New pow implementation
The new implementation is provided under !__OBSOLETE_MATH, it uses ISO C99 code. With default settings the worst case error in nearest rounding mode is 0.54 ULP with inlined fma and fma contraction. It uses a 4 KB lookup table in addition to the table in exp_data.c, on aarch64 .text+.rodata size of libm.a is increased by 2295 bytes. Improvements on Cortex-A72: latency: 3.3x thruput: 4.9x
This commit is contained in:
Szabolcs Nagy
Corinna Vinschen
parent
07e2c32828
commit
b99d49e506
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@ -13,7 +13,8 @@ src = s_finite.c s_copysign.c s_modf.c s_scalbn.c \
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s_lrint.c s_llrint.c \
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s_lround.c s_llround.c s_nearbyint.c s_remquo.c s_round.c s_scalbln.c \
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s_signbit.c s_trunc.c \
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exp.c exp2.c exp_data.c math_err.c log.c log_data.c log2.c log2_data.c
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exp.c exp2.c exp_data.c math_err.c log.c log_data.c log2.c log2_data.c \
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pow.c pow_log_data.c
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fsrc = sf_finite.c sf_copysign.c sf_modf.c sf_scalbn.c \
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sf_cbrt.c sf_exp10.c sf_expm1.c sf_ilogb.c \
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@ -97,7 +97,8 @@ am__objects_1 = lib_a-s_finite.$(OBJEXT) lib_a-s_copysign.$(OBJEXT) \
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lib_a-exp2.$(OBJEXT) lib_a-exp_data.$(OBJEXT) \
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lib_a-math_err.$(OBJEXT) lib_a-log.$(OBJEXT) \
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lib_a-log_data.$(OBJEXT) lib_a-log2.$(OBJEXT) \
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lib_a-log2_data.$(OBJEXT)
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lib_a-log2_data.$(OBJEXT) lib_a-pow.$(OBJEXT) \
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lib_a-pow_log_data.$(OBJEXT)
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am__objects_2 = lib_a-sf_finite.$(OBJEXT) lib_a-sf_copysign.$(OBJEXT) \
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lib_a-sf_modf.$(OBJEXT) lib_a-sf_scalbn.$(OBJEXT) \
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lib_a-sf_cbrt.$(OBJEXT) lib_a-sf_exp10.$(OBJEXT) \
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@ -168,7 +169,7 @@ am__objects_5 = s_finite.lo s_copysign.lo s_modf.lo s_scalbn.lo \
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s_lround.lo s_llround.lo s_nearbyint.lo s_remquo.lo s_round.lo \
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s_scalbln.lo s_signbit.lo s_trunc.lo exp.lo exp2.lo \
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exp_data.lo math_err.lo log.lo log_data.lo log2.lo \
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log2_data.lo
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log2_data.lo pow.lo pow_log_data.lo
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am__objects_6 = sf_finite.lo sf_copysign.lo sf_modf.lo sf_scalbn.lo \
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sf_cbrt.lo sf_exp10.lo sf_expm1.lo sf_ilogb.lo sf_infinity.lo \
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sf_isinf.lo sf_isinff.lo sf_isnan.lo sf_isnanf.lo sf_log1p.lo \
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@ -359,7 +360,8 @@ src = s_finite.c s_copysign.c s_modf.c s_scalbn.c \
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s_lrint.c s_llrint.c \
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s_lround.c s_llround.c s_nearbyint.c s_remquo.c s_round.c s_scalbln.c \
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s_signbit.c s_trunc.c \
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exp.c exp2.c exp_data.c math_err.c log.c log_data.c log2.c log2_data.c
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exp.c exp2.c exp_data.c math_err.c log.c log_data.c log2.c log2_data.c \
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pow.c pow_log_data.c
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fsrc = sf_finite.c sf_copysign.c sf_modf.c sf_scalbn.c \
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sf_cbrt.c sf_exp10.c sf_expm1.c sf_ilogb.c \
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@ -751,6 +753,18 @@ lib_a-log2_data.o: log2_data.c
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lib_a-log2_data.obj: log2_data.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-log2_data.obj `if test -f 'log2_data.c'; then $(CYGPATH_W) 'log2_data.c'; else $(CYGPATH_W) '$(srcdir)/log2_data.c'; fi`
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lib_a-pow.o: pow.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-pow.o `test -f 'pow.c' || echo '$(srcdir)/'`pow.c
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lib_a-pow.obj: pow.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-pow.obj `if test -f 'pow.c'; then $(CYGPATH_W) 'pow.c'; else $(CYGPATH_W) '$(srcdir)/pow.c'; fi`
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lib_a-pow_log_data.o: pow_log_data.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-pow_log_data.o `test -f 'pow_log_data.c' || echo '$(srcdir)/'`pow_log_data.c
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lib_a-pow_log_data.obj: pow_log_data.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-pow_log_data.obj `if test -f 'pow_log_data.c'; then $(CYGPATH_W) 'pow_log_data.c'; else $(CYGPATH_W) '$(srcdir)/pow_log_data.c'; fi`
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lib_a-sf_finite.o: sf_finite.c
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$(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-sf_finite.o `test -f 'sf_finite.c' || echo '$(srcdir)/'`sf_finite.c
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@ -233,6 +233,7 @@ extern float __ieee754_scalbf __P((float,float));
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and there are no observable error handling side effects. */
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# define __ieee754_exp(x) exp(x)
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# define __ieee754_log(x) log(x)
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# define __ieee754_pow(x,y) pow(x,y)
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# define __ieee754_expf(x) expf(x)
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# define __ieee754_logf(x) logf(x)
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# define __ieee754_powf(x,y) powf(x,y)
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@ -147,6 +147,15 @@ issignalingf_inline (float x)
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return 2 * (ix ^ 0x00400000) > 2u * 0x7fc00000;
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}
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static inline int
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issignaling_inline (double x)
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{
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uint64_t ix = asuint64 (x);
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if (!IEEE_754_2008_SNAN)
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return (ix & 0x7ff8000000000000) == 0x7ff8000000000000;
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return 2 * (ix ^ 0x0008000000000000) > 2 * 0x7ff8000000000000ULL;
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}
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#if __aarch64__ && __GNUC__
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/* Prevent the optimization of a floating-point expression. */
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static inline float
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@ -354,4 +363,14 @@ extern const struct log2_data {
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#endif
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} __log2_data HIDDEN;
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#define POW_LOG_TABLE_BITS 7
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#define POW_LOG_POLY_ORDER 8
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extern const struct pow_log_data {
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double ln2hi;
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double ln2lo;
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double poly[POW_LOG_POLY_ORDER - 1]; /* First coefficient is 1. */
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/* Note: the pad field is unused, but allows slightly faster indexing. */
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struct {double invc, pad, logc, logctail;} tab[1 << POW_LOG_TABLE_BITS];
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} __pow_log_data HIDDEN;
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#endif
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@ -0,0 +1,377 @@
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/* Double-precision x^y function.
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Copyright (c) 2018 Arm Ltd. All rights reserved.
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SPDX-License-Identifier: BSD-3-Clause
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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1. Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. The name of the company may not be used to endorse or promote
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products derived from this software without specific prior written
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permission.
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THIS SOFTWARE IS PROVIDED BY Arm LTD ``AS IS AND ANY EXPRESS OR IMPLIED
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WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
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MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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IN NO EVENT SHALL Arm LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
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TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
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#include "fdlibm.h"
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#if !__OBSOLETE_MATH
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#include <math.h>
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#include <stdint.h>
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#include "math_config.h"
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/*
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Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
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relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
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ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
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*/
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#define T __pow_log_data.tab
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#define A __pow_log_data.poly
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#define Ln2hi __pow_log_data.ln2hi
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#define Ln2lo __pow_log_data.ln2lo
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#define N (1 << POW_LOG_TABLE_BITS)
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#define OFF 0x3fe6955500000000
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static inline uint32_t
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top12 (double x)
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{
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return asuint64 (x) >> 52;
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}
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static inline double_t
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log_inline (uint64_t ix, double_t *tail)
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{
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
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uint64_t iz, tmp;
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int k, i;
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/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
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The range is split into N subintervals.
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The ith subinterval contains z and c is near its center. */
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tmp = ix - OFF;
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i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
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k = (int64_t) tmp >> 52; /* arithmetic shift */
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iz = ix - (tmp & 0xfffULL << 52);
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z = asdouble (iz);
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kd = (double_t) k;
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/* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
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invc = T[i].invc;
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logc = T[i].logc;
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logctail = T[i].logctail;
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/* r = z/c - 1, arranged to be exact. */
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#if __HAVE_FAST_FMA
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r = fma (z, invc, -1.0);
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#else
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double_t zhi = asdouble (iz & (-1ULL << 32));
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double_t zlo = z - zhi;
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double_t rhi = zhi * invc - 1.0;
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double_t rlo = zlo * invc;
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r = rhi + rlo;
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#endif
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/* k*Ln2 + log(c) + r. */
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t1 = kd * Ln2hi + logc;
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t2 = t1 + r;
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lo1 = kd * Ln2lo + logctail;
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lo2 = t1 - t2 + r;
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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double_t ar, ar2, ar3, lo3, lo4;
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ar = A[0] * r; /* A[0] = -0.5. */
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ar2 = r * ar;
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ar3 = r * ar2;
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/* k*Ln2 + log(c) + r + A[0]*r*r. */
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#if __HAVE_FAST_FMA
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hi = t2 + ar2;
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lo3 = fma (ar, r, -ar2);
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lo4 = t2 - hi + ar2;
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#else
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double_t arhi = A[0] * rhi;
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double_t arhi2 = rhi * arhi;
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hi = t2 + arhi2;
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lo3 = rlo * (ar + arhi);
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lo4 = t2 - hi + arhi2;
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#endif
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/* p = log1p(r) - r - A[0]*r*r. */
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#if POW_LOG_POLY_ORDER == 8
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p = ar3*(A[1] + r*A[2] + ar2*(A[3] + r*A[4] + ar2*(A[5] + r*A[6])));
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#endif
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lo = lo1 + lo2 + lo3 + lo4 + p;
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y = hi + lo;
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*tail = hi - y + lo;
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return y;
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}
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#undef N
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#undef T
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#define N (1 << EXP_TABLE_BITS)
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#define InvLn2N __exp_data.invln2N
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#define NegLn2hiN __exp_data.negln2hiN
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#define NegLn2loN __exp_data.negln2loN
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#define Shift __exp_data.shift
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#define T __exp_data.tab
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#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
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#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
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#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
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#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
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#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
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static inline double
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specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
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{
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double_t scale, y;
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if ((ki & 0x80000000) == 0)
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{
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/* k > 0, the exponent of scale might have overflowed by <= 460. */
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sbits -= 1009ull << 52;
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scale = asdouble (sbits);
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y = 0x1p1009 * (scale + scale * tmp);
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return check_oflow (y);
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}
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/* k < 0, need special care in the subnormal range. */
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sbits += 1022ull << 52;
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/* Note: sbits is signed scale. */
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scale = asdouble (sbits);
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y = scale + scale * tmp;
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if (fabs (y) < 1.0)
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{
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/* Round y to the right precision before scaling it into the subnormal
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range to avoid double rounding that can cause 0.5+E/2 ulp error where
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E is the worst-case ulp error outside the subnormal range. So this
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is only useful if the goal is better than 1 ulp worst-case error. */
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double_t hi, lo, one = 1.0;
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if (y < 0.0)
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one = -1.0;
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lo = scale - y + scale * tmp;
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hi = one + y;
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lo = one - hi + y + lo;
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y = eval_as_double (hi + lo) - one;
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/* Fix the sign of 0. */
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if (y == 0.0)
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y = asdouble (sbits & 0x8000000000000000);
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/* The underflow exception needs to be signaled explicitly. */
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force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
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}
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y = 0x1p-1022 * y;
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return check_uflow (y);
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}
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#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
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static inline double
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exp_inline (double x, double xtail, uint32_t sign_bias)
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{
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uint32_t abstop;
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uint64_t ki, idx, top, sbits;
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t kd, z, r, r2, scale, tail, tmp;
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abstop = top12 (x) & 0x7ff;
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if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
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{
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if (abstop - top12 (0x1p-54) >= 0x80000000)
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{
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/* Avoid spurious underflow for tiny x. */
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/* Note: 0 is common input. */
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double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
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return sign_bias ? -one : one;
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}
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if (abstop >= top12 (1024.0))
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{
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/* Note: inf and nan are already handled. */
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if (asuint64 (x) >> 63)
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return __math_uflow (sign_bias);
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else
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return __math_oflow (sign_bias);
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}
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/* Large x is special cased below. */
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abstop = 0;
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}
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/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
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/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
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z = InvLn2N * x;
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#if TOINT_INTRINSICS
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kd = roundtoint (z);
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ki = converttoint (z);
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#elif EXP_USE_TOINT_NARROW
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/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
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kd = eval_as_double (z + Shift);
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ki = asuint64 (kd) >> 16;
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kd = (double_t) (int32_t) ki;
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#else
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/* z - kd is in [-1, 1] in non-nearest rounding modes. */
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kd = eval_as_double (z + Shift);
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ki = asuint64 (kd);
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kd -= Shift;
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#endif
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r = x + kd*NegLn2hiN + kd*NegLn2loN;
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/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
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r += xtail;
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/* 2^(k/N) ~= scale * (1 + tail). */
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idx = 2*(ki % N);
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top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
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tail = asdouble (T[idx]);
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/* This is only a valid scale when -1023*N < k < 1024*N. */
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sbits = T[idx + 1] + top;
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/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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r2 = r*r;
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/* Without fma the worst case error is 0.25/N ulp larger. */
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/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
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#if EXP_POLY_ORDER == 4
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tmp = tail + r + r2*C2 + r*r2*(C3 + r*C4);
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#elif EXP_POLY_ORDER == 5
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tmp = tail + r + r2*(C2 + r*C3) + r2*r2*(C4 + r*C5);
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#elif EXP_POLY_ORDER == 6
|
||||
tmp = tail + r + r2*(0.5 + r*C3) + r2*r2*(C4 + r*C5 + r2*C6);
|
||||
#endif
|
||||
if (unlikely (abstop == 0))
|
||||
return specialcase (tmp, sbits, ki);
|
||||
scale = asdouble (sbits);
|
||||
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
|
||||
is no spurious underflow here even without fma. */
|
||||
return scale + scale * tmp;
|
||||
}
|
||||
|
||||
/* Returns 0 if not int, 1 if odd int, 2 if even int. */
|
||||
static inline int
|
||||
checkint (uint64_t iy)
|
||||
{
|
||||
int e = iy >> 52 & 0x7ff;
|
||||
if (e < 0x3ff)
|
||||
return 0;
|
||||
if (e > 0x3ff + 52)
|
||||
return 2;
|
||||
if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
|
||||
return 0;
|
||||
if (iy & (1ULL << (0x3ff + 52 - e)))
|
||||
return 1;
|
||||
return 2;
|
||||
}
|
||||
|
||||
static inline int
|
||||
zeroinfnan (uint64_t i)
|
||||
{
|
||||
return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
|
||||
}
|
||||
|
||||
double
|
||||
pow (double x, double y)
|
||||
{
|
||||
uint32_t sign_bias = 0;
|
||||
uint64_t ix, iy;
|
||||
uint32_t topx, topy;
|
||||
|
||||
ix = asuint64 (x);
|
||||
iy = asuint64 (y);
|
||||
topx = top12 (x);
|
||||
topy = top12 (y);
|
||||
if (unlikely (topx - 0x001 >= 0x7ff - 0x001
|
||||
|| (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
|
||||
{
|
||||
/* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
|
||||
and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */
|
||||
/* Special cases: (x < 0x1p-126 or inf or nan) or
|
||||
(|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */
|
||||
if (unlikely (zeroinfnan (iy)))
|
||||
{
|
||||
if (2 * iy == 0)
|
||||
return issignaling_inline (x) ? x + y : 1.0;
|
||||
if (ix == asuint64 (1.0))
|
||||
return issignaling_inline (y) ? x + y : 1.0;
|
||||
if (2 * ix > 2 * asuint64 (INFINITY)
|
||||
|| 2 * iy > 2 * asuint64 (INFINITY))
|
||||
return x + y;
|
||||
if (2 * ix == 2 * asuint64 (1.0))
|
||||
return 1.0;
|
||||
if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
|
||||
return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
|
||||
return y * y;
|
||||
}
|
||||
if (unlikely (zeroinfnan (ix)))
|
||||
{
|
||||
double_t x2 = x * x;
|
||||
if (ix >> 63 && checkint (iy) == 1)
|
||||
{
|
||||
x2 = -x2;
|
||||
sign_bias = 1;
|
||||
}
|
||||
if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
|
||||
return __math_divzero (sign_bias);
|
||||
/* Without the barrier some versions of clang hoist the 1/x2 and
|
||||
thus division by zero exception can be signaled spuriously. */
|
||||
return iy >> 63 ? opt_barrier_double (1 / x2) : x2;
|
||||
}
|
||||
/* Here x and y are non-zero finite. */
|
||||
if (ix >> 63)
|
||||
{
|
||||
/* Finite x < 0. */
|
||||
int yint = checkint (iy);
|
||||
if (yint == 0)
|
||||
return __math_invalid (x);
|
||||
if (yint == 1)
|
||||
sign_bias = SIGN_BIAS;
|
||||
ix &= 0x7fffffffffffffff;
|
||||
topx &= 0x7ff;
|
||||
}
|
||||
if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
|
||||
{
|
||||
/* Note: sign_bias == 0 here because y is not odd. */
|
||||
if (ix == asuint64 (1.0))
|
||||
return 1.0;
|
||||
if ((topy & 0x7ff) < 0x3be)
|
||||
{
|
||||
/* |y| < 2^-65, x^y ~= 1 + y*log(x). */
|
||||
if (WANT_ROUNDING)
|
||||
return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
|
||||
else
|
||||
return 1.0;
|
||||
}
|
||||
return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
|
||||
: __math_uflow (0);
|
||||
}
|
||||
if (topx == 0)
|
||||
{
|
||||
/* Normalize subnormal x so exponent becomes negative. */
|
||||
ix = asuint64 (x * 0x1p52);
|
||||
ix &= 0x7fffffffffffffff;
|
||||
ix -= 52ULL << 52;
|
||||
}
|
||||
}
|
||||
|
||||
double_t lo;
|
||||
double_t hi = log_inline (ix, &lo);
|
||||
double_t ehi, elo;
|
||||
#if __HAVE_FAST_FMA
|
||||
ehi = y * hi;
|
||||
elo = y * lo + fma (y, hi, -ehi);
|
||||
#else
|
||||
double_t yhi = asdouble (iy & -1ULL << 27);
|
||||
double_t ylo = y - yhi;
|
||||
double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
|
||||
double_t llo = hi - lhi + lo;
|
||||
ehi = yhi * lhi;
|
||||
elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */
|
||||
#endif
|
||||
return exp_inline (ehi, elo, sign_bias);
|
||||
}
|
||||
#endif
|
||||
|
|
@ -0,0 +1,187 @@
|
|||
/* Data for the log part of pow.
|
||||
Copyright (c) 2018 Arm Ltd. All rights reserved.
|
||||
|
||||
SPDX-License-Identifier: BSD-3-Clause
|
||||
|
||||
Redistribution and use in source and binary forms, with or without
|
||||
modification, are permitted provided that the following conditions
|
||||
are met:
|
||||
1. Redistributions of source code must retain the above copyright
|
||||
notice, this list of conditions and the following disclaimer.
|
||||
2. Redistributions in binary form must reproduce the above copyright
|
||||
notice, this list of conditions and the following disclaimer in the
|
||||
documentation and/or other materials provided with the distribution.
|
||||
3. The name of the company may not be used to endorse or promote
|
||||
products derived from this software without specific prior written
|
||||
permission.
|
||||
|
||||
THIS SOFTWARE IS PROVIDED BY Arm LTD ``AS IS AND ANY EXPRESS OR IMPLIED
|
||||
WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
|
||||
MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
|
||||
IN NO EVENT SHALL Arm LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
|
||||
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
|
||||
TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
|
||||
PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
|
||||
LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
|
||||
NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
|
||||
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
|
||||
|
||||
#include "fdlibm.h"
|
||||
#if !__OBSOLETE_MATH
|
||||
|
||||
#include "math_config.h"
|
||||
|
||||
#define N (1 << POW_LOG_TABLE_BITS)
|
||||
|
||||
const struct pow_log_data __pow_log_data = {
|
||||
.ln2hi = 0x1.62e42fefa3800p-1,
|
||||
.ln2lo = 0x1.ef35793c76730p-45,
|
||||
.poly = {
|
||||
#if N == 128 && POW_LOG_POLY_ORDER == 8
|
||||
// relative error: 0x1.11922ap-70
|
||||
// in -0x1.6bp-8 0x1.6bp-8
|
||||
// Coefficients are scaled to match the scaling during evaluation.
|
||||
-0x1p-1,
|
||||
0x1.555555555556p-2 * -2,
|
||||
-0x1.0000000000006p-2 * -2,
|
||||
0x1.999999959554ep-3 * 4,
|
||||
-0x1.555555529a47ap-3 * 4,
|
||||
0x1.2495b9b4845e9p-3 * -8,
|
||||
-0x1.0002b8b263fc3p-3 * -8,
|
||||
#endif
|
||||
},
|
||||
.tab = {
|
||||
#if N == 128
|
||||
#define A(a,b,c) {a,0,b,c},
|
||||
A(0x1.6a00000000000p+0, -0x1.62c82f2b9c800p-2, 0x1.ab42428375680p-48)
|
||||
A(0x1.6800000000000p+0, -0x1.5d1bdbf580800p-2, -0x1.ca508d8e0f720p-46)
|
||||
A(0x1.6600000000000p+0, -0x1.5767717455800p-2, -0x1.362a4d5b6506dp-45)
|
||||
A(0x1.6400000000000p+0, -0x1.51aad872df800p-2, -0x1.684e49eb067d5p-49)
|
||||
A(0x1.6200000000000p+0, -0x1.4be5f95777800p-2, -0x1.41b6993293ee0p-47)
|
||||
A(0x1.6000000000000p+0, -0x1.4618bc21c6000p-2, 0x1.3d82f484c84ccp-46)
|
||||
A(0x1.5e00000000000p+0, -0x1.404308686a800p-2, 0x1.c42f3ed820b3ap-50)
|
||||
A(0x1.5c00000000000p+0, -0x1.3a64c55694800p-2, 0x1.0b1c686519460p-45)
|
||||
A(0x1.5a00000000000p+0, -0x1.347dd9a988000p-2, 0x1.5594dd4c58092p-45)
|
||||
A(0x1.5800000000000p+0, -0x1.2e8e2bae12000p-2, 0x1.67b1e99b72bd8p-45)
|
||||
A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
|
||||
A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
|
||||
A(0x1.5400000000000p+0, -0x1.22941fbcf7800p-2, -0x1.65a242853da76p-46)
|
||||
A(0x1.5200000000000p+0, -0x1.1c898c1699800p-2, -0x1.fafbc68e75404p-46)
|
||||
A(0x1.5000000000000p+0, -0x1.1675cababa800p-2, 0x1.f1fc63382a8f0p-46)
|
||||
A(0x1.4e00000000000p+0, -0x1.1058bf9ae4800p-2, -0x1.6a8c4fd055a66p-45)
|
||||
A(0x1.4c00000000000p+0, -0x1.0a324e2739000p-2, -0x1.c6bee7ef4030ep-47)
|
||||
A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
|
||||
A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
|
||||
A(0x1.4800000000000p+0, -0x1.fb9186d5e4000p-3, 0x1.d572aab993c87p-47)
|
||||
A(0x1.4600000000000p+0, -0x1.ef0adcbdc6000p-3, 0x1.b26b79c86af24p-45)
|
||||
A(0x1.4400000000000p+0, -0x1.e27076e2af000p-3, -0x1.72f4f543fff10p-46)
|
||||
A(0x1.4200000000000p+0, -0x1.d5c216b4fc000p-3, 0x1.1ba91bbca681bp-45)
|
||||
A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
|
||||
A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
|
||||
A(0x1.3e00000000000p+0, -0x1.bc286742d9000p-3, 0x1.94eb0318bb78fp-46)
|
||||
A(0x1.3c00000000000p+0, -0x1.af3c94e80c000p-3, 0x1.a4e633fcd9066p-52)
|
||||
A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
|
||||
A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
|
||||
A(0x1.3800000000000p+0, -0x1.9525a9cf45000p-3, -0x1.ad1d904c1d4e3p-45)
|
||||
A(0x1.3600000000000p+0, -0x1.87fa06520d000p-3, 0x1.bbdbf7fdbfa09p-45)
|
||||
A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
|
||||
A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
|
||||
A(0x1.3200000000000p+0, -0x1.6d60fe719d000p-3, -0x1.0e46aa3b2e266p-46)
|
||||
A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
|
||||
A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
|
||||
A(0x1.2e00000000000p+0, -0x1.526e5e3a1b000p-3, -0x1.0de8b90075b8fp-45)
|
||||
A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
|
||||
A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
|
||||
A(0x1.2a00000000000p+0, -0x1.371fc201e9000p-3, 0x1.178864d27543ap-48)
|
||||
A(0x1.2800000000000p+0, -0x1.29552f81ff000p-3, -0x1.48d301771c408p-45)
|
||||
A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
|
||||
A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
|
||||
A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
|
||||
A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
|
||||
A(0x1.2200000000000p+0, -0x1.fec9131dbe000p-4, -0x1.575545ca333f2p-45)
|
||||
A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
|
||||
A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
|
||||
A(0x1.1e00000000000p+0, -0x1.c5e548f5bc000p-4, -0x1.d0c57585fbe06p-46)
|
||||
A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
|
||||
A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
|
||||
A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
|
||||
A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
|
||||
A(0x1.1800000000000p+0, -0x1.6f0d28ae56000p-4, -0x1.69737c93373dap-45)
|
||||
A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
|
||||
A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
|
||||
A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
|
||||
A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
|
||||
A(0x1.1200000000000p+0, -0x1.16536eea38000p-4, 0x1.47c5e768fa309p-46)
|
||||
A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
|
||||
A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
|
||||
A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
|
||||
A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
|
||||
A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
|
||||
A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
|
||||
A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
|
||||
A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
|
||||
A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
|
||||
A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
|
||||
A(0x1.0600000000000p+0, -0x1.7b91b07d58000p-6, -0x1.88d5493faa639p-45)
|
||||
A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
|
||||
A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
|
||||
A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
|
||||
A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
|
||||
A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
|
||||
A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
|
||||
A(0x1.fc00000000000p-1, 0x1.0101575890000p-7, -0x1.0c76b999d2be8p-46)
|
||||
A(0x1.f800000000000p-1, 0x1.0205658938000p-6, -0x1.3dc5b06e2f7d2p-45)
|
||||
A(0x1.f400000000000p-1, 0x1.8492528c90000p-6, -0x1.aa0ba325a0c34p-45)
|
||||
A(0x1.f000000000000p-1, 0x1.0415d89e74000p-5, 0x1.111c05cf1d753p-47)
|
||||
A(0x1.ec00000000000p-1, 0x1.466aed42e0000p-5, -0x1.c167375bdfd28p-45)
|
||||
A(0x1.e800000000000p-1, 0x1.894aa149fc000p-5, -0x1.97995d05a267dp-46)
|
||||
A(0x1.e400000000000p-1, 0x1.ccb73cdddc000p-5, -0x1.a68f247d82807p-46)
|
||||
A(0x1.e200000000000p-1, 0x1.eea31c006c000p-5, -0x1.e113e4fc93b7bp-47)
|
||||
A(0x1.de00000000000p-1, 0x1.1973bd1466000p-4, -0x1.5325d560d9e9bp-45)
|
||||
A(0x1.da00000000000p-1, 0x1.3bdf5a7d1e000p-4, 0x1.cc85ea5db4ed7p-45)
|
||||
A(0x1.d600000000000p-1, 0x1.5e95a4d97a000p-4, -0x1.c69063c5d1d1ep-45)
|
||||
A(0x1.d400000000000p-1, 0x1.700d30aeac000p-4, 0x1.c1e8da99ded32p-49)
|
||||
A(0x1.d000000000000p-1, 0x1.9335e5d594000p-4, 0x1.3115c3abd47dap-45)
|
||||
A(0x1.cc00000000000p-1, 0x1.b6ac88dad6000p-4, -0x1.390802bf768e5p-46)
|
||||
A(0x1.ca00000000000p-1, 0x1.c885801bc4000p-4, 0x1.646d1c65aacd3p-45)
|
||||
A(0x1.c600000000000p-1, 0x1.ec739830a2000p-4, -0x1.dc068afe645e0p-45)
|
||||
A(0x1.c400000000000p-1, 0x1.fe89139dbe000p-4, -0x1.534d64fa10afdp-45)
|
||||
A(0x1.c000000000000p-1, 0x1.1178e8227e000p-3, 0x1.1ef78ce2d07f2p-45)
|
||||
A(0x1.be00000000000p-1, 0x1.1aa2b7e23f000p-3, 0x1.ca78e44389934p-45)
|
||||
A(0x1.ba00000000000p-1, 0x1.2d1610c868000p-3, 0x1.39d6ccb81b4a1p-47)
|
||||
A(0x1.b800000000000p-1, 0x1.365fcb0159000p-3, 0x1.62fa8234b7289p-51)
|
||||
A(0x1.b400000000000p-1, 0x1.4913d8333b000p-3, 0x1.5837954fdb678p-45)
|
||||
A(0x1.b200000000000p-1, 0x1.527e5e4a1b000p-3, 0x1.633e8e5697dc7p-45)
|
||||
A(0x1.ae00000000000p-1, 0x1.6574ebe8c1000p-3, 0x1.9cf8b2c3c2e78p-46)
|
||||
A(0x1.ac00000000000p-1, 0x1.6f0128b757000p-3, -0x1.5118de59c21e1p-45)
|
||||
A(0x1.aa00000000000p-1, 0x1.7898d85445000p-3, -0x1.c661070914305p-46)
|
||||
A(0x1.a600000000000p-1, 0x1.8beafeb390000p-3, -0x1.73d54aae92cd1p-47)
|
||||
A(0x1.a400000000000p-1, 0x1.95a5adcf70000p-3, 0x1.7f22858a0ff6fp-47)
|
||||
A(0x1.a000000000000p-1, 0x1.a93ed3c8ae000p-3, -0x1.8724350562169p-45)
|
||||
A(0x1.9e00000000000p-1, 0x1.b31d8575bd000p-3, -0x1.c358d4eace1aap-47)
|
||||
A(0x1.9c00000000000p-1, 0x1.bd087383be000p-3, -0x1.d4bc4595412b6p-45)
|
||||
A(0x1.9a00000000000p-1, 0x1.c6ffbc6f01000p-3, -0x1.1ec72c5962bd2p-48)
|
||||
A(0x1.9600000000000p-1, 0x1.db13db0d49000p-3, -0x1.aff2af715b035p-45)
|
||||
A(0x1.9400000000000p-1, 0x1.e530effe71000p-3, 0x1.212276041f430p-51)
|
||||
A(0x1.9200000000000p-1, 0x1.ef5ade4dd0000p-3, -0x1.a211565bb8e11p-51)
|
||||
A(0x1.9000000000000p-1, 0x1.f991c6cb3b000p-3, 0x1.bcbecca0cdf30p-46)
|
||||
A(0x1.8c00000000000p-1, 0x1.07138604d5800p-2, 0x1.89cdb16ed4e91p-48)
|
||||
A(0x1.8a00000000000p-1, 0x1.0c42d67616000p-2, 0x1.7188b163ceae9p-45)
|
||||
A(0x1.8800000000000p-1, 0x1.1178e8227e800p-2, -0x1.c210e63a5f01cp-45)
|
||||
A(0x1.8600000000000p-1, 0x1.16b5ccbacf800p-2, 0x1.b9acdf7a51681p-45)
|
||||
A(0x1.8400000000000p-1, 0x1.1bf99635a6800p-2, 0x1.ca6ed5147bdb7p-45)
|
||||
A(0x1.8200000000000p-1, 0x1.214456d0eb800p-2, 0x1.a87deba46baeap-47)
|
||||
A(0x1.7e00000000000p-1, 0x1.2bef07cdc9000p-2, 0x1.a9cfa4a5004f4p-45)
|
||||
A(0x1.7c00000000000p-1, 0x1.314f1e1d36000p-2, -0x1.8e27ad3213cb8p-45)
|
||||
A(0x1.7a00000000000p-1, 0x1.36b6776be1000p-2, 0x1.16ecdb0f177c8p-46)
|
||||
A(0x1.7800000000000p-1, 0x1.3c25277333000p-2, 0x1.83b54b606bd5cp-46)
|
||||
A(0x1.7600000000000p-1, 0x1.419b423d5e800p-2, 0x1.8e436ec90e09dp-47)
|
||||
A(0x1.7400000000000p-1, 0x1.4718dc271c800p-2, -0x1.f27ce0967d675p-45)
|
||||
A(0x1.7200000000000p-1, 0x1.4c9e09e173000p-2, -0x1.e20891b0ad8a4p-45)
|
||||
A(0x1.7000000000000p-1, 0x1.522ae0738a000p-2, 0x1.ebe708164c759p-45)
|
||||
A(0x1.6e00000000000p-1, 0x1.57bf753c8d000p-2, 0x1.fadedee5d40efp-46)
|
||||
A(0x1.6c00000000000p-1, 0x1.5d5bddf596000p-2, -0x1.a0b2a08a465dcp-47)
|
||||
#endif
|
||||
},
|
||||
};
|
||||
#endif /* __OBSOLETE_MATH */
|
||||
|
|
@ -58,6 +58,7 @@
|
|||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
#if __OBSOLETE_MATH
|
||||
|
||||
#ifndef _DOUBLE_IS_32BITS
|
||||
|
||||
|
|
@ -313,3 +314,4 @@ ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
|
|||
}
|
||||
|
||||
#endif /* defined(_DOUBLE_IS_32BITS) */
|
||||
#endif /* __OBSOLETE_MATH */
|
||||
|
|
|
|||
|
|
@ -52,6 +52,7 @@ PORTABILITY
|
|||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
#if __OBSOLETE_MATH
|
||||
#include <errno.h>
|
||||
|
||||
#ifndef _DOUBLE_IS_32BITS
|
||||
|
|
@ -207,7 +208,7 @@ PORTABILITY
|
|||
}
|
||||
|
||||
#endif /* defined(_DOUBLE_IS_32BITS) */
|
||||
|
||||
#endif /* __OBSOLETE_MATH */
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue