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Fix powl inaccuracy for x86_64 and x86 (bug 13881)
- From: "Joseph S. Myers" <joseph at codesourcery dot com>
- To: <libc-alpha at sourceware dot org>
- Date: Wed, 28 Nov 2012 03:57:54 +0000
- Subject: Fix powl inaccuracy for x86_64 and x86 (bug 13881)
This patch fixes bug 13881, inaccuracy of powl on x86 / x86_64, by
using a C helper function to implement the computation of the log of
the result with the extra precision required for powl (that's
complicated enough that at least initially using a C function seems
much simpler than direct implementation in both x86 and x86_64
assembly). That log is computed using a reasonably straightforward
power series approach, after reducing the argument to something within
+/- 1/25 of 1, that may not be optimally efficient but is at least not
too complicated to implement and where the few precomputed values
needed are easy to compute with MPFR.
Tested x86_64 and x86 and ulps updated accordingly.
(In the long chains of dependencies for fixing open "math" bugs,
having reasonably accurate pow for all floating-point formats should
help fix tgamma inaccuracy (bugs 2546 / 2560 / 5159), where the
present code uses exp (lgamma) which is inaccurate for the same reason
as exp (y * log (x)) is inaccurate for pow. But that will need to
wait for 2.18.)
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.
diff --git a/math/libm-test.inc b/math/libm-test.inc
index c026511..2159064 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 ac4842c..7e29775 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 $2, %dh
jz 292f
// x is negative. If y is an odd integer, negate the result.
+#ifdef PIC
+ LOAD_PIC_REG (cx)
+#endif
fldt 24(%esp) // y : abs(result)
fld %st // y : y : abs(result)
fabs // |y| : y : abs(result)
diff --git a/sysdeps/i386/fpu/libm-test-ulps b/sysdeps/i386/fpu/libm-test-ulps
index 8be0086..5b595bc 100644
--- a/sysdeps/i386/fpu/libm-test-ulps
+++ b/sysdeps/i386/fpu/libm-test-ulps
@@ -2480,6 +2480,11 @@ ifloat: 1
ildouble: 1
ldouble: 1
+# pow
+Test "pow (0x0.ffffffp0, -0x1p24) == 2.7182819094701610539628664526874952929416":
+ildouble: 1
+ldouble: 1
+
# pow_downward
Test "pow_downward (1.0625, 1.125) == 1.070582293028761362162622578677070098674":
double: 1
@@ -3782,6 +3787,10 @@ ifloat: 1
ildouble: 1
ldouble: 1
+Function: "pow":
+ildouble: 1
+ldouble: 1
+
Function: "pow_downward":
double: 1
float: 1
diff --git a/sysdeps/x86/fpu/Makefile b/sysdeps/x86/fpu/Makefile
new file mode 100644
index 0000000..8054380
--- /dev/null
+++ b/sysdeps/x86/fpu/Makefile
@@ -0,0 +1,3 @@
+ifeq ($(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 0000000..3f69b08
--- /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 1b37185..ff96cec 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 f33dfed..9e7a8ad 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
--
Joseph S. Myers
joseph@codesourcery.com