From 9feaa15d988bcd48a399b8616eb234de805e7b5e Mon Sep 17 00:00:00 2001
From: Szabolcs Nagy <szabolcs.nagy@arm.com>
Date: Wed, 6 Sep 2017 17:42:00 +0100
Subject: [PATCH 1/4] expf
---
math/Makefile | 2 +-
sysdeps/aarch64/fpu/math_private.h | 20 ++
sysdeps/i386/fpu/e_exp2f_data.c | 1 +
sysdeps/i386/fpu/math_errf.c | 1 +
sysdeps/ia64/fpu/e_exp2f_data.c | 1 +
sysdeps/ia64/fpu/math_errf.c | 1 +
sysdeps/ieee754/flt-32/e_exp2f.c | 170 ++++++----------
sysdeps/ieee754/flt-32/e_exp2f_data.c | 44 +++++
sysdeps/ieee754/flt-32/e_expf.c | 185 ++++++++----------
sysdeps/ieee754/flt-32/math_config.h | 114 +++++++++++
sysdeps/ieee754/flt-32/math_errf.c | 76 ++++++++
sysdeps/ieee754/flt-32/t_exp2f.h | 351 ----------------------------------
12 files changed, 399 insertions(+), 567 deletions(-)
create mode 100644 sysdeps/i386/fpu/e_exp2f_data.c
create mode 100644 sysdeps/i386/fpu/math_errf.c
create mode 100644 sysdeps/ia64/fpu/e_exp2f_data.c
create mode 100644 sysdeps/ia64/fpu/math_errf.c
create mode 100644 sysdeps/ieee754/flt-32/e_exp2f_data.c
create mode 100644 sysdeps/ieee754/flt-32/math_config.h
create mode 100644 sysdeps/ieee754/flt-32/math_errf.c
delete mode 100644 sysdeps/ieee754/flt-32/t_exp2f.h
@@ -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
@@ -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
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
new file mode 100644
@@ -0,0 +1 @@
+/* Not needed. */
@@ -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 = (double) (xd + SHIFT); /* Rounding to double precision is required. */
+ 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)
new file mode 100644
@@ -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
+ },
+};
@@ -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 = (double) (z + SHIFT); /* Rounding to double precision is required. */
+ 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)
new file mode 100644
@@ -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;
+
+#endif
new file mode 100644
@@ -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);
+}
deleted file mode 100644
@@ -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 */
-};
--
2.11.0