Remove slow paths from exp

Remove the __slowexp code, so exp is no longer correctly rounded.  The
result is computed to about 70 bits precision so the worst case ulp
error is about 0.500007 in nearest rounding mode.

	* manual/probes.texi: Remove slowexp probes.
	* math/Makefile: Remove slowexp.
	* sysdeps/generic/math_private.h (__slowexp): Remove.
	* sysdeps/ieee754/dbl-64/e_exp.c (__ieee754_exp): Remove __slowexp and
	document error bounds.
	* sysdeps/i386/fpu/slowexp.c: Remove.
	* sysdeps/ia64/fpu/slowexp.c: Remove.
	* sysdeps/ieee754/dbl-64/slowexp.c: Remove.
	* sysdeps/ieee754/dbl-64/uexp.h (err_0): Remove.
	* sysdeps/m68k/m680x0/fpu/slowexp.c: Remove.
	* sysdeps/powerpc/power4/fpu/Makefile (CPPFLAGS-slowexp.c): Remove.
	* sysdeps/x86_64/fpu/multiarch/Makefile: Remove slowexp-fma.
	* sysdeps/x86_64/fpu/multiarch/e_exp-avx.c (__slowexp): Remove.
	* sysdeps/x86_64/fpu/multiarch/e_exp-fma.c (__slowexp): Remove.
	* sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c (__slowexp): Remove.
	* sysdeps/x86_64/fpu/multiarch/slowexp-avx.c: Remove.
	* sysdeps/x86_64/fpu/multiarch/slowexp-fma.c: Remove.
	* sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c: Remove.

18 files changed, 43 insertions, 189 deletions

diff --git a/ChangeLog b/ChangeLog
index 9efcaca587..a6bbee5a91 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,24 @@
+2018-02-12  Szabolcs Nagy  <szabolcs.nagy@arm.com>
+
+	* manual/probes.texi: Remove slowexp probes.
+	* math/Makefile: Remove slowexp.
+	* sysdeps/generic/math_private.h (__slowexp): Remove.
+	* sysdeps/ieee754/dbl-64/e_exp.c (__ieee754_exp): Remove __slowexp and
+	document error bounds.
+	* sysdeps/i386/fpu/slowexp.c: Remove.
+	* sysdeps/ia64/fpu/slowexp.c: Remove.
+	* sysdeps/ieee754/dbl-64/slowexp.c: Remove.
+	* sysdeps/ieee754/dbl-64/uexp.h (err_0): Remove.
+	* sysdeps/m68k/m680x0/fpu/slowexp.c: Remove.
+	* sysdeps/powerpc/power4/fpu/Makefile (CPPFLAGS-slowexp.c): Remove.
+	* sysdeps/x86_64/fpu/multiarch/Makefile: Remove slowexp-fma.
+	* sysdeps/x86_64/fpu/multiarch/e_exp-avx.c (__slowexp): Remove.
+	* sysdeps/x86_64/fpu/multiarch/e_exp-fma.c (__slowexp): Remove.
+	* sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c (__slowexp): Remove.
+	* sysdeps/x86_64/fpu/multiarch/slowexp-avx.c: Remove.
+	* sysdeps/x86_64/fpu/multiarch/slowexp-fma.c: Remove.
+	* sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c: Remove.
+
 2018-02-12  Wilco Dijkstra  <wdijkstr@arm.com>
 
 	[BZ #13932]
diff --git a/manual/probes.texi b/manual/probes.texi
index 9584838546..fa6e38f785 100644
--- a/manual/probes.texi
+++ b/manual/probes.texi
@@ -258,20 +258,6 @@ Unless explicitly mentioned otherwise, a precision of 1 implies 24 bits of
 precision in the mantissa of the multiple precision number.  Hence, a precision
 level of 32 implies 768 bits of precision in the mantissa.
 
-@deftp Probe slowexp_p6 (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{exp} function is called with an
-input that results in multiple precision computation with precision
-6.  Argument @var{$arg1} is the input value and @var{$arg2} is the
-computed output.
-@end deftp
-
-@deftp Probe slowexp_p32 (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{exp} function is called with an
-input that results in multiple precision computation with precision
-32.  Argument @var{$arg1} is the input value and @var{$arg2} is the
-computed output.
-@end deftp
-
 @deftp Probe slowatan2 (int @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
 This probe is triggered when the @code{atan2} function is called with
 an input that results in multiple precision computation.  Argument
diff --git a/math/Makefile b/math/Makefile
index 8152011c22..8d4f998ce5 100644
--- a/math/Makefile
+++ b/math/Makefile
@@ -124,7 +124,7 @@ type-ldouble-yes := ldouble
 # double support
 type-double-suffix :=
 type-double-routines := branred doasin dosincos mpa mpatan2	\
-		       mpatan mpexp mplog mpsqrt mptan sincos32 slowexp	\
+		       mpatan mpexp mplog mpsqrt mptan sincos32	\
 		       sincostab k_rem_pio2
 
 # float support
diff --git a/sysdeps/generic/math_private.h b/sysdeps/generic/math_private.h
index 07c97fa5e3..ecb37394fb 100644
--- a/sysdeps/generic/math_private.h
+++ b/sysdeps/generic/math_private.h
@@ -261,7 +261,6 @@ extern double __sin32 (double __x, double __res, double __res1);
 extern double __cos32 (double __x, double __res, double __res1);
 extern double __mpsin (double __x, double __dx, bool __range_reduce);
 extern double __mpcos (double __x, double __dx, bool __range_reduce);
-extern double __slowexp (double __x);
 extern void __docos (double __x, double __dx, double __v[]);
 
 #ifndef math_opt_barrier
diff --git a/sysdeps/i386/fpu/slowexp.c b/sysdeps/i386/fpu/slowexp.c
deleted file mode 100644
index 1cc8931700..0000000000
--- a/sysdeps/i386/fpu/slowexp.c
+++ /dev/null
@@ -1 +0,0 @@
-/* Not needed.  */
diff --git a/sysdeps/ia64/fpu/slowexp.c b/sysdeps/ia64/fpu/slowexp.c
deleted file mode 100644
index 1cc8931700..0000000000
--- a/sysdeps/ia64/fpu/slowexp.c
+++ /dev/null
@@ -1 +0,0 @@
-/* Not needed.  */
diff --git a/sysdeps/ieee754/dbl-64/e_exp.c b/sysdeps/ieee754/dbl-64/e_exp.c
index 7a9daa5858..1bcb382c77 100644
--- a/sysdeps/ieee754/dbl-64/e_exp.c
+++ b/sysdeps/ieee754/dbl-64/e_exp.c
@@ -23,10 +23,10 @@
 /*           exp1                                                          */
 /*                                                                         */
 /* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h                       */
-/*              mpa.c mpexp.x slowexp.c                                    */
+/*              mpa.c mpexp.x                                              */
 /*                                                                         */
 /* An ultimate exp routine. Given an IEEE double machine number x          */
-/* it computes the correctly rounded (to nearest) value of e^x             */
+/* it computes an almost correctly rounded (to nearest) value of e^x       */
 /* Assumption: Machine arithmetic operations are performed in              */
 /* round to nearest mode of IEEE 754 standard.                             */
 /*                                                                         */
@@ -46,10 +46,6 @@
 # define SECTION
 #endif
 
-double __slowexp (double);
-
-/* An ultimate exp routine. Given an IEEE double machine number x it computes
-   the correctly rounded (to nearest) value of e^x.  */
 double
 SECTION
 __ieee754_exp (double x)
@@ -93,17 +89,10 @@ __ieee754_exp (double x)
 
 	rem = (bet + bet * eps) + al * eps;
 	res = al + rem;
-	cor = (al - res) + rem;
-	if (res == (res + cor * err_0))
-	  {
-	    retval = res * binexp.x;
-	    goto ret;
-	  }
-	else
-	  {
-	    retval = __slowexp (x);
-	    goto ret;
-	  }			/*if error is over bound */
+	/* Maximum relative error is 7.8e-22 (70.1 bits).
+	   Maximum ULP error is 0.500007.  */
+	retval = res * binexp.x;
+	goto ret;
       }
 
     if (n <= smallint)
@@ -166,38 +155,22 @@ __ieee754_exp (double x)
 	if (ex >= -1022)
 	  {
 	    binexp.i[HIGH_HALF] = (1023 + ex) << 20;
-	    if (res == (res + cor * err_0))
-	      {
-		retval = res * binexp.x;
-		goto ret;
-	      }
-	    else
-	      {
-		retval = __slowexp (x);
-		goto check_uflow_ret;
-	      }			/*if error is over bound */
+	    /* Does not underflow: res >= 1.0, binexp >= 0x1p-1022
+	       Maximum relative error is 7.8e-22 (70.1 bits).
+	       Maximum ULP error is 0.500007.  */
+	    retval = res * binexp.x;
+	    goto ret;
 	  }
 	ex = -(1022 + ex);
 	binexp.i[HIGH_HALF] = (1023 - ex) << 20;
 	res *= binexp.x;
 	cor *= binexp.x;
-	eps = 1.0000000001 + err_0 * binexp.x;
 	t = 1.0 + res;
 	y = ((1.0 - t) + res) + cor;
 	res = t + y;
-	cor = (t - res) + y;
-	if (res == (res + eps * cor))
-	  {
-	    binexp.i[HIGH_HALF] = 0x00100000;
-	    retval = (res - 1.0) * binexp.x;
-	    goto check_uflow_ret;
-	  }
-	else
-	  {
-	    retval = __slowexp (x);
-	    goto check_uflow_ret;
-	  }			/*   if error is over bound    */
-      check_uflow_ret:
+	/* Maximum ULP error is 0.5000035.  */
+	binexp.i[HIGH_HALF] = 0x00100000;
+	retval = (res - 1.0) * binexp.x;
 	if (retval < DBL_MIN)
 	  {
 	    double force_underflow = tiny * tiny;
@@ -210,10 +183,9 @@ __ieee754_exp (double x)
     else
       {
 	binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
-	if (res == (res + cor * err_0))
-	  retval = res * binexp.x * t256.x;
-	else
-	  retval = __slowexp (x);
+	/* Maximum relative error is 7.8e-22 (70.1 bits).
+	   Maximum ULP error is 0.500007.  */
+	retval = res * binexp.x * t256.x;
 	if (isinf (retval))
 	  goto ret_huge;
 	else
diff --git a/sysdeps/ieee754/dbl-64/slowexp.c b/sysdeps/ieee754/dbl-64/slowexp.c
deleted file mode 100644
index ba5617466c..0000000000
--- a/sysdeps/ieee754/dbl-64/slowexp.c
+++ /dev/null
@@ -1,86 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2018 Free Software Foundation, Inc.
- *
- * This program 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.
- *
- * This program 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 this program; if not, see <http://www.gnu.org/licenses/>.
- */
-/**************************************************************************/
-/*  MODULE_NAME:slowexp.c                                                 */
-/*                                                                        */
-/*  FUNCTION:slowexp                                                      */
-/*                                                                        */
-/*  FILES NEEDED:mpa.h                                                    */
-/*               mpa.c mpexp.c                                            */
-/*                                                                        */
-/*Converting from double precision to Multi-precision and calculating     */
-/* e^x                                                                    */
-/**************************************************************************/
-#include <math_private.h>
-
-#include <stap-probe.h>
-
-#ifndef USE_LONG_DOUBLE_FOR_MP
-# include "mpa.h"
-void __mpexp (mp_no *x, mp_no *y, int p);
-#endif
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-/*Converting from double precision to Multi-precision and calculating  e^x */
-double
-SECTION
-__slowexp (double x)
-{
-#ifndef USE_LONG_DOUBLE_FOR_MP
-  double w, z, res, eps = 3.0e-26;
-  int p;
-  mp_no mpx, mpy, mpz, mpw, mpeps, mpcor;
-
-  /* Use the multiple precision __MPEXP function to compute the exponential
-     First at 144 bits and if it is not accurate enough, at 768 bits.  */
-  p = 6;
-  __dbl_mp (x, &mpx, p);
-  __mpexp (&mpx, &mpy, p);
-  __dbl_mp (eps, &mpeps, p);
-  __mul (&mpeps, &mpy, &mpcor, p);
-  __add (&mpy, &mpcor, &mpw, p);
-  __sub (&mpy, &mpcor, &mpz, p);
-  __mp_dbl (&mpw, &w, p);
-  __mp_dbl (&mpz, &z, p);
-  if (w == z)
-    {
-      /* Track how often we get to the slow exp code plus
-	 its input/output values.  */
-      LIBC_PROBE (slowexp_p6, 2, &x, &w);
-      return w;
-    }
-  else
-    {
-      p = 32;
-      __dbl_mp (x, &mpx, p);
-      __mpexp (&mpx, &mpy, p);
-      __mp_dbl (&mpy, &res, p);
-
-      /* Track how often we get to the uber-slow exp code plus
-	 its input/output values.  */
-      LIBC_PROBE (slowexp_p32, 2, &x, &res);
-      return res;
-    }
-#else
-  return (double) __ieee754_expl((long double)x);
-#endif
-}
diff --git a/sysdeps/ieee754/dbl-64/uexp.h b/sysdeps/ieee754/dbl-64/uexp.h
index 2edf530b69..64ab2c8fca 100644
--- a/sysdeps/ieee754/dbl-64/uexp.h
+++ b/sysdeps/ieee754/dbl-64/uexp.h
@@ -29,8 +29,7 @@
 
 #include "mydefs.h"
 
-const static double zero = 0.0, hhuge = 1.0e300, tiny = 1.0e-300,
-err_0 = 1.000014;
+const static double zero = 0.0, hhuge = 1.0e300, tiny = 1.0e-300;
 const static int4 bigint = 0x40862002,
              badint = 0x40876000,smallint = 0x3C8fffff;
 const static int4 hugeint = 0x7FFFFFFF, infint = 0x7ff00000;
diff --git a/sysdeps/m68k/m680x0/fpu/slowexp.c b/sysdeps/m68k/m680x0/fpu/slowexp.c
deleted file mode 100644
index 1cc8931700..0000000000
--- a/sysdeps/m68k/m680x0/fpu/slowexp.c
+++ /dev/null
@@ -1 +0,0 @@
-/* Not needed.  */
diff --git a/sysdeps/powerpc/power4/fpu/Makefile b/sysdeps/powerpc/power4/fpu/Makefile
index fa1b070a00..f487ed6014 100644
--- a/sysdeps/powerpc/power4/fpu/Makefile
+++ b/sysdeps/powerpc/power4/fpu/Makefile
@@ -2,5 +2,4 @@
 
 ifeq ($(subdir),math)
 CFLAGS-mpa.c += --param max-unroll-times=4 -funroll-loops -fpeel-loops
-CPPFLAGS-slowexp.c += -DUSE_LONG_DOUBLE_FOR_MP=1
 endif
diff --git a/sysdeps/x86_64/fpu/multiarch/Makefile b/sysdeps/x86_64/fpu/multiarch/Makefile
index 9391eb5511..9a0283850a 100644
--- a/sysdeps/x86_64/fpu/multiarch/Makefile
+++ b/sysdeps/x86_64/fpu/multiarch/Makefile
@@ -10,7 +10,7 @@ libm-sysdep_routines += s_ceil-sse4_1 s_ceilf-sse4_1 s_floor-sse4_1 \
 
 libm-sysdep_routines += e_exp-fma e_log-fma e_pow-fma s_atan-fma \
 			e_asin-fma e_atan2-fma s_sin-fma s_tan-fma \
-			mplog-fma mpa-fma slowexp-fma \
+			mplog-fma mpa-fma \
 			sincos32-fma doasin-fma dosincos-fma \
 			mpexp-fma \
 			mpatan2-fma mpatan-fma mpsqrt-fma mptan-fma
@@ -31,7 +31,6 @@ CFLAGS-mpsqrt-fma.c = -mfma -mavx2
 CFLAGS-mptan-fma.c = -mfma -mavx2
 CFLAGS-s_atan-fma.c = -mfma -mavx2
 CFLAGS-sincos32-fma.c = -mfma -mavx2
-CFLAGS-slowexp-fma.c = -mfma -mavx2
 CFLAGS-s_sin-fma.c = -mfma -mavx2
 CFLAGS-s_tan-fma.c = -mfma -mavx2
 
@@ -51,7 +50,7 @@ CFLAGS-s_sincosf-fma.c = -mfma -mavx2
 
 libm-sysdep_routines += e_exp-fma4 e_log-fma4 e_pow-fma4 s_atan-fma4 \
 			e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4 \
-			mplog-fma4 mpa-fma4 slowexp-fma4 \
+			mplog-fma4 mpa-fma4 \
 			sincos32-fma4 doasin-fma4 dosincos-fma4 \
 			mpexp-fma4 \
 			mpatan2-fma4 mpatan-fma4 mpsqrt-fma4 mptan-fma4
@@ -72,13 +71,12 @@ CFLAGS-mpsqrt-fma4.c = -mfma4
 CFLAGS-mptan-fma4.c = -mfma4
 CFLAGS-s_atan-fma4.c = -mfma4
 CFLAGS-sincos32-fma4.c = -mfma4
-CFLAGS-slowexp-fma4.c = -mfma4
 CFLAGS-s_sin-fma4.c = -mfma4
 CFLAGS-s_tan-fma4.c = -mfma4
 
 libm-sysdep_routines += e_exp-avx e_log-avx s_atan-avx \
 			e_atan2-avx s_sin-avx s_tan-avx \
-			mplog-avx mpa-avx slowexp-avx \
+			mplog-avx mpa-avx \
 			mpexp-avx
 
 CFLAGS-e_atan2-avx.c = -msse2avx -DSSE2AVX
@@ -89,7 +87,6 @@ CFLAGS-mpexp-avx.c = -msse2avx -DSSE2AVX
 CFLAGS-mplog-avx.c = -msse2avx -DSSE2AVX
 CFLAGS-s_atan-avx.c = -msse2avx -DSSE2AVX
 CFLAGS-s_sin-avx.c = -msse2avx -DSSE2AVX
-CFLAGS-slowexp-avx.c = -msse2avx -DSSE2AVX
 CFLAGS-s_tan-avx.c = -msse2avx -DSSE2AVX
 endif
 
diff --git a/sysdeps/x86_64/fpu/multiarch/e_exp-avx.c b/sysdeps/x86_64/fpu/multiarch/e_exp-avx.c
index ee5dd6d2dc..afd917442a 100644
--- a/sysdeps/x86_64/fpu/multiarch/e_exp-avx.c
+++ b/sysdeps/x86_64/fpu/multiarch/e_exp-avx.c
@@ -1,6 +1,5 @@
 #define __ieee754_exp __ieee754_exp_avx
 #define __exp1 __exp1_avx
-#define __slowexp __slowexp_avx
 #define SECTION __attribute__ ((section (".text.avx")))
 
 #include <sysdeps/ieee754/dbl-64/e_exp.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/e_exp-fma.c b/sysdeps/x86_64/fpu/multiarch/e_exp-fma.c
index 6e0fdb7941..765b1b9dd3 100644
--- a/sysdeps/x86_64/fpu/multiarch/e_exp-fma.c
+++ b/sysdeps/x86_64/fpu/multiarch/e_exp-fma.c
@@ -1,6 +1,5 @@
 #define __ieee754_exp __ieee754_exp_fma
 #define __exp1 __exp1_fma
-#define __slowexp __slowexp_fma
 #define SECTION __attribute__ ((section (".text.fma")))
 
 #include <sysdeps/ieee754/dbl-64/e_exp.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c b/sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c
index ae6eb67603..9ac7acad28 100644
--- a/sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c
+++ b/sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c
@@ -1,6 +1,5 @@
 #define __ieee754_exp __ieee754_exp_fma4
 #define __exp1 __exp1_fma4
-#define __slowexp __slowexp_fma4
 #define SECTION __attribute__ ((section (".text.fma4")))
 
 #include <sysdeps/ieee754/dbl-64/e_exp.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/slowexp-avx.c b/sysdeps/x86_64/fpu/multiarch/slowexp-avx.c
deleted file mode 100644
index d01c6d71a4..0000000000
--- a/sysdeps/x86_64/fpu/multiarch/slowexp-avx.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_avx
-#define __add __add_avx
-#define __dbl_mp __dbl_mp_avx
-#define __mpexp __mpexp_avx
-#define __mul __mul_avx
-#define __sub __sub_avx
-#define SECTION __attribute__ ((section (".text.avx")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/slowexp-fma.c b/sysdeps/x86_64/fpu/multiarch/slowexp-fma.c
deleted file mode 100644
index 6fffca1a93..0000000000
--- a/sysdeps/x86_64/fpu/multiarch/slowexp-fma.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_fma
-#define __add __add_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mpexp __mpexp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>
diff --git a/sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c b/sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c
deleted file mode 100644
index 3bcde84233..0000000000
--- a/sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c
+++ /dev/null
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_fma4
-#define __add __add_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mpexp __mpexp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>