
-      /* ln(2) */
-#undef M_LN2
-      static const double M_LN2 = .6931471805599452862;
-
-      int tval;
-      double x22, t, result, dx;
-      float n, delta;
-      union ieee754_double ex2_u;
-
-      {
-	SET_RESTORE_ROUND_NOEXF (FE_TONEAREST);
-
-	/* Calculate n.  */
-	n = x * M_1_LN2 + THREEp22;
-	n -= THREEp22;
-	dx = x - n*M_LN2;
-
-	/* Calculate t/512.  */
-	t = dx + THREEp42;
-	t -= THREEp42;
-	dx -= t;
-
-	/* Compute tval = t.  */
-	tval = (int) (t * 512.0);
-
-	if (t >= 0)
-	  delta = - __exp_deltatable[tval];
-	else
-	  delta = __exp_deltatable[-tval];
-
-	/* Compute ex2 = 2^n e^(t/512+delta[t]).  */
-	ex2_u.d = __exp_atable[tval+177];
-	ex2_u.ieee.exponent += (int) n;
-
-	/* Approximate e^(dx+delta) - 1, using a second-degree polynomial,
-	   with maximum error in [-2^-10-2^-28,2^-10+2^-28]
-	   less than 5e-11.  */
-	x22 = (0.5000000496709180453 * dx + 1.0000001192102037084) * dx + delta;
-      }
-
-      /* Return result.  */
-      result = x22 * ex2_u.d + ex2_u.d;
-      return (float) result;
+      /* |x| >= 88 or x is nan.  */
+      if (asuint (x) == asuint (-INFINITY))
+	return 0.0f;
+      if (abstop >= top12 (INFINITY))
+	return x + x;
+      if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */
+	return __math_oflowf (0);
+      if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */
+	return __math_uflowf (0);
+#if WANT_ERRNO_UFLOW
+      if (x < -0x1.9d1d9ep6f) /* x < log(0x1p-149) ~= -103.28 */
+	return __math_may_uflowf (0);
+#endif
     }
-  /* Exceptional cases:  */
-  else if (isless (x, himark))
-    {
-      if (isinf (x))
-	/* e^-inf == 0, with no error.  */
-	return 0;
-      else
-	/* Underflow */
-	return TWOM100 * TWOM100;
-    }
-  else
-    /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
-    return TWO127*x;
+
+  /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k.  */
+  z = InvLn2N * xd;
+
+  /* Round and convert z to int, the result is in [-150*N, 128*N] and
+     ideally ties-to-even rule is used, otherwise the magnitude of r
+     can be bigger which gives larger approximation error.  */
+#if TOINT_INTRINSICS
+  kd = roundtoint (z);
+  ki = converttoint (z);
+#elif TOINT_RINT
+  kd = rint (z);
+  ki = (long) kd;
+#elif TOINT_SHIFT
+# define SHIFT __exp2f_data.shift
+  kd = (double) (z + SHIFT); /* Rounding to double precision is required.  */
+  ki = asuint64 (kd);
+  kd -= SHIFT;
+#endif
+  r = z - kd;
+
+  /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
+  t = T[ki % N];
+  t += ki << (52 - EXP2F_TABLE_BITS);
+  s = asdouble (t);
+  z = C[0] * r + C[1];
+  r2 = r * r;
+  y = C[2] * r + 1;
+  y = z * r2 + y;
+  y = y * s;
+  return (float) y;
 }
 strong_alias (__ieee754_expf, __expf_finite)
diff --git a/sysdeps/ieee754/flt-32/math_config.h b/sysdeps/ieee754/flt-32/math_config.h
new file mode 100644
index 0000000000..1759c447c8
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/math_config.h
@@ -0,0 +1,114 @@ 
+/* Configuration for math routines.
+   Copyright (C) 2017 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#ifndef _MATH_CONFIG_H
+#define _MATH_CONFIG_H
+
+#include <math.h>
+#include <math_private.h>
+#include <stdint.h>
+
+#ifndef WANT_ROUNDING
+/* Correct special case results in non-nearest rounding modes.  */
+# define WANT_ROUNDING 1
+#endif
+#ifndef WANT_ERRNO
+/* Set errno according to ISO C with (math_errhandling & MATH_ERRNO) != 0.  */
+# define WANT_ERRNO 1
+#endif
+#ifndef WANT_ERRNO_UFLOW
+/* Set errno to ERANGE if result underflows to 0 (in all rounding modes).  */
+# define WANT_ERRNO_UFLOW (WANT_ROUNDING && WANT_ERRNO)
+#endif
+
+#ifndef TOINT_INTRINSICS
+# define TOINT_INTRINSICS 0
+#endif
+#ifndef TOINT_RINT
+# define TOINT_RINT 0
+#endif
+#ifndef TOINT_SHIFT
+# define TOINT_SHIFT 1
+#endif
+
+static inline uint32_t
+asuint (float f)
+{
+  union
+  {
+    float f;
+    uint32_t i;
+  } u = {f};
+  return u.i;
+}
+
+static inline float
+asfloat (uint32_t i)
+{
+  union
+  {
+    uint32_t i;
+    float f;
+  } u = {i};
+  return u.f;
+}
+
+static inline uint64_t
+asuint64 (double f)
+{
+  union
+  {
+    double f;
+    uint64_t i;
+  } u = {f};
+  return u.i;
+}
+
+static inline double
+asdouble (uint64_t i)
+{
+  union
+  {
+    uint64_t i;
+    double f;
+  } u = {i};
+  return u.f;
+}
+
+#define NOINLINE __attribute__ ((noinline))
+
+attribute_hidden float __math_oflowf (unsigned long);
+attribute_hidden float __math_uflowf (unsigned long);
+attribute_hidden float __math_may_uflowf (unsigned long);
+attribute_hidden float __math_divzerof (unsigned long);
+attribute_hidden float __math_invalidf (float);
+
+/* Shared between expf, exp2f and powf.  */
+#define EXP2F_TABLE_BITS 5
+#define EXP2F_POLY_ORDER 3
+extern const struct exp2f_data
+{
+  uint64_t tab[1 << EXP2F_TABLE_BITS];
+  double shift_scaled;
+  double poly[EXP2F_POLY_ORDER];
+  double shift;
+  double invln2_scaled;
+  double poly_scaled[EXP2F_POLY_ORDER];
+} __exp2f_data;
+
+#endif
diff --git a/sysdeps/ieee754/flt-32/math_errf.c b/sysdeps/ieee754/flt-32/math_errf.c
new file mode 100644
index 0000000000..ab546e24cf
--- /dev/null
+++ b/sysdeps/ieee754/flt-32/math_errf.c
@@ -0,0 +1,76 @@ 
+/* Single-precision math error handling.
+   Copyright (C) 2017 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include "math_config.h"
+
+#if WANT_ERRNO
+# include <errno.h>
+/* NOINLINE reduces code size.  */
+NOINLINE static float
+with_errnof (float y, int e)
+{
+  errno = e;
+  return y;
+}
+#else
+# define with_errnof(x, e) (x)
+#endif
+
+/* NOINLINE prevents fenv semantics breaking optimizations.  */
+NOINLINE static float
+xflowf (unsigned long sign, float y)
+{
+  y = (sign ? -y : y) * y;
+  return with_errnof (y, ERANGE);
+}
+
+attribute_hidden float
+__math_uflowf (unsigned long sign)
+{
+  return xflowf (sign, 0x1p-95f);
+}
+
+#if WANT_ERRNO_UFLOW
+/* Underflows to zero in some non-nearest rounding mode, setting errno
+   is valid even if the result is non-zero, but in the subnormal range.  */
+attribute_hidden float
+__math_may_uflowf (unsigned long sign)
+{
+  return xflowf (sign, 0x1.4p-75f);
+}
+#endif
+
+attribute_hidden float
+__math_oflowf (unsigned long sign)
+{
+  return xflowf (sign, 0x1p97f);
+}
+
+attribute_hidden float
+__math_divzerof (unsigned long sign)
+{
+  float y = 0;
+  return with_errnof ((sign ? -1 : 1) / y, ERANGE);
+}
+
+attribute_hidden float
+__math_invalidf (float x)
+{
+  float y = (x - x) / (x - x);
+  return isnan (x) ? y : with_errnof (y, EDOM);
+}
diff --git a/sysdeps/ieee754/flt-32/t_exp2f.h b/sysdeps/ieee754/flt-32/t_exp2f.h
deleted file mode 100644
index aecabcc372..0000000000
--- a/sysdeps/ieee754/flt-32/t_exp2f.h
+++ /dev/null
@@ -1,351 +0,0 @@ 
-/* Accurate tables for exp2f().
-   Copyright (C) 1998-2017 Free Software Foundation, Inc.
-   This file is part of the GNU C Library.
-   Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
-
-   The GNU C Library is free software; you can redistribute it and/or
-   modify it under the terms of the GNU Lesser General Public
-   License as published by the Free Software Foundation; either
-   version 2.1 of the License, or (at your option) any later version.
-
-   The GNU C Library is distributed in the hope that it will be useful,
-   but WITHOUT ANY WARRANTY; without even the implied warranty of
-   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-   Lesser General Public License for more details.
-
-   You should have received a copy of the GNU Lesser General Public
-   License along with the GNU C Library; if not, see
-   <http://www.gnu.org/licenses/>.  */
-
-/* This table has the property that, for all integers -128 <= i <= 127,
-   exp(i/256.0 + __exp2f_deltatable[i-128]) == __exp2f_atable[i+128] + r
-   for some -2^-35 < r < 2^-35 (abs(r) < 2^-36 if i <= 0); and that
-   __exp2f_deltatable[i+128] == t * 2^-30
-   for integer t so that abs(t) <= 43447 * 2^0.  */
-
-#define W30 (9.31322575e-10)
-static const float __exp2f_deltatable[256] = {
-      -810*W30,       283*W30,     -1514*W30,      1304*W30,
-     -1148*W30,       -98*W30,      -744*W30,      -156*W30,
-      -419*W30,      -155*W30,       474*W30,       167*W30,
-     -1984*W30,      -826*W30,       692*W30,       781*W30,
-      -578*W30,      -411*W30,      -129*W30,     -1500*W30,
-       654*W30,      -141*W30,      -816*W30,       -53*W30,
-       148*W30,       493*W30,     -2214*W30,       760*W30,
-       260*W30,       750*W30,     -1300*W30,      1424*W30,
-     -1445*W30,      -339*W30,      -680*W30,      -349*W30,
-      -922*W30,       531*W30,       193*W30,     -2892*W30,
-       290*W30,     -2145*W30,      -276*W30,       485*W30,
-      -695*W30,       215*W30,     -7093*W30,       412*W30,
-     -4596*W30,       367*W30,       592*W30,      -615*W30,
-       -97*W30,     -1066*W30,       972*W30,      -226*W30,
-      -625*W30,      -374*W30,     -5647*W30,      -180*W30,
-     20349*W30,      -447*W30,       111*W30,     -4164*W30,
-       -87*W30,       -21*W30,      -251*W30,        66*W30,
-      -517*W30,      2093*W30,      -263*W30,       182*W30,
-      -601*W30,       475*W30,      -483*W30,     -1251*W30,
-      -373*W30,      1471*W30,       -92*W30,      -215*W30,
-       -97*W30,      -190*W30,         0*W30,      -290*W30,
-     -2647*W30,      1940*W30,      -582*W30,        28*W30,
-       833*W30,      1493*W30,        34*W30,       321*W30,
-      3327*W30,       -35*W30,       177*W30,      -135*W30,
-      -796*W30,      -428*W30,       129*W30,      9332*W30,
-       -12*W30,       -69*W30,     -1743*W30,      6508*W30,
-       -60*W30,       359*W30,     43447*W30,        15*W30,
-       -23*W30,      -305*W30,      -375*W30,      -652*W30,
-       667*W30,       269*W30,     -1575*W30,       185*W30,
-      -329*W30,       200*W30,      6002*W30,       163*W30,
-      -647*W30,        19*W30,      -603*W30,      -755*W30,
-       742*W30,      -438*W30,      3587*W30,      2560*W30,
-         0*W30,      -520*W30,      -241*W30,      -299*W30,
-     -1270*W30,      -991*W30,     -1138*W30,       255*W30,
-     -1192*W30,      1722*W30,      1023*W30,      3700*W30,
-     -1388*W30,     -1551*W30,     -2549*W30,        27*W30,
-       282*W30,       673*W30,       113*W30,      1561*W30,
-        72*W30,       873*W30,        87*W30,      -395*W30,
-      -433*W30,       629*W30,      3440*W30,      -284*W30,
-      -592*W30,      -103*W30,       -46*W30,     -3844*W30,
-      1712*W30,       303*W30,      1555*W30,      -631*W30,
-     -1400*W30,      -961*W30,      -854*W30,      -276*W30,
-       407*W30,       833*W30,      -345*W30,     -1501*W30,
-       121*W30,     -1581*W30,       400*W30,       150*W30,
-      1224*W30,      -139*W30,      -563*W30,       879*W30,
-       933*W30,      2939*W30,       788*W30,       211*W30,
-       530*W30,      -192*W30,       706*W30,    -13347*W30,
-      1065*W30,         3*W30,       111*W30,      -208*W30,
-      -360*W30,      -532*W30,      -291*W30,       483*W30,
-       987*W30,       -33*W30,     -1373*W30,      -166*W30,
-     -1174*W30,     -3955*W30,      1601*W30,      -280*W30,
-      1405*W30,       600*W30,     -1659*W30,       -23*W30,
-       390*W30,       449*W30,       570*W30,    -13143*W30,
-        -9*W30,     -1646*W30,      1201*W30,       294*W30,
-      2181*W30,     -1173*W30,      1388*W30,     -4504*W30,
-       190*W30,     -2304*W30,       211*W30,       239*W30,
-        48*W30,      -817*W30,      1018*W30,      1828*W30,
-      -663*W30,      1408*W30,       408*W30,       -36*W30,
-      1295*W30,      -230*W30,      1341*W30,         9*W30,
-        40*W30,       705*W30,       186*W30,       376*W30,
-       557*W30,      5866*W30,       363*W30,     -1558*W30,
-       718*W30,       669*W30,      1369*W30,     -2972*W30,
-      -468*W30,      -121*W30,      -219*W30,       667*W30,
-     29954*W30,       366*W30,        48*W30,      -203*W30
-};
-
-static const float __exp2f_atable[256] /* __attribute__((mode(SF))) */ = {
- 0.707106411447, /* 0x0.b504ecfff */
- 0.709024071690, /* 0x0.b58299fff */
- 0.710945606239, /* 0x0.b60088000 */
- 0.712874472142, /* 0x0.b67ef1000 */
- 0.714806139464, /* 0x0.b6fd88fff */
- 0.716744661340, /* 0x0.b77c94000 */
- 0.718687653549, /* 0x0.b7fbea000 */
- 0.720636486992, /* 0x0.b87ba1fff */
- 0.722590208040, /* 0x0.b8fbabfff */
- 0.724549472323, /* 0x0.b97c12fff */
- 0.726514220228, /* 0x0.b9fcd5fff */
- 0.728483855735, /* 0x0.ba7deb000 */
- 0.730457961549, /* 0x0.baff4afff */
- 0.732438981522, /* 0x0.bb811efff */
- 0.734425544748, /* 0x0.bc0350000 */
- 0.736416816713, /* 0x0.bc85d0000 */
- 0.738412797450, /* 0x0.bd089efff */
- 0.740414917465, /* 0x0.bd8bd4fff */
- 0.742422521111, /* 0x0.be0f66fff */
- 0.744434773914, /* 0x0.be9346fff */
- 0.746454179287, /* 0x0.bf179f000 */
- 0.748477637755, /* 0x0.bf9c3afff */
- 0.750506639473, /* 0x0.c02133fff */
- 0.752541840064, /* 0x0.c0a694fff */
- 0.754582285889, /* 0x0.c12c4e000 */
- 0.756628334525, /* 0x0.c1b265000 */
- 0.758678436269, /* 0x0.c238bffff */
- 0.760736882681, /* 0x0.c2bfa6fff */
- 0.762799203401, /* 0x0.c346cf000 */
- 0.764867603790, /* 0x0.c3ce5d000 */
- 0.766940355298, /* 0x0.c45633fff */
- 0.769021093841, /* 0x0.c4de90fff */
- 0.771104693409, /* 0x0.c5671dfff */
- 0.773195922364, /* 0x0.c5f02afff */
- 0.775292098512, /* 0x0.c6798afff */
- 0.777394294745, /* 0x0.c70350000 */
- 0.779501736166, /* 0x0.c78d6d000 */
- 0.781615912910, /* 0x0.c817fafff */
- 0.783734917628, /* 0x0.c8a2d9fff */
- 0.785858273516, /* 0x0.c92e02000 */
- 0.787990570071, /* 0x0.c9b9c0000 */
- 0.790125787245, /* 0x0.ca45aefff */
- 0.792268991467, /* 0x0.cad223fff */
- 0.794417440881, /* 0x0.cb5ef0fff */
- 0.796570718287, /* 0x0.cbec0efff */
- 0.798730909811, /* 0x0.cc79a0fff */
- 0.800892710672, /* 0x0.cd074dfff */
- 0.803068041795, /* 0x0.cd95ddfff */
- 0.805242776881, /* 0x0.ce2464000 */
- 0.807428598393, /* 0x0.ceb3a3fff */
- 0.809617877002, /* 0x0.cf431dfff */
- 0.811812341211, /* 0x0.cfd2eefff */
- 0.814013659956, /* 0x0.d06333000 */
- 0.816220164311, /* 0x0.d0f3ce000 */
- 0.818434238424, /* 0x0.d184e7fff */
- 0.820652604094, /* 0x0.d21649fff */
- 0.822877407074, /* 0x0.d2a818000 */
- 0.825108587751, /* 0x0.d33a51000 */
- 0.827342867839, /* 0x0.d3ccbdfff */
- 0.829588949684, /* 0x0.d45ff1000 */
- 0.831849217401, /* 0x0.d4f411fff */
- 0.834093391880, /* 0x0.d58724fff */
- 0.836355149750, /* 0x0.d61b5f000 */
- 0.838620424257, /* 0x0.d6afd3fff */
- 0.840896368027, /* 0x0.d744fc000 */
- 0.843176305293, /* 0x0.d7da66fff */
- 0.845462262643, /* 0x0.d87037000 */
- 0.847754716864, /* 0x0.d90673fff */
- 0.850052893157, /* 0x0.d99d10fff */
- 0.852359056469, /* 0x0.da3433fff */
- 0.854668736446, /* 0x0.dacb91fff */
- 0.856986224651, /* 0x0.db6373000 */
- 0.859309315673, /* 0x0.dbfbb1fff */
- 0.861639738080, /* 0x0.dc946bfff */
- 0.863975346095, /* 0x0.dd2d7d000 */
- 0.866317391394, /* 0x0.ddc6f9fff */
- 0.868666708472, /* 0x0.de60f1000 */
- 0.871022939695, /* 0x0.defb5c000 */
- 0.873383641229, /* 0x0.df9611fff */
- 0.875751554968, /* 0x0.e03141000 */
- 0.878126025200, /* 0x0.e0ccde000 */
- 0.880506813521, /* 0x0.e168e4fff */
- 0.882894217966, /* 0x0.e2055afff */
- 0.885287821299, /* 0x0.e2a239000 */
- 0.887686729423, /* 0x0.e33f6ffff */
- 0.890096127973, /* 0x0.e3dd56fff */
- 0.892507970338, /* 0x0.e47b67000 */
- 0.894928157336, /* 0x0.e51a03000 */
- 0.897355020043, /* 0x0.e5b90efff */
- 0.899788379682, /* 0x0.e65888000 */
- 0.902227103705, /* 0x0.e6f85afff */
- 0.904673457151, /* 0x0.e798ae000 */
- 0.907128036008, /* 0x0.e8398afff */
- 0.909585535528, /* 0x0.e8da99000 */
- 0.912051796915, /* 0x0.e97c3a000 */
- 0.914524436003, /* 0x0.ea1e46000 */
- 0.917003571999, /* 0x0.eac0bf000 */
- 0.919490039339, /* 0x0.eb63b2fff */
- 0.921983361257, /* 0x0.ec071a000 */
- 0.924488604054, /* 0x0.ecab48fff */
- 0.926989555360, /* 0x0.ed4f30000 */
- 0.929502844812, /* 0x0.edf3e6000 */
- 0.932021975503, /* 0x0.ee98fdfff */
- 0.934553921208, /* 0x0.ef3eecfff */
- 0.937083780759, /* 0x0.efe4b8fff */
- 0.939624726786, /* 0x0.f08b3f000 */
- 0.942198514924, /* 0x0.f133ebfff */
- 0.944726586343, /* 0x0.f1d99a000 */
- 0.947287976728, /* 0x0.f28176fff */
- 0.949856162070, /* 0x0.f329c5fff */
- 0.952431440345, /* 0x0.f3d28bfff */
- 0.955013573175, /* 0x0.f47bc5000 */
- 0.957603693021, /* 0x0.f52584000 */
- 0.960199773321, /* 0x0.f5cfa7000 */
- 0.962801992906, /* 0x0.f67a31000 */
- 0.965413510788, /* 0x0.f72556fff */
- 0.968030691152, /* 0x0.f7d0dc000 */
- 0.970655620084, /* 0x0.f87ce2fff */
- 0.973290979849, /* 0x0.f92998fff */
- 0.975926160805, /* 0x0.f9d64bfff */
- 0.978571653370, /* 0x0.fa83ac000 */
- 0.981225252139, /* 0x0.fb3193fff */
- 0.983885228626, /* 0x0.fbdfe6fff */
- 0.986552715296, /* 0x0.fc8eb7fff */
- 0.989228487027, /* 0x0.fd3e14000 */
- 0.991909801964, /* 0x0.fdedcd000 */
- 0.994601726545, /* 0x0.fe9e38000 */
- 0.997297704209, /* 0x0.ff4ee6fff */
- 1.000000000000, /* 0x1.000000000 */
- 1.002710938457, /* 0x1.00b1aa000 */
- 1.005429744692, /* 0x1.0163d7ffe */
- 1.008155703526, /* 0x1.02167dffe */
- 1.010888457284, /* 0x1.02c995fff */
- 1.013629436498, /* 0x1.037d38000 */
- 1.016377568250, /* 0x1.043152000 */
- 1.019134163841, /* 0x1.04e5f9ffe */
- 1.021896362316, /* 0x1.059b00000 */
- 1.024668931945, /* 0x1.0650b3ffe */
- 1.027446627635, /* 0x1.0706be001 */
- 1.030234098408, /* 0x1.07bd6bffe */
- 1.033023953416, /* 0x1.087441ffe */
- 1.035824656494, /* 0x1.092bce000 */
- 1.038632392900, /* 0x1.09e3d0001 */
- 1.041450142840, /* 0x1.0a9c79ffe */
- 1.044273972530, /* 0x1.0b558a001 */
- 1.047105550795, /* 0x1.0c0f1c001 */
- 1.049944162390, /* 0x1.0cc924001 */
- 1.052791833895, /* 0x1.0d83c4001 */
- 1.055645227426, /* 0x1.0e3ec3fff */
- 1.058507919326, /* 0x1.0efa60001 */
- 1.061377286898, /* 0x1.0fb66bfff */
- 1.064254641510, /* 0x1.1072fdffe */
- 1.067140102389, /* 0x1.113018000 */
- 1.070034146304, /* 0x1.11edc1fff */
- 1.072937250162, /* 0x1.12ac04001 */
- 1.075843691823, /* 0x1.136a7dfff */
- 1.078760385496, /* 0x1.1429a3ffe */
- 1.081685543070, /* 0x1.14e958000 */
- 1.084618330005, /* 0x1.15a98c000 */
- 1.087556362176, /* 0x1.166a18001 */
- 1.090508937863, /* 0x1.172b98001 */
- 1.093464612954, /* 0x1.17ed4bfff */
- 1.096430182434, /* 0x1.18afa5ffe */
- 1.099401354802, /* 0x1.19725e000 */
- 1.102381587017, /* 0x1.1a35adfff */
- 1.105370759965, /* 0x1.1af994000 */
- 1.108367800686, /* 0x1.1bbdfdffe */
- 1.111373305331, /* 0x1.1c82f6000 */
- 1.114387035385, /* 0x1.1d4878001 */
- 1.117408752440, /* 0x1.1e0e7ffff */
- 1.120437502874, /* 0x1.1ed4fe000 */
- 1.123474478729, /* 0x1.1f9c06000 */
- 1.126521706601, /* 0x1.2063ba001 */
- 1.129574775716, /* 0x1.212bd0001 */
- 1.132638812065, /* 0x1.21f49e000 */
- 1.135709524130, /* 0x1.22bddbffe */
- 1.138789534565, /* 0x1.2387b5fff */
- 1.141876101508, /* 0x1.2451fe000 */
- 1.144971728301, /* 0x1.251cddffe */
- 1.148077130296, /* 0x1.25e861ffe */
- 1.151189923305, /* 0x1.26b462001 */
- 1.154312610610, /* 0x1.278107ffe */
- 1.157440662410, /* 0x1.284e08001 */
- 1.160578370109, /* 0x1.291baa001 */
- 1.163725256932, /* 0x1.29e9e6000 */
- 1.166879892324, /* 0x1.2ab8a3ffe */
- 1.170044302935, /* 0x1.2b8805fff */
- 1.173205971694, /* 0x1.2c5739ffe */
- 1.176397800428, /* 0x1.2d2867ffe */
- 1.179586529747, /* 0x1.2df962001 */
- 1.182784795737, /* 0x1.2ecafbffe */
- 1.185991406414, /* 0x1.2f9d21ffe */
- 1.189206838636, /* 0x1.306fdc001 */
- 1.192430973067, /* 0x1.314328000 */
- 1.195664167430, /* 0x1.32170c001 */
- 1.198906540890, /* 0x1.32eb8a001 */
- 1.202157497408, /* 0x1.33c098000 */
- 1.205416083326, /* 0x1.349625fff */
- 1.208683252332, /* 0x1.356c43fff */
- 1.211961269402, /* 0x1.364318001 */
- 1.215246438983, /* 0x1.371a64000 */
- 1.218539118740, /* 0x1.37f22dffe */
- 1.221847295770, /* 0x1.38cafc000 */
- 1.225158572187, /* 0x1.39a3fdfff */
- 1.228481650325, /* 0x1.3a7dc5ffe */
- 1.231811761846, /* 0x1.3b5803fff */
- 1.235149741144, /* 0x1.3c32c5ffe */
- 1.238499879811, /* 0x1.3d0e53ffe */
- 1.241858124726, /* 0x1.3dea69fff */
- 1.245225191102, /* 0x1.3ec713fff */
- 1.248601436624, /* 0x1.3fa458000 */
- 1.251975655584, /* 0x1.40817a001 */
- 1.255380749731, /* 0x1.4160a2001 */
- 1.258783102010, /* 0x1.423f9bffe */
- 1.262198328973, /* 0x1.431f6e000 */
- 1.265619754780, /* 0x1.43ffa7fff */
- 1.269052743928, /* 0x1.44e0a4001 */
- 1.272490739830, /* 0x1.45c1f4000 */
- 1.275942921659, /* 0x1.46a432001 */
- 1.279397487615, /* 0x1.478697ffe */
- 1.282870173427, /* 0x1.486a2dffe */
- 1.286346316319, /* 0x1.494dfdffe */
- 1.289836049094, /* 0x1.4a32b2001 */
- 1.293333172770, /* 0x1.4b17e1ffe */
- 1.296839594835, /* 0x1.4bfdadfff */
- 1.300354957560, /* 0x1.4ce40fffe */
- 1.303882122055, /* 0x1.4dcb38001 */
- 1.307417988757, /* 0x1.4eb2f1ffe */
- 1.310960650439, /* 0x1.4f9b1dfff */
- 1.314516782746, /* 0x1.50842bfff */
- 1.318079948424, /* 0x1.516daffff */
- 1.321653246888, /* 0x1.5257de000 */
- 1.325237751030, /* 0x1.5342c8001 */
- 1.328829526907, /* 0x1.542e2c000 */
- 1.332433700535, /* 0x1.551a5fffe */
- 1.336045145966, /* 0x1.56070dffe */
- 1.339667558645, /* 0x1.56f473ffe */
- 1.343300342533, /* 0x1.57e287ffe */
- 1.346941947961, /* 0x1.58d130001 */
- 1.350594043714, /* 0x1.59c087ffe */
- 1.354256033883, /* 0x1.5ab085fff */
- 1.357932448365, /* 0x1.5ba175ffe */
- 1.361609339707, /* 0x1.5c926dfff */
- 1.365299344044, /* 0x1.5d8441ffe */
- 1.369003057507, /* 0x1.5e76fc001 */
- 1.372714757920, /* 0x1.5f6a3c000 */
- 1.376437187179, /* 0x1.605e2fffe */
- 1.380165219333, /* 0x1.615282001 */
- 1.383909463864, /* 0x1.6247e3ffe */
- 1.387661933907, /* 0x1.633dd0000 */
- 1.391424179060, /* 0x1.64345fffe */
- 1.395197510706, /* 0x1.652ba9fff */
- 1.399006724329, /* 0x1.66254dffe */
- 1.402773022651, /* 0x1.671c22000 */
- 1.406576037403, /* 0x1.68155dfff */
- 1.410389423392, /* 0x1.690f48001 */
-};
-- 
2.11.0