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GET /api/patches/813335/?format=api
{ "id": 813335, "url": "http://patchwork.ozlabs.org/api/patches/813335/?format=api", "web_url": "http://patchwork.ozlabs.org/project/glibc/patch/59B90C2E.9050806@arm.com/", "project": { "id": 41, "url": "http://patchwork.ozlabs.org/api/projects/41/?format=api", "name": "GNU C Library", "link_name": "glibc", "list_id": "libc-alpha.sourceware.org", "list_email": "libc-alpha@sourceware.org", "web_url": "", "scm_url": "", "webscm_url": "", "list_archive_url": "", "list_archive_url_format": "", "commit_url_format": "" }, "msgid": "<59B90C2E.9050806@arm.com>", "list_archive_url": null, "date": "2017-09-13T10:45:02", "name": "[1/4,v2] Optimized generic expf and exp2f with wrappers", "commit_ref": null, "pull_url": null, "state": "new", "archived": false, "hash": "c31ade538fc81604662f2489b20a1009afd7179c", "submitter": { "id": 65667, "url": "http://patchwork.ozlabs.org/api/people/65667/?format=api", "name": "Szabolcs Nagy", "email": "szabolcs.nagy@arm.com" }, "delegate": null, "mbox": "http://patchwork.ozlabs.org/project/glibc/patch/59B90C2E.9050806@arm.com/mbox/", "series": [ { "id": 2878, "url": "http://patchwork.ozlabs.org/api/series/2878/?format=api", "web_url": "http://patchwork.ozlabs.org/project/glibc/list/?series=2878", "date": "2017-09-13T10:43:43", "name": "Optimized generic expf and exp2f", "version": 1, "mbox": "http://patchwork.ozlabs.org/series/2878/mbox/" } ], "comments": "http://patchwork.ozlabs.org/api/patches/813335/comments/", "check": "pending", "checks": "http://patchwork.ozlabs.org/api/patches/813335/checks/", "tags": {}, "related": [], "headers": { "Return-Path": "<libc-alpha-return-84540-incoming=patchwork.ozlabs.org@sourceware.org>", "X-Original-To": "incoming@patchwork.ozlabs.org", "Delivered-To": [ "patchwork-incoming@bilbo.ozlabs.org", "mailing list libc-alpha@sourceware.org" ], "Authentication-Results": [ "ozlabs.org;\n\tspf=pass (mailfrom) smtp.mailfrom=sourceware.org\n\t(client-ip=209.132.180.131; helo=sourceware.org;\n\tenvelope-from=libc-alpha-return-84540-incoming=patchwork.ozlabs.org@sourceware.org;\n\treceiver=<UNKNOWN>)", "ozlabs.org; dkim=pass (1024-bit key;\n\tsecure) header.d=sourceware.org header.i=@sourceware.org\n\theader.b=\"Og8f2RVM\"; dkim-atps=neutral", "sourceware.org; auth=none", "spf=none (sender IP is )\n\tsmtp.mailfrom=Szabolcs.Nagy@arm.com; " ], "Received": [ "from sourceware.org (server1.sourceware.org [209.132.180.131])\n\t(using TLSv1.2 with cipher ECDHE-RSA-AES256-GCM-SHA384 (256/256\n\tbits)) (No client certificate requested)\n\tby ozlabs.org (Postfix) with ESMTPS id 3xsdcH32xLz9sPk\n\tfor <incoming@patchwork.ozlabs.org>;\n\tWed, 13 Sep 2017 20:45:51 +1000 (AEST)", "(qmail 19146 invoked by alias); 13 Sep 2017 10:45:38 -0000", "(qmail 3407 invoked by uid 89); 13 Sep 2017 10:45:27 -0000" ], "DomainKey-Signature": "a=rsa-sha1; c=nofws; d=sourceware.org; h=list-id\n\t:list-unsubscribe:list-subscribe:list-archive:list-post\n\t:list-help:sender:message-id:date:from:mime-version:to:cc\n\t:subject:references:in-reply-to:content-type; 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Linux x86_64;\n\trv:31.0) Gecko/20100101 Thunderbird/31.8.0", "MIME-Version": "1.0", "To": "GNU C Library <libc-alpha@sourceware.org>", "CC": "nd@arm.com", "Subject": "[PATCH 1/4 v2] Optimized generic expf and exp2f with wrappers", "References": "<59B90BDF.7000503@arm.com>", "In-Reply-To": "<59B90BDF.7000503@arm.com>", "Content-Type": "multipart/mixed;\n\tboundary=\"------------030504040404060706010306\"", "X-ClientProxiedBy": "AM4PR07CA0033.eurprd07.prod.outlook.com\n\t(2603:10a6:205:1::46) To DB6PR0802MB2485.eurprd08.prod.outlook.com\n\t(2603:10a6:4:9b::23)", "X-MS-PublicTrafficType": "Email", "X-MS-Office365-Filtering-Correlation-Id": "21bece6b-7c95-441d-d71a-08d4fa947ed3", "X-MS-Office365-Filtering-HT": "Tenant", "X-Microsoft-Antispam": "UriScan:; BCL:0; PCL:0;\n\tRULEID:(300000500095)(300135000095)(300000501095)(300135300095)(300000502095)(300135100095)(22001)(2017030254152)(48565401081)(300000503095)(300135400095)(49563074)(201703131423075)(201703031133081)(201702281549075)(300000504095)(300135200095)(300000505095)(300135600095)(300000506095)(300135500095);\n\tSRVR:DB6PR0802MB2485; 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SFP:1101; SCL:1; SRVR:DB6PR0802MB2485;\n\tH:[10.2.206.69]; FPR:; SPF:None; PTR:InfoNoRecords; MX:1; A:1;\n\tLANG:en; ", "Received-SPF": "None (protection.outlook.com: arm.com does not designate\n\tpermitted sender hosts)", "SpamDiagnosticOutput": "1:99", "SpamDiagnosticMetadata": "NSPM", "X-OriginatorOrg": "arm.com", "X-MS-Exchange-CrossTenant-OriginalArrivalTime": "13 Sep 2017 10:45:05.2561\n\t(UTC)", "X-MS-Exchange-CrossTenant-FromEntityHeader": "Hosted", "X-MS-Exchange-Transport-CrossTenantHeadersStamped": "DB6PR0802MB2485" }, "content": "Based on new expf and exp2f code from\nhttps://github.com/ARM-software/optimized-routines/\n\nwith the new expf benchmark\nhttps://sourceware.org/ml/libc-alpha/2017-08/msg01126.html\naarch64 expf reciprocal-throughput: 3.3x faster\naarch64 expf latency: 1.7x faster\naarch64 libm.so size: -8byte data, -2140byte text\nexpf/exp2f worst case nearest rounding ulp error: 0.502\n\nError checks are inline and errno setting is in separate\ntail called functions, but the wrappers are kept in this\npatch to handle the _LIB_VERSION==_SVID_ case. (So e.g.\nerrno is set twice for expf calls and once __expf_finite\ncalls now on targets where the new code is used.)\n\nDouble precision arithmetics is used which is expected\nto be faster on most targets (including soft-float) than\nusing single precision and it is easier to get good\nprecision result with it.\n\nConst data is kept in a separate translation unit which\ncomplicates maintenance a bit, but is expected to give\ngood code for literal loads on most targets and allows\nsharing data across expf, exp2f and powf. (This data is\ndisabled on i386 and ia64 which have their own expf, exp2f\nand powf code.)\n\nSome configuration is in a new math_config.h the settings\nmay need further discussion.\n\nSome details may need target specific tweaks:\n- best convert and round to int operation in the arg\nreduction may be different across targets.\n- code was optimized on fma target, optimal polynomial\neval may be different without fma.\n- gcc does not always generate good code for fp bit\nrepresentation access via unions or it may be inherently\nslow on some target.\n\nThe libm-test-ulps will need adjustment because..\n- The argument reduction ideally uses nearest rounded rint,\nbut that is not efficient on most targets, so the polynomial\ncan get evaluated on a wider interval in non-nearest\nrounding mode making 1 ulp errors common in that case.\n- The polynomial is evaluated such that it has 1 ulp error\non negative tiny inputs with upward rounding, but in\nexchange the evaluation is better pipelined.\n\n\nv2:\n- keep the compat wrappers for now.\n- cosmetic changes to math_config.h.\n- disable math_errf.c and e_exp2f_data.c on i386 and ia64.\n- remove unused t_exp2f.h\n\n2017-09-13 Szabolcs Nagy <szabolcs.nagy@arm.com>\n\n\t* math/Makefile (type-float-routines): Add math_errf and e_exp2f_data.\n\t* sysdeps/aarch64/fpu/math_private.h (TOINT_INTRINSICS): Define.\n\t(roundtoint, converttoint): Likewise.\n\t* sysdeps/i386/fpu/e_exp2f_data.c: New file.\n\t* sysdeps/i386/fpu/math_errf.c: New file.\n\t* sysdeps/ia64/fpu/e_exp2f_data.c: New file.\n\t* sysdeps/ia64/fpu/math_errf.c: New file.\n\t* sysdeps/ieee754/flt-32/e_expf.c: New implementation.\n\t* sysdeps/ieee754/flt-32/e_exp2f.c: New implementation.\n\t* sysdeps/ieee754/flt-32/e_exp2f_data.c: New file.\n\t* sysdeps/ieee754/flt-32/math_config.h: New file.\n\t* sysdeps/ieee754/flt-32/math_errf.c: New file.\n\t* sysdeps/ieee754/flt-32/t_exp2f.h: Remove.", "diff": "From 9feaa15d988bcd48a399b8616eb234de805e7b5e Mon Sep 17 00:00:00 2001\nFrom: Szabolcs Nagy <szabolcs.nagy@arm.com>\nDate: Wed, 6 Sep 2017 17:42:00 +0100\nSubject: [PATCH 1/4] expf\n\n---\n math/Makefile | 2 +-\n sysdeps/aarch64/fpu/math_private.h | 20 ++\n sysdeps/i386/fpu/e_exp2f_data.c | 1 +\n sysdeps/i386/fpu/math_errf.c | 1 +\n sysdeps/ia64/fpu/e_exp2f_data.c | 1 +\n sysdeps/ia64/fpu/math_errf.c | 1 +\n sysdeps/ieee754/flt-32/e_exp2f.c | 170 ++++++----------\n sysdeps/ieee754/flt-32/e_exp2f_data.c | 44 +++++\n sysdeps/ieee754/flt-32/e_expf.c | 185 ++++++++----------\n sysdeps/ieee754/flt-32/math_config.h | 114 +++++++++++\n sysdeps/ieee754/flt-32/math_errf.c | 76 ++++++++\n sysdeps/ieee754/flt-32/t_exp2f.h | 351 ----------------------------------\n 12 files changed, 399 insertions(+), 567 deletions(-)\n create mode 100644 sysdeps/i386/fpu/e_exp2f_data.c\n create mode 100644 sysdeps/i386/fpu/math_errf.c\n create mode 100644 sysdeps/ia64/fpu/e_exp2f_data.c\n create mode 100644 sysdeps/ia64/fpu/math_errf.c\n create mode 100644 sysdeps/ieee754/flt-32/e_exp2f_data.c\n create mode 100644 sysdeps/ieee754/flt-32/math_config.h\n create mode 100644 sysdeps/ieee754/flt-32/math_errf.c\n delete mode 100644 sysdeps/ieee754/flt-32/t_exp2f.h\n\ndiff --git a/math/Makefile b/math/Makefile\nindex 0601f3ac43..04586156f8 100644\n--- a/math/Makefile\n+++ b/math/Makefile\n@@ -115,7 +115,7 @@ type-double-routines := branred doasin dosincos halfulp mpa mpatan2\t\\\n \n # float support\n type-float-suffix := f\n-type-float-routines := k_rem_pio2f\n+type-float-routines := k_rem_pio2f math_errf e_exp2f_data\n \n # _Float128 support\n type-float128-suffix := f128\ndiff --git a/sysdeps/aarch64/fpu/math_private.h b/sysdeps/aarch64/fpu/math_private.h\nindex 807111ea5a..775237ffb2 100644\n--- a/sysdeps/aarch64/fpu/math_private.h\n+++ b/sysdeps/aarch64/fpu/math_private.h\n@@ -319,6 +319,26 @@ libc_feresetround_noex_aarch64_ctx (struct rm_ctx *ctx)\n #define libc_feresetround_noexf_ctx\tlibc_feresetround_noex_aarch64_ctx\n #define libc_feresetround_noexl_ctx\tlibc_feresetround_noex_aarch64_ctx\n \n+/* Hack: only include the large arm_neon.h when needed. */\n+#ifdef _MATH_CONFIG_H\n+#include <arm_neon.h>\n+\n+/* ACLE intrinsics for frintn and fcvtns instructions. */\n+#define TOINT_INTRINSICS 1\n+\n+static inline double_t\n+roundtoint (double_t x)\n+{\n+ return vget_lane_f64 (vrndn_f64 (vld1_f64 (&x)), 0);\n+}\n+\n+static inline uint64_t\n+converttoint (double_t x)\n+{\n+ return vcvtnd_s64_f64 (x);\n+}\n+#endif\n+\n #include_next <math_private.h>\n \n #endif\ndiff --git a/sysdeps/i386/fpu/e_exp2f_data.c b/sysdeps/i386/fpu/e_exp2f_data.c\nnew file mode 100644\nindex 0000000000..1cc8931700\n--- /dev/null\n+++ b/sysdeps/i386/fpu/e_exp2f_data.c\n@@ -0,0 +1 @@\n+/* Not needed. */\ndiff --git a/sysdeps/i386/fpu/math_errf.c b/sysdeps/i386/fpu/math_errf.c\nnew file mode 100644\nindex 0000000000..1cc8931700\n--- /dev/null\n+++ b/sysdeps/i386/fpu/math_errf.c\n@@ -0,0 +1 @@\n+/* Not needed. */\ndiff --git a/sysdeps/ia64/fpu/e_exp2f_data.c b/sysdeps/ia64/fpu/e_exp2f_data.c\nnew file mode 100644\nindex 0000000000..1cc8931700\n--- /dev/null\n+++ b/sysdeps/ia64/fpu/e_exp2f_data.c\n@@ -0,0 +1 @@\n+/* Not needed. */\ndiff --git a/sysdeps/ia64/fpu/math_errf.c b/sysdeps/ia64/fpu/math_errf.c\nnew file mode 100644\nindex 0000000000..1cc8931700\n--- /dev/null\n+++ b/sysdeps/ia64/fpu/math_errf.c\n@@ -0,0 +1 @@\n+/* Not needed. */\ndiff --git a/sysdeps/ieee754/flt-32/e_exp2f.c b/sysdeps/ieee754/flt-32/e_exp2f.c\nindex 567d3ff6d0..be53e5c699 100644\n--- a/sysdeps/ieee754/flt-32/e_exp2f.c\n+++ b/sysdeps/ieee754/flt-32/e_exp2f.c\n@@ -1,7 +1,6 @@\n-/* Single-precision floating point 2^x.\n- Copyright (C) 1997-2017 Free Software Foundation, Inc.\n+/* Single-precision 2^x function.\n+ Copyright (C) 2017 Free Software Foundation, Inc.\n This file is part of the GNU C Library.\n- Contributed by Geoffrey Keating <geoffk@ozemail.com.au>\n \n The GNU C Library is free software; you can redistribute it and/or\n modify it under the terms of the GNU Lesser General Public\n@@ -17,116 +16,73 @@\n License along with the GNU C Library; if not, see\n <http://www.gnu.org/licenses/>. */\n \n-/* The basic design here is from\n- Shmuel Gal and Boris Bachelis, \"An Accurate Elementary Mathematical\n- Library for the IEEE Floating Point Standard\", ACM Trans. Math. Soft.,\n- 17 (1), March 1991, pp. 26-45.\n- It has been slightly modified to compute 2^x instead of e^x, and for\n- single-precision.\n- */\n-#ifndef _GNU_SOURCE\n-# define _GNU_SOURCE\n-#endif\n-#include <stdlib.h>\n-#include <float.h>\n-#include <ieee754.h>\n #include <math.h>\n-#include <fenv.h>\n-#include <inttypes.h>\n-#include <math_private.h>\n-\n-#include \"t_exp2f.h\"\n-\n-static const float TWOM100 = 7.88860905e-31;\n-static const float TWO127 = 1.7014118346e+38;\n+#include <stdint.h>\n+#include \"math_config.h\"\n+\n+/*\n+EXP2F_TABLE_BITS = 5\n+EXP2F_POLY_ORDER = 3\n+\n+ULP error: 0.502 (nearest rounding.)\n+Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.)\n+Wrong count: 168353 (all nearest rounding wrong results with fma.)\n+Non-nearest ULP error: 1 (rounded ULP error)\n+*/\n+\n+#define N (1 << EXP2F_TABLE_BITS)\n+#define T __exp2f_data.tab\n+#define C __exp2f_data.poly\n+#define SHIFT __exp2f_data.shift_scaled\n+\n+static inline uint32_t\n+top12 (float x)\n+{\n+ return asuint (x) >> 20;\n+}\n \n float\n __ieee754_exp2f (float x)\n {\n- static const float himark = (float) FLT_MAX_EXP;\n- static const float lomark = (float) (FLT_MIN_EXP - FLT_MANT_DIG - 1);\n-\n- /* Check for usual case. */\n- if (isless (x, himark) && isgreaterequal (x, lomark))\n+ uint32_t abstop;\n+ uint64_t ki, t;\n+ /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */\n+ double_t kd, xd, z, r, r2, y, s;\n+\n+ xd = (double_t) x;\n+ abstop = top12 (x) & 0x7ff;\n+ if (__glibc_unlikely (abstop >= top12 (128.0f)))\n {\n- static const float THREEp14 = 49152.0;\n- int tval, unsafe;\n- float rx, x22, result;\n- union ieee754_float ex2_u, scale_u;\n-\n- if (fabsf (x) < FLT_EPSILON / 4.0f)\n-\treturn 1.0f + x;\n-\n- {\n-\tSET_RESTORE_ROUND_NOEXF (FE_TONEAREST);\n-\n-\t/* 1. Argument reduction.\n-\t Choose integers ex, -128 <= t < 128, and some real\n-\t -1/512 <= x1 <= 1/512 so that\n-\t x = ex + t/512 + x1.\n-\n-\t First, calculate rx = ex + t/256. */\n-\trx = x + THREEp14;\n-\trx -= THREEp14;\n-\tx -= rx; /* Compute x=x1. */\n-\t/* Compute tval = (ex*256 + t)+128.\n-\t Now, t = (tval mod 256)-128 and ex=tval/256 [that's mod, NOT %;\n-\t and /-round-to-nearest not the usual c integer /]. */\n-\ttval = (int) (rx * 256.0f + 128.0f);\n-\n-\t/* 2. Adjust for accurate table entry.\n-\t Find e so that\n-\t x = ex + t/256 + e + x2\n-\t where -7e-4 < e < 7e-4, and\n-\t (float)(2^(t/256+e))\n-\t is accurate to one part in 2^-64. */\n-\n-\t/* 'tval & 255' is the same as 'tval%256' except that it's always\n-\t positive.\n-\t Compute x = x2. */\n-\tx -= __exp2f_deltatable[tval & 255];\n-\n-\t/* 3. Compute ex2 = 2^(t/255+e+ex). */\n-\tex2_u.f = __exp2f_atable[tval & 255];\n-\ttval >>= 8;\n-\t/* x2 is an integer multiple of 2^-30; avoid intermediate\n-\t underflow from the calculation of x22 * x. */\n-\tunsafe = abs(tval) >= -FLT_MIN_EXP - 32;\n-\tex2_u.ieee.exponent += tval >> unsafe;\n-\tscale_u.f = 1.0;\n-\tscale_u.ieee.exponent += tval - (tval >> unsafe);\n-\n-\t/* 4. Approximate 2^x2 - 1, using a second-degree polynomial,\n-\t with maximum error in [-2^-9 - 2^-14, 2^-9 + 2^-14]\n-\t less than 1.3e-10. */\n-\n-\tx22 = (.24022656679f * x + .69314736128f) * ex2_u.f;\n- }\n-\n- /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex). */\n- result = x22 * x + ex2_u.f;\n-\n- if (!unsafe)\n-\treturn result;\n- else\n-\t{\n-\t result *= scale_u.f;\n-\t math_check_force_underflow_nonneg (result);\n-\t return result;\n-\t}\n- }\n- /* Exceptional cases: */\n- else if (isless (x, himark))\n- {\n- if (isinf (x))\n-\t/* e^-inf == 0, with no error. */\n-\treturn 0;\n- else\n-\t/* Underflow */\n-\treturn TWOM100 * TWOM100;\n+ /* |x| >= 128 or x is nan. */\n+ if (asuint (x) == asuint (-INFINITY))\n+\treturn 0.0f;\n+ if (abstop >= top12 (INFINITY))\n+\treturn x + x;\n+ if (x > 0.0f)\n+\treturn __math_oflowf (0);\n+ if (x <= -150.0f)\n+\treturn __math_uflowf (0);\n+#if WANT_ERRNO_UFLOW\n+ if (x < -149.0f)\n+\treturn __math_may_uflowf (0);\n+#endif\n }\n- else\n- /* Return x, if x is a NaN or Inf; or overflow, otherwise. */\n- return TWO127*x;\n+\n+ /* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */\n+ kd = (double) (xd + SHIFT); /* Rounding to double precision is required. */\n+ ki = asuint64 (kd);\n+ kd -= SHIFT; /* k/N for int k. */\n+ r = xd - kd;\n+\n+ /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */\n+ t = T[ki % N];\n+ t += ki << (52 - EXP2F_TABLE_BITS);\n+ s = asdouble (t);\n+ z = C[0] * r + C[1];\n+ r2 = r * r;\n+ y = C[2] * r + 1;\n+ y = z * r2 + y;\n+ y = y * s;\n+ return (float) y;\n }\n strong_alias (__ieee754_exp2f, __exp2f_finite)\ndiff --git a/sysdeps/ieee754/flt-32/e_exp2f_data.c b/sysdeps/ieee754/flt-32/e_exp2f_data.c\nnew file mode 100644\nindex 0000000000..390dcae333\n--- /dev/null\n+++ b/sysdeps/ieee754/flt-32/e_exp2f_data.c\n@@ -0,0 +1,44 @@\n+/* Shared data between expf, exp2f and powf.\n+ Copyright (C) 2017 Free Software Foundation, Inc.\n+ This file is part of the GNU C Library.\n+\n+ The GNU C Library is free software; you can redistribute it and/or\n+ modify it under the terms of the GNU Lesser General Public\n+ License as published by the Free Software Foundation; either\n+ version 2.1 of the License, or (at your option) any later version.\n+\n+ The GNU C Library is distributed in the hope that it will be useful,\n+ but WITHOUT ANY WARRANTY; without even the implied warranty of\n+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\n+ Lesser General Public License for more details.\n+\n+ You should have received a copy of the GNU Lesser General Public\n+ License along with the GNU C Library; if not, see\n+ <http://www.gnu.org/licenses/>. */\n+\n+#include \"math_config.h\"\n+\n+#define N (1 << EXP2F_TABLE_BITS)\n+\n+const struct exp2f_data __exp2f_data = {\n+ /* tab[i] = uint(2^(i/N)) - (i << 52-BITS)\n+ used for computing 2^(k/N) for an int |k| < 150 N as\n+ double(tab[k%N] + (k << 52-BITS)) */\n+ .tab = {\n+0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51,\n+0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1,\n+0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d,\n+0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585,\n+0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13,\n+0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d,\n+0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069,\n+0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540,\n+ },\n+ .shift_scaled = 0x1.8p+52 / N,\n+ .poly = { 0x1.c6af84b912394p-5, 0x1.ebfce50fac4f3p-3, 0x1.62e42ff0c52d6p-1 },\n+ .shift = 0x1.8p+52,\n+ .invln2_scaled = 0x1.71547652b82fep+0 * N,\n+ .poly_scaled = {\n+0x1.c6af84b912394p-5/N/N/N, 0x1.ebfce50fac4f3p-3/N/N, 0x1.62e42ff0c52d6p-1/N\n+ },\n+};\ndiff --git a/sysdeps/ieee754/flt-32/e_expf.c b/sysdeps/ieee754/flt-32/e_expf.c\nindex 782072f213..b8244a82cf 100644\n--- a/sysdeps/ieee754/flt-32/e_expf.c\n+++ b/sysdeps/ieee754/flt-32/e_expf.c\n@@ -1,7 +1,6 @@\n-/* Single-precision floating point e^x.\n- Copyright (C) 1997-2017 Free Software Foundation, Inc.\n+/* Single-precision e^x function.\n+ Copyright (C) 2017 Free Software Foundation, Inc.\n This file is part of the GNU C Library.\n- Contributed by Geoffrey Keating <geoffk@ozemail.com.au>\n \n The GNU C Library is free software; you can redistribute it and/or\n modify it under the terms of the GNU Lesser General Public\n@@ -17,117 +16,87 @@\n License along with the GNU C Library; if not, see\n <http://www.gnu.org/licenses/>. */\n \n-/* How this works:\n-\n- The input value, x, is written as\n-\n- x = n * ln(2) + t/512 + delta[t] + x;\n-\n- where:\n- - n is an integer, 127 >= n >= -150;\n- - t is an integer, 177 >= t >= -177\n- - delta is based on a table entry, delta[t] < 2^-28\n- - x is whatever is left, |x| < 2^-10\n-\n- Then e^x is approximated as\n-\n- e^x = 2^n ( e^(t/512 + delta[t])\n-\t + ( e^(t/512 + delta[t])\n-\t\t * ( p(x + delta[t] + n * ln(2)) - delta ) ) )\n-\n- where\n- - p(x) is a polynomial approximating e(x)-1;\n- - e^(t/512 + delta[t]) is obtained from a table.\n-\n- The table used is the same one as for the double precision version;\n- since we have the table, we might as well use it.\n-\n- It turns out to be faster to do calculations in double precision than\n- to perform an 'accurate table method' expf, because of the range reduction\n- overhead (compare exp2f).\n- */\n-#include <float.h>\n-#include <ieee754.h>\n #include <math.h>\n-#include <fenv.h>\n-#include <inttypes.h>\n-#include <math_private.h>\n-\n-extern const float __exp_deltatable[178];\n-extern const double __exp_atable[355] /* __attribute__((mode(DF))) */;\n-\n-static const float TWOM100 = 7.88860905e-31;\n-static const float TWO127 = 1.7014118346e+38;\n+#include <stdint.h>\n+#include \"math_config.h\"\n+\n+/*\n+EXP2F_TABLE_BITS = 5\n+EXP2F_POLY_ORDER = 3\n+\n+ULP error: 0.502 (nearest rounding.)\n+Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.)\n+Wrong count: 170635 (all nearest rounding wrong results with fma.)\n+Non-nearest ULP error: 1 (rounded ULP error)\n+*/\n+\n+#define N (1 << EXP2F_TABLE_BITS)\n+#define InvLn2N __exp2f_data.invln2_scaled\n+#define T __exp2f_data.tab\n+#define C __exp2f_data.poly_scaled\n+\n+static inline uint32_t\n+top12 (float x)\n+{\n+ return asuint (x) >> 20;\n+}\n \n float\n __ieee754_expf (float x)\n {\n- static const float himark = 88.72283935546875;\n- static const float lomark = -103.972084045410;\n- /* Check for usual case. */\n- if (isless (x, himark) && isgreater (x, lomark))\n+ uint32_t abstop;\n+ uint64_t ki, t;\n+ /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */\n+ double_t kd, xd, z, r, r2, y, s;\n+\n+ xd = (double_t) x;\n+ abstop = top12 (x) & 0x7ff;\n+ if (__glibc_unlikely (abstop >= top12 (88.0f)))\n {\n- static const float THREEp42 = 13194139533312.0;\n- static const float THREEp22 = 12582912.0;\n- /* 1/ln(2). */\n-#undef M_1_LN2\n- static const float M_1_LN2 = 1.44269502163f;\n- /* ln(2) */\n-#undef M_LN2\n- static const double M_LN2 = .6931471805599452862;\n-\n- int tval;\n- double x22, t, result, dx;\n- float n, delta;\n- union ieee754_double ex2_u;\n-\n- {\n-\tSET_RESTORE_ROUND_NOEXF (FE_TONEAREST);\n-\n-\t/* Calculate n. */\n-\tn = x * M_1_LN2 + THREEp22;\n-\tn -= THREEp22;\n-\tdx = x - n*M_LN2;\n-\n-\t/* Calculate t/512. */\n-\tt = dx + THREEp42;\n-\tt -= THREEp42;\n-\tdx -= t;\n-\n-\t/* Compute tval = t. */\n-\ttval = (int) (t * 512.0);\n-\n-\tif (t >= 0)\n-\t delta = - __exp_deltatable[tval];\n-\telse\n-\t delta = __exp_deltatable[-tval];\n-\n-\t/* Compute ex2 = 2^n e^(t/512+delta[t]). */\n-\tex2_u.d = __exp_atable[tval+177];\n-\tex2_u.ieee.exponent += (int) n;\n-\n-\t/* Approximate e^(dx+delta) - 1, using a second-degree polynomial,\n-\t with maximum error in [-2^-10-2^-28,2^-10+2^-28]\n-\t less than 5e-11. */\n-\tx22 = (0.5000000496709180453 * dx + 1.0000001192102037084) * dx + delta;\n- }\n-\n- /* Return result. */\n- result = x22 * ex2_u.d + ex2_u.d;\n- return (float) result;\n+ /* |x| >= 88 or x is nan. */\n+ if (asuint (x) == asuint (-INFINITY))\n+\treturn 0.0f;\n+ if (abstop >= top12 (INFINITY))\n+\treturn x + x;\n+ if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */\n+\treturn __math_oflowf (0);\n+ if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */\n+\treturn __math_uflowf (0);\n+#if WANT_ERRNO_UFLOW\n+ if (x < -0x1.9d1d9ep6f) /* x < log(0x1p-149) ~= -103.28 */\n+\treturn __math_may_uflowf (0);\n+#endif\n }\n- /* Exceptional cases: */\n- else if (isless (x, himark))\n- {\n- if (isinf (x))\n-\t/* e^-inf == 0, with no error. */\n-\treturn 0;\n- else\n-\t/* Underflow */\n-\treturn TWOM100 * TWOM100;\n- }\n- else\n- /* Return x, if x is a NaN or Inf; or overflow, otherwise. */\n- return TWO127*x;\n+\n+ /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */\n+ z = InvLn2N * xd;\n+\n+ /* Round and convert z to int, the result is in [-150*N, 128*N] and\n+ ideally ties-to-even rule is used, otherwise the magnitude of r\n+ can be bigger which gives larger approximation error. */\n+#if TOINT_INTRINSICS\n+ kd = roundtoint (z);\n+ ki = converttoint (z);\n+#elif TOINT_RINT\n+ kd = rint (z);\n+ ki = (long) kd;\n+#elif TOINT_SHIFT\n+# define SHIFT __exp2f_data.shift\n+ kd = (double) (z + SHIFT); /* Rounding to double precision is required. */\n+ ki = asuint64 (kd);\n+ kd -= SHIFT;\n+#endif\n+ r = z - kd;\n+\n+ /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */\n+ t = T[ki % N];\n+ t += ki << (52 - EXP2F_TABLE_BITS);\n+ s = asdouble (t);\n+ z = C[0] * r + C[1];\n+ r2 = r * r;\n+ y = C[2] * r + 1;\n+ y = z * r2 + y;\n+ y = y * s;\n+ return (float) y;\n }\n strong_alias (__ieee754_expf, __expf_finite)\ndiff --git a/sysdeps/ieee754/flt-32/math_config.h b/sysdeps/ieee754/flt-32/math_config.h\nnew file mode 100644\nindex 0000000000..1759c447c8\n--- /dev/null\n+++ b/sysdeps/ieee754/flt-32/math_config.h\n@@ -0,0 +1,114 @@\n+/* Configuration for math routines.\n+ Copyright (C) 2017 Free Software Foundation, Inc.\n+ This file is part of the GNU C Library.\n+\n+ The GNU C Library is free software; you can redistribute it and/or\n+ modify it under the terms of the GNU Lesser General Public\n+ License as published by the Free Software Foundation; either\n+ version 2.1 of the License, or (at your option) any later version.\n+\n+ The GNU C Library is distributed in the hope that it will be useful,\n+ but WITHOUT ANY WARRANTY; without even the implied warranty of\n+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\n+ Lesser General Public License for more details.\n+\n+ You should have received a copy of the GNU Lesser General Public\n+ License along with the GNU C Library; if not, see\n+ <http://www.gnu.org/licenses/>. */\n+\n+#ifndef _MATH_CONFIG_H\n+#define _MATH_CONFIG_H\n+\n+#include <math.h>\n+#include <math_private.h>\n+#include <stdint.h>\n+\n+#ifndef WANT_ROUNDING\n+/* Correct special case results in non-nearest rounding modes. */\n+# define WANT_ROUNDING 1\n+#endif\n+#ifndef WANT_ERRNO\n+/* Set errno according to ISO C with (math_errhandling & MATH_ERRNO) != 0. */\n+# define WANT_ERRNO 1\n+#endif\n+#ifndef WANT_ERRNO_UFLOW\n+/* Set errno to ERANGE if result underflows to 0 (in all rounding modes). */\n+# define WANT_ERRNO_UFLOW (WANT_ROUNDING && WANT_ERRNO)\n+#endif\n+\n+#ifndef TOINT_INTRINSICS\n+# define TOINT_INTRINSICS 0\n+#endif\n+#ifndef TOINT_RINT\n+# define TOINT_RINT 0\n+#endif\n+#ifndef TOINT_SHIFT\n+# define TOINT_SHIFT 1\n+#endif\n+\n+static inline uint32_t\n+asuint (float f)\n+{\n+ union\n+ {\n+ float f;\n+ uint32_t i;\n+ } u = {f};\n+ return u.i;\n+}\n+\n+static inline float\n+asfloat (uint32_t i)\n+{\n+ union\n+ {\n+ uint32_t i;\n+ float f;\n+ } u = {i};\n+ return u.f;\n+}\n+\n+static inline uint64_t\n+asuint64 (double f)\n+{\n+ union\n+ {\n+ double f;\n+ uint64_t i;\n+ } u = {f};\n+ return u.i;\n+}\n+\n+static inline double\n+asdouble (uint64_t i)\n+{\n+ union\n+ {\n+ uint64_t i;\n+ double f;\n+ } u = {i};\n+ return u.f;\n+}\n+\n+#define NOINLINE __attribute__ ((noinline))\n+\n+attribute_hidden float __math_oflowf (unsigned long);\n+attribute_hidden float __math_uflowf (unsigned long);\n+attribute_hidden float __math_may_uflowf (unsigned long);\n+attribute_hidden float __math_divzerof (unsigned long);\n+attribute_hidden float __math_invalidf (float);\n+\n+/* Shared between expf, exp2f and powf. */\n+#define EXP2F_TABLE_BITS 5\n+#define EXP2F_POLY_ORDER 3\n+extern const struct exp2f_data\n+{\n+ uint64_t tab[1 << EXP2F_TABLE_BITS];\n+ double shift_scaled;\n+ double poly[EXP2F_POLY_ORDER];\n+ double shift;\n+ double invln2_scaled;\n+ double poly_scaled[EXP2F_POLY_ORDER];\n+} __exp2f_data;\n+\n+#endif\ndiff --git a/sysdeps/ieee754/flt-32/math_errf.c b/sysdeps/ieee754/flt-32/math_errf.c\nnew file mode 100644\nindex 0000000000..ab546e24cf\n--- /dev/null\n+++ b/sysdeps/ieee754/flt-32/math_errf.c\n@@ -0,0 +1,76 @@\n+/* Single-precision math error handling.\n+ Copyright (C) 2017 Free Software Foundation, Inc.\n+ This file is part of the GNU C Library.\n+\n+ The GNU C Library is free software; you can redistribute it and/or\n+ modify it under the terms of the GNU Lesser General Public\n+ License as published by the Free Software Foundation; either\n+ version 2.1 of the License, or (at your option) any later version.\n+\n+ The GNU C Library is distributed in the hope that it will be useful,\n+ but WITHOUT ANY WARRANTY; without even the implied warranty of\n+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\n+ Lesser General Public License for more details.\n+\n+ You should have received a copy of the GNU Lesser General Public\n+ License along with the GNU C Library; if not, see\n+ <http://www.gnu.org/licenses/>. */\n+\n+#include \"math_config.h\"\n+\n+#if WANT_ERRNO\n+# include <errno.h>\n+/* NOINLINE reduces code size. */\n+NOINLINE static float\n+with_errnof (float y, int e)\n+{\n+ errno = e;\n+ return y;\n+}\n+#else\n+# define with_errnof(x, e) (x)\n+#endif\n+\n+/* NOINLINE prevents fenv semantics breaking optimizations. */\n+NOINLINE static float\n+xflowf (unsigned long sign, float y)\n+{\n+ y = (sign ? -y : y) * y;\n+ return with_errnof (y, ERANGE);\n+}\n+\n+attribute_hidden float\n+__math_uflowf (unsigned long sign)\n+{\n+ return xflowf (sign, 0x1p-95f);\n+}\n+\n+#if WANT_ERRNO_UFLOW\n+/* Underflows to zero in some non-nearest rounding mode, setting errno\n+ is valid even if the result is non-zero, but in the subnormal range. */\n+attribute_hidden float\n+__math_may_uflowf (unsigned long sign)\n+{\n+ return xflowf (sign, 0x1.4p-75f);\n+}\n+#endif\n+\n+attribute_hidden float\n+__math_oflowf (unsigned long sign)\n+{\n+ return xflowf (sign, 0x1p97f);\n+}\n+\n+attribute_hidden float\n+__math_divzerof (unsigned long sign)\n+{\n+ float y = 0;\n+ return with_errnof ((sign ? -1 : 1) / y, ERANGE);\n+}\n+\n+attribute_hidden float\n+__math_invalidf (float x)\n+{\n+ float y = (x - x) / (x - x);\n+ return isnan (x) ? y : with_errnof (y, EDOM);\n+}\ndiff --git a/sysdeps/ieee754/flt-32/t_exp2f.h b/sysdeps/ieee754/flt-32/t_exp2f.h\ndeleted file mode 100644\nindex aecabcc372..0000000000\n--- a/sysdeps/ieee754/flt-32/t_exp2f.h\n+++ /dev/null\n@@ -1,351 +0,0 @@\n-/* Accurate tables for exp2f().\n- Copyright (C) 1998-2017 Free Software Foundation, Inc.\n- This file is part of the GNU C Library.\n- Contributed by Geoffrey Keating <geoffk@ozemail.com.au>\n-\n- The GNU C Library is free software; you can redistribute it and/or\n- modify it under the terms of the GNU Lesser General Public\n- License as published by the Free Software Foundation; either\n- version 2.1 of the License, or (at your option) any later version.\n-\n- The GNU C Library is distributed in the hope that it will be useful,\n- but WITHOUT ANY WARRANTY; without even the implied warranty of\n- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\n- Lesser General Public License for more details.\n-\n- You should have received a copy of the GNU Lesser General Public\n- License along with the GNU C Library; if not, see\n- <http://www.gnu.org/licenses/>. */\n-\n-/* This table has the property that, for all integers -128 <= i <= 127,\n- exp(i/256.0 + __exp2f_deltatable[i-128]) == __exp2f_atable[i+128] + r\n- for some -2^-35 < r < 2^-35 (abs(r) < 2^-36 if i <= 0); and that\n- __exp2f_deltatable[i+128] == t * 2^-30\n- for integer t so that abs(t) <= 43447 * 2^0. */\n-\n-#define W30 (9.31322575e-10)\n-static const float __exp2f_deltatable[256] = {\n- -810*W30, 283*W30, -1514*W30, 1304*W30,\n- -1148*W30, -98*W30, -744*W30, -156*W30,\n- -419*W30, -155*W30, 474*W30, 167*W30,\n- -1984*W30, -826*W30, 692*W30, 781*W30,\n- -578*W30, -411*W30, -129*W30, -1500*W30,\n- 654*W30, -141*W30, -816*W30, -53*W30,\n- 148*W30, 493*W30, -2214*W30, 760*W30,\n- 260*W30, 750*W30, -1300*W30, 1424*W30,\n- -1445*W30, -339*W30, -680*W30, -349*W30,\n- -922*W30, 531*W30, 193*W30, -2892*W30,\n- 290*W30, -2145*W30, -276*W30, 485*W30,\n- -695*W30, 215*W30, -7093*W30, 412*W30,\n- -4596*W30, 367*W30, 592*W30, -615*W30,\n- -97*W30, -1066*W30, 972*W30, -226*W30,\n- -625*W30, -374*W30, -5647*W30, -180*W30,\n- 20349*W30, -447*W30, 111*W30, -4164*W30,\n- -87*W30, -21*W30, -251*W30, 66*W30,\n- -517*W30, 2093*W30, -263*W30, 182*W30,\n- -601*W30, 475*W30, -483*W30, -1251*W30,\n- -373*W30, 1471*W30, -92*W30, -215*W30,\n- -97*W30, -190*W30, 0*W30, -290*W30,\n- -2647*W30, 1940*W30, -582*W30, 28*W30,\n- 833*W30, 1493*W30, 34*W30, 321*W30,\n- 3327*W30, -35*W30, 177*W30, -135*W30,\n- -796*W30, -428*W30, 129*W30, 9332*W30,\n- -12*W30, -69*W30, -1743*W30, 6508*W30,\n- -60*W30, 359*W30, 43447*W30, 15*W30,\n- -23*W30, -305*W30, -375*W30, -652*W30,\n- 667*W30, 269*W30, -1575*W30, 185*W30,\n- -329*W30, 200*W30, 6002*W30, 163*W30,\n- -647*W30, 19*W30, -603*W30, -755*W30,\n- 742*W30, -438*W30, 3587*W30, 2560*W30,\n- 0*W30, -520*W30, -241*W30, -299*W30,\n- -1270*W30, -991*W30, -1138*W30, 255*W30,\n- -1192*W30, 1722*W30, 1023*W30, 3700*W30,\n- -1388*W30, -1551*W30, -2549*W30, 27*W30,\n- 282*W30, 673*W30, 113*W30, 1561*W30,\n- 72*W30, 873*W30, 87*W30, -395*W30,\n- -433*W30, 629*W30, 3440*W30, -284*W30,\n- -592*W30, -103*W30, -46*W30, -3844*W30,\n- 1712*W30, 303*W30, 1555*W30, -631*W30,\n- -1400*W30, -961*W30, -854*W30, -276*W30,\n- 407*W30, 833*W30, -345*W30, -1501*W30,\n- 121*W30, -1581*W30, 400*W30, 150*W30,\n- 1224*W30, -139*W30, -563*W30, 879*W30,\n- 933*W30, 2939*W30, 788*W30, 211*W30,\n- 530*W30, -192*W30, 706*W30, -13347*W30,\n- 1065*W30, 3*W30, 111*W30, -208*W30,\n- -360*W30, -532*W30, -291*W30, 483*W30,\n- 987*W30, -33*W30, -1373*W30, -166*W30,\n- -1174*W30, -3955*W30, 1601*W30, -280*W30,\n- 1405*W30, 600*W30, -1659*W30, -23*W30,\n- 390*W30, 449*W30, 570*W30, -13143*W30,\n- -9*W30, -1646*W30, 1201*W30, 294*W30,\n- 2181*W30, -1173*W30, 1388*W30, -4504*W30,\n- 190*W30, -2304*W30, 211*W30, 239*W30,\n- 48*W30, -817*W30, 1018*W30, 1828*W30,\n- -663*W30, 1408*W30, 408*W30, -36*W30,\n- 1295*W30, -230*W30, 1341*W30, 9*W30,\n- 40*W30, 705*W30, 186*W30, 376*W30,\n- 557*W30, 5866*W30, 363*W30, -1558*W30,\n- 718*W30, 669*W30, 1369*W30, -2972*W30,\n- -468*W30, -121*W30, -219*W30, 667*W30,\n- 29954*W30, 366*W30, 48*W30, -203*W30\n-};\n-\n-static const float __exp2f_atable[256] /* __attribute__((mode(SF))) */ = {\n- 0.707106411447, /* 0x0.b504ecfff */\n- 0.709024071690, /* 0x0.b58299fff */\n- 0.710945606239, /* 0x0.b60088000 */\n- 0.712874472142, /* 0x0.b67ef1000 */\n- 0.714806139464, /* 0x0.b6fd88fff */\n- 0.716744661340, /* 0x0.b77c94000 */\n- 0.718687653549, /* 0x0.b7fbea000 */\n- 0.720636486992, /* 0x0.b87ba1fff */\n- 0.722590208040, /* 0x0.b8fbabfff */\n- 0.724549472323, /* 0x0.b97c12fff */\n- 0.726514220228, /* 0x0.b9fcd5fff */\n- 0.728483855735, /* 0x0.ba7deb000 */\n- 0.730457961549, /* 0x0.baff4afff */\n- 0.732438981522, /* 0x0.bb811efff */\n- 0.734425544748, /* 0x0.bc0350000 */\n- 0.736416816713, /* 0x0.bc85d0000 */\n- 0.738412797450, /* 0x0.bd089efff */\n- 0.740414917465, /* 0x0.bd8bd4fff */\n- 0.742422521111, /* 0x0.be0f66fff */\n- 0.744434773914, /* 0x0.be9346fff */\n- 0.746454179287, /* 0x0.bf179f000 */\n- 0.748477637755, /* 0x0.bf9c3afff */\n- 0.750506639473, /* 0x0.c02133fff */\n- 0.752541840064, /* 0x0.c0a694fff */\n- 0.754582285889, /* 0x0.c12c4e000 */\n- 0.756628334525, /* 0x0.c1b265000 */\n- 0.758678436269, /* 0x0.c238bffff */\n- 0.760736882681, /* 0x0.c2bfa6fff */\n- 0.762799203401, /* 0x0.c346cf000 */\n- 0.764867603790, /* 0x0.c3ce5d000 */\n- 0.766940355298, /* 0x0.c45633fff */\n- 0.769021093841, /* 0x0.c4de90fff */\n- 0.771104693409, /* 0x0.c5671dfff */\n- 0.773195922364, /* 0x0.c5f02afff */\n- 0.775292098512, /* 0x0.c6798afff */\n- 0.777394294745, /* 0x0.c70350000 */\n- 0.779501736166, /* 0x0.c78d6d000 */\n- 0.781615912910, /* 0x0.c817fafff */\n- 0.783734917628, /* 0x0.c8a2d9fff */\n- 0.785858273516, /* 0x0.c92e02000 */\n- 0.787990570071, /* 0x0.c9b9c0000 */\n- 0.790125787245, /* 0x0.ca45aefff */\n- 0.792268991467, /* 0x0.cad223fff */\n- 0.794417440881, /* 0x0.cb5ef0fff */\n- 0.796570718287, /* 0x0.cbec0efff */\n- 0.798730909811, /* 0x0.cc79a0fff */\n- 0.800892710672, /* 0x0.cd074dfff */\n- 0.803068041795, /* 0x0.cd95ddfff */\n- 0.805242776881, /* 0x0.ce2464000 */\n- 0.807428598393, /* 0x0.ceb3a3fff */\n- 0.809617877002, /* 0x0.cf431dfff */\n- 0.811812341211, /* 0x0.cfd2eefff */\n- 0.814013659956, /* 0x0.d06333000 */\n- 0.816220164311, /* 0x0.d0f3ce000 */\n- 0.818434238424, /* 0x0.d184e7fff */\n- 0.820652604094, /* 0x0.d21649fff */\n- 0.822877407074, /* 0x0.d2a818000 */\n- 0.825108587751, /* 0x0.d33a51000 */\n- 0.827342867839, /* 0x0.d3ccbdfff */\n- 0.829588949684, /* 0x0.d45ff1000 */\n- 0.831849217401, /* 0x0.d4f411fff */\n- 0.834093391880, /* 0x0.d58724fff */\n- 0.836355149750, /* 0x0.d61b5f000 */\n- 0.838620424257, /* 0x0.d6afd3fff */\n- 0.840896368027, /* 0x0.d744fc000 */\n- 0.843176305293, /* 0x0.d7da66fff */\n- 0.845462262643, /* 0x0.d87037000 */\n- 0.847754716864, /* 0x0.d90673fff */\n- 0.850052893157, /* 0x0.d99d10fff */\n- 0.852359056469, /* 0x0.da3433fff */\n- 0.854668736446, /* 0x0.dacb91fff */\n- 0.856986224651, /* 0x0.db6373000 */\n- 0.859309315673, /* 0x0.dbfbb1fff */\n- 0.861639738080, /* 0x0.dc946bfff */\n- 0.863975346095, /* 0x0.dd2d7d000 */\n- 0.866317391394, /* 0x0.ddc6f9fff */\n- 0.868666708472, /* 0x0.de60f1000 */\n- 0.871022939695, /* 0x0.defb5c000 */\n- 0.873383641229, /* 0x0.df9611fff */\n- 0.875751554968, /* 0x0.e03141000 */\n- 0.878126025200, /* 0x0.e0ccde000 */\n- 0.880506813521, /* 0x0.e168e4fff */\n- 0.882894217966, /* 0x0.e2055afff */\n- 0.885287821299, /* 0x0.e2a239000 */\n- 0.887686729423, /* 0x0.e33f6ffff */\n- 0.890096127973, /* 0x0.e3dd56fff */\n- 0.892507970338, /* 0x0.e47b67000 */\n- 0.894928157336, /* 0x0.e51a03000 */\n- 0.897355020043, /* 0x0.e5b90efff */\n- 0.899788379682, /* 0x0.e65888000 */\n- 0.902227103705, /* 0x0.e6f85afff */\n- 0.904673457151, /* 0x0.e798ae000 */\n- 0.907128036008, /* 0x0.e8398afff */\n- 0.909585535528, /* 0x0.e8da99000 */\n- 0.912051796915, /* 0x0.e97c3a000 */\n- 0.914524436003, /* 0x0.ea1e46000 */\n- 0.917003571999, /* 0x0.eac0bf000 */\n- 0.919490039339, /* 0x0.eb63b2fff */\n- 0.921983361257, /* 0x0.ec071a000 */\n- 0.924488604054, /* 0x0.ecab48fff */\n- 0.926989555360, /* 0x0.ed4f30000 */\n- 0.929502844812, /* 0x0.edf3e6000 */\n- 0.932021975503, /* 0x0.ee98fdfff */\n- 0.934553921208, /* 0x0.ef3eecfff */\n- 0.937083780759, /* 0x0.efe4b8fff */\n- 0.939624726786, /* 0x0.f08b3f000 */\n- 0.942198514924, /* 0x0.f133ebfff */\n- 0.944726586343, /* 0x0.f1d99a000 */\n- 0.947287976728, /* 0x0.f28176fff */\n- 0.949856162070, /* 0x0.f329c5fff */\n- 0.952431440345, /* 0x0.f3d28bfff */\n- 0.955013573175, /* 0x0.f47bc5000 */\n- 0.957603693021, /* 0x0.f52584000 */\n- 0.960199773321, /* 0x0.f5cfa7000 */\n- 0.962801992906, /* 0x0.f67a31000 */\n- 0.965413510788, /* 0x0.f72556fff */\n- 0.968030691152, /* 0x0.f7d0dc000 */\n- 0.970655620084, /* 0x0.f87ce2fff */\n- 0.973290979849, /* 0x0.f92998fff */\n- 0.975926160805, /* 0x0.f9d64bfff */\n- 0.978571653370, /* 0x0.fa83ac000 */\n- 0.981225252139, /* 0x0.fb3193fff */\n- 0.983885228626, /* 0x0.fbdfe6fff */\n- 0.986552715296, /* 0x0.fc8eb7fff */\n- 0.989228487027, /* 0x0.fd3e14000 */\n- 0.991909801964, /* 0x0.fdedcd000 */\n- 0.994601726545, /* 0x0.fe9e38000 */\n- 0.997297704209, /* 0x0.ff4ee6fff */\n- 1.000000000000, /* 0x1.000000000 */\n- 1.002710938457, /* 0x1.00b1aa000 */\n- 1.005429744692, /* 0x1.0163d7ffe */\n- 1.008155703526, /* 0x1.02167dffe */\n- 1.010888457284, /* 0x1.02c995fff */\n- 1.013629436498, /* 0x1.037d38000 */\n- 1.016377568250, /* 0x1.043152000 */\n- 1.019134163841, /* 0x1.04e5f9ffe */\n- 1.021896362316, /* 0x1.059b00000 */\n- 1.024668931945, /* 0x1.0650b3ffe */\n- 1.027446627635, /* 0x1.0706be001 */\n- 1.030234098408, /* 0x1.07bd6bffe */\n- 1.033023953416, /* 0x1.087441ffe */\n- 1.035824656494, /* 0x1.092bce000 */\n- 1.038632392900, /* 0x1.09e3d0001 */\n- 1.041450142840, /* 0x1.0a9c79ffe */\n- 1.044273972530, /* 0x1.0b558a001 */\n- 1.047105550795, /* 0x1.0c0f1c001 */\n- 1.049944162390, /* 0x1.0cc924001 */\n- 1.052791833895, /* 0x1.0d83c4001 */\n- 1.055645227426, /* 0x1.0e3ec3fff */\n- 1.058507919326, /* 0x1.0efa60001 */\n- 1.061377286898, /* 0x1.0fb66bfff */\n- 1.064254641510, /* 0x1.1072fdffe */\n- 1.067140102389, /* 0x1.113018000 */\n- 1.070034146304, /* 0x1.11edc1fff */\n- 1.072937250162, /* 0x1.12ac04001 */\n- 1.075843691823, /* 0x1.136a7dfff */\n- 1.078760385496, /* 0x1.1429a3ffe */\n- 1.081685543070, /* 0x1.14e958000 */\n- 1.084618330005, /* 0x1.15a98c000 */\n- 1.087556362176, /* 0x1.166a18001 */\n- 1.090508937863, /* 0x1.172b98001 */\n- 1.093464612954, /* 0x1.17ed4bfff */\n- 1.096430182434, /* 0x1.18afa5ffe */\n- 1.099401354802, /* 0x1.19725e000 */\n- 1.102381587017, /* 0x1.1a35adfff */\n- 1.105370759965, /* 0x1.1af994000 */\n- 1.108367800686, /* 0x1.1bbdfdffe */\n- 1.111373305331, /* 0x1.1c82f6000 */\n- 1.114387035385, /* 0x1.1d4878001 */\n- 1.117408752440, /* 0x1.1e0e7ffff */\n- 1.120437502874, /* 0x1.1ed4fe000 */\n- 1.123474478729, /* 0x1.1f9c06000 */\n- 1.126521706601, /* 0x1.2063ba001 */\n- 1.129574775716, /* 0x1.212bd0001 */\n- 1.132638812065, /* 0x1.21f49e000 */\n- 1.135709524130, /* 0x1.22bddbffe */\n- 1.138789534565, /* 0x1.2387b5fff */\n- 1.141876101508, /* 0x1.2451fe000 */\n- 1.144971728301, /* 0x1.251cddffe */\n- 1.148077130296, /* 0x1.25e861ffe */\n- 1.151189923305, /* 0x1.26b462001 */\n- 1.154312610610, /* 0x1.278107ffe */\n- 1.157440662410, /* 0x1.284e08001 */\n- 1.160578370109, /* 0x1.291baa001 */\n- 1.163725256932, /* 0x1.29e9e6000 */\n- 1.166879892324, /* 0x1.2ab8a3ffe */\n- 1.170044302935, /* 0x1.2b8805fff */\n- 1.173205971694, /* 0x1.2c5739ffe */\n- 1.176397800428, /* 0x1.2d2867ffe */\n- 1.179586529747, /* 0x1.2df962001 */\n- 1.182784795737, /* 0x1.2ecafbffe */\n- 1.185991406414, /* 0x1.2f9d21ffe */\n- 1.189206838636, /* 0x1.306fdc001 */\n- 1.192430973067, /* 0x1.314328000 */\n- 1.195664167430, /* 0x1.32170c001 */\n- 1.198906540890, /* 0x1.32eb8a001 */\n- 1.202157497408, /* 0x1.33c098000 */\n- 1.205416083326, /* 0x1.349625fff */\n- 1.208683252332, /* 0x1.356c43fff */\n- 1.211961269402, /* 0x1.364318001 */\n- 1.215246438983, /* 0x1.371a64000 */\n- 1.218539118740, /* 0x1.37f22dffe */\n- 1.221847295770, /* 0x1.38cafc000 */\n- 1.225158572187, /* 0x1.39a3fdfff */\n- 1.228481650325, /* 0x1.3a7dc5ffe */\n- 1.231811761846, /* 0x1.3b5803fff */\n- 1.235149741144, /* 0x1.3c32c5ffe */\n- 1.238499879811, /* 0x1.3d0e53ffe */\n- 1.241858124726, /* 0x1.3dea69fff */\n- 1.245225191102, /* 0x1.3ec713fff */\n- 1.248601436624, /* 0x1.3fa458000 */\n- 1.251975655584, /* 0x1.40817a001 */\n- 1.255380749731, /* 0x1.4160a2001 */\n- 1.258783102010, /* 0x1.423f9bffe */\n- 1.262198328973, /* 0x1.431f6e000 */\n- 1.265619754780, /* 0x1.43ffa7fff */\n- 1.269052743928, /* 0x1.44e0a4001 */\n- 1.272490739830, /* 0x1.45c1f4000 */\n- 1.275942921659, /* 0x1.46a432001 */\n- 1.279397487615, /* 0x1.478697ffe */\n- 1.282870173427, /* 0x1.486a2dffe */\n- 1.286346316319, /* 0x1.494dfdffe */\n- 1.289836049094, /* 0x1.4a32b2001 */\n- 1.293333172770, /* 0x1.4b17e1ffe */\n- 1.296839594835, /* 0x1.4bfdadfff */\n- 1.300354957560, /* 0x1.4ce40fffe */\n- 1.303882122055, /* 0x1.4dcb38001 */\n- 1.307417988757, /* 0x1.4eb2f1ffe */\n- 1.310960650439, /* 0x1.4f9b1dfff */\n- 1.314516782746, /* 0x1.50842bfff */\n- 1.318079948424, /* 0x1.516daffff */\n- 1.321653246888, /* 0x1.5257de000 */\n- 1.325237751030, /* 0x1.5342c8001 */\n- 1.328829526907, /* 0x1.542e2c000 */\n- 1.332433700535, /* 0x1.551a5fffe */\n- 1.336045145966, /* 0x1.56070dffe */\n- 1.339667558645, /* 0x1.56f473ffe */\n- 1.343300342533, /* 0x1.57e287ffe */\n- 1.346941947961, /* 0x1.58d130001 */\n- 1.350594043714, /* 0x1.59c087ffe */\n- 1.354256033883, /* 0x1.5ab085fff */\n- 1.357932448365, /* 0x1.5ba175ffe */\n- 1.361609339707, /* 0x1.5c926dfff */\n- 1.365299344044, /* 0x1.5d8441ffe */\n- 1.369003057507, /* 0x1.5e76fc001 */\n- 1.372714757920, /* 0x1.5f6a3c000 */\n- 1.376437187179, /* 0x1.605e2fffe */\n- 1.380165219333, /* 0x1.615282001 */\n- 1.383909463864, /* 0x1.6247e3ffe */\n- 1.387661933907, /* 0x1.633dd0000 */\n- 1.391424179060, /* 0x1.64345fffe */\n- 1.395197510706, /* 0x1.652ba9fff */\n- 1.399006724329, /* 0x1.66254dffe */\n- 1.402773022651, /* 0x1.671c22000 */\n- 1.406576037403, /* 0x1.68155dfff */\n- 1.410389423392, /* 0x1.690f48001 */\n-};\n-- \n2.11.0\n\n", "prefixes": [ "1/4", "v2" ] }