diff options
Diffstat (limited to 'sysdeps/ieee754/flt-32')
| -rw-r--r-- | sysdeps/ieee754/flt-32/e_exp2f.c | 170 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/e_exp2f_data.c | 44 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/e_expf.c | 185 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/math_config.h | 114 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/math_errf.c | 76 | ||||
| -rw-r--r-- | sysdeps/ieee754/flt-32/t_exp2f.h | 351 |
6 files changed, 374 insertions, 566 deletions
diff --git a/sysdeps/ieee754/flt-32/e_exp2f.c b/sysdeps/ieee754/flt-32/e_exp2f.c index 567d3ff6d0..72b7d8829f 100644 --- a/sysdeps/ieee754/flt-32/e_exp2f.c +++ b/sysdeps/ieee754/flt-32/e_exp2f.c @@ -1,7 +1,6 @@ -/* Single-precision floating point 2^x. - Copyright (C) 1997-2017 Free Software Foundation, Inc. +/* Single-precision 2^x function. + Copyright (C) 2017 Free Software Foundation, Inc. This file is part of the GNU C Library. - Contributed by Geoffrey Keating <geoffk@ozemail.com.au> The GNU C Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public @@ -17,116 +16,73 @@ License along with the GNU C Library; if not, see <http://www.gnu.org/licenses/>. */ -/* The basic design here is from - Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical - Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft., - 17 (1), March 1991, pp. 26-45. - It has been slightly modified to compute 2^x instead of e^x, and for - single-precision. - */ -#ifndef _GNU_SOURCE -# define _GNU_SOURCE -#endif -#include <stdlib.h> -#include <float.h> -#include <ieee754.h> #include <math.h> -#include <fenv.h> -#include <inttypes.h> -#include <math_private.h> - -#include "t_exp2f.h" - -static const float TWOM100 = 7.88860905e-31; -static const float TWO127 = 1.7014118346e+38; +#include <stdint.h> +#include "math_config.h" + +/* +EXP2F_TABLE_BITS = 5 +EXP2F_POLY_ORDER = 3 + +ULP error: 0.502 (nearest rounding.) +Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.) +Wrong count: 168353 (all nearest rounding wrong results with fma.) +Non-nearest ULP error: 1 (rounded ULP error) +*/ + +#define N (1 << EXP2F_TABLE_BITS) +#define T __exp2f_data.tab +#define C __exp2f_data.poly +#define SHIFT __exp2f_data.shift_scaled + +static inline uint32_t +top12 (float x) +{ + return asuint (x) >> 20; +} float __ieee754_exp2f (float x) { - static const float himark = (float) FLT_MAX_EXP; - static const float lomark = (float) (FLT_MIN_EXP - FLT_MANT_DIG - 1); - - /* Check for usual case. */ - if (isless (x, himark) && isgreaterequal (x, lomark)) + uint32_t abstop; + uint64_t ki, t; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, xd, z, r, r2, y, s; + + xd = (double_t) x; + abstop = top12 (x) & 0x7ff; + if (__glibc_unlikely (abstop >= top12 (128.0f))) { - static const float THREEp14 = 49152.0; - int tval, unsafe; - float rx, x22, result; - union ieee754_float ex2_u, scale_u; - - if (fabsf (x) < FLT_EPSILON / 4.0f) - return 1.0f + x; - - { - SET_RESTORE_ROUND_NOEXF (FE_TONEAREST); - - /* 1. Argument reduction. - Choose integers ex, -128 <= t < 128, and some real - -1/512 <= x1 <= 1/512 so that - x = ex + t/512 + x1. - - First, calculate rx = ex + t/256. */ - rx = x + THREEp14; - rx -= THREEp14; - x -= rx; /* Compute x=x1. */ - /* Compute tval = (ex*256 + t)+128. - Now, t = (tval mod 256)-128 and ex=tval/256 [that's mod, NOT %; - and /-round-to-nearest not the usual c integer /]. */ - tval = (int) (rx * 256.0f + 128.0f); - - /* 2. Adjust for accurate table entry. - Find e so that - x = ex + t/256 + e + x2 - where -7e-4 < e < 7e-4, and - (float)(2^(t/256+e)) - is accurate to one part in 2^-64. */ - - /* 'tval & 255' is the same as 'tval%256' except that it's always - positive. - Compute x = x2. */ - x -= __exp2f_deltatable[tval & 255]; - - /* 3. Compute ex2 = 2^(t/255+e+ex). */ - ex2_u.f = __exp2f_atable[tval & 255]; - tval >>= 8; - /* x2 is an integer multiple of 2^-30; avoid intermediate - underflow from the calculation of x22 * x. */ - unsafe = abs(tval) >= -FLT_MIN_EXP - 32; - ex2_u.ieee.exponent += tval >> unsafe; - scale_u.f = 1.0; - scale_u.ieee.exponent += tval - (tval >> unsafe); - - /* 4. Approximate 2^x2 - 1, using a second-degree polynomial, - with maximum error in [-2^-9 - 2^-14, 2^-9 + 2^-14] - less than 1.3e-10. */ - - x22 = (.24022656679f * x + .69314736128f) * ex2_u.f; - } - - /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex). */ - result = x22 * x + ex2_u.f; - - if (!unsafe) - return result; - else - { - result *= scale_u.f; - math_check_force_underflow_nonneg (result); - return result; - } - } - /* Exceptional cases: */ - else if (isless (x, himark)) - { - if (isinf (x)) - /* e^-inf == 0, with no error. */ - return 0; - else - /* Underflow */ - return TWOM100 * TWOM100; + /* |x| >= 128 or x is nan. */ + if (asuint (x) == asuint (-INFINITY)) + return 0.0f; + if (abstop >= top12 (INFINITY)) + return x + x; + if (x > 0.0f) + return __math_oflowf (0); + if (x <= -150.0f) + return __math_uflowf (0); +#if WANT_ERRNO_UFLOW + if (x < -149.0f) + return __math_may_uflowf (0); +#endif } - else - /* Return x, if x is a NaN or Inf; or overflow, otherwise. */ - return TWO127*x; + + /* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */ + kd = math_narrow_eval ((double) (xd + SHIFT)); /* Needs to be double. */ + ki = asuint64 (kd); + kd -= SHIFT; /* k/N for int k. */ + r = xd - kd; + + /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ + t = T[ki % N]; + t += ki << (52 - EXP2F_TABLE_BITS); + s = asdouble (t); + z = C[0] * r + C[1]; + r2 = r * r; + y = C[2] * r + 1; + y = z * r2 + y; + y = y * s; + return (float) y; } strong_alias (__ieee754_exp2f, __exp2f_finite) diff --git a/sysdeps/ieee754/flt-32/e_exp2f_data.c b/sysdeps/ieee754/flt-32/e_exp2f_data.c new file mode 100644 index 0000000000..390dcae333 --- /dev/null +++ b/sysdeps/ieee754/flt-32/e_exp2f_data.c @@ -0,0 +1,44 @@ +/* Shared data between expf, exp2f and powf. + Copyright (C) 2017 Free Software Foundation, Inc. + This file is part of the GNU C Library. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, see + <http://www.gnu.org/licenses/>. */ + +#include "math_config.h" + +#define N (1 << EXP2F_TABLE_BITS) + +const struct exp2f_data __exp2f_data = { + /* tab[i] = uint(2^(i/N)) - (i << 52-BITS) + used for computing 2^(k/N) for an int |k| < 150 N as + double(tab[k%N] + (k << 52-BITS)) */ + .tab = { +0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51, +0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1, +0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d, +0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585, +0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13, +0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d, +0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069, +0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540, + }, + .shift_scaled = 0x1.8p+52 / N, + .poly = { 0x1.c6af84b912394p-5, 0x1.ebfce50fac4f3p-3, 0x1.62e42ff0c52d6p-1 }, + .shift = 0x1.8p+52, + .invln2_scaled = 0x1.71547652b82fep+0 * N, + .poly_scaled = { +0x1.c6af84b912394p-5/N/N/N, 0x1.ebfce50fac4f3p-3/N/N, 0x1.62e42ff0c52d6p-1/N + }, +}; diff --git a/sysdeps/ieee754/flt-32/e_expf.c b/sysdeps/ieee754/flt-32/e_expf.c index 782072f213..12239e1862 100644 --- a/sysdeps/ieee754/flt-32/e_expf.c +++ b/sysdeps/ieee754/flt-32/e_expf.c @@ -1,7 +1,6 @@ -/* Single-precision floating point e^x. - Copyright (C) 1997-2017 Free Software Foundation, Inc. +/* Single-precision e^x function. + Copyright (C) 2017 Free Software Foundation, Inc. This file is part of the GNU C Library. - Contributed by Geoffrey Keating <geoffk@ozemail.com.au> The GNU C Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public @@ -17,117 +16,87 @@ License along with the GNU C Library; if not, see <http://www.gnu.org/licenses/>. */ -/* How this works: - - The input value, x, is written as - - x = n * ln(2) + t/512 + delta[t] + x; - - where: - - n is an integer, 127 >= n >= -150; - - t is an integer, 177 >= t >= -177 - - delta is based on a table entry, delta[t] < 2^-28 - - x is whatever is left, |x| < 2^-10 - - Then e^x is approximated as - - e^x = 2^n ( e^(t/512 + delta[t]) - + ( e^(t/512 + delta[t]) - * ( p(x + delta[t] + n * ln(2)) - delta ) ) ) - - where - - p(x) is a polynomial approximating e(x)-1; - - e^(t/512 + delta[t]) is obtained from a table. - - The table used is the same one as for the double precision version; - since we have the table, we might as well use it. - - It turns out to be faster to do calculations in double precision than - to perform an 'accurate table method' expf, because of the range reduction - overhead (compare exp2f). - */ -#include <float.h> -#include <ieee754.h> #include <math.h> -#include <fenv.h> -#include <inttypes.h> -#include <math_private.h> - -extern const float __exp_deltatable[178]; -extern const double __exp_atable[355] /* __attribute__((mode(DF))) */; - -static const float TWOM100 = 7.88860905e-31; -static const float TWO127 = 1.7014118346e+38; +#include <stdint.h> +#include "math_config.h" + +/* +EXP2F_TABLE_BITS = 5 +EXP2F_POLY_ORDER = 3 + +ULP error: 0.502 (nearest rounding.) +Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.) +Wrong count: 170635 (all nearest rounding wrong results with fma.) +Non-nearest ULP error: 1 (rounded ULP error) +*/ + +#define N (1 << EXP2F_TABLE_BITS) +#define InvLn2N __exp2f_data.invln2_scaled +#define T __exp2f_data.tab +#define C __exp2f_data.poly_scaled + +static inline uint32_t +top12 (float x) +{ + return asuint (x) >> 20; +} float __ieee754_expf (float x) { - static const float himark = 88.72283935546875; - static const float lomark = -103.972084045410; - /* Check for usual case. */ - if (isless (x, himark) && isgreater (x, lomark)) + uint32_t abstop; + uint64_t ki, t; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, xd, z, r, r2, y, s; + + xd = (double_t) x; + abstop = top12 (x) & 0x7ff; + if (__glibc_unlikely (abstop >= top12 (88.0f))) { - static const float THREEp42 = 13194139533312.0; - static const float THREEp22 = 12582912.0; - /* 1/ln(2). */ -#undef M_1_LN2 - static const float M_1_LN2 = 1.44269502163f; - /* ln(2) */ -#undef M_LN2 - static const double M_LN2 = .6931471805599452862; - - int tval; - double x22, t, result, dx; - float n, delta; - union ieee754_double ex2_u; - - { - SET_RESTORE_ROUND_NOEXF (FE_TONEAREST); - - /* Calculate n. */ - n = x * M_1_LN2 + THREEp22; - n -= THREEp22; - dx = x - n*M_LN2; - - /* Calculate t/512. */ - t = dx + THREEp42; - t -= THREEp42; - dx -= t; - - /* Compute tval = t. */ - tval = (int) (t * 512.0); - - if (t >= 0) - delta = - __exp_deltatable[tval]; - else - delta = __exp_deltatable[-tval]; - - /* Compute ex2 = 2^n e^(t/512+delta[t]). */ - ex2_u.d = __exp_atable[tval+177]; - ex2_u.ieee.exponent += (int) n; - - /* Approximate e^(dx+delta) - 1, using a second-degree polynomial, - with maximum error in [-2^-10-2^-28,2^-10+2^-28] - less than 5e-11. */ - x22 = (0.5000000496709180453 * dx + 1.0000001192102037084) * dx + delta; - } - - /* Return result. */ - result = x22 * ex2_u.d + ex2_u.d; - return (float) result; + /* |x| >= 88 or x is nan. */ + if (asuint (x) == asuint (-INFINITY)) + return 0.0f; + if (abstop >= top12 (INFINITY)) + return x + x; + if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */ + return __math_oflowf (0); + if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */ + return __math_uflowf (0); +#if WANT_ERRNO_UFLOW + if (x < -0x1.9d1d9ep6f) /* x < log(0x1p-149) ~= -103.28 */ + return __math_may_uflowf (0); +#endif } - /* Exceptional cases: */ - else if (isless (x, himark)) - { - if (isinf (x)) - /* e^-inf == 0, with no error. */ - return 0; - else - /* Underflow */ - return TWOM100 * TWOM100; - } - else - /* Return x, if x is a NaN or Inf; or overflow, otherwise. */ - return TWO127*x; + + /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */ + z = InvLn2N * xd; + + /* Round and convert z to int, the result is in [-150*N, 128*N] and + ideally ties-to-even rule is used, otherwise the magnitude of r + can be bigger which gives larger approximation error. */ +#if TOINT_INTRINSICS + kd = roundtoint (z); + ki = converttoint (z); +#elif TOINT_RINT + kd = rint (z); + ki = (long) kd; +#elif TOINT_SHIFT +# define SHIFT __exp2f_data.shift + kd = math_narrow_eval ((double) (z + SHIFT)); /* Needs to be double. */ + ki = asuint64 (kd); + kd -= SHIFT; +#endif + r = z - kd; + + /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */ + t = T[ki % N]; + t += ki << (52 - EXP2F_TABLE_BITS); + s = asdouble (t); + z = C[0] * r + C[1]; + r2 = r * r; + y = C[2] * r + 1; + y = z * r2 + y; + y = y * s; + return (float) y; } strong_alias (__ieee754_expf, __expf_finite) diff --git a/sysdeps/ieee754/flt-32/math_config.h b/sysdeps/ieee754/flt-32/math_config.h new file mode 100644 index 0000000000..31f0470612 --- /dev/null +++ b/sysdeps/ieee754/flt-32/math_config.h @@ -0,0 +1,114 @@ +/* Configuration for math routines. + Copyright (C) 2017 Free Software Foundation, Inc. + This file is part of the GNU C Library. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, see + <http://www.gnu.org/licenses/>. */ + +#ifndef _MATH_CONFIG_H +#define _MATH_CONFIG_H + +#include <math.h> +#include <math_private.h> +#include <stdint.h> + +#ifndef WANT_ROUNDING +/* Correct special case results in non-nearest rounding modes. */ +# define WANT_ROUNDING 1 +#endif +#ifndef WANT_ERRNO +/* Set errno according to ISO C with (math_errhandling & MATH_ERRNO) != 0. */ +# define WANT_ERRNO 1 +#endif +#ifndef WANT_ERRNO_UFLOW +/* Set errno to ERANGE if result underflows to 0 (in all rounding modes). */ +# define WANT_ERRNO_UFLOW (WANT_ROUNDING && WANT_ERRNO) +#endif + +#ifndef TOINT_INTRINSICS +# define TOINT_INTRINSICS 0 +#endif +#ifndef TOINT_RINT +# define TOINT_RINT 0 +#endif +#ifndef TOINT_SHIFT +# define TOINT_SHIFT 1 +#endif + +static inline uint32_t +asuint (float f) +{ + union + { + float f; + uint32_t i; + } u = {f}; + return u.i; +} + +static inline float +asfloat (uint32_t i) +{ + union + { + uint32_t i; + float f; + } u = {i}; + return u.f; +} + +static inline uint64_t +asuint64 (double f) +{ + union + { + double f; + uint64_t i; + } u = {f}; + return u.i; +} + +static inline double +asdouble (uint64_t i) +{ + union + { + uint64_t i; + double f; + } u = {i}; + return u.f; +} + +#define NOINLINE __attribute__ ((noinline)) + +attribute_hidden float __math_oflowf (unsigned long); +attribute_hidden float __math_uflowf (unsigned long); +attribute_hidden float __math_may_uflowf (unsigned long); +attribute_hidden float __math_divzerof (unsigned long); +attribute_hidden float __math_invalidf (float); + +/* Shared between expf, exp2f and powf. */ +#define EXP2F_TABLE_BITS 5 +#define EXP2F_POLY_ORDER 3 +extern const struct exp2f_data +{ + uint64_t tab[1 << EXP2F_TABLE_BITS]; + double shift_scaled; + double poly[EXP2F_POLY_ORDER]; + double shift; + double invln2_scaled; + double poly_scaled[EXP2F_POLY_ORDER]; +} __exp2f_data attribute_hidden; + +#endif diff --git a/sysdeps/ieee754/flt-32/math_errf.c b/sysdeps/ieee754/flt-32/math_errf.c new file mode 100644 index 0000000000..ab546e24cf --- /dev/null +++ b/sysdeps/ieee754/flt-32/math_errf.c @@ -0,0 +1,76 @@ +/* Single-precision math error handling. + Copyright (C) 2017 Free Software Foundation, Inc. + This file is part of the GNU C Library. + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, see + <http://www.gnu.org/licenses/>. */ + +#include "math_config.h" + +#if WANT_ERRNO +# include <errno.h> +/* NOINLINE reduces code size. */ +NOINLINE static float +with_errnof (float y, int e) +{ + errno = e; + return y; +} +#else +# define with_errnof(x, e) (x) +#endif + +/* NOINLINE prevents fenv semantics breaking optimizations. */ +NOINLINE static float +xflowf (unsigned long sign, float y) +{ + y = (sign ? -y : y) * y; + return with_errnof (y, ERANGE); +} + +attribute_hidden float +__math_uflowf (unsigned long sign) +{ + return xflowf (sign, 0x1p-95f); +} + +#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. */ +attribute_hidden float +__math_may_uflowf (unsigned long sign) +{ + return xflowf (sign, 0x1.4p-75f); +} +#endif + +attribute_hidden float +__math_oflowf (unsigned long sign) +{ + return xflowf (sign, 0x1p97f); +} + +attribute_hidden float +__math_divzerof (unsigned long sign) +{ + float y = 0; + return with_errnof ((sign ? -1 : 1) / y, ERANGE); +} + +attribute_hidden float +__math_invalidf (float x) +{ + float y = (x - x) / (x - x); + return isnan (x) ? y : with_errnof (y, EDOM); +} diff --git a/sysdeps/ieee754/flt-32/t_exp2f.h b/sysdeps/ieee754/flt-32/t_exp2f.h deleted file mode 100644 index aecabcc372..0000000000 --- a/sysdeps/ieee754/flt-32/t_exp2f.h +++ /dev/null @@ -1,351 +0,0 @@ -/* Accurate tables for exp2f(). - Copyright (C) 1998-2017 Free Software Foundation, Inc. - This file is part of the GNU C Library. - Contributed by Geoffrey Keating <geoffk@ozemail.com.au> - - The GNU C Library is free software; you can redistribute it and/or - modify it under the terms of the GNU Lesser General Public - License as published by the Free Software Foundation; either - version 2.1 of the License, or (at your option) any later version. - - The GNU C Library is distributed in the hope that it will be useful, - but WITHOUT ANY WARRANTY; without even the implied warranty of - MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU - Lesser General Public License for more details. - - You should have received a copy of the GNU Lesser General Public - License along with the GNU C Library; if not, see - <http://www.gnu.org/licenses/>. */ - -/* This table has the property that, for all integers -128 <= i <= 127, - exp(i/256.0 + __exp2f_deltatable[i-128]) == __exp2f_atable[i+128] + r - for some -2^-35 < r < 2^-35 (abs(r) < 2^-36 if i <= 0); and that - __exp2f_deltatable[i+128] == t * 2^-30 - for integer t so that abs(t) <= 43447 * 2^0. */ - -#define W30 (9.31322575e-10) -static const float __exp2f_deltatable[256] = { - -810*W30, 283*W30, -1514*W30, 1304*W30, - -1148*W30, -98*W30, -744*W30, -156*W30, - -419*W30, -155*W30, 474*W30, 167*W30, - -1984*W30, -826*W30, 692*W30, 781*W30, - -578*W30, -411*W30, -129*W30, -1500*W30, - 654*W30, -141*W30, -816*W30, -53*W30, - 148*W30, 493*W30, -2214*W30, 760*W30, - 260*W30, 750*W30, -1300*W30, 1424*W30, - -1445*W30, -339*W30, -680*W30, -349*W30, - -922*W30, 531*W30, 193*W30, -2892*W30, - 290*W30, -2145*W30, -276*W30, 485*W30, - -695*W30, 215*W30, -7093*W30, 412*W30, - -4596*W30, 367*W30, 592*W30, -615*W30, - -97*W30, -1066*W30, 972*W30, -226*W30, - -625*W30, -374*W30, -5647*W30, -180*W30, - 20349*W30, -447*W30, 111*W30, -4164*W30, - -87*W30, -21*W30, -251*W30, 66*W30, - -517*W30, 2093*W30, -263*W30, 182*W30, - -601*W30, 475*W30, -483*W30, -1251*W30, - -373*W30, 1471*W30, -92*W30, -215*W30, - -97*W30, -190*W30, 0*W30, -290*W30, - -2647*W30, 1940*W30, -582*W30, 28*W30, - 833*W30, 1493*W30, 34*W30, 321*W30, - 3327*W30, -35*W30, 177*W30, -135*W30, - -796*W30, -428*W30, 129*W30, 9332*W30, - -12*W30, -69*W30, -1743*W30, 6508*W30, - -60*W30, 359*W30, 43447*W30, 15*W30, - -23*W30, -305*W30, -375*W30, -652*W30, - 667*W30, 269*W30, -1575*W30, 185*W30, - -329*W30, 200*W30, 6002*W30, 163*W30, - -647*W30, 19*W30, -603*W30, -755*W30, - 742*W30, -438*W30, 3587*W30, 2560*W30, - 0*W30, -520*W30, -241*W30, -299*W30, - -1270*W30, -991*W30, -1138*W30, 255*W30, - -1192*W30, 1722*W30, 1023*W30, 3700*W30, - -1388*W30, -1551*W30, -2549*W30, 27*W30, - 282*W30, 673*W30, 113*W30, 1561*W30, - 72*W30, 873*W30, 87*W30, -395*W30, - -433*W30, 629*W30, 3440*W30, -284*W30, - -592*W30, -103*W30, -46*W30, -3844*W30, - 1712*W30, 303*W30, 1555*W30, -631*W30, - -1400*W30, -961*W30, -854*W30, -276*W30, - 407*W30, 833*W30, -345*W30, -1501*W30, - 121*W30, -1581*W30, 400*W30, 150*W30, - 1224*W30, -139*W30, -563*W30, 879*W30, - 933*W30, 2939*W30, 788*W30, 211*W30, - 530*W30, -192*W30, 706*W30, -13347*W30, - 1065*W30, 3*W30, 111*W30, -208*W30, - -360*W30, -532*W30, -291*W30, 483*W30, - 987*W30, -33*W30, -1373*W30, -166*W30, - -1174*W30, -3955*W30, 1601*W30, -280*W30, - 1405*W30, 600*W30, -1659*W30, -23*W30, - 390*W30, 449*W30, 570*W30, -13143*W30, - -9*W30, -1646*W30, 1201*W30, 294*W30, - 2181*W30, -1173*W30, 1388*W30, -4504*W30, - 190*W30, -2304*W30, 211*W30, 239*W30, - 48*W30, -817*W30, 1018*W30, 1828*W30, - -663*W30, 1408*W30, 408*W30, -36*W30, - 1295*W30, -230*W30, 1341*W30, 9*W30, - 40*W30, 705*W30, 186*W30, 376*W30, - 557*W30, 5866*W30, 363*W30, -1558*W30, - 718*W30, 669*W30, 1369*W30, -2972*W30, - -468*W30, -121*W30, -219*W30, 667*W30, - 29954*W30, 366*W30, 48*W30, -203*W30 -}; - -static const float __exp2f_atable[256] /* __attribute__((mode(SF))) */ = { - 0.707106411447, /* 0x0.b504ecfff */ - 0.709024071690, /* 0x0.b58299fff */ - 0.710945606239, /* 0x0.b60088000 */ - 0.712874472142, /* 0x0.b67ef1000 */ - 0.714806139464, /* 0x0.b6fd88fff */ - 0.716744661340, /* 0x0.b77c94000 */ - 0.718687653549, /* 0x0.b7fbea000 */ - 0.720636486992, /* 0x0.b87ba1fff */ - 0.722590208040, /* 0x0.b8fbabfff */ - 0.724549472323, /* 0x0.b97c12fff */ - 0.726514220228, /* 0x0.b9fcd5fff */ - 0.728483855735, /* 0x0.ba7deb000 */ - 0.730457961549, /* 0x0.baff4afff */ - 0.732438981522, /* 0x0.bb811efff */ - 0.734425544748, /* 0x0.bc0350000 */ - 0.736416816713, /* 0x0.bc85d0000 */ - 0.738412797450, /* 0x0.bd089efff */ - 0.740414917465, /* 0x0.bd8bd4fff */ - 0.742422521111, /* 0x0.be0f66fff */ - 0.744434773914, /* 0x0.be9346fff */ - 0.746454179287, /* 0x0.bf179f000 */ - 0.748477637755, /* 0x0.bf9c3afff */ - 0.750506639473, /* 0x0.c02133fff */ - 0.752541840064, /* 0x0.c0a694fff */ - 0.754582285889, /* 0x0.c12c4e000 */ - 0.756628334525, /* 0x0.c1b265000 */ - 0.758678436269, /* 0x0.c238bffff */ - 0.760736882681, /* 0x0.c2bfa6fff */ - 0.762799203401, /* 0x0.c346cf000 */ - 0.764867603790, /* 0x0.c3ce5d000 */ - 0.766940355298, /* 0x0.c45633fff */ - 0.769021093841, /* 0x0.c4de90fff */ - 0.771104693409, /* 0x0.c5671dfff */ - 0.773195922364, /* 0x0.c5f02afff */ - 0.775292098512, /* 0x0.c6798afff */ - 0.777394294745, /* 0x0.c70350000 */ - 0.779501736166, /* 0x0.c78d6d000 */ - 0.781615912910, /* 0x0.c817fafff */ - 0.783734917628, /* 0x0.c8a2d9fff */ - 0.785858273516, /* 0x0.c92e02000 */ - 0.787990570071, /* 0x0.c9b9c0000 */ - 0.790125787245, /* 0x0.ca45aefff */ - 0.792268991467, /* 0x0.cad223fff */ - 0.794417440881, /* 0x0.cb5ef0fff */ - 0.796570718287, /* 0x0.cbec0efff */ - 0.798730909811, /* 0x0.cc79a0fff */ - 0.800892710672, /* 0x0.cd074dfff */ - 0.803068041795, /* 0x0.cd95ddfff */ - 0.805242776881, /* 0x0.ce2464000 */ - 0.807428598393, /* 0x0.ceb3a3fff */ - 0.809617877002, /* 0x0.cf431dfff */ - 0.811812341211, /* 0x0.cfd2eefff */ - 0.814013659956, /* 0x0.d06333000 */ - 0.816220164311, /* 0x0.d0f3ce000 */ - 0.818434238424, /* 0x0.d184e7fff */ - 0.820652604094, /* 0x0.d21649fff */ - 0.822877407074, /* 0x0.d2a818000 */ - 0.825108587751, /* 0x0.d33a51000 */ - 0.827342867839, /* 0x0.d3ccbdfff */ - 0.829588949684, /* 0x0.d45ff1000 */ - 0.831849217401, /* 0x0.d4f411fff */ - 0.834093391880, /* 0x0.d58724fff */ - 0.836355149750, /* 0x0.d61b5f000 */ - 0.838620424257, /* 0x0.d6afd3fff */ - 0.840896368027, /* 0x0.d744fc000 */ - 0.843176305293, /* 0x0.d7da66fff */ - 0.845462262643, /* 0x0.d87037000 */ - 0.847754716864, /* 0x0.d90673fff */ - 0.850052893157, /* 0x0.d99d10fff */ - 0.852359056469, /* 0x0.da3433fff */ - 0.854668736446, /* 0x0.dacb91fff */ - 0.856986224651, /* 0x0.db6373000 */ - 0.859309315673, /* 0x0.dbfbb1fff */ - 0.861639738080, /* 0x0.dc946bfff */ - 0.863975346095, /* 0x0.dd2d7d000 */ - 0.866317391394, /* 0x0.ddc6f9fff */ - 0.868666708472, /* 0x0.de60f1000 */ - 0.871022939695, /* 0x0.defb5c000 */ - 0.873383641229, /* 0x0.df9611fff */ - 0.875751554968, /* 0x0.e03141000 */ - 0.878126025200, /* 0x0.e0ccde000 */ - 0.880506813521, /* 0x0.e168e4fff */ - 0.882894217966, /* 0x0.e2055afff */ - 0.885287821299, /* 0x0.e2a239000 */ - 0.887686729423, /* 0x0.e33f6ffff */ - 0.890096127973, /* 0x0.e3dd56fff */ - 0.892507970338, /* 0x0.e47b67000 */ - 0.894928157336, /* 0x0.e51a03000 */ - 0.897355020043, /* 0x0.e5b90efff */ - 0.899788379682, /* 0x0.e65888000 */ - 0.902227103705, /* 0x0.e6f85afff */ - 0.904673457151, /* 0x0.e798ae000 */ - 0.907128036008, /* 0x0.e8398afff */ - 0.909585535528, /* 0x0.e8da99000 */ - 0.912051796915, /* 0x0.e97c3a000 */ - 0.914524436003, /* 0x0.ea1e46000 */ - 0.917003571999, /* 0x0.eac0bf000 */ - 0.919490039339, /* 0x0.eb63b2fff */ - 0.921983361257, /* 0x0.ec071a000 */ - 0.924488604054, /* 0x0.ecab48fff */ - 0.926989555360, /* 0x0.ed4f30000 */ - 0.929502844812, /* 0x0.edf3e6000 */ - 0.932021975503, /* 0x0.ee98fdfff */ - 0.934553921208, /* 0x0.ef3eecfff */ - 0.937083780759, /* 0x0.efe4b8fff */ - 0.939624726786, /* 0x0.f08b3f000 */ - 0.942198514924, /* 0x0.f133ebfff */ - 0.944726586343, /* 0x0.f1d99a000 */ - 0.947287976728, /* 0x0.f28176fff */ - 0.949856162070, /* 0x0.f329c5fff */ - 0.952431440345, /* 0x0.f3d28bfff */ - 0.955013573175, /* 0x0.f47bc5000 */ - 0.957603693021, /* 0x0.f52584000 */ - 0.960199773321, /* 0x0.f5cfa7000 */ - 0.962801992906, /* 0x0.f67a31000 */ - 0.965413510788, /* 0x0.f72556fff */ - 0.968030691152, /* 0x0.f7d0dc000 */ - 0.970655620084, /* 0x0.f87ce2fff */ - 0.973290979849, /* 0x0.f92998fff */ - 0.975926160805, /* 0x0.f9d64bfff */ - 0.978571653370, /* 0x0.fa83ac000 */ - 0.981225252139, /* 0x0.fb3193fff */ - 0.983885228626, /* 0x0.fbdfe6fff */ - 0.986552715296, /* 0x0.fc8eb7fff */ - 0.989228487027, /* 0x0.fd3e14000 */ - 0.991909801964, /* 0x0.fdedcd000 */ - 0.994601726545, /* 0x0.fe9e38000 */ - 0.997297704209, /* 0x0.ff4ee6fff */ - 1.000000000000, /* 0x1.000000000 */ - 1.002710938457, /* 0x1.00b1aa000 */ - 1.005429744692, /* 0x1.0163d7ffe */ - 1.008155703526, /* 0x1.02167dffe */ - 1.010888457284, /* 0x1.02c995fff */ - 1.013629436498, /* 0x1.037d38000 */ - 1.016377568250, /* 0x1.043152000 */ - 1.019134163841, /* 0x1.04e5f9ffe */ - 1.021896362316, /* 0x1.059b00000 */ - 1.024668931945, /* 0x1.0650b3ffe */ - 1.027446627635, /* 0x1.0706be001 */ - 1.030234098408, /* 0x1.07bd6bffe */ - 1.033023953416, /* 0x1.087441ffe */ - 1.035824656494, /* 0x1.092bce000 */ - 1.038632392900, /* 0x1.09e3d0001 */ - 1.041450142840, /* 0x1.0a9c79ffe */ - 1.044273972530, /* 0x1.0b558a001 */ - 1.047105550795, /* 0x1.0c0f1c001 */ - 1.049944162390, /* 0x1.0cc924001 */ - 1.052791833895, /* 0x1.0d83c4001 */ - 1.055645227426, /* 0x1.0e3ec3fff */ - 1.058507919326, /* 0x1.0efa60001 */ - 1.061377286898, /* 0x1.0fb66bfff */ - 1.064254641510, /* 0x1.1072fdffe */ - 1.067140102389, /* 0x1.113018000 */ - 1.070034146304, /* 0x1.11edc1fff */ - 1.072937250162, /* 0x1.12ac04001 */ - 1.075843691823, /* 0x1.136a7dfff */ - 1.078760385496, /* 0x1.1429a3ffe */ - 1.081685543070, /* 0x1.14e958000 */ - 1.084618330005, /* 0x1.15a98c000 */ - 1.087556362176, /* 0x1.166a18001 */ - 1.090508937863, /* 0x1.172b98001 */ - 1.093464612954, /* 0x1.17ed4bfff */ - 1.096430182434, /* 0x1.18afa5ffe */ - 1.099401354802, /* 0x1.19725e000 */ - 1.102381587017, /* 0x1.1a35adfff */ - 1.105370759965, /* 0x1.1af994000 */ - 1.108367800686, /* 0x1.1bbdfdffe */ - 1.111373305331, /* 0x1.1c82f6000 */ - 1.114387035385, /* 0x1.1d4878001 */ - 1.117408752440, /* 0x1.1e0e7ffff */ - 1.120437502874, /* 0x1.1ed4fe000 */ - 1.123474478729, /* 0x1.1f9c06000 */ - 1.126521706601, /* 0x1.2063ba001 */ - 1.129574775716, /* 0x1.212bd0001 */ - 1.132638812065, /* 0x1.21f49e000 */ - 1.135709524130, /* 0x1.22bddbffe */ - 1.138789534565, /* 0x1.2387b5fff */ - 1.141876101508, /* 0x1.2451fe000 */ - 1.144971728301, /* 0x1.251cddffe */ - 1.148077130296, /* 0x1.25e861ffe */ - 1.151189923305, /* 0x1.26b462001 */ - 1.154312610610, /* 0x1.278107ffe */ - 1.157440662410, /* 0x1.284e08001 */ - 1.160578370109, /* 0x1.291baa001 */ - 1.163725256932, /* 0x1.29e9e6000 */ - 1.166879892324, /* 0x1.2ab8a3ffe */ - 1.170044302935, /* 0x1.2b8805fff */ - 1.173205971694, /* 0x1.2c5739ffe */ - 1.176397800428, /* 0x1.2d2867ffe */ - 1.179586529747, /* 0x1.2df962001 */ - 1.182784795737, /* 0x1.2ecafbffe */ - 1.185991406414, /* 0x1.2f9d21ffe */ - 1.189206838636, /* 0x1.306fdc001 */ - 1.192430973067, /* 0x1.314328000 */ - 1.195664167430, /* 0x1.32170c001 */ - 1.198906540890, /* 0x1.32eb8a001 */ - 1.202157497408, /* 0x1.33c098000 */ - 1.205416083326, /* 0x1.349625fff */ - 1.208683252332, /* 0x1.356c43fff */ - 1.211961269402, /* 0x1.364318001 */ - 1.215246438983, /* 0x1.371a64000 */ - 1.218539118740, /* 0x1.37f22dffe */ - 1.221847295770, /* 0x1.38cafc000 */ - 1.225158572187, /* 0x1.39a3fdfff */ - 1.228481650325, /* 0x1.3a7dc5ffe */ - 1.231811761846, /* 0x1.3b5803fff */ - 1.235149741144, /* 0x1.3c32c5ffe */ - 1.238499879811, /* 0x1.3d0e53ffe */ - 1.241858124726, /* 0x1.3dea69fff */ - 1.245225191102, /* 0x1.3ec713fff */ - 1.248601436624, /* 0x1.3fa458000 */ - 1.251975655584, /* 0x1.40817a001 */ - 1.255380749731, /* 0x1.4160a2001 */ - 1.258783102010, /* 0x1.423f9bffe */ - 1.262198328973, /* 0x1.431f6e000 */ - 1.265619754780, /* 0x1.43ffa7fff */ - 1.269052743928, /* 0x1.44e0a4001 */ - 1.272490739830, /* 0x1.45c1f4000 */ - 1.275942921659, /* 0x1.46a432001 */ - 1.279397487615, /* 0x1.478697ffe */ - 1.282870173427, /* 0x1.486a2dffe */ - 1.286346316319, /* 0x1.494dfdffe */ - 1.289836049094, /* 0x1.4a32b2001 */ - 1.293333172770, /* 0x1.4b17e1ffe */ - 1.296839594835, /* 0x1.4bfdadfff */ - 1.300354957560, /* 0x1.4ce40fffe */ - 1.303882122055, /* 0x1.4dcb38001 */ - 1.307417988757, /* 0x1.4eb2f1ffe */ - 1.310960650439, /* 0x1.4f9b1dfff */ - 1.314516782746, /* 0x1.50842bfff */ - 1.318079948424, /* 0x1.516daffff */ - 1.321653246888, /* 0x1.5257de000 */ - 1.325237751030, /* 0x1.5342c8001 */ - 1.328829526907, /* 0x1.542e2c000 */ - 1.332433700535, /* 0x1.551a5fffe */ - 1.336045145966, /* 0x1.56070dffe */ - 1.339667558645, /* 0x1.56f473ffe */ - 1.343300342533, /* 0x1.57e287ffe */ - 1.346941947961, /* 0x1.58d130001 */ - 1.350594043714, /* 0x1.59c087ffe */ - 1.354256033883, /* 0x1.5ab085fff */ - 1.357932448365, /* 0x1.5ba175ffe */ - 1.361609339707, /* 0x1.5c926dfff */ - 1.365299344044, /* 0x1.5d8441ffe */ - 1.369003057507, /* 0x1.5e76fc001 */ - 1.372714757920, /* 0x1.5f6a3c000 */ - 1.376437187179, /* 0x1.605e2fffe */ - 1.380165219333, /* 0x1.615282001 */ - 1.383909463864, /* 0x1.6247e3ffe */ - 1.387661933907, /* 0x1.633dd0000 */ - 1.391424179060, /* 0x1.64345fffe */ - 1.395197510706, /* 0x1.652ba9fff */ - 1.399006724329, /* 0x1.66254dffe */ - 1.402773022651, /* 0x1.671c22000 */ - 1.406576037403, /* 0x1.68155dfff */ - 1.410389423392, /* 0x1.690f48001 */ -}; |