Fix powl inaccuracy for x86_64 and x86 (bug 13881).

9 files changed, 334 insertions, 74 deletions

diff --git a/ChangeLog b/ChangeLog
index 7df91e5798..d93d0a35fa 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,21 @@
+2012-11-28  Joseph Myers  <joseph@codesourcery.com>
+
+	[BZ #13881]
+	* sysdeps/x86/fpu/powl_helper.c: New file.
+	* sysdeps/x86/fpu/Makefile: Likewise.
+	* sysdeps/i386/fpu/e_powl.S (limit): Remove object.
+	(p3): New object.
+	(__ieee754_powl): Use __powl_helper for finite arguments except
+	integer exponents below 8.
+	* sysdeps/x86_64/fpu/e_powl.S (limit): Remove object.
+	(p3): New object.
+	(__ieee754_powl): Use __powl_helper for finite arguments except
+	integer exponents below 8.
+	* math/libm-test.inc (pow_test): Add more tests and enable some
+	previously disabled tests.
+	* sysdeps/i386/fpu/libm-test-ulps: Update.
+	* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
+
 2012-11-28  Siddhesh Poyarekar  <siddhesh@redhat.com>
 	    Carlos O'Donell  <carlos_odonell@mentor.com>
 
diff --git a/NEWS b/NEWS
index e8c44adc25..1d4a979179 100644
--- a/NEWS
+++ b/NEWS
@@ -12,16 +12,17 @@ Version 2.17
   1349, 3439, 3479, 3665, 5044, 5246, 5298, 5400, 6530, 6778, 6808, 9685,
   9914, 10014, 10038, 10631, 10873, 11438, 11607, 11638, 11741, 12140,
   13412, 13542, 13601, 13603, 13604, 13629, 13679, 13696, 13698, 13717,
-  13741, 13759, 13763, 13939, 13950, 13952, 13966, 14042, 14047, 14090,
-  14150, 14151, 14152, 14154, 14157, 14166, 14173, 14195, 14237, 14251,
-  14252, 14283, 14298, 14303, 14307, 14328, 14331, 14336, 14337, 14347,
-  14349, 14368, 14376, 14417, 14459, 14476, 14477, 14501, 14505, 14510,
-  14516, 14518, 14519, 14530, 14532, 14538, 14543, 14544, 14545, 14557,
-  14562, 14568, 14576, 14579, 14583, 14587, 14595, 14602, 14610, 14621,
-  14638, 14645, 14648, 14652, 14660, 14661, 14669, 14672, 14683, 14694,
-  14716, 14719, 14743, 14767, 14783, 14784, 14785, 14793, 14796, 14797,
-  14801, 14805, 14807, 14809, 14811, 14815, 14821, 14822, 14824, 14828,
-  14831, 14835, 14838, 14856, 14863, 14865, 14866, 14868, 14869, 14871.
+  13741, 13759, 13763, 13881, 13939, 13950, 13952, 13966, 14042, 14047,
+  14090, 14150, 14151, 14152, 14154, 14157, 14166, 14173, 14195, 14237,
+  14251, 14252, 14283, 14298, 14303, 14307, 14328, 14331, 14336, 14337,
+  14347, 14349, 14368, 14376, 14417, 14459, 14476, 14477, 14501, 14505,
+  14510, 14516, 14518, 14519, 14530, 14532, 14538, 14543, 14544, 14545,
+  14557, 14562, 14568, 14576, 14579, 14583, 14587, 14595, 14602, 14610,
+  14621, 14638, 14645, 14648, 14652, 14660, 14661, 14669, 14672, 14683,
+  14694, 14716, 14719, 14743, 14767, 14783, 14784, 14785, 14793, 14796,
+  14797, 14801, 14805, 14807, 14809, 14811, 14815, 14821, 14822, 14824,
+  14828, 14831, 14835, 14838, 14856, 14863, 14865, 14866, 14868, 14869,
+  14871.
 
 * Port to ARM AArch64 contributed by Linaro.
 
diff --git a/math/libm-test.inc b/math/libm-test.inc
index c0265113ca..21590645f9 100644
--- a/math/libm-test.inc
+++ b/math/libm-test.inc
@@ -8431,7 +8431,6 @@ pow_test (void)
 #endif
   TEST_ff_f (pow, -min_value, max_value, plus_zero, UNDERFLOW_EXCEPTION);
 
-#ifndef TEST_LDOUBLE /* Bug 13881.  */
   TEST_ff_f (pow, 0x0.ffffffp0, 10, 0.999999403953712118183885036774764444747L);
   TEST_ff_f (pow, 0x0.ffffffp0, 100, 0.999994039553108359406305079606228341585L);
   TEST_ff_f (pow, 0x0.ffffffp0, 1000, 0.9999403971297699052276650144650733772182L);
@@ -8447,16 +8446,44 @@ pow_test (void)
   TEST_ff_f (pow, 0x1.000002p0, 0x1p24, 7.3890552180866447284268641248075832310141L);
   TEST_ff_f (pow, 0x1.000002p0, 0x1.234566p29, 4.2107033006507495188536371520637025716256e+31L);
   TEST_ff_f (pow, 0x1.000002p0, -0x1.234566p29, 2.3749001736727769098946062325205705312166e-32L);
-#endif
 
-  /* Bug 13881: powl inaccurate so these tests disabled for long double.  */
-#if !defined TEST_FLOAT && !defined TEST_LDOUBLE
+#if !defined TEST_FLOAT
   TEST_ff_f (pow, 0x0.fffffffffffff8p0L, 0x1.23456789abcdfp62L, 1.0118762747827252817436395051178295138220e-253L);
   TEST_ff_f (pow, 0x0.fffffffffffff8p0L, -0x1.23456789abcdfp62L, 9.8826311568054561811190162420900667121992e+252L);
   TEST_ff_f (pow, 0x1.0000000000001p0L, 0x1.23456789abcdfp61L, 9.8826311568044974397135026217687399395481e+252L);
   TEST_ff_f (pow, 0x1.0000000000001p0L, -0x1.23456789abcdfp61L, 1.0118762747828234466621210689458255908670e-253L);
 #endif
 
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 64 && LDBL_MAX_EXP >= 16384
+  TEST_ff_f (pow, 0x0.ffffffffffffffffp0L, 0x1.23456789abcdef0ep77L, 1.2079212226420368189981778807634890018840e-4048L);
+  TEST_ff_f (pow, 0x0.ffffffffffffffffp0L, -0x1.23456789abcdef0ep77L, 8.2786855736563746280496724205839522148001e+4047L);
+  TEST_ff_f (pow, 0x1.0000000000000002p0L, 0x1.23456789abcdef0ep76L, 8.2786855736563683535324500168799315131570e+4047L);
+  TEST_ff_f (pow, 0x1.0000000000000002p0L, -0x1.23456789abcdef0ep76L, 1.2079212226420377344964713407722652880280e-4048L);
+#endif
+
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 113
+  TEST_ff_f (pow, 0x0.ffffffffffffffffffffffffffff8p0L, 0x1.23456789abcdef0123456789abcdp126L, 1.2079212226420440237790185999151440179953e-4048L);
+  TEST_ff_f (pow, 0x0.ffffffffffffffffffffffffffff8p0L, -0x1.23456789abcdef0123456789abcdp126L, 8.2786855736563252489063231915535105363602e+4047L);
+  TEST_ff_f (pow, 0x1.0000000000000000000000000001p0L, 0x1.23456789abcdef0123456789abcdp125L, 8.2786855736563252489063231915423647547782e+4047L);
+  TEST_ff_f (pow, 0x1.0000000000000000000000000001p0L, -0x1.23456789abcdef0123456789abcdp125L, 1.2079212226420440237790185999167702696503e-4048L);
+#endif
+
+#if defined TEST_LDOUBLE && LDBL_MAX_EXP >= 16384
+  TEST_ff_f (pow, 1e4932L, 0.75L, 1e3699L);
+  TEST_ff_f (pow, 1e4928L, 0.75L, 1e3696L);
+  TEST_ff_f (pow, 1e4924L, 0.75L, 1e3693L);
+  TEST_ff_f (pow, 1e4920L, 0.75L, 1e3690L);
+  TEST_ff_f (pow, 10.0L, 4932.0L, 1e4932L);
+  TEST_ff_f (pow, 10.0L, 4931.0L, 1e4931L);
+  TEST_ff_f (pow, 10.0L, 4930.0L, 1e4930L);
+  TEST_ff_f (pow, 10.0L, 4929.0L, 1e4929L);
+  TEST_ff_f (pow, 10.0L, -4931.0L, 1e-4931L);
+  TEST_ff_f (pow, 10.0L, -4930.0L, 1e-4930L);
+  TEST_ff_f (pow, 10.0L, -4929.0L, 1e-4929L);
+  TEST_ff_f (pow, 1e27L, 182.0L, 1e4914L);
+  TEST_ff_f (pow, 1e27L, -182.0L, 1e-4914L);
+#endif
+
   TEST_ff_f (pow, min_subnorm_value, min_subnorm_value, 1.0L);
   TEST_ff_f (pow, min_subnorm_value, -min_subnorm_value, 1.0L);
   TEST_ff_f (pow, max_value, min_subnorm_value, 1.0L);
diff --git a/sysdeps/i386/fpu/e_powl.S b/sysdeps/i386/fpu/e_powl.S
index ac4842cf63..7e297756ff 100644
--- a/sysdeps/i386/fpu/e_powl.S
+++ b/sysdeps/i386/fpu/e_powl.S
@@ -26,9 +26,9 @@
 	.type one,@object
 one:	.double 1.0
 	ASM_SIZE_DIRECTIVE(one)
-	.type limit,@object
-limit:	.double 0.29
-	ASM_SIZE_DIRECTIVE(limit)
+	.type p3,@object
+p3:	.byte 0, 0, 0, 0, 0, 0, 0x20, 0x40
+	ASM_SIZE_DIRECTIVE(p3)
 	.type p63,@object
 p63:	.byte 0, 0, 0, 0, 0, 0, 0xe0, 0x43
 	ASM_SIZE_DIRECTIVE(p63)
@@ -141,7 +141,15 @@ ENTRY(__ieee754_powl)
 	fchs			// -0x1p-79 : x
 	jmp	3f
 
-9:	/* OK, we have an integer value for y.  */
+9:	/* OK, we have an integer value for y.  Unless very small
+	   (we use < 8), use the algorithm for real exponent to avoid
+	   accumulation of errors.  */
+	fld	%st		// y : y : x
+	fabs			// |y| : y : x
+	fcompl	MO(p3)		// y : x
+	fnstsw
+	sahf
+	jnc	2f
 	popl	%eax
 	cfi_adjust_cfa_offset (-4)
 	popl	%edx
@@ -182,7 +190,7 @@ ENTRY(__ieee754_powl)
 
 	cfi_adjust_cfa_offset (8)
 	.align ALIGNARG(4)
-2:	// y is a large integer (absolute value at least 1L<<63), but
+2:	// y is a large integer (absolute value at least 8), but
 	// may be odd unless at least 1L<<64.  So it may be necessary
 	// to adjust the sign of a negative result afterwards.
 	fxch			// x : y
@@ -205,34 +213,21 @@ ENTRY(__ieee754_powl)
 	fchs			// -(1L<<78) : |x|
 	.align ALIGNARG(4)
 3:	/* y is a real number.  */
-	fxch			// x : y
-	fldl	MO(one)		// 1.0 : x : y
-	fldl	MO(limit)	// 0.29 : 1.0 : x : y
-	fld	%st(2)		// x : 0.29 : 1.0 : x : y
-	fsub	%st(2)		// x-1 : 0.29 : 1.0 : x : y
-	fabs			// |x-1| : 0.29 : 1.0 : x : y
-	fucompp			// 1.0 : x : y
-	fnstsw
-	fxch			// x : 1.0 : y
-	sahf
-	ja	7f
-	fsub	%st(1)		// x-1 : 1.0 : y
-	fyl2xp1			// log2(x) : y
-	jmp	8f
-
-7:	fyl2x			// log2(x) : y
-8:	fmul	%st(1)		// y*log2(x) : y
-	fst	%st(1)		// y*log2(x) : y*log2(x)
-	frndint			// int(y*log2(x)) : y*log2(x)
-	fsubr	%st, %st(1)	// int(y*log2(x)) : fract(y*log2(x))
-	fxch			// fract(y*log2(x)) : int(y*log2(x))
-	f2xm1			// 2^fract(y*log2(x))-1 : int(y*log2(x))
-	faddl	MO(one)		// 2^fract(y*log2(x)) : int(y*log2(x))
-	fscale			// 2^fract(y*log2(x))*2^int(y*log2(x)) : int(y*log2(x))
-	fstp	%st(1)		// 2^fract(y*log2(x))*2^int(y*log2(x))
+	subl	$28, %esp
+	cfi_adjust_cfa_offset (28)
+	fstpt	12(%esp)	// x
+	fstpt	(%esp)		// <empty>
+	mov	%edx, 24(%esp)
+	call	HIDDEN_JUMPTARGET (__powl_helper)	// <result>
+	mov	24(%esp), %edx
+	addl	$28, %esp
+	cfi_adjust_cfa_offset (-28)
 	testb	$2, %dh
 	jz	292f
 	// x is negative.  If y is an odd integer, negate the result.
+#ifdef	PIC
+	LOAD_PIC_REG (cx)
+#endif
 	fldt	24(%esp)	// y : abs(result)
 	fld	%st		// y : y : abs(result)
 	fabs			// |y| : y : abs(result)
diff --git a/sysdeps/i386/fpu/libm-test-ulps b/sysdeps/i386/fpu/libm-test-ulps
index 8be00860c9..5b595bc566 100644
--- a/sysdeps/i386/fpu/libm-test-ulps
+++ b/sysdeps/i386/fpu/libm-test-ulps
@@ -2480,6 +2480,11 @@ ifloat: 1
 ildouble: 1
 ldouble: 1
 
+# pow
+Test "pow (0x0.ffffffp0, -0x1p24) == 2.7182819094701610539628664526874952929416":
+ildouble: 1
+ldouble: 1
+
 # pow_downward
 Test "pow_downward (1.0625, 1.125) == 1.070582293028761362162622578677070098674":
 double: 1
@@ -3782,6 +3787,10 @@ ifloat: 1
 ildouble: 1
 ldouble: 1
 
+Function: "pow":
+ildouble: 1
+ldouble: 1
+
 Function: "pow_downward":
 double: 1
 float: 1
diff --git a/sysdeps/x86/fpu/Makefile b/sysdeps/x86/fpu/Makefile
new file mode 100644
index 0000000000..8054380477
--- /dev/null
+++ b/sysdeps/x86/fpu/Makefile
@@ -0,0 +1,3 @@
+ifeq ($(subdir),math)
+libm-support += powl_helper
+endif
diff --git a/sysdeps/x86/fpu/powl_helper.c b/sysdeps/x86/fpu/powl_helper.c
new file mode 100644
index 0000000000..3f69b08a1b
--- /dev/null
+++ b/sysdeps/x86/fpu/powl_helper.c
@@ -0,0 +1,211 @@
+/* Implement powl for x86 using extra-precision log.
+   Copyright (C) 2012 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.h>
+#include <math_private.h>
+
+/* High parts and low parts of -log (k/16), for integer k from 12 to
+   24.  */
+
+static const long double powl_log_table[] =
+  {
+    0x4.9a58844d36e49e1p-4L, -0x1.0522624fd558f574p-68L,
+    0x3.527da7915b3c6de4p-4L, 0x1.7d4ef4b901b99b9ep-68L,
+    0x2.22f1d044fc8f7bc8p-4L, -0x1.8e97c071a42fc388p-68L,
+    0x1.08598b59e3a0688ap-4L, 0x3.fd9bf503372c12fcp-72L,
+    -0x0p+0L, 0x0p+0L,
+    -0xf.85186008b15330cp-8L, 0x1.9b47488a6687672cp-72L,
+    -0x1.e27076e2af2e5e9ep-4L, -0xa.87ffe1fe9e155dcp-72L,
+    -0x2.bfe60e14f27a791p-4L, 0x1.83bebf1bdb88a032p-68L,
+    -0x3.91fef8f353443584p-4L, -0xb.b03de5ff734495cp-72L,
+    -0x4.59d72aeae98380e8p-4L, 0xc.e0aa3be4747dc1p-72L,
+    -0x5.1862f08717b09f4p-4L, -0x2.decdeccf1cd10578p-68L,
+    -0x5.ce75fdaef401a738p-4L, -0x9.314feb4fbde5aaep-72L,
+    -0x6.7cc8fb2fe612fcbp-4L, 0x2.5ca2642feb779f98p-68L,
+  };
+
+/* High 32 bits of log2 (e), and remainder rounded to 64 bits.  */
+static const long double log2e_hi = 0x1.71547652p+0L;
+static const long double log2e_lo = 0xb.82fe1777d0ffda1p-36L;
+
+/* Given a number with high part HI and low part LO, add the number X
+   to it and store the result in *RHI and *RLO.  It is given that
+   either |X| < |0.7 * HI|, or HI == LO == 0, and that the values are
+   small enough that no overflow occurs.  The result does not need to
+   be exact to 128 bits; 78-bit accuracy of the final accumulated
+   result suffices.  */
+
+static inline void
+acc_split (long double *rhi, long double *rlo, long double hi, long double lo,
+	   long double x)
+{
+  long double thi = hi + x;
+  long double tlo = (hi - thi) + x + lo;
+  *rhi = thi + tlo;
+  *rlo = (thi - *rhi) + tlo;
+}
+
+extern long double __powl_helper (long double x, long double y);
+libm_hidden_proto (__powl_helper)
+
+/* Given X a value that is finite and nonzero, or a NaN, and only
+   negative if Y is not an integer, and Y a finite nonzero value with
+   0x1p-79 <= |Y| <= 0x1p78, compute X to the power Y.  */
+
+long double
+__powl_helper (long double x, long double y)
+{
+  if (isnan (x) || x < 0)
+    return __ieee754_expl (y * __ieee754_logl (x));
+
+  /* We need to compute Y * log2 (X) to at least 64 bits after the
+     point for normal results (that is, to at least 78 bits
+     precision).  */
+  int x_int_exponent;
+  long double x_frac;
+  x_frac = __frexpl (x, &x_int_exponent);
+  if (x_frac <= 0x0.aaaaaaaaaaaaaaaap0L) /* 2.0L / 3.0L, rounded down */
+    {
+      x_frac *= 2.0;
+      x_int_exponent--;
+    }
+
+  long double log_x_frac_hi, log_x_frac_lo;
+  /* Determine an initial approximation to log (X_FRAC) using
+     POWL_LOG_TABLE, and multiply by a value K/16 to reduce to an
+     interval (24/25, 26/25).  */
+  int k = (int) ((16.0L / x_frac) + 0.5L);
+  log_x_frac_hi = powl_log_table[2 * k - 24];
+  log_x_frac_lo = powl_log_table[2 * k - 23];
+  long double x_frac_low;
+  if (k == 16)
+    x_frac_low = 0.0L;
+  else
+    {
+      /* Mask off low 5 bits of X_FRAC so the multiplication by K/16
+	 is exact.  These bits are small enough that they can be
+	 corrected for by adding log2 (e) * X_FRAC_LOW to the final
+	 result.  */
+      int32_t se;
+      u_int32_t i0, i1;
+      GET_LDOUBLE_WORDS (se, i0, i1, x_frac);
+      x_frac_low = x_frac;
+      i1 &= 0xffffffe0;
+      SET_LDOUBLE_WORDS (x_frac, se, i0, i1);
+      x_frac_low -= x_frac;
+      x_frac_low /= x_frac;
+      x_frac *= k / 16.0L;
+    }
+
+  /* Now compute log (X_FRAC) for X_FRAC in (24/25, 26/25).  Separate
+     W = X_FRAC - 1 into high 16 bits and remaining bits, so that
+     multiplications for low-order power series terms are exact.  The
+     remaining bits are small enough that adding a 64-bit value of
+     log2 (1 + W_LO / (1 + W_HI)) will be a sufficient correction for
+     them.  */
+  long double w = x_frac - 1;
+  long double w_hi, w_lo;
+  int32_t se;
+  u_int32_t i0, i1;
+  GET_LDOUBLE_WORDS (se, i0, i1, w);
+  i0 &= 0xffff0000;
+  i1 = 0;
+  SET_LDOUBLE_WORDS (w_hi, se, i0, i1);
+  w_lo = w - w_hi;
+  long double wp = w_hi;
+  acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo, wp);
+  wp *= -w_hi;
+  acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
+	     wp / 2.0L);
+  wp *= -w_hi;
+  acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
+	     wp * 0x0.5555p0L); /* -W_HI**3 / 3, high part.  */
+  acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
+	     wp * 0x0.5555555555555555p-16L); /* -W_HI**3 / 3, low part.  */
+  wp *= -w_hi;
+  acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
+	     wp / 4.0L);
+  /* Subsequent terms are small enough that they only need be computed
+     to 64 bits.  */
+  for (int i = 5; i <= 17; i++)
+    {
+      wp *= -w_hi;
+      acc_split (&log_x_frac_hi, &log_x_frac_lo, log_x_frac_hi, log_x_frac_lo,
+		 wp / i);
+    }
+
+  /* Convert LOG_X_FRAC_HI + LOG_X_FRAC_LO to a base-2 logarithm.  */
+  long double log2_x_frac_hi, log2_x_frac_lo;
+  long double log_x_frac_hi32, log_x_frac_lo64;
+  GET_LDOUBLE_WORDS (se, i0, i1, log_x_frac_hi);
+  i1 = 0;
+  SET_LDOUBLE_WORDS (log_x_frac_hi32, se, i0, i1);
+  log_x_frac_lo64 = (log_x_frac_hi - log_x_frac_hi32) + log_x_frac_lo;
+  long double log2_x_frac_hi1 = log_x_frac_hi32 * log2e_hi;
+  long double log2_x_frac_lo1
+    = log_x_frac_lo64 * log2e_hi + log_x_frac_hi * log2e_lo;
+  log2_x_frac_hi = log2_x_frac_hi1 + log2_x_frac_lo1;
+  log2_x_frac_lo = (log2_x_frac_hi1 - log2_x_frac_hi) + log2_x_frac_lo1;
+
+  /* Correct for the masking off of W_LO.  */
+  long double log2_1p_w_lo;
+  asm ("fyl2xp1"
+       : "=t" (log2_1p_w_lo)
+       : "0" (w_lo / (1.0L + w_hi)), "u" (1.0L)
+       : "st(1)");
+  acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
+	     log2_1p_w_lo);
+
+  /* Correct for the masking off of X_FRAC_LOW.  */
+  acc_split (&log2_x_frac_hi, &log2_x_frac_lo, log2_x_frac_hi, log2_x_frac_lo,
+	     x_frac_low * M_LOG2El);
+
+  /* Add the integer and fractional parts of the base-2 logarithm.  */
+  long double log2_x_hi, log2_x_lo;
+  log2_x_hi = x_int_exponent + log2_x_frac_hi;
+  log2_x_lo = ((x_int_exponent - log2_x_hi) + log2_x_frac_hi) + log2_x_frac_lo;
+
+  /* Compute the base-2 logarithm of the result.  */
+  long double log2_res_hi, log2_res_lo;
+  long double log2_x_hi32, log2_x_lo64;
+  GET_LDOUBLE_WORDS (se, i0, i1, log2_x_hi);
+  i1 = 0;
+  SET_LDOUBLE_WORDS (log2_x_hi32, se, i0, i1);
+  log2_x_lo64 = (log2_x_hi - log2_x_hi32) + log2_x_lo;
+  long double y_hi32, y_lo32;
+  GET_LDOUBLE_WORDS (se, i0, i1, y);
+  i1 = 0;
+  SET_LDOUBLE_WORDS (y_hi32, se, i0, i1);
+  y_lo32 = y - y_hi32;
+  log2_res_hi = log2_x_hi32 * y_hi32;
+  log2_res_lo = log2_x_hi32 * y_lo32 + log2_x_lo64 * y;
+
+  /* Split the base-2 logarithm of the result into integer and
+     fractional parts.  */
+  long double log2_res_int = __roundl (log2_res_hi);
+  long double log2_res_frac = log2_res_hi - log2_res_int + log2_res_lo;
+
+  /* Compute the final result.  */
+  long double res;
+  asm ("f2xm1" : "=t" (res) : "0" (log2_res_frac));
+  res += 1.0L;
+  asm ("fscale" : "=t" (res) : "0" (res), "u" (log2_res_int));
+  return res;
+}
+
+libm_hidden_def (__powl_helper)
diff --git a/sysdeps/x86_64/fpu/e_powl.S b/sysdeps/x86_64/fpu/e_powl.S
index 1b3718522d..ff96cec68a 100644
--- a/sysdeps/x86_64/fpu/e_powl.S
+++ b/sysdeps/x86_64/fpu/e_powl.S
@@ -26,9 +26,9 @@
 	.type one,@object
 one:	.double 1.0
 	ASM_SIZE_DIRECTIVE(one)
-	.type limit,@object
-limit:	.double 0.29
-	ASM_SIZE_DIRECTIVE(limit)
+	.type p3,@object
+p3:	.byte 0, 0, 0, 0, 0, 0, 0x20, 0x40
+	ASM_SIZE_DIRECTIVE(p3)
 	.type p63,@object
 p63:	.byte 0, 0, 0, 0, 0, 0, 0xe0, 0x43
 	ASM_SIZE_DIRECTIVE(p63)
@@ -131,7 +131,15 @@ ENTRY(__ieee754_powl)
 	fchs			// -0x1p-79 : x
 	jmp	3f
 
-9:	/* OK, we have an integer value for y.  */
+9:	/* OK, we have an integer value for y.  Unless very small
+	   (we use < 8), use the algorithm for real exponent to avoid
+	   accumulation of errors.  */
+	fldl	MO(p3)		// 8 : y : x
+	fld	%st(1)		// y : 8 : y : x
+	fabs			// |y| : 8 : y : x
+	fcomip	%st(1), %st	// 8 : y : x
+	fstp	%st(0)		// y : x
+	jnc	2f
 	mov	-8(%rsp),%eax
 	mov	-4(%rsp),%edx
 	orl	$0, %edx
@@ -167,7 +175,7 @@ ENTRY(__ieee754_powl)
 	ret
 
 	.align ALIGNARG(4)
-2:	// y is a large integer (absolute value at least 1L<<63), but
+2:	// y is a large integer (absolute value at least 8), but
 	// may be odd unless at least 1L<<64.  So it may be necessary
 	// to adjust the sign of a negative result afterwards.
 	fxch			// x : y
@@ -190,31 +198,15 @@ ENTRY(__ieee754_powl)
 	fchs			// -(1L<<78) : |x|
 	.align ALIGNARG(4)
 3:	/* y is a real number.  */
-	fxch			// x : y
-	fldl	MO(one)		// 1.0 : x : y
-	fldl	MO(limit)	// 0.29 : 1.0 : x : y
-	fld	%st(2)		// x : 0.29 : 1.0 : x : y
-	fsub	%st(2)		// x-1 : 0.29 : 1.0 : x : y
-	fabs			// |x-1| : 0.29 : 1.0 : x : y
-	fucompp			// 1.0 : x : y
-	fnstsw
-	fxch			// x : 1.0 : y
-	test	$0x4500,%eax
-	jz	7f
-	fsub	%st(1)		// x-1 : 1.0 : y
-	fyl2xp1			// log2(x) : y
-	jmp	8f
-
-7:	fyl2x			// log2(x) : y
-8:	fmul	%st(1)		// y*log2(x) : y
-	fst	%st(1)		// y*log2(x) : y*log2(x)
-	frndint			// int(y*log2(x)) : y*log2(x)
-	fsubr	%st, %st(1)	// int(y*log2(x)) : fract(y*log2(x))
-	fxch			// fract(y*log2(x)) : int(y*log2(x))
-	f2xm1			// 2^fract(y*log2(x))-1 : int(y*log2(x))
-	faddl	MO(one)		// 2^fract(y*log2(x)) : int(y*log2(x))
-	fscale			// 2^fract(y*log2(x))*2^int(y*log2(x)) : int(y*log2(x))
-	fstp	%st(1)		// 2^fract(y*log2(x))*2^int(y*log2(x))
+	subq	$40, %rsp
+	cfi_adjust_cfa_offset (40)
+	fstpt	16(%rsp)	// x
+	fstpt	(%rsp)		// <empty>
+	mov	%edx, 32(%rsp)
+	call	HIDDEN_JUMPTARGET (__powl_helper)	// <result>
+	mov	32(%rsp), %edx
+	addq	$40, %rsp
+	cfi_adjust_cfa_offset (-40)
 	testb	$2, %dh
 	jz	292f
 	// x is negative.  If y is an odd integer, negate the result.
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index f33dfed3df..9e7a8adac3 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -2398,6 +2398,8 @@ ifloat: 1
 Test "pow (0x0.ffffffp0, -0x1p24) == 2.7182819094701610539628664526874952929416":
 float: 1
 ifloat: 1
+ildouble: 1
+ldouble: 1
 Test "pow (0x0.ffffffp0, 0x1p24) == 0.3678794302077803437135155590023422899744":
 float: 1
 ifloat: 1
@@ -3575,6 +3577,8 @@ ifloat: 1
 Function: "pow":
 float: 1
 ifloat: 1
+ildouble: 1
+ldouble: 1
 
 Function: "pow_downward":
 float: 1