diff options
| -rw-r--r-- | ChangeLog | 18 | ||||
| -rw-r--r-- | NEWS | 2 | ||||
| -rw-r--r-- | math/Makefile | 2 | ||||
| -rw-r--r-- | sysdeps/aarch64/fpu/math_private.h | 20 | ||||
| -rw-r--r-- | sysdeps/i386/fpu/e_exp2f_data.c | 1 | ||||
| -rw-r--r-- | sysdeps/i386/fpu/math_errf.c | 1 | ||||
| -rw-r--r-- | sysdeps/ia64/fpu/e_exp2f_data.c | 1 | ||||
| -rw-r--r-- | sysdeps/ia64/fpu/math_errf.c | 1 | ||||
| -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 | ||||
| -rw-r--r-- | sysdeps/m68k/m680x0/fpu/e_exp2f_data.c | 1 | ||||
| -rw-r--r-- | sysdeps/m68k/m680x0/fpu/math_errf.c | 1 |
16 files changed, 421 insertions, 567 deletions
@@ -1,3 +1,21 @@ +2017-09-25 Szabolcs Nagy <szabolcs.nagy@arm.com> + + * math/Makefile (type-float-routines): Add math_errf and e_exp2f_data. + * sysdeps/aarch64/fpu/math_private.h (TOINT_INTRINSICS): Define. + (roundtoint, converttoint): Likewise. + * sysdeps/ieee754/flt-32/e_expf.c: New implementation. + * sysdeps/ieee754/flt-32/e_exp2f.c: New implementation. + * sysdeps/ieee754/flt-32/e_exp2f_data.c: New file. + * sysdeps/ieee754/flt-32/math_config.h: New file. + * sysdeps/ieee754/flt-32/math_errf.c: New file. + * sysdeps/ieee754/flt-32/t_exp2f.h: Remove. + * sysdeps/i386/fpu/e_exp2f_data.c: New file. + * sysdeps/i386/fpu/math_errf.c: New file. + * sysdeps/ia64/fpu/e_exp2f_data.c: New file. + * sysdeps/ia64/fpu/math_errf.c: New file. + * sysdeps/m68k/m680x0/fpu/e_exp2f_data.c: New file. + * sysdeps/m68k/m680x0/fpu/math_errf.c: New file. + 2017-09-24 Samuel Thibault <samuel.thibault@ens-lyon.org> * sysdeps/mach/hurd/dl-sysdep.c (check_no_hidden): New macro. @@ -14,6 +14,8 @@ Major new features: * Optimized x86-64 trunc and truncf for processors with SSE4.1. +* Optimized generic expf and exp2f. + * In order to support faster and safer process termination the malloc API family of functions will no longer print a failure address and stack backtrace after detecting heap corruption. The goal is to minimize the diff --git a/math/Makefile b/math/Makefile index 0601f3ac43..04586156f8 100644 --- a/math/Makefile +++ b/math/Makefile @@ -115,7 +115,7 @@ type-double-routines := branred doasin dosincos halfulp mpa mpatan2 \ # float support type-float-suffix := f -type-float-routines := k_rem_pio2f +type-float-routines := k_rem_pio2f math_errf e_exp2f_data # _Float128 support type-float128-suffix := f128 diff --git a/sysdeps/aarch64/fpu/math_private.h b/sysdeps/aarch64/fpu/math_private.h index 807111ea5a..be10488f5d 100644 --- a/sysdeps/aarch64/fpu/math_private.h +++ b/sysdeps/aarch64/fpu/math_private.h @@ -319,6 +319,26 @@ libc_feresetround_noex_aarch64_ctx (struct rm_ctx *ctx) #define libc_feresetround_noexf_ctx libc_feresetround_noex_aarch64_ctx #define libc_feresetround_noexl_ctx libc_feresetround_noex_aarch64_ctx +/* Hack: only include the large arm_neon.h when needed. */ +#ifdef _MATH_CONFIG_H +# include <arm_neon.h> + +/* ACLE intrinsics for frintn and fcvtns instructions. */ +# define TOINT_INTRINSICS 1 + +static inline double_t +roundtoint (double_t x) +{ + return vget_lane_f64 (vrndn_f64 (vld1_f64 (&x)), 0); +} + +static inline uint64_t +converttoint (double_t x) +{ + return vcvtnd_s64_f64 (x); +} +#endif + #include_next <math_private.h> #endif diff --git a/sysdeps/i386/fpu/e_exp2f_data.c b/sysdeps/i386/fpu/e_exp2f_data.c new file mode 100644 index 0000000000..1cc8931700 --- /dev/null +++ b/sysdeps/i386/fpu/e_exp2f_data.c @@ -0,0 +1 @@ +/* Not needed. */ diff --git a/sysdeps/i386/fpu/math_errf.c b/sysdeps/i386/fpu/math_errf.c new file mode 100644 index 0000000000..1cc8931700 --- /dev/null +++ b/sysdeps/i386/fpu/math_errf.c @@ -0,0 +1 @@ +/* Not needed. */ diff --git a/sysdeps/ia64/fpu/e_exp2f_data.c b/sysdeps/ia64/fpu/e_exp2f_data.c new file mode 100644 index 0000000000..1cc8931700 --- /dev/null +++ b/sysdeps/ia64/fpu/e_exp2f_data.c @@ -0,0 +1 @@ +/* Not needed. */ diff --git a/sysdeps/ia64/fpu/math_errf.c b/sysdeps/ia64/fpu/math_errf.c new file mode 100644 index 0000000000..1cc8931700 --- /dev/null +++ b/sysdeps/ia64/fpu/math_errf.c @@ -0,0 +1 @@ +/* Not needed. */ 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}; |