diff options
| -rw-r--r-- | newlib/libm/common/Makefile.am | 3 | ||||
| -rw-r--r-- | newlib/libm/common/Makefile.in | 34 | ||||
| -rw-r--r-- | newlib/libm/common/exp.c | 165 | ||||
| -rw-r--r-- | newlib/libm/common/exp2.c | 152 | ||||
| -rw-r--r-- | newlib/libm/common/exp_data.c | 611 | ||||
| -rw-r--r-- | newlib/libm/common/fdlibm.h | 1 | ||||
| -rw-r--r-- | newlib/libm/common/math_config.h | 70 | ||||
| -rw-r--r-- | newlib/libm/common/math_err.c | 101 | ||||
| -rw-r--r-- | newlib/libm/math/e_exp.c | 2 | ||||
| -rw-r--r-- | newlib/libm/math/w_exp.c | 2 | ||||
| -rw-r--r-- | newlib/libm/math/w_exp2.c | 2 |
11 files changed, 1138 insertions, 5 deletions
diff --git a/newlib/libm/common/Makefile.am b/newlib/libm/common/Makefile.am index 97b24315b..d7c479117 100644 --- a/newlib/libm/common/Makefile.am +++ b/newlib/libm/common/Makefile.am @@ -12,7 +12,8 @@ src = s_finite.c s_copysign.c s_modf.c s_scalbn.c \ s_fdim.c s_fma.c s_fmax.c s_fmin.c s_fpclassify.c \ s_lrint.c s_llrint.c \ s_lround.c s_llround.c s_nearbyint.c s_remquo.c s_round.c s_scalbln.c \ - s_signbit.c s_trunc.c + s_signbit.c s_trunc.c \ + exp.c exp2.c exp_data.c math_err.c fsrc = sf_finite.c sf_copysign.c sf_modf.c sf_scalbn.c \ sf_cbrt.c sf_exp10.c sf_expm1.c sf_ilogb.c \ diff --git a/newlib/libm/common/Makefile.in b/newlib/libm/common/Makefile.in index c944ac790..c35ada3ec 100644 --- a/newlib/libm/common/Makefile.in +++ b/newlib/libm/common/Makefile.in @@ -93,7 +93,9 @@ am__objects_1 = lib_a-s_finite.$(OBJEXT) lib_a-s_copysign.$(OBJEXT) \ lib_a-s_llround.$(OBJEXT) lib_a-s_nearbyint.$(OBJEXT) \ lib_a-s_remquo.$(OBJEXT) lib_a-s_round.$(OBJEXT) \ lib_a-s_scalbln.$(OBJEXT) lib_a-s_signbit.$(OBJEXT) \ - lib_a-s_trunc.$(OBJEXT) + lib_a-s_trunc.$(OBJEXT) lib_a-exp.$(OBJEXT) \ + lib_a-exp2.$(OBJEXT) lib_a-exp_data.$(OBJEXT) \ + lib_a-math_err.$(OBJEXT) am__objects_2 = lib_a-sf_finite.$(OBJEXT) lib_a-sf_copysign.$(OBJEXT) \ lib_a-sf_modf.$(OBJEXT) lib_a-sf_scalbn.$(OBJEXT) \ lib_a-sf_cbrt.$(OBJEXT) lib_a-sf_exp10.$(OBJEXT) \ @@ -162,7 +164,8 @@ am__objects_5 = s_finite.lo s_copysign.lo s_modf.lo s_scalbn.lo \ s_log2.lo s_matherr.lo s_lib_ver.lo s_fdim.lo s_fma.lo \ s_fmax.lo s_fmin.lo s_fpclassify.lo s_lrint.lo s_llrint.lo \ s_lround.lo s_llround.lo s_nearbyint.lo s_remquo.lo s_round.lo \ - s_scalbln.lo s_signbit.lo s_trunc.lo + s_scalbln.lo s_signbit.lo s_trunc.lo exp.lo exp2.lo \ + exp_data.lo math_err.lo am__objects_6 = sf_finite.lo sf_copysign.lo sf_modf.lo sf_scalbn.lo \ sf_cbrt.lo sf_exp10.lo sf_expm1.lo sf_ilogb.lo sf_infinity.lo \ sf_isinf.lo sf_isinff.lo sf_isnan.lo sf_isnanf.lo sf_log1p.lo \ @@ -352,7 +355,8 @@ src = s_finite.c s_copysign.c s_modf.c s_scalbn.c \ s_fdim.c s_fma.c s_fmax.c s_fmin.c s_fpclassify.c \ s_lrint.c s_llrint.c \ s_lround.c s_llround.c s_nearbyint.c s_remquo.c s_round.c s_scalbln.c \ - s_signbit.c s_trunc.c + s_signbit.c s_trunc.c \ + exp.c exp2.c exp_data.c math_err.c fsrc = sf_finite.c sf_copysign.c sf_modf.c sf_scalbn.c \ sf_cbrt.c sf_exp10.c sf_expm1.c sf_ilogb.c \ @@ -696,6 +700,30 @@ lib_a-s_trunc.o: s_trunc.c lib_a-s_trunc.obj: s_trunc.c $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-s_trunc.obj `if test -f 's_trunc.c'; then $(CYGPATH_W) 's_trunc.c'; else $(CYGPATH_W) '$(srcdir)/s_trunc.c'; fi` +lib_a-exp.o: exp.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp.o `test -f 'exp.c' || echo '$(srcdir)/'`exp.c + +lib_a-exp.obj: exp.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp.obj `if test -f 'exp.c'; then $(CYGPATH_W) 'exp.c'; else $(CYGPATH_W) '$(srcdir)/exp.c'; fi` + +lib_a-exp2.o: exp2.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp2.o `test -f 'exp2.c' || echo '$(srcdir)/'`exp2.c + +lib_a-exp2.obj: exp2.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp2.obj `if test -f 'exp2.c'; then $(CYGPATH_W) 'exp2.c'; else $(CYGPATH_W) '$(srcdir)/exp2.c'; fi` + +lib_a-exp_data.o: exp_data.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp_data.o `test -f 'exp_data.c' || echo '$(srcdir)/'`exp_data.c + +lib_a-exp_data.obj: exp_data.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-exp_data.obj `if test -f 'exp_data.c'; then $(CYGPATH_W) 'exp_data.c'; else $(CYGPATH_W) '$(srcdir)/exp_data.c'; fi` + +lib_a-math_err.o: math_err.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-math_err.o `test -f 'math_err.c' || echo '$(srcdir)/'`math_err.c + +lib_a-math_err.obj: math_err.c + $(CC) $(DEFS) $(DEFAULT_INCLUDES) $(INCLUDES) $(AM_CPPFLAGS) $(CPPFLAGS) $(lib_a_CFLAGS) $(CFLAGS) -c -o lib_a-math_err.obj `if test -f 'math_err.c'; then $(CYGPATH_W) 'math_err.c'; else $(CYGPATH_W) '$(srcdir)/math_err.c'; fi` + lib_a-sf_finite.o: sf_finite.c $(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 diff --git a/newlib/libm/common/exp.c b/newlib/libm/common/exp.c new file mode 100644 index 000000000..f295b6173 --- /dev/null +++ b/newlib/libm/common/exp.c @@ -0,0 +1,165 @@ +/* Double-precision e^x function. + 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.h> +#include <stdint.h> +#include "math_config.h" + +#define N (1 << EXP_TABLE_BITS) +#define InvLn2N __exp_data.invln2N +#define NegLn2hiN __exp_data.negln2hiN +#define NegLn2loN __exp_data.negln2loN +#define Shift __exp_data.shift +#define T __exp_data.tab +#define C2 __exp_data.poly[5 - EXP_POLY_ORDER] +#define C3 __exp_data.poly[6 - EXP_POLY_ORDER] +#define C4 __exp_data.poly[7 - EXP_POLY_ORDER] +#define C5 __exp_data.poly[8 - EXP_POLY_ORDER] +#define C6 __exp_data.poly[9 - EXP_POLY_ORDER] + +static inline double +specialcase (double_t tmp, uint64_t sbits, uint64_t ki) +{ + double_t scale, y; + + if ((ki & 0x80000000) == 0) + { + /* k > 0, the exponent of scale might have overflowed by <= 460. */ + sbits -= 1009ull << 52; + scale = asdouble (sbits); + y = 0x1p1009 * (scale + scale * tmp); + return check_oflow (y); + } + /* k < 0, need special care in the subnormal range. */ + sbits += 1022ull << 52; + scale = asdouble (sbits); + y = scale + scale * tmp; + if (y < 1.0) + { + /* Round y to the right precision before scaling it into the subnormal + range to avoid double rounding that can cause 0.5+E/2 ulp error where + E is the worst-case ulp error outside the subnormal range. So this + is only useful if the goal is better than 1 ulp worst-case error. */ + double_t hi, lo; + lo = scale - y + scale * tmp; + hi = 1.0 + y; + lo = 1.0 - hi + y + lo; + y = eval_as_double (hi + lo) - 1.0; + /* Avoid -0.0 with downward rounding. */ + if (WANT_ROUNDING && y == 0.0) + y = 0.0; + /* The underflow exception needs to be signaled explicitly. */ + force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); + } + y = 0x1p-1022 * y; + return check_uflow (y); +} + +static inline uint32_t +top12 (double x) +{ + return asuint64 (x) >> 52; +} + +double +exp (double x) +{ + uint32_t abstop; + uint64_t ki, idx, top, sbits; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, z, r, r2, scale, tail, tmp; + + abstop = top12 (x) & 0x7ff; + if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) + { + if (abstop - top12 (0x1p-54) >= 0x80000000) + /* Avoid spurious underflow for tiny x. */ + /* Note: 0 is common input. */ + return WANT_ROUNDING ? 1.0 + x : 1.0; + if (abstop >= top12 (1024.0)) + { + if (asuint64 (x) == asuint64 (-INFINITY)) + return 0.0; + if (abstop >= top12 (INFINITY)) + return 1.0 + x; + if (asuint64 (x) >> 63) + return __math_uflow (0); + else + return __math_oflow (0); + } + /* Large x is special cased below. */ + abstop = 0; + } + + /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ + /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ + z = InvLn2N * x; +#if TOINT_INTRINSICS + kd = roundtoint (z); + ki = converttoint (z); +#elif EXP_USE_TOINT_NARROW + /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */ + kd = eval_as_double (z + Shift); + ki = asuint64 (kd) >> 16; + kd = (double_t) (int32_t) ki; +#else + /* z - kd is in [-1, 1] in non-nearest rounding modes. */ + kd = eval_as_double (z + Shift); + ki = asuint64 (kd); + kd -= Shift; +#endif + r = x + kd*NegLn2hiN + kd*NegLn2loN; + /* 2^(k/N) ~= scale * (1 + tail). */ + idx = 2*(ki % N); + top = ki << (52 - EXP_TABLE_BITS); + tail = asdouble (T[idx]); + /* This is only a valid scale when -1023*N < k < 1024*N. */ + sbits = T[idx + 1] + top; + /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ + /* Evaluation is optimized assuming superscalar pipelined execution. */ + r2 = r*r; + /* Without fma the worst case error is 0.25/N ulp larger. */ + /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ +#if EXP_POLY_ORDER == 4 + tmp = tail + r + r2*C2 + r*r2*(C3 + r*C4); +#elif EXP_POLY_ORDER == 5 + tmp = tail + r + r2*(C2 + r*C3) + r2*r2*(C4 + r*C5); +#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; +} +#endif diff --git a/newlib/libm/common/exp2.c b/newlib/libm/common/exp2.c new file mode 100644 index 000000000..6579e16fd --- /dev/null +++ b/newlib/libm/common/exp2.c @@ -0,0 +1,152 @@ +/* Double-precision 2^x function. + 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.h> +#include <stdint.h> +#include "math_config.h" + +#define N (1 << EXP_TABLE_BITS) +#define Shift __exp_data.exp2_shift +#define T __exp_data.tab +#define C1 __exp_data.exp2_poly[0] +#define C2 __exp_data.exp2_poly[1] +#define C3 __exp_data.exp2_poly[2] +#define C4 __exp_data.exp2_poly[3] +#define C5 __exp_data.exp2_poly[4] +#define C6 __exp_data.exp2_poly[5] + +static inline double +specialcase (double_t tmp, uint64_t sbits, uint64_t ki) +{ + double_t scale, y; + + if ((ki & 0x80000000) == 0) + { + /* k > 0, the exponent of scale might have overflowed by 1. */ + sbits -= 1ull << 52; + scale = asdouble (sbits); + y = 2 * (scale + scale * tmp); + return check_oflow (y); + } + /* k < 0, need special care in the subnormal range. */ + sbits += 1022ull << 52; + scale = asdouble (sbits); + y = scale + scale * tmp; + if (y < 1.0) + { + /* Round y to the right precision before scaling it into the subnormal + range to avoid double rounding that can cause 0.5+E/2 ulp error where + E is the worst-case ulp error outside the subnormal range. So this + is only useful if the goal is better than 1 ulp worst-case error. */ + double_t hi, lo; + lo = scale - y + scale * tmp; + hi = 1.0 + y; + lo = 1.0 - hi + y + lo; + y = eval_as_double (hi + lo) - 1.0; + /* Avoid -0.0 with downward rounding. */ + if (WANT_ROUNDING && y == 0.0) + y = 0.0; + /* The underflow exception needs to be signaled explicitly. */ + force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022); + } + y = 0x1p-1022 * y; + return check_uflow (y); +} + +static inline uint32_t +top12 (double x) +{ + return asuint64 (x) >> 52; +} + +double +exp2 (double x) +{ + uint32_t abstop; + uint64_t ki, idx, top, sbits; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, r, r2, scale, tail, tmp; + + abstop = top12 (x) & 0x7ff; + if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54))) + { + if (abstop - top12 (0x1p-54) >= 0x80000000) + /* Avoid spurious underflow for tiny x. */ + /* Note: 0 is common input. */ + return WANT_ROUNDING ? 1.0 + x : 1.0; + if (abstop >= top12 (1024.0)) + { + if (asuint64 (x) == asuint64 (-INFINITY)) + return 0.0; + if (abstop >= top12 (INFINITY)) + return 1.0 + x; + if (!(asuint64 (x) >> 63)) + return __math_oflow (0); + else if (asuint64 (x) >= asuint64 (-1075.0)) + return __math_uflow (0); + } + if (2 * asuint64 (x) > 2 * asuint64 (928.0)) + /* Large x is special cased below. */ + abstop = 0; + } + + /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */ + /* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */ + kd = eval_as_double (x + Shift); + ki = asuint64 (kd); /* k. */ + kd -= Shift; /* k/N for int k. */ + r = x - kd; + /* 2^(k/N) ~= scale * (1 + tail). */ + idx = 2*(ki % N); + top = ki << (52 - EXP_TABLE_BITS); + tail = asdouble (T[idx]); + /* This is only a valid scale when -1023*N < k < 1024*N. */ + sbits = T[idx + 1] + top; + /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */ + /* Evaluation is optimized assuming superscalar pipelined execution. */ + r2 = r*r; + /* Without fma the worst case error is 0.5/N ulp larger. */ + /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */ +#if EXP2_POLY_ORDER == 4 + tmp = tail + r*C1 + r2*C2 + r*r2*(C3 + r*C4); +#elif EXP2_POLY_ORDER == 5 + tmp = tail + r*C1 + r2*(C2 + r*C3) + r2*r2*(C4 + r*C5); +#elif EXP2_POLY_ORDER == 6 + tmp = tail + r*C1 + 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^-65 and scale > 2^-928, so there + is no spurious underflow here even without fma. */ + return scale + scale * tmp; +} +#endif diff --git a/newlib/libm/common/exp_data.c b/newlib/libm/common/exp_data.c new file mode 100644 index 000000000..3b60f87ef --- /dev/null +++ b/newlib/libm/common/exp_data.c @@ -0,0 +1,611 @@ +/* Shared data between exp, exp2 and 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 << EXP_TABLE_BITS) + +const struct exp_data __exp_data = { +// N/ln2 +.invln2N = 0x1.71547652b82fep0 * N, +// -ln2/N +#if N == 64 +.negln2hiN = -0x1.62e42fefa0000p-7, +.negln2loN = -0x1.cf79abc9e3b3ap-46, +#elif N == 128 +.negln2hiN = -0x1.62e42fefa0000p-8, +.negln2loN = -0x1.cf79abc9e3b3ap-47, +#elif N == 256 +.negln2hiN = -0x1.62e42fefc0000p-9, +.negln2loN = 0x1.c610ca86c3899p-45, +#endif +// Used for rounding when !TOINT_INTRINSICS +#if EXP_USE_TOINT_NARROW +.shift = 0x1800000000.8p0, +#else +.shift = 0x1.8p52, +#endif +// exp polynomial coefficients. +.poly = { +#if N == 64 && EXP_POLY_ORDER == 5 && !EXP_POLY_WIDE +// abs error: 1.5543*2^-60 +// ulp error: 0.529 (0.533 without fma) +// if |x| < ln2/128+eps +// abs error if |x| < ln2/64: 1.7157*2^-50 +0x1.fffffffffdbcdp-2, +0x1.555555555444cp-3, +0x1.555573c6a9f7dp-5, +0x1.1111266d28935p-7, +#elif N == 64 && EXP_POLY_ORDER == 6 && EXP_POLY_WIDE +// abs error: 1.6735*2^-64 +// ulp error: 0.518 (0.522 without fma) +// if |x| < ln2/64 +0x1.5555555548f9ap-3, +0x1.555555554bf5dp-5, +0x1.11115b75f0f4dp-7, +0x1.6c171a6b6303ep-10, +#elif N == 128 && EXP_POLY_ORDER == 5 && !EXP_POLY_WIDE +// abs error: 1.555*2^-66 +// ulp error: 0.509 (0.511 without fma) +// if |x| < ln2/256+eps +// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65 +// abs error if |x| < ln2/128: 1.7145*2^-56 +0x1.ffffffffffdbdp-2, +0x1.555555555543cp-3, +0x1.55555cf172b91p-5, +0x1.1111167a4d017p-7, +#elif N == 128 && EXP_POLY_ORDER == 5 && EXP_POLY_WIDE +// abs error: 1.5542*2^-60 +// ulp error: 0.521 (0.523 without fma) +// if |x| < ln2/128 +0x1.fffffffffdbcep-2, +0x1.55555555543c2p-3, +0x1.555573c64f2e3p-5, +0x1.111126b4eff73p-7, +#elif N == 128 && EXP_POLY_ORDER == 6 && EXP_POLY_WIDE +// abs error: 1.6861*2^-71 +// ulp error: 0.509 (0.511 without fma) +// if |x| < ln2/128 +0x1.55555555548fdp-3, +0x1.555555555658fp-5, +0x1.111123a859bb6p-7, +0x1.6c16ba6920cabp-10, +#elif N == 256 && EXP_POLY_ORDER == 4 && !EXP_POLY_WIDE +// abs error: 1.43*2^-58 +// ulp error: 0.549 (0.550 without fma) +// if |x| < ln2/512 +0x1p0, // unused +0x1.fffffffffffd4p-2, +0x1.5555571d6ef9p-3, +0x1.5555576a5adcep-5, +#elif N == 256 && EXP_POLY_ORDER == 5 && EXP_POLY_WIDE +// abs error: 1.5547*2^-66 +// ulp error: 0.505 (0.506 without fma) +// if |x| < ln2/256 +0x1.ffffffffffdbdp-2, +0x1.555555555543cp-3, +0x1.55555cf16e1edp-5, +0x1.1111167a4b553p-7, +#endif +}, +.exp2_shift = 0x1.8p52 / N, +// exp2 polynomial coefficients. +.exp2_poly = { +#if N == 64 && EXP2_POLY_ORDER == 6 && EXP2_POLY_WIDE +// abs error: 1.3054*2^-63 +// ulp error: 0.515 +// if |x| < 1/64 +0x1.62e42fefa39efp-1, +0x1.ebfbdff82c58fp-3, +0x1.c6b08d7045cf1p-5, +0x1.3b2ab6fb8fd0ep-7, +0x1.5d884afec48d7p-10, +0x1.43097dc684ae1p-13, +#elif N == 128 && EXP2_POLY_ORDER == 5 && !EXP2_POLY_WIDE +// abs error: 1.2195*2^-65 +// ulp error: 0.507 (0.511 without fma) +// if |x| < 1/256 +// abs error if |x| < 1/128: 1.9941*2^-56 +0x1.62e42fefa39efp-1, +0x1.ebfbdff82c424p-3, +0x1.c6b08d70cf4b5p-5, +0x1.3b2abd24650ccp-7, +0x1.5d7e09b4e3a84p-10, +#elif N == 256 && EXP2_POLY_ORDER == 5 && EXP2_POLY_WIDE +// abs error: 1.2195*2^-65 +// ulp error: 0.504 (0.508 without fma) +// if |x| < 1/256 +0x1.62e42fefa39efp-1, +0x1.ebfbdff82c424p-3, +0x1.c6b08d70cf4b5p-5, +0x1.3b2abd24650ccp-7, +0x1.5d7e09b4e3a84p-10, +#endif +}, +// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N) +// tab[2*k] = asuint64(T[k]) +// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N +.tab = { +#if N == 64 +0x0, 0x3ff0000000000000, +0xbc7160139cd8dc5d, 0x3fefec9a3e778061, +0x3c8cd2523567f613, 0x3fefd9b0d3158574, +0x3c60f74e61e6c861, 0x3fefc74518759bc8, +0x3c979aa65d837b6d, 0x3fefb5586cf9890f, +0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2, +0xbc9556522a2fbd0e, 0x3fef9301d0125b51, +0xbc91c923b9d5f416, 0x3fef829aaea92de0, +0xbc801b15eaa59348, 0x3fef72b83c7d517b, +0x3c8b898c3f1353bf, 0x3fef635beb6fcb75, +0x3c9aecf73e3a2f60, 0x3fef54873168b9aa, +0x3c8a6f4144a6c38d, 0x3fef463b88628cd6, +0x3c968efde3a8a894, 0x3fef387a6e756238, +0x3c80472b981fe7f2, 0x3fef2b4565e27cdd, +0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1, +0x3c8b3782720c0ab4, 0x3fef1285a6e4030b, +0x3c834d754db0abb6, 0x3fef06fe0a31b715, +0x3c8fdd395dd3f84a, 0x3feefc08b26416ff, +0xbc924aedcc4b5068, 0x3feef1a7373aa9cb, +0xbc71d1e83e9436d2, 0x3feee7db34e59ff7, +0x3c859f48a72a4c6d, 0x3feedea64c123422, +0xbc58a78f4817895b, 0x3feed60a21f72e2a, +0x3c4363ed60c2ac11, 0x3feece086061892d, +0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0, +0x3c7690cebb7aafb0, 0x3feebfdad5362a27, +0xbc8f94340071a38e, 0x3feeb9b2769d2ca7, +0xbc78dec6bd0f385f, 0x3feeb42b569d4f82, +0x3c93350518fdd78e, 0x3feeaf4736b527da, +0x3c9063e1e21c5409, 0x3feeab07dd485429, +0x3c9432e62b64c035, 0x3feea76f15ad2148, +0xbc8c33c53bef4da8, 0x3feea47eb03a5585, +0xbc93cedd78565858, 0x3feea23882552225, +0xbc93b3efbf5e2228, 0x3feea09e667f3bcd, +0xbc6367efb86da9ee, 0x3fee9fb23c651a2f, +0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74, +0xbc8619321e55e68a, 0x3fee9feb564267c9, +0xbc7b32dcb94da51d, 0x3feea11473eb0187, +0x3c65ebe1abd66c55, 0x3feea2f336cf4e62, +0xbc9369b6f13b3734, 0x3feea589994cce13, +0xbc94d450d872576e, 0x3feea8d99b4492ed, +0x3c8db72fc1f0eab4, 0x3feeace5422aa0db, +0x3c7bf68359f35f44, 0x3feeb1ae99157736, +0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5, +0xbc92434322f4f9aa, 0x3feebd829fde4e50, +0x3c71affc2b91ce27, 0x3feec49182a3f090, +0xbc87c50422622263, 0x3feecc667b5de565, +0xbc91bbd1d3bcbb15, 0x3feed503b23e255d, +0x3c8469846e735ab3, 0x3feede6b5579fdbf, +0x3c8c1a7792cb3387, 0x3feee89f995ad3ad, +0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb, +0xbc68d6f438ad9334, 0x3feeff76f2fb5e47, +0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2, +0x3c736eae30af0cb3, 0x3fef199bdd85529c, +0x3c84e08fd10959ac, 0x3fef27f12e57d14b, +0x3c676b2c6c921968, 0x3fef3720dcef9069, +0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c, +0x3c74a385a63d07a7, 0x3fef5818dcfba487, +0x3c8e5a50d5c192ac, 0x3fef69e603db3285, +0xbc82d52107b43e1f, 0x3fef7c97337b9b5f, +0x3c74b604603a88d3, 0x3fef902ee78b3ff6, +0xbc8ff7128fd391f0, 0x3fefa4afa2a490da, +0x3c8ec3bc41aa2008, 0x3fefba1bee615a27, +0x3c8a64a931d185ee, 0x3fefd0765b6e4540, +0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8, +#elif N == 128 +0x0, 0x3ff0000000000000, +0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335, +0xbc7160139cd8dc5d, 0x3fefec9a3e778061, +0xbc905e7a108766d1, 0x3fefe315e86e7f85, +0x3c8cd2523567f613, 0x3fefd9b0d3158574, +0xbc8bce8023f98efa, 0x3fefd06b29ddf6de, +0x3c60f74e61e6c861, 0x3fefc74518759bc8, +0x3c90a3e45b33d399, 0x3fefbe3ecac6f383, +0x3c979aa65d837b6d, 0x3fefb5586cf9890f, +0x3c8eb51a92fdeffc, 0x3fefac922b7247f7, +0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2, +0xbc6a033489906e0b, 0x3fef9b66affed31b, +0xbc9556522a2fbd0e, 0x3fef9301d0125b51, +0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc, +0xbc91c923b9d5f416, 0x3fef829aaea92de0, +0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51, +0xbc801b15eaa59348, 0x3fef72b83c7d517b, +0xbc8f1ff055de323d, 0x3fef6af9388c8dea, +0x3c8b898c3f1353bf, 0x3fef635beb6fcb75, +0xbc96d99c7611eb26, 0x3fef5be084045cd4, +0x3c9aecf73e3a2f60, 0x3fef54873168b9aa, +0xbc8fe782cb86389d, 0x3fef4d5022fcd91d, +0x3c8a6f4144a6c38d, 0x3fef463b88628cd6, +0x3c807a05b0e4047d, 0x3fef3f49917ddc96, +0x3c968efde3a8a894, 0x3fef387a6e756238, +0x3c875e18f274487d, 0x3fef31ce4fb2a63f, +0x3c80472b981fe7f2, 0x3fef2b4565e27cdd, +0xbc96b87b3f71085e, 0x3fef24dfe1f56381, +0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1, +0xbc3d219b1a6fbffa, 0x3fef187fd0dad990, +0x3c8b3782720c0ab4, 0x3fef1285a6e4030b, +0x3c6e149289cecb8f, 0x3fef0cafa93e2f56, +0x3c834d754db0abb6, 0x3fef06fe0a31b715, +0x3c864201e2ac744c, 0x3fef0170fc4cd831, +0x3c8fdd395dd3f84a, 0x3feefc08b26416ff, +0xbc86a3803b8e5b04, 0x3feef6c55f929ff1, +0xbc924aedcc4b5068, 0x3feef1a7373aa9cb, +0xbc9907f81b512d8e, 0x3feeecae6d05d866, +0xbc71d1e83e9436d2, 0x3feee7db34e59ff7, +0xbc991919b3ce1b15, 0x3feee32dc313a8e5, +0x3c859f48a72a4c6d, 0x3feedea64c123422, +0xbc9312607a28698a, 0x3feeda4504ac801c, +0xbc58a78f4817895b, 0x3feed60a21f72e2a, +0xbc7c2c9b67499a1b, 0x3feed1f5d950a897, +0x3c4363ed60c2ac11, 0x3feece086061892d, +0x3c9666093b0664ef, 0x3feeca41ed1d0057, +0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0, +0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de, +0x3c7690cebb7aafb0, 0x3feebfdad5362a27, +0x3c931dbdeb54e077, 0x3feebcb299fddd0d, +0xbc8f94340071a38e, 0x3feeb9b2769d2ca7, +0xbc87deccdc93a349, 0x3feeb6daa2cf6642, +0xbc78dec6bd0f385f, 0x3feeb42b569d4f82, +0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f, +0x3c93350518fdd78e, 0x3feeaf4736b527da, +0x3c7b98b72f8a9b05, 0x3feead12d497c7fd, +0x3c9063e1e21c5409, 0x3feeab07dd485429, +0x3c34c7855019c6ea, 0x3feea9268a5946b7, +0x3c9432e62b64c035, 0x3feea76f15ad2148, +0xbc8ce44a6199769f, 0x3feea5e1b976dc09, +0xbc8c33c53bef4da8, 0x3feea47eb03a5585, +0xbc845378892be9ae, 0x3feea34634ccc320, +0xbc93cedd78565858, 0x3feea23882552225, +0x3c5710aa807e1964, 0x3feea155d44ca973, +0xbc93b3efbf5e2228, 0x3feea09e667f3bcd, +0xbc6a12ad8734b982, 0x3feea012750bdabf, +0xbc6367efb86da9ee, 0x3fee9fb23c651a2f, +0xbc80dc3d54e08851, 0x3fee9f7df9519484, +0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74, +0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174, +0xbc8619321e55e68a, 0x3fee9feb564267c9, +0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f, +0xbc7b32dcb94da51d, 0x3feea11473eb0187, +0x3c94ecfd5467c06b, 0x3feea1ed0130c132, +0x3c65ebe1abd66c55, 0x3feea2f336cf4e62, +0xbc88a1c52fb3cf42, 0x3feea427543e1a12, +0xbc9369b6f13b3734, 0x3feea589994cce13, +0xbc805e843a19ff1e, 0x3feea71a4623c7ad, +0xbc94d450d872576e, 0x3feea8d99b4492ed, +0x3c90ad675b0e8a00, 0x3feeaac7d98a6699, +0x3c8db72fc1f0eab4, 0x3feeace5422aa0db, +0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c, +0x3c7bf68359f35f44, 0x3feeb1ae99157736, +0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6, +0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5, +0xbc6c23f97c90b959, 0x3feeba44cbc8520f, +0xbc92434322f4f9aa, 0x3feebd829fde4e50, +0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba, +0x3c71affc2b91ce27, 0x3feec49182a3f090, +0x3c6dd235e10a73bb, 0x3feec86319e32323, +0xbc87c50422622263, 0x3feecc667b5de565, +0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33, +0xbc91bbd1d3bcbb15, 0x3feed503b23e255d, +0x3c90cc319cee31d2, 0x3feed99e1330b358, +0x3c8469846e735ab3, 0x3feede6b5579fdbf, +0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a, +0x3c8c1a7792cb3387, 0x3feee89f995ad3ad, +0xbc907b8f4ad1d9fa, 0x3feeee07298db666, +0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb, +0xbc90a40e3da6f640, 0x3feef9728de5593a, +0xbc68d6f438ad9334, 0x3feeff76f2fb5e47, +0xbc91eee26b588a35, 0x3fef05b030a1064a, +0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2, +0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09, +0x3c736eae30af0cb3, 0x3fef199bdd85529c, +0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a, +0x3c84e08fd10959ac, 0x3fef27f12e57d14b, +0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5, +0x3c676b2c6c921968, 0x3fef3720dcef9069, +0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa, +0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c, +0xbc900dae3875a949, 0x3fef4f87080d89f2, +0x3c74a385a63d07a7, 0x3fef5818dcfba487, +0xbc82919e2040220f, 0x3fef60e316c98398, +0x3c8e5a50d5c192ac, 0x3fef69e603db3285, +0x3c843a59ac016b4b, 0x3fef7321f301b460, +0xbc82d52107b43e1f, 0x3fef7c97337b9b5f, +0xbc892ab93b470dc9, 0x3fef864614f5a129, +0x3c74b604603a88d3, 0x3fef902ee78b3ff6, +0x3c83c5ec519d7271, 0x3fef9a51fbc74c83, +0xbc8ff7128fd391f0, 0x3fefa4afa2a490da, +0xbc8dae98e223747d, 0x3fefaf482d8e67f1, +0x3c8ec3bc41aa2008, 0x3fefba1bee615a27, +0x3c842b94c3a9eb32, 0x3fefc52b376bba97, +0x3c8a64a931d185ee, 0x3fefd0765b6e4540, +0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14, +0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8, +0x3c5305c14160cc89, 0x3feff3c22b8f71f1, +#elif N == 256 +0x0, 0x3ff0000000000000, +0xbc84e82fc61851ac, 0x3feffb1afa5abcbf, +0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335, +0xbc82985dd8521d32, 0x3feff168143b0281, +0xbc7160139cd8dc5d, 0x3fefec9a3e778061, +0x3c651e617061bfbd, 0x3fefe7d42e11bbcc, +0xbc905e7a108766d1, 0x3fefe315e86e7f85, +0x3c845fad437fa426, 0x3fefde5f72f654b1, +0x3c8cd2523567f613, 0x3fefd9b0d3158574, +0xbc954529642b232f, 0x3fefd50a0e3c1f89, +0xbc8bce8023f98efa, 0x3fefd06b29ddf6de, +0x3c8293708ef5c32e, 0x3fefcbd42b72a836, +0x3c60f74e61e6c861, 0x3fefc74518759bc8, +0xbc95b9280905b2a4, 0x3fefc2bdf66607e0, +0x3c90a3e45b33d399, 0x3fefbe3ecac6f383, +0x3c84f31f32c4b7e7, 0x3fefb9c79b1f3919, +0x3c979aa65d837b6d, 0x3fefb5586cf9890f, +0x3c9407fb30d06420, 0x3fefb0f145e46c85, +0x3c8eb51a92fdeffc, 0x3fefac922b7247f7, +0xbc9a5d04b3b9911b, 0x3fefa83b23395dec, +0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2, +0xbc937a01f0739546, 0x3fef9fa55fdfa9c5, +0xbc6a033489906e0b, 0x3fef9b66affed31b, +0x3c8b8268b04ef0a5, 0x3fef973028d7233e, +0xbc9556522a2fbd0e, 0x3fef9301d0125b51, +0xbc9ac46e44a2ebcc, 0x3fef8edbab5e2ab6, +0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc, +0xbc65704e90c9f860, 0x3fef86a814f204ab, +0xbc91c923b9d5f416, 0x3fef829aaea92de0, +0xbc897cea57e46280, 0x3fef7e95934f312e, +0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51, +0x3c56f01429e2b9d2, 0x3fef76a45471c3c2, +0xbc801b15eaa59348, 0x3fef72b83c7d517b, +0x3c6e653b2459034b, 0x3fef6ed48695bbc0, +0xbc8f1ff055de323d, 0x3fef6af9388c8dea, +0x3c92cc7ea345b7dc, 0x3fef672658375d2f, +0x3c8b898c3f1353bf, 0x3fef635beb6fcb75, +0x3c957bfb2876ea9e, 0x3fef5f99f8138a1c, +0xbc96d99c7611eb26, 0x3fef5be084045cd4, +0x3c8cdc1873af2155, 0x3fef582f95281c6b, +0x3c9aecf73e3a2f60, 0x3fef54873168b9aa, +0xbc9493684653a131, 0x3fef50e75eb44027, +0xbc8fe782cb86389d, 0x3fef4d5022fcd91d, +0xbc98e2899077520a, 0x3fef49c18438ce4d, +0x3c8a6f4144a6c38d, 0x3fef463b88628cd6, +0x3c9120fcd4f59273, 0x3fef42be3578a819, +0x3c807a05b0e4047d, 0x3fef3f49917ddc96, +0x3c89b788c188c9b8, 0x3fef3bdda27912d1, +0x3c968efde3a8a894, 0x3fef387a6e756238, +0x3c877afbca90ef84, 0x3fef351ffb82140a, +0x3c875e18f274487d, 0x3fef31ce4fb2a63f, +0x3c91512f082876ee, 0x3fef2e85711ece75, +0x3c80472b981fe7f2, 0x3fef2b4565e27cdd, +0x3c9a02f0c7d75ec6, 0x3fef280e341ddf29, +0xbc96b87b3f71085e, 0x3fef24dfe1f56381, +0xbc803297e78260bf, 0x3fef21ba7591bb70, +0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1, +0xbc95b77e5ccd9fbf, 0x3fef1b8a66d10f13, +0xbc3d219b1a6fbffa, 0x3fef187fd0dad990, +0xbc91e75c40b4251e, 0x3fef157e39771b2f, +0x3c8b3782720c0ab4, 0x3fef1285a6e4030b, +0x3c98a911f1f7785a, 0x3fef0f961f641589, +0x3c6e149289cecb8f, 0x3fef0cafa93e2f56, +0xbc61e7c998db7dbb, 0x3fef09d24abd886b, +0x3c834d754db0abb6, 0x3fef06fe0a31b715, +0x3c85425c11faadf4, 0x3fef0432edeeb2fd, +0x3c864201e2ac744c, 0x3fef0170fc4cd831, +0xbc979517a03e2847, 0x3feefeb83ba8ea32, +0x3c8fdd395dd3f84a, 0x3feefc08b26416ff, +0xbc800e2a46da4bee, 0x3feef96266e3fa2d, +0xbc86a3803b8e5b04, 0x3feef6c55f929ff1, +0xbc87430803972b34, 0x3feef431a2de883b, +0xbc924aedcc4b5068, 0x3feef1a7373aa9cb, +0xbc954de30ae02d94, 0x3feeef26231e754a, +0xbc9907f81b512d8e, 0x3feeecae6d05d866, +0xbc94f2487e1c03ec, 0x3feeea401b7140ef, +0xbc71d1e83e9436d2, 0x3feee7db34e59ff7, +0x3c914a5432fcb2f4, 0x3feee57fbfec6cf4, +0xbc991919b3ce1b15, 0x3feee32dc313a8e5, +0x3c79c3bba5562a2f, 0x3feee0e544ede173, +0x3c859f48a72a4c6d, 0x3feedea64c123422, +0xbc85a71612e21658, 0x3feedc70df1c5175, +0xbc9312607a28698a, 0x3feeda4504ac801c, +0x3c86421f6f1d24d6, 0x3feed822c367a024, +0xbc58a78f4817895b, 0x3feed60a21f72e2a, +0xbc9348a6815fce65, 0x3feed3fb2709468a, +0xbc7c2c9b67499a1b, 0x3feed1f5d950a897, +0x3c835c43984d9871, 0x3feecffa3f84b9d4, +0x3c4363ed60c2ac11, 0x3feece086061892d, +0xbc632afc8d9473a0, 0x3feecc2042a7d232, +0x3c9666093b0664ef, 0x3feeca41ed1d0057, +0xbc95fc5e44de020e, 0x3feec86d668b3237, +0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0, +0xbc7ea0148327c42f, 0x3feec4e1e192aed2, +0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de, +0xbc7a843ad1a88022, 0x3feec17dea6db7d7, +0x3c7690cebb7aafb0, 0x3feebfdad5362a27, +0x3c892ca3bf144e63, 0x3feebe41b817c114, +0x3c931dbdeb54e077, 0x3feebcb299fddd0d, +0xbc902c99b04aa8b0, 0x3feebb2d81d8abff, +0xbc8f94340071a38e, 0x3feeb9b2769d2ca7, +0x3c73e34f67e67118, 0x3feeb8417f4531ee, +0xbc87deccdc93a349, 0x3feeb6daa2cf6642, +0xbc75a3b1197ba0f0, 0x3feeb57de83f4eef, +0xbc78dec6bd0f385f, 0x3feeb42b569d4f82, +0x3c81bd2888075068, 0x3feeb2e2f4f6ad27, +0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f, +0xbc896be8ae89ef8f, 0x3feeb070dde910d2, +0x3c93350518fdd78e, 0x3feeaf4736b527da, +0xbc88e6ac90348602, 0x3feeae27dbe2c4cf, +0x3c7b98b72f8a9b05, 0x3feead12d497c7fd, +0xbc91af7f1365c3ac, 0x3feeac0827ff07cc, +0x3c9063e1e21c5409, 0x3feeab07dd485429, +0xbc943a3540d1898a, 0x3feeaa11fba87a03, +0x3c34c7855019c6ea, 0x3feea9268a5946b7, +0xbc951f58ddaa8090, 0x3feea84590998b93, +0x3c9432e62b64c035, 0x3feea76f15ad2148, +0xbc82e1648e50a17c, 0x3feea6a320dceb71, +0xbc8ce44a6199769f, 0x3feea5e1b976dc09, +0x3c95f30eda98a575, 0x3feea52ae6cdf6f4, +0xbc8c33c53bef4da8, 0x3feea47eb03a5585, +0x3c917ecda8a72159, 0x3feea3dd1d1929fd, +0xbc845378892be9ae, 0x3feea34634ccc320, +0xbc9345f3cee1ae6e, 0x3feea2b9febc8fb7, +0xbc93cedd78565858, 0x3feea23882552225, +0xbc85c33fdf910406, 0x3feea1c1c70833f6, +0x3c5710aa807e1964, 0x3feea155d44ca973, +0x3c81079ab5789604, 0x3feea0f4b19e9538, +0xbc93b3efbf5e2228, 0x3feea09e667f3bcd, +0x3c727df161cd7778, 0x3feea052fa75173e, +0xbc6a12ad8734b982, 0x3feea012750bdabf, +0x3c93f9924a05b767, 0x3fee9fdcddd47645, +0xbc6367efb86da9ee, 0x3fee9fb23c651a2f, +0xbc87557939a8b5ef, 0x3fee9f9298593ae5, +0xbc80dc3d54e08851, 0x3fee9f7df9519484, +0x3c51ed2f56fa9d1a, 0x3fee9f7466f42e87, +0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74, +0xbc88e67a9006c909, 0x3fee9f8286ead08a, +0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174, +0x3c86597566977ac8, 0x3fee9fbd35d7cbfd, +0xbc8619321e55e68a, 0x3fee9feb564267c9, +0x3c92c0b7028a5c3a, 0x3feea024b1ab6e09, +0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f, +0x3c8a30faf49cc78c, 0x3feea0b938ac1cf6, +0xbc7b32dcb94da51d, 0x3feea11473eb0187, +0xbc92dad3519d7b5b, 0x3feea17b0976cfdb, +0x3c94ecfd5467c06b, 0x3feea1ed0130c132, +0x3c87d51410fd15c2, 0x3feea26a62ff86f0, +0x3c65ebe1abd66c55, 0x3feea2f336cf4e62, +0xbc760a3629969871, 0x3feea3878491c491, +0xbc88a1c52fb3cf42, 0x3feea427543e1a12, +0x3c8b18c6e3fdef5d, 0x3feea4d2add106d9, +0xbc9369b6f13b3734, 0x3feea589994cce13, +0x3c90ec1ddcb1390a, 0x3feea64c1eb941f7, +0xbc805e843a19ff1e, 0x3feea71a4623c7ad, +0xbc522cea4f3afa1e, 0x3feea7f4179f5b21, +0xbc94d450d872576e, 0x3feea8d99b4492ed, +0x3c7c88549b958471, 0x3feea9cad931a436, +0x3c90ad675b0e8a00, 0x3feeaac7d98a6699, +0x3c931143962f7877, 0x3feeabd0a478580f, +0x3c8db72fc1f0eab4, 0x3feeace5422aa0db, +0x3c93e9e96f112479, 0x3feeae05bad61778, +0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c, +0xbc8dac42a4a38df0, 0x3feeb06a5e0866d9, +0x3c7bf68359f35f44, 0x3feeb1ae99157736, +0x3c8b99dd98b1ed84, 0x3feeb2fed0282c8a, +0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6, +0xbc7885ad50cbb750, 0x3feeb5c353aa2fe2, +0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5, +0xbc82d5e85f3e0301, 0x3feeb8b82b5f98e5, +0xbc6c23f97c90b959, 0x3feeba44cbc8520f, +0xbc51669428996971, 0x3feebbdd9a7670b3, +0xbc92434322f4f9aa, 0x3feebd829fde4e50, +0x3c71f2b2c1c4c014, 0x3feebf33e47a22a2, +0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba, +0xbc9294f304f166b6, 0x3feec2bb4d53fe0d, +0x3c71affc2b91ce27, 0x3feec49182a3f090, +0xbc8a1e58414c07d3, 0x3feec674194bb8d5, +0x3c6dd235e10a73bb, 0x3feec86319e32323, +0xbc79740b58a20091, 0x3feeca5e8d07f29e, +0xbc87c50422622263, 0x3feecc667b5de565, +0x3c9165830a2b96c2, 0x3feece7aed8eb8bb, +0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33, +0xbc903d5cbe27874b, 0x3feed2c980460ad8, +0xbc91bbd1d3bcbb15, 0x3feed503b23e255d, +0x3c5986178980fce0, 0x3feed74a8af46052, +0x3c90cc319cee31d2, 0x3feed99e1330b358, +0xbc89472975b1f2a5, 0x3feedbfe53c12e59, +0x3c8469846e735ab3, 0x3feede6b5579fdbf, +0x3c7d8157a34b7e7f, 0x3feee0e521356eba, +0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a, +0x3c8c8a4e231ebb7d, 0x3feee5ff3a3c2774, +0x3c8c1a7792cb3387, 0x3feee89f995ad3ad, +0xbc888c8d11a142e5, 0x3feeeb4ce622f2ff, +0xbc907b8f4ad1d9fa, 0x3feeee07298db666, +0x3c889c2ea41433c7, 0x3feef0ce6c9a8952, +0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb, +0xbc7274aedac8ff80, 0x3feef68415b749b1, +0xbc90a40e3da6f640, 0x3feef9728de5593a, +0x3c85c620ce76df06, 0x3feefc6e29f1c52a, +0xbc68d6f438ad9334, 0x3feeff76f2fb5e47, +0xbc8fda52e1b51e41, 0x3fef028cf22749e4, +0xbc91eee26b588a35, 0x3fef05b030a1064a, +0xbc32141a7b3e2cd8, 0x3fef08e0b79a6f1f, +0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2, +0xbc302899507554e5, 0x3fef0f69c3f3a207, +0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09, +0xbc80dda2d4c0010c, 0x3fef16286141b33d, +0x3c736eae30af0cb3, 0x3fef199bdd85529c, +0xbc8a007daadf8d68, 0x3fef1d1cd9fa652c, +0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a, +0x3c836909391181d3, 0x3fef244778fafb22, +0x3c84e08fd10959ac, 0x3fef27f12e57d14b, +0xbc811cd7dbdf9547, 0x3fef2ba88988c933, +0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5, +0xbc7ac28b7bef6621, 0x3fef33405751c4db, +0x3c676b2c6c921968, 0x3fef3720dcef9069, +0xbc7030587207b9e1, 0x3fef3b0f2e6d1675, +0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa, +0xbc8cc734592af7fc, 0x3fef43155b5bab74, +0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c, +0x3c87752a44f587e8, 0x3fef4b532b08c968, +0xbc900dae3875a949, 0x3fef4f87080d89f2, +0x3c85b66fefeef52e, 0x3fef53c8eacaa1d6, +0x3c74a385a63d07a7, 0x3fef5818dcfba487, +0x3c5159d9d908a96e, 0x3fef5c76e862e6d3, +0xbc82919e2040220f, 0x3fef60e316c98398, +0x3c8c254d16117a68, 0x3fef655d71ff6075, +0x3c8e5a50d5c192ac, 0x3fef69e603db3285, +0xbc8d8c329fbd0e03, 0x3fef6e7cd63a8315, +0x3c843a59ac016b4b, 0x3fef7321f301b460, +0xbc8ea6e6fbd5f2a6, 0x3fef77d5641c0658, +0xbc82d52107b43e1f, 0x3fef7c97337b9b5f, +0xbc63e8e3eab2cbb4, 0x3fef81676b197d17, +0xbc892ab93b470dc9, 0x3fef864614f5a129, +0xbc8b7966cd0d2cd9, 0x3fef8b333b16ee12, +0x3c74b604603a88d3, 0x3fef902ee78b3ff6, +0xbc776caa4c2ff1cf, 0x3fef953924676d76, +0x3c83c5ec519d7271, 0x3fef9a51fbc74c83, +0xbc81d5fc525d9940, 0x3fef9f7977cdb740, +0xbc8ff7128fd391f0, 0x3fefa4afa2a490da, +0x3c855cd8aaea3d21, 0x3fefa9f4867cca6e, +0xbc8dae98e223747d, 0x3fefaf482d8e67f1, +0x3c8269947c2bed4a, 0x3fefb4aaa2188510, +0x3c8ec3bc41aa2008, 0x3fefba1bee615a27, +0xbc83b6137e9afe9e, 0x3fefbf9c1cb6412a, +0x3c842b94c3a9eb32, 0x3fefc52b376bba97, +0xbc69fa74878ba7c7, 0x3fefcac948dd7274, +0x3c8a64a931d185ee, 0x3fefd0765b6e4540, +0x3c901f3a75ee0efe, 0x3fefd632798844f8, +0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14, +0xbc516a9ce6ed84fa, 0x3fefe1d802243c89, +0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8, +0xbc699c7db2effc76, 0x3fefedba3692d514, +0x3c5305c14160cc89, 0x3feff3c22b8f71f1, +0x3c64b458677f9840, 0x3feff9d96b2a23d9, +#endif +}, +}; +#endif /* __OBSOLETE_MATH */ diff --git a/newlib/libm/common/fdlibm.h b/newlib/libm/common/fdlibm.h index 4523e8b2a..f6c3740e3 100644 --- a/newlib/libm/common/fdlibm.h +++ b/newlib/libm/common/fdlibm.h @@ -231,6 +231,7 @@ extern float __ieee754_scalbf __P((float,float)); This is valid as long as there are no namespace issues (the extern symbol is reserved whenever the caller is reserved) and there are no observable error handling side effects. */ +# define __ieee754_exp(x) exp(x) # define __ieee754_expf(x) expf(x) # define __ieee754_logf(x) logf(x) # define __ieee754_powf(x,y) powf(x,y) diff --git a/newlib/libm/common/math_config.h b/newlib/libm/common/math_config.h index e4b53fcdd..0fa1de8a2 100644 --- a/newlib/libm/common/math_config.h +++ b/newlib/libm/common/math_config.h @@ -147,8 +147,21 @@ issignalingf_inline (float x) return 2 * (ix ^ 0x00400000) > 2u * 0x7fc00000; } -/* Force the evaluation of a floating-point expression for its side-effect. */ #if __aarch64__ && __GNUC__ +/* Prevent the optimization of a floating-point expression. */ +static inline float +opt_barrier_float (float x) +{ + __asm__ __volatile__ ("" : "+w" (x)); + return x; +} +static inline double +opt_barrier_double (double x) +{ + __asm__ __volatile__ ("" : "+w" (x)); + return x; +} +/* Force the evaluation of a floating-point expression for its side-effect. */ static inline void force_eval_float (float x) { @@ -160,6 +173,18 @@ force_eval_double (double x) __asm__ __volatile__ ("" : "+w" (x)); } #else +static inline float +opt_barrier_float (float x) +{ + volatile float y = x; + return y; +} +static inline double +opt_barrier_double (double x) +{ + volatile double y = x; + return y; +} static inline void force_eval_float (float x) { @@ -199,11 +224,33 @@ eval_as_double (double x) # define unlikely(x) (x) #endif +/* Error handling tail calls for special cases, with sign argument. */ HIDDEN float __math_oflowf (uint32_t); HIDDEN float __math_uflowf (uint32_t); HIDDEN float __math_may_uflowf (uint32_t); HIDDEN float __math_divzerof (uint32_t); +HIDDEN double __math_oflow (uint32_t); +HIDDEN double __math_uflow (uint32_t); +HIDDEN double __math_may_uflow (uint32_t); +HIDDEN double __math_divzero (uint32_t); +/* Error handling using input checking. */ HIDDEN float __math_invalidf (float); +HIDDEN double __math_invalid (double); +/* Error handling using output checking, only for errno setting. */ +HIDDEN double __math_check_oflow (double); +HIDDEN double __math_check_uflow (double); + +static inline double +check_oflow (double x) +{ + return WANT_ERRNO ? __math_check_oflow (x) : x; +} + +static inline double +check_uflow (double x) +{ + return WANT_ERRNO ? __math_check_uflow (x) : x; +} /* Shared between expf, exp2f and powf. */ #define EXP2F_TABLE_BITS 5 @@ -258,4 +305,25 @@ extern const struct powf_log2_data double poly[POWF_LOG2_POLY_ORDER]; } __powf_log2_data HIDDEN; +#define EXP_TABLE_BITS 7 +#define EXP_POLY_ORDER 5 +/* Use polynomial that is optimized for a wider input range. This may be + needed for good precision in non-nearest rounding and !TOINT_INTRINSICS. */ +#define EXP_POLY_WIDE 0 +/* Use close to nearest rounding toint when !TOINT_INTRINSICS. This may be + needed for good precision in non-nearest rouning and !EXP_POLY_WIDE. */ +#define EXP_USE_TOINT_NARROW 0 +#define EXP2_POLY_ORDER 5 +#define EXP2_POLY_WIDE 0 +extern const struct exp_data { + double invln2N; + double shift; + double negln2hiN; + double negln2loN; + double poly[4]; /* Last four coefficients. */ + double exp2_shift; + double exp2_poly[EXP2_POLY_ORDER]; + uint64_t tab[2*(1 << EXP_TABLE_BITS)]; +} __exp_data HIDDEN; + #endif diff --git a/newlib/libm/common/math_err.c b/newlib/libm/common/math_err.c new file mode 100644 index 000000000..e9d2a5c82 --- /dev/null +++ b/newlib/libm/common/math_err.c @@ -0,0 +1,101 @@ +/* Double-precision math error handling. + 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 "math_config.h" + +#if WANT_ERRNO +#include <errno.h> +/* NOINLINE reduces code size and avoids making math functions non-leaf + when the error handling is inlined. */ +NOINLINE static double +with_errno (double y, int e) +{ + errno = e; + return y; +} +#else +#define with_errno(x, e) (x) +#endif + +/* NOINLINE reduces code size. */ +NOINLINE static double +xflow (uint32_t sign, double y) +{ + y = opt_barrier_double (sign ? -y : y) * y; + return with_errno (y, ERANGE); +} + +HIDDEN double +__math_uflow (uint32_t sign) +{ + return xflow (sign, 0x1p-767); +} + +#if WANT_ERRNO_UFLOW +/* Underflows to zero in some non-nearest rounding mode, setting errno + is valid even if the result is non-zero, but in the subnormal range. */ +HIDDEN double +__math_may_uflow (uint32_t sign) +{ + return xflow (sign, 0x1.8p-538); +} +#endif + +HIDDEN double +__math_oflow (uint32_t sign) +{ + return xflow (sign, 0x1p769); +} + +HIDDEN double +__math_divzero (uint32_t sign) +{ + double y = opt_barrier_double (sign ? -1.0 : 1.0) / 0.0; + return with_errno (y, ERANGE); +} + +HIDDEN double +__math_invalid (double x) +{ + double y = (x - x) / (x - x); + return isnan (x) ? y : with_errno (y, EDOM); +} + +/* Check result and set errno if necessary. */ + +HIDDEN double +__math_check_uflow (double y) +{ + return y == 0.0 ? with_errno (y, ERANGE) : y; +} + +HIDDEN double +__math_check_oflow (double y) +{ + return isinf (y) ? with_errno (y, ERANGE) : y; +} diff --git a/newlib/libm/math/e_exp.c b/newlib/libm/math/e_exp.c index e7455eef6..81ea64dfb 100644 --- a/newlib/libm/math/e_exp.c +++ b/newlib/libm/math/e_exp.c @@ -75,6 +75,7 @@ */ #include "fdlibm.h" +#if __OBSOLETE_MATH #ifndef _DOUBLE_IS_32BITS @@ -164,3 +165,4 @@ P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */ } #endif /* defined(_DOUBLE_IS_32BITS) */ +#endif /* __OBSOLETE_MATH */ diff --git a/newlib/libm/math/w_exp.c b/newlib/libm/math/w_exp.c index 9cd607e4e..0c4467712 100644 --- a/newlib/libm/math/w_exp.c +++ b/newlib/libm/math/w_exp.c @@ -53,6 +53,7 @@ PORTABILITY */ #include "fdlibm.h" +#if __OBSOLETE_MATH #include <errno.h> #ifndef _DOUBLE_IS_32BITS @@ -126,3 +127,4 @@ u_threshold= -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */ } #endif /* defined(_DOUBLE_IS_32BITS) */ +#endif /* __OBSOLETE_MATH */ diff --git a/newlib/libm/math/w_exp2.c b/newlib/libm/math/w_exp2.c index 3e41dfc79..c48bdf699 100644 --- a/newlib/libm/math/w_exp2.c +++ b/newlib/libm/math/w_exp2.c @@ -52,6 +52,7 @@ PORTABILITY */ #include "fdlibm.h" +#if __OBSOLETE_MATH #include <errno.h> #include <math.h> @@ -68,3 +69,4 @@ PORTABILITY } #endif /* defined(_DOUBLE_IS_32BITS) */ +#endif /* __OBSOLETE_MATH */ |