[10/17] math: Use exp2m1f from CORE-MATH
Commit Message
The CORE-MATH implementation is correctly rounded (for any rounding mode)
and shows better performance compared to the generic exp2m1f.
The code was adapted to glibc style and to use the definition of
math_config.h (to handle errno, overflow, and underflow). The
only change is to handle FLT_MAX_EXP for FE_DOWNWARD or FE_TOWARDZERO.
The benchmark inputs are based on exp2f ones.
Benchtest on x64_64 (Ryzen 9 5900X, gcc 14.2.1), aarch64 (Neoverse-N1,
gcc 13.3.1), and powerpc (POWER10, gcc 13.2.1):
Latency master patched improvement
x86_64 40.6042 48.7104 -19.96%
x86_64v2 40.7506 35.9032 11.90%
x86_64v3 35.2301 31.7956 9.75%
i686 102.094 94.6657 7.28%
aarch64 18.2704 15.1387 17.14%
power10 11.9444 8.2402 31.01%
reciprocal-throughput master patched improvement
x86_64 20.8683 16.1428 22.64%
x86_64v2 19.5076 10.4474 46.44%
x86_64v3 19.2106 10.4014 45.86%
i686 56.4054 59.3004 -5.13%
aarch64 12.0781 7.3953 38.77%
power10 6.5306 5.9388 9.06%
The generic implementation calls __ieee754_exp2f and x86_64 provides
an optimized ifunc version (built with -mfma -mavx2, not correctly
rounded). This explains the performance difference for x86_64.
Same for i686, where the ABI provides an optimized __ieee754_exp2f
version built with '-msse2 -mfpmath=sse'. When built wth same
flags, the new algorithm shows a better performance:
master patched improvement
latency 102.094 91.2823 10.59%
reciprocal-throughput 56.4054 52.7984 6.39%
Signed-off-by: Alexei Sibidanov <sibid@uvic.ca>
Signed-off-by: Paul Zimmermann <Paul.Zimmermann@inria.fr>
Signed-off-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
---
SHARED-FILES | 4 +
sysdeps/aarch64/libm-test-ulps | 4 -
sysdeps/arc/fpu/libm-test-ulps | 4 -
sysdeps/arc/nofpu/libm-test-ulps | 1 -
sysdeps/arm/libm-test-ulps | 4 -
sysdeps/hppa/fpu/libm-test-ulps | 4 -
sysdeps/i386/fpu/libm-test-ulps | 4 -
.../i386/i686/fpu/multiarch/libm-test-ulps | 4 -
sysdeps/ieee754/flt-32/s_exp2m1f.c | 192 ++++++++++++++++++
sysdeps/loongarch/lp64/libm-test-ulps | 4 -
sysdeps/m68k/m680x0/fpu/libm-test-ulps | 4 -
sysdeps/mips/mips32/libm-test-ulps | 4 -
sysdeps/mips/mips64/libm-test-ulps | 4 -
sysdeps/powerpc/fpu/libm-test-ulps | 4 -
sysdeps/powerpc/nofpu/libm-test-ulps | 4 -
sysdeps/riscv/nofpu/libm-test-ulps | 1 -
sysdeps/riscv/rvd/libm-test-ulps | 4 -
sysdeps/s390/fpu/libm-test-ulps | 4 -
sysdeps/sparc/fpu/libm-test-ulps | 4 -
sysdeps/x86_64/fpu/libm-test-ulps | 4 -
20 files changed, 196 insertions(+), 66 deletions(-)
create mode 100644 sysdeps/ieee754/flt-32/s_exp2m1f.c
Comments
LGTM
Reviewed-by: DJ Delorie <dj@redhat.com>
Adhemerval Zanella <adhemerval.zanella@linaro.org> writes:
> diff --git a/SHARED-FILES b/SHARED-FILES
> +sysdeps/ieee754/flt-32/s_exp2m1f.c
> + (file src/binary32/exp2m1/exp2m1f.c in CORE_MATH)
> + - The code was adapted to use glibc code style and internal
> + functions to handle errno, overflow, and underflow.
Ok.
> diff --git a/sysdeps/aarch64/libm-test-ulps b/sysdeps/aarch64/libm-test-ulps
> diff --git a/sysdeps/arc/fpu/libm-test-ulps b/sysdeps/arc/fpu/libm-test-ulps
Ok.
> diff --git a/sysdeps/arc/nofpu/libm-test-ulps b/sysdeps/arc/nofpu/libm-test-ulps
>
> Function: "exp2m1":
> double: 2
> -float: 2
>
> Function: "expm1":
> double: 1
Ok?
> diff --git a/sysdeps/arm/libm-test-ulps b/sysdeps/arm/libm-test-ulps
> diff --git a/sysdeps/hppa/fpu/libm-test-ulps b/sysdeps/hppa/fpu/libm-test-ulps
> diff --git a/sysdeps/i386/fpu/libm-test-ulps b/sysdeps/i386/fpu/libm-test-ulps
> diff --git a/sysdeps/i386/i686/fpu/multiarch/libm-test-ulps b/sysdeps/i386/i686/fpu/multiarch/libm-test-ulps
Ok.
> diff --git a/sysdeps/ieee754/flt-32/s_exp2m1f.c b/sysdeps/ieee754/flt-32/s_exp2m1f.c
> +/* Correctly-rounded base-2 exponent function biased by 1 for binary32 value.
> +
> +Copyright (c) 2022-2024 Alexei Sibidanov.
> +
> +The original version of this file was copied from the CORE-MATH
> +project (file src/binary32/exp2m1/exp2m1f.c, revision bc385c2).
> +
> +Permission is hereby granted, free of charge, to any person obtaining a copy
> +of this software and associated documentation files (the "Software"), to deal
> +in the Software without restriction, including without limitation the rights
> +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
> +copies of the Software, and to permit persons to whom the Software is
> +furnished to do so, subject to the following conditions:
> +
> +The above copyright notice and this permission notice shall be included in all
> +copies or substantial portions of the Software.
> +
> +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
> +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
> +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
> +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
> +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
> +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
> +SOFTWARE.
> +*/
> +
> +#include <fenv.h>
> +#include <math.h>
> +#include "math_config.h"
> +#include <libm-alias-float.h>
> +#include <math-narrow-eval.h>
> +#include <float.h>
Ok.
> +float
> +__exp2m1f (float x)
> +{
> + double z = x;
> + uint32_t ux = asuint (x);
> + uint32_t ax = ux & (~0u >> 1);
> + if (__glibc_unlikely (ux >= 0xc1c80000u))
> + { /* x <= -25 */
> + if (ax > (0xffu << 23))
> + return x + x; /* nan */
> + return (ux == 0xff800000) ? -0x1p+0 : -0x1p+0 + 0x1p-26f;
> + }
Ok.
> + else if (__glibc_unlikely (ax >= 0x43000000u))
> + { /* x >= 128 */
> + if (ax >= asuint (INFINITY))
> + return x + x; /* +Inf or NaN */
> + /* exp2m1 (MAX_EXP) should not overflow when rounding towards zero
> + or towards -Inf. We round FLT_MAX + 2^103 which is in the middle
> + between FLT_MAX and 2^128 (the next number with unbounded range). */
> + float ret = math_narrow_eval (FLT_MAX + 0x1p103f);
> + if (x == FLT_MAX_EXP && ret == FLT_MAX)
> + return ret;
> + return __math_oflowf (0);
> + }
Ok.
> + else if (__glibc_unlikely (ax < 0x3df95f1fu))
> + { /* |x| < 8.44e-2/log(2) */
> + double z2 = z * z, r;
> + if (__glibc_unlikely (ax < 0x3d67a4ccu))
> + { /* |x| < 3.92e-2/log(2) */
> + if (__glibc_unlikely (ax < 0x3caa2feeu))
> + { /* |x| < 1.44e-2/log(2) */
> + if (__glibc_unlikely (ax < 0x3bac1405u))
> + { /* |x| < 3.64e-3/log(2) */
> + if (__glibc_unlikely (ax < 0x3a358876u))
> + { /* |x| < 4.8e-4/log(2) */
> + if (__glibc_unlikely (ax < 0x37d32ef6u))
> + { /* |x| < 1.745e-5/log(2) */
> + if (__glibc_unlikely (ax < 0x331fdd82u))
> + { /* |x| < 2.58e-8/log(2) */
> + if (__glibc_unlikely (ax < 0x2538aa3bu))
> + /* |x| < 1.60171e-16 */
> + r = 0x1.62e42fefa39efp-1;
> + else
> + r = 0x1.62e42fefa39fp-1
> + + z * 0x1.ebfbdff82c58fp-3;
> + }
> + else
> + {
> + if (__glibc_unlikely (ux == 0xb3d85005u))
> + return -0x1.2bdf76p-24 - 0x1.8p-77;
> + if (__glibc_unlikely (ux == 0x3338428du))
> + return 0x1.fee08ap-26 + 0x1p-80;
> + static const double c[] =
> + {
> + 0x1.62e42fefa39efp-1, 0x1.ebfbdff8548fdp-3,
> + 0x1.c6b08d704a06dp-5
> + };
> + r = c[0] + z * (c[1] + z * c[2]);
> + }
> + }
> + else
> + {
> + if (__glibc_unlikely (ux == 0x388bca4fu))
> + return 0x1.839702p-15 - 0x1.8p-68;
> + static const double c[] =
> + {
> + 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c58fp-3,
> + 0x1.c6b08dc82b347p-5, 0x1.3b2ab6fbad172p-7
> + };
> + r = (c[0] + z * c[1]) + z2 * (c[2] + z * c[3]);
> + }
> + }
> + else
> + {
> + static const double c[] =
> + {
> + 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c068p-3,
> + 0x1.c6b08d704a6dcp-5, 0x1.3b2ac262c3eedp-7,
> + 0x1.5d87fe7af779ap-10
> + };
> + r = (c[0] + z * c[1])
> + + z2 * (c[2] + z * (c[3] + z * c[4]));
> + }
> + }
> + else
> + {
> + static const double c[] =
> + {
> + 0x1.62e42fefa39fp-1, 0x1.ebfbdff82c58dp-3,
> + 0x1.c6b08d7011d13p-5, 0x1.3b2ab6fbd267dp-7,
> + 0x1.5d88a81cea49ep-10, 0x1.430912ea9b963p-13
> + };
> + r = (c[0] + z * c[1])
> + + z2 * ((c[2] + z * c[3]) + z2 * (c[4] + z * c[5]));
> + }
> + }
> + else
> + {
> + static const double c[] =
> + {
> + 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c639p-3,
> + 0x1.c6b08d7049f1cp-5, 0x1.3b2ab6f5243bdp-7,
> + 0x1.5d87fe80a9e6cp-10, 0x1.430d0b9257fa8p-13,
> + 0x1.ffcbfc4cf0952p-17
> + };
> + r = (c[0] + z * c[1])
> + + z2 * ((c[2] + z * c[3])
> + + z2 * (c[4] + z * (c[5] + z * c[6])));
> + }
> + }
> + else
> + {
> + static const double c[] =
> + {
> + 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c591p-3,
> + 0x1.c6b08d704cf6bp-5, 0x1.3b2ab6fba00cep-7,
> + 0x1.5d87fdfdaadb4p-10, 0x1.4309137333066p-13,
> + 0x1.ffe5e90daf7ddp-17, 0x1.62c0220eed731p-20
> + };
> + r = ((c[0] + z * c[1]) + z2 * (c[2] + z * c[3]))
> + + (z2 * z2) * ((c[4] + z * c[5]) + z2 * (c[6] + z * c[7]));
> + }
> + r *= z;
> + return r;
> + }
Ok.
> + else
> + {
> + static const double c[] =
> + {
> + 0x1.62e42fefa398bp-5, 0x1.ebfbdff84555ap-11,
> + 0x1.c6b08d4ad86d3p-17, 0x1.3b2ad1b1716a2p-23,
> + 0x1.5d7472718ce9dp-30, 0x1.4a1d7f457ac56p-37
> + };
> + static const double tb[] =
> + {
> + 0x1p+0, 0x1.0b5586cf9890fp+0, 0x1.172b83c7d517bp+0,
> + 0x1.2387a6e756238p+0, 0x1.306fe0a31b715p+0, 0x1.3dea64c123422p+0,
> + 0x1.4bfdad5362a27p+0, 0x1.5ab07dd485429p+0, 0x1.6a09e667f3bcdp+0,
> + 0x1.7a11473eb0187p+0, 0x1.8ace5422aa0dap+0, 0x1.9c49182a3f09p+0,
> + 0x1.ae89f995ad3adp+0, 0x1.c199bdd85529cp+0, 0x1.d5818dcfba487p+0,
> + 0x1.ea4afa2a490dap+0
> + };
> + double a = 16.0 * z;
> + double ia = floor (a);
> + double h = a - ia;
> + double h2 = h * h;
> + int64_t i = ia, j = i & 0xf, e = i - j;
> + e >>= 4;
> + double s = tb[j];
> + s *= asdouble ((e + 0x3ffull) << 52);
> + double c0 = c[0] + h * c[1];
> + double c2 = c[2] + h * c[3];
> + double c4 = c[4] + h * c[5];
> + c0 += h2 * (c2 + h2 * c4);
> + double w = s * h;
> + return (s - 1.0) + w * c0;
> + }
> +}
> +libm_alias_float (__exp2m1, exp2m1)
Ok.
> diff --git a/sysdeps/loongarch/lp64/libm-test-ulps b/sysdeps/loongarch/lp64/libm-test-ulps
> diff --git a/sysdeps/m68k/m680x0/fpu/libm-test-ulps b/sysdeps/m68k/m680x0/fpu/libm-test-ulps
> diff --git a/sysdeps/mips/mips32/libm-test-ulps b/sysdeps/mips/mips32/libm-test-ulps
> diff --git a/sysdeps/mips/mips64/libm-test-ulps b/sysdeps/mips/mips64/libm-test-ulps
> diff --git a/sysdeps/powerpc/fpu/libm-test-ulps b/sysdeps/powerpc/fpu/libm-test-ulps
> diff --git a/sysdeps/powerpc/nofpu/libm-test-ulps b/sysdeps/powerpc/nofpu/libm-test-ulps
Ok.
> diff --git a/sysdeps/riscv/nofpu/libm-test-ulps b/sysdeps/riscv/nofpu/libm-test-ulps
> index 2271e1ee6c..11f891988e 100644
> --- a/sysdeps/riscv/nofpu/libm-test-ulps
> +++ b/sysdeps/riscv/nofpu/libm-test-ulps
> @@ -995,7 +995,6 @@ ldouble: 2
>
> Function: "exp2m1":
> double: 1
> -float: 1
> ldouble: 1
>
> Function: "exp_downward":
Ok?
> diff --git a/sysdeps/riscv/rvd/libm-test-ulps b/sysdeps/riscv/rvd/libm-test-ulps
> diff --git a/sysdeps/s390/fpu/libm-test-ulps b/sysdeps/s390/fpu/libm-test-ulps
> diff --git a/sysdeps/sparc/fpu/libm-test-ulps b/sysdeps/sparc/fpu/libm-test-ulps
> diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
Ok.
@@ -235,3 +235,7 @@ sysdeps/ieee754/flt-32/s_exp10m1f.c
(file src/binary32/exp10m1/exp10m1f.c in CORE_MATH)
- The code was adapted to use glibc code style and internal
functions to handle errno, overflow, and underflow.
+sysdeps/ieee754/flt-32/s_exp2m1f.c
+ (file src/binary32/exp2m1/exp2m1f.c in CORE_MATH)
+ - The code was adapted to use glibc code style and internal
+ functions to handle errno, overflow, and underflow.
@@ -1127,22 +1127,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_advsimd":
@@ -818,19 +818,15 @@ float: 1
Function: "exp2m1":
double: 2
-float: 2
Function: "exp2m1_downward":
double: 3
-float: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
Function: "exp2m1_upward":
double: 3
-float: 3
Function: "exp_downward":
double: 1
@@ -200,7 +200,6 @@ double: 1
Function: "exp2m1":
double: 2
-float: 2
Function: "expm1":
double: 1
@@ -807,19 +807,15 @@ float: 1
Function: "exp2m1":
double: 2
-float: 2
Function: "exp2m1_downward":
double: 3
-float: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
Function: "exp2m1_upward":
double: 3
-float: 3
Function: "exp_downward":
double: 1
@@ -826,19 +826,15 @@ float: 1
Function: "exp2m1":
double: 2
-float: 2
Function: "exp2m1_downward":
double: 3
-float: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
Function: "exp2m1_upward":
double: 3
-float: 3
Function: "exp_downward":
double: 1
@@ -1194,25 +1194,21 @@ ldouble: 1
Function: "exp2m1":
double: 2
-float: 1
float128: 2
ldouble: 3
Function: "exp2m1_downward":
double: 3
-float: 3
float128: 3
ldouble: 6
Function: "exp2m1_towardzero":
double: 3
-float: 2
float128: 4
ldouble: 5
Function: "exp2m1_upward":
double: 3
-float: 3
float128: 5
ldouble: 6
@@ -1197,25 +1197,21 @@ ldouble: 1
Function: "exp2m1":
double: 2
-float: 1
float128: 2
ldouble: 3
Function: "exp2m1_downward":
double: 3
-float: 3
float128: 3
ldouble: 6
Function: "exp2m1_towardzero":
double: 3
-float: 2
float128: 4
ldouble: 6
Function: "exp2m1_upward":
double: 3
-float: 3
float128: 5
ldouble: 6
new file mode 100644
@@ -0,0 +1,192 @@
+/* Correctly-rounded base-2 exponent function biased by 1 for binary32 value.
+
+Copyright (c) 2022-2024 Alexei Sibidanov.
+
+The original version of this file was copied from the CORE-MATH
+project (file src/binary32/exp2m1/exp2m1f.c, revision bc385c2).
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+*/
+
+#include <fenv.h>
+#include <math.h>
+#include "math_config.h"
+#include <libm-alias-float.h>
+#include <math-narrow-eval.h>
+#include <float.h>
+
+float
+__exp2m1f (float x)
+{
+ double z = x;
+ uint32_t ux = asuint (x);
+ uint32_t ax = ux & (~0u >> 1);
+ if (__glibc_unlikely (ux >= 0xc1c80000u))
+ { /* x <= -25 */
+ if (ax > (0xffu << 23))
+ return x + x; /* nan */
+ return (ux == 0xff800000) ? -0x1p+0 : -0x1p+0 + 0x1p-26f;
+ }
+ else if (__glibc_unlikely (ax >= 0x43000000u))
+ { /* x >= 128 */
+ if (ax >= asuint (INFINITY))
+ return x + x; /* +Inf or NaN */
+ /* exp2m1 (MAX_EXP) should not overflow when rounding towards zero
+ or towards -Inf. We round FLT_MAX + 2^103 which is in the middle
+ between FLT_MAX and 2^128 (the next number with unbounded range). */
+ float ret = math_narrow_eval (FLT_MAX + 0x1p103f);
+ if (x == FLT_MAX_EXP && ret == FLT_MAX)
+ return ret;
+ return __math_oflowf (0);
+ }
+ else if (__glibc_unlikely (ax < 0x3df95f1fu))
+ { /* |x| < 8.44e-2/log(2) */
+ double z2 = z * z, r;
+ if (__glibc_unlikely (ax < 0x3d67a4ccu))
+ { /* |x| < 3.92e-2/log(2) */
+ if (__glibc_unlikely (ax < 0x3caa2feeu))
+ { /* |x| < 1.44e-2/log(2) */
+ if (__glibc_unlikely (ax < 0x3bac1405u))
+ { /* |x| < 3.64e-3/log(2) */
+ if (__glibc_unlikely (ax < 0x3a358876u))
+ { /* |x| < 4.8e-4/log(2) */
+ if (__glibc_unlikely (ax < 0x37d32ef6u))
+ { /* |x| < 1.745e-5/log(2) */
+ if (__glibc_unlikely (ax < 0x331fdd82u))
+ { /* |x| < 2.58e-8/log(2) */
+ if (__glibc_unlikely (ax < 0x2538aa3bu))
+ /* |x| < 1.60171e-16 */
+ r = 0x1.62e42fefa39efp-1;
+ else
+ r = 0x1.62e42fefa39fp-1
+ + z * 0x1.ebfbdff82c58fp-3;
+ }
+ else
+ {
+ if (__glibc_unlikely (ux == 0xb3d85005u))
+ return -0x1.2bdf76p-24 - 0x1.8p-77;
+ if (__glibc_unlikely (ux == 0x3338428du))
+ return 0x1.fee08ap-26 + 0x1p-80;
+ static const double c[] =
+ {
+ 0x1.62e42fefa39efp-1, 0x1.ebfbdff8548fdp-3,
+ 0x1.c6b08d704a06dp-5
+ };
+ r = c[0] + z * (c[1] + z * c[2]);
+ }
+ }
+ else
+ {
+ if (__glibc_unlikely (ux == 0x388bca4fu))
+ return 0x1.839702p-15 - 0x1.8p-68;
+ static const double c[] =
+ {
+ 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c58fp-3,
+ 0x1.c6b08dc82b347p-5, 0x1.3b2ab6fbad172p-7
+ };
+ r = (c[0] + z * c[1]) + z2 * (c[2] + z * c[3]);
+ }
+ }
+ else
+ {
+ static const double c[] =
+ {
+ 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c068p-3,
+ 0x1.c6b08d704a6dcp-5, 0x1.3b2ac262c3eedp-7,
+ 0x1.5d87fe7af779ap-10
+ };
+ r = (c[0] + z * c[1])
+ + z2 * (c[2] + z * (c[3] + z * c[4]));
+ }
+ }
+ else
+ {
+ static const double c[] =
+ {
+ 0x1.62e42fefa39fp-1, 0x1.ebfbdff82c58dp-3,
+ 0x1.c6b08d7011d13p-5, 0x1.3b2ab6fbd267dp-7,
+ 0x1.5d88a81cea49ep-10, 0x1.430912ea9b963p-13
+ };
+ r = (c[0] + z * c[1])
+ + z2 * ((c[2] + z * c[3]) + z2 * (c[4] + z * c[5]));
+ }
+ }
+ else
+ {
+ static const double c[] =
+ {
+ 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c639p-3,
+ 0x1.c6b08d7049f1cp-5, 0x1.3b2ab6f5243bdp-7,
+ 0x1.5d87fe80a9e6cp-10, 0x1.430d0b9257fa8p-13,
+ 0x1.ffcbfc4cf0952p-17
+ };
+ r = (c[0] + z * c[1])
+ + z2 * ((c[2] + z * c[3])
+ + z2 * (c[4] + z * (c[5] + z * c[6])));
+ }
+ }
+ else
+ {
+ static const double c[] =
+ {
+ 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c591p-3,
+ 0x1.c6b08d704cf6bp-5, 0x1.3b2ab6fba00cep-7,
+ 0x1.5d87fdfdaadb4p-10, 0x1.4309137333066p-13,
+ 0x1.ffe5e90daf7ddp-17, 0x1.62c0220eed731p-20
+ };
+ r = ((c[0] + z * c[1]) + z2 * (c[2] + z * c[3]))
+ + (z2 * z2) * ((c[4] + z * c[5]) + z2 * (c[6] + z * c[7]));
+ }
+ r *= z;
+ return r;
+ }
+ else
+ {
+ static const double c[] =
+ {
+ 0x1.62e42fefa398bp-5, 0x1.ebfbdff84555ap-11,
+ 0x1.c6b08d4ad86d3p-17, 0x1.3b2ad1b1716a2p-23,
+ 0x1.5d7472718ce9dp-30, 0x1.4a1d7f457ac56p-37
+ };
+ static const double tb[] =
+ {
+ 0x1p+0, 0x1.0b5586cf9890fp+0, 0x1.172b83c7d517bp+0,
+ 0x1.2387a6e756238p+0, 0x1.306fe0a31b715p+0, 0x1.3dea64c123422p+0,
+ 0x1.4bfdad5362a27p+0, 0x1.5ab07dd485429p+0, 0x1.6a09e667f3bcdp+0,
+ 0x1.7a11473eb0187p+0, 0x1.8ace5422aa0dap+0, 0x1.9c49182a3f09p+0,
+ 0x1.ae89f995ad3adp+0, 0x1.c199bdd85529cp+0, 0x1.d5818dcfba487p+0,
+ 0x1.ea4afa2a490dap+0
+ };
+ double a = 16.0 * z;
+ double ia = floor (a);
+ double h = a - ia;
+ double h2 = h * h;
+ int64_t i = ia, j = i & 0xf, e = i - j;
+ e >>= 4;
+ double s = tb[j];
+ s *= asdouble ((e + 0x3ffull) << 52);
+ double c0 = c[0] + h * c[1];
+ double c2 = c[2] + h * c[3];
+ double c4 = c[4] + h * c[5];
+ c0 += h2 * (c2 + h2 * c4);
+ double w = s * h;
+ return (s - 1.0) + w * c0;
+ }
+}
+libm_alias_float (__exp2m1, exp2m1)
@@ -1010,22 +1010,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_downward":
@@ -903,19 +903,15 @@ double: 1
Function: "exp2m1":
double: 1
-float: 1
Function: "exp2m1_downward":
double: 2
-float: 1
Function: "exp2m1_towardzero":
double: 2
-float: 1
Function: "exp2m1_upward":
double: 1
-float: 1
Function: "exp_upward":
double: 1
@@ -807,19 +807,15 @@ float: 1
Function: "exp2m1":
double: 2
-float: 2
Function: "exp2m1_downward":
double: 3
-float: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
Function: "exp2m1_upward":
double: 3
-float: 3
Function: "exp_downward":
double: 1
@@ -1014,22 +1014,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_downward":
@@ -1227,25 +1227,21 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
float128: 2
ldouble: 3
Function: "exp2m1_downward":
double: 3
-float: 3
float128: 3
ldouble: 7
Function: "exp2m1_towardzero":
double: 3
-float: 2
float128: 4
ldouble: 8
Function: "exp2m1_upward":
double: 3
-float: 3
float128: 5
ldouble: 9
@@ -1023,22 +1023,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 7
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 8
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 9
Function: "exp_downward":
@@ -995,7 +995,6 @@ ldouble: 2
Function: "exp2m1":
double: 1
-float: 1
ldouble: 1
Function: "exp_downward":
@@ -1010,22 +1010,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_downward":
@@ -1011,22 +1011,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_downward":
@@ -1014,22 +1014,18 @@ ldouble: 2
Function: "exp2m1":
double: 2
-float: 2
ldouble: 2
Function: "exp2m1_downward":
double: 3
-float: 3
ldouble: 3
Function: "exp2m1_towardzero":
double: 3
-float: 2
ldouble: 4
Function: "exp2m1_upward":
double: 3
-float: 3
ldouble: 5
Function: "exp_downward":
@@ -1495,25 +1495,21 @@ float: 1
Function: "exp2m1":
double: 2
-float: 2
float128: 2
ldouble: 3
Function: "exp2m1_downward":
double: 3
-float: 3
float128: 3
ldouble: 6
Function: "exp2m1_towardzero":
double: 3
-float: 2
float128: 4
ldouble: 5
Function: "exp2m1_upward":
double: 3
-float: 3
float128: 5
ldouble: 6