From patchwork Fri Feb 2 17:44:12 2018 Content-Type: text/plain; charset="utf-8" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit X-Patchwork-Submitter: Wilco Dijkstra X-Patchwork-Id: 25777 Received: (qmail 71501 invoked by alias); 2 Feb 2018 17:44:24 -0000 Mailing-List: contact libc-alpha-help@sourceware.org; run by ezmlm Precedence: bulk List-Id: List-Unsubscribe: List-Subscribe: List-Archive: List-Post: List-Help: , Sender: libc-alpha-owner@sourceware.org Delivered-To: mailing list libc-alpha@sourceware.org Received: (qmail 71451 invoked by uid 89); 2 Feb 2018 17:44:18 -0000 Authentication-Results: sourceware.org; auth=none X-Virus-Found: No X-Spam-SWARE-Status: No, score=-25.2 required=5.0 tests=AWL, BAYES_00, GIT_PATCH_0, GIT_PATCH_1, GIT_PATCH_2, GIT_PATCH_3, RCVD_IN_DNSWL_NONE, SPF_HELO_PASS, SPF_PASS autolearn=ham version=3.3.2 spammy=III, 003, ridiculous, mpy X-HELO: EUR02-VE1-obe.outbound.protection.outlook.com From: Wilco Dijkstra To: "libc-alpha@sourceware.org" CC: nd Subject: [PATCH] Remove slow paths from log Date: Fri, 2 Feb 2018 17:44:12 +0000 Message-ID: authentication-results: spf=none (sender IP is ) smtp.mailfrom=Wilco.Dijkstra@arm.com; x-ms-publictraffictype: Email x-microsoft-exchange-diagnostics: 1; DB6PR0801MB1720; 6:7OmLqamnfvTvkbz3KDcJh3oduBtmMWs/bbyCHn1Ap6TpGEWJddRD3BpSp+gbszTRbKrt3PeoWT/dq+GGcYHn47bg6s3QtTMcuBOgDXFH8EGZfF2W4B5cOfqAg6L8PVAqpy1jfr+Ro7TWzQeksFJgvbztYEBNKnijNmjh4ltQ5dP8xgi12w7vXSr7Vn0JVlAeSwO9Sd7nf2kinU3ZrvHJVsIlUk9zFxP/mkJAotim6KjreEudcwEkqeaBgb4M1rkZD6oQWElRoy9pwB+kOlnOg68otFoL1JD48DmCDYzaNfSjmUmm6PhpywaEsN6TMyiFS8flTAmTVkJrdCWJhwn5E97GWb8M3tsaS8C3AqIR28bgivvpQ9gkbk+gK/RNrSWk; 5:DjI4AwPsVmrg2MHBMWXyBKqWEUXWZV0SwN7UNSsB2EzHZYisCclCpGn4lNkHxLXxYsSy2/qleMbgQxaDmZE5D9CIUhCeei6vXIVotlWTa5CZ+75H80wGSHT4bXvNE5l+djI7sevfNOy59BAPetQtq+0ddmnR0DRavaU+E0lYmxk=; 24:SJkhBeW6Z2CGrvItlcjYnc0+71g/Vhi9+A2m9xyzFxR5w0bOgw45URQdtBSmLOZlY9HfdMouLIMTBqo7VwKaOtNCNJvsJwfQsNHUITp9n8U=; 7:3sGLkLrKGSQOPn57RAFeRY2eHadqvo9lX0NChLJhVvS+E8jRVV4OyrcdGyu6sZtZEf/rUHHhcLKqg30DjMGefYQohduhxInqMBWgO6pMUoRwvxpOpjbPr+NNDjbfTqrG6JKr68vzBHCibT9xy+jK17RE0bmwx7HccUk7aRivwL+isG2Yw8Jm/RAAKariqpGVnhVpP5zE4TFroJEsWyG74o91iGibTOxnV/i2en/LRv/kuquxhDXNlU59Xsi3lJ+U x-ms-exchange-antispam-srfa-diagnostics: SSOS; x-ms-office365-filtering-ht: Tenant x-ms-office365-filtering-correlation-id: a0119ee3-8228-43fd-628b-08d56a649200 x-microsoft-antispam: UriScan:; BCL:0; PCL:0; RULEID:(7020095)(4652020)(48565401081)(5600026)(4604075)(3008032)(2017052603307)(7153060)(7193020); SRVR:DB6PR0801MB1720; x-ms-traffictypediagnostic: DB6PR0801MB1720: nodisclaimer: True x-microsoft-antispam-prvs: x-exchange-antispam-report-test: UriScan:(180628864354917); x-exchange-antispam-report-cfa-test: BCL:0; PCL:0; RULEID:(6040501)(2401047)(8121501046)(5005006)(10201501046)(3231101)(2400082)(944501161)(93006095)(93001095)(3002001)(6055026)(6041288)(20161123562045)(20161123564045)(201703131423095)(201702281528075)(20161123555045)(201703061421075)(201703061406153)(20161123560045)(20161123558120)(6072148)(201708071742011); SRVR:DB6PR0801MB1720; BCL:0; PCL:0; RULEID:; SRVR:DB6PR0801MB1720; x-forefront-prvs: 05715BE7FD x-forefront-antispam-report: SFV:NSPM; SFS:(10009020)(39860400002)(39380400002)(366004)(376002)(396003)(346002)(54534003)(189003)(199004)(377424004)(316002)(68736007)(9686003)(26005)(97736004)(3280700002)(99286004)(3660700001)(6506007)(59450400001)(305945005)(33656002)(478600001)(55016002)(102836004)(6436002)(72206003)(4326008)(2351001)(86362001)(575784001)(5640700003)(25786009)(53936002)(7696005)(106356001)(81156014)(5660300001)(8676002)(8936002)(14454004)(66066001)(81166006)(2906002)(105586002)(3846002)(6116002)(2501003)(5250100002)(7736002)(6916009)(2900100001)(74316002); DIR:OUT; SFP:1101; SCL:1; SRVR:DB6PR0801MB1720; H:DB6PR0801MB2053.eurprd08.prod.outlook.com; FPR:; SPF:None; PTR:InfoNoRecords; MX:1; A:1; LANG:en; received-spf: None (protection.outlook.com: arm.com does not designate permitted sender hosts) x-microsoft-antispam-message-info: gPplE3spzFs+RPxK91aWfobnyiQzwtLb7y1cDoX1uKw41LqNikC8P8NmzVbi0/tkQFX3so34LssUebyI2HL7zQ== spamdiagnosticoutput: 1:99 spamdiagnosticmetadata: NSPM MIME-Version: 1.0 X-OriginatorOrg: arm.com X-MS-Exchange-CrossTenant-Network-Message-Id: a0119ee3-8228-43fd-628b-08d56a649200 X-MS-Exchange-CrossTenant-originalarrivaltime: 02 Feb 2018 17:44:12.0784 (UTC) X-MS-Exchange-CrossTenant-fromentityheader: Hosted X-MS-Exchange-CrossTenant-id: f34e5979-57d9-4aaa-ad4d-b122a662184d X-MS-Exchange-Transport-CrossTenantHeadersStamped: DB6PR0801MB1720 Remove the slow paths from log. Like several other double precision math functions, log is exactly rounded. This is not required from math functions and causes major overheads as it requires multiple fallbacks using higher precision arithmetic if a result is close to 0.5ULP. Ridiculous slowdowns of up to 100000x have been reported when the highest precision path triggers. Interestingly removing the slow paths makes hardly any difference in practice: the worst case error is still ~0.5ULP, and exp(log(x)) shows identical results before/after on many millions of random cases. All GLIBC math tests pass on AArch64 and x64 with no change in ULP error. OK for commit? ChangeLog: 2018-02-02 Wilco Dijkstra * sysdeps/ieee754/dbl-64/e_log.c (__ieee754_log): Remove slow paths. * sysdeps/ieee754/dbl-64/ulog.h: Remove unused declarations. diff --git a/sysdeps/ieee754/dbl-64/e_log.c b/sysdeps/ieee754/dbl-64/e_log.c index 6a18ebb904fc42a69ed72d79f6db646addf46054..cd8ac3534d642bfb936483696c1ff7f880319538 100644 --- a/sysdeps/ieee754/dbl-64/e_log.c +++ b/sysdeps/ieee754/dbl-64/e_log.c @@ -23,11 +23,10 @@ /* FUNCTION:ulog */ /* */ /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h ulog.h */ -/* mpexp.c mplog.c mpa.c */ /* ulog.tbl */ /* */ /* An ultimate log routine. Given an IEEE double machine number x */ -/* it computes the correctly rounded (to nearest) value of log(x). */ +/* it computes the rounded (to nearest) value of log(x). */ /* Assumption: Machine arithmetic operations are performed in */ /* round to nearest mode of IEEE 754 standard. */ /* */ @@ -40,34 +39,26 @@ #include "MathLib.h" #include #include -#include #ifndef SECTION # define SECTION #endif -void __mplog (mp_no *, mp_no *, int); - /*********************************************************************/ -/* An ultimate log routine. Given an IEEE double machine number x */ -/* it computes the correctly rounded (to nearest) value of log(x). */ +/* An ultimate log routine. Given an IEEE double machine number x */ +/* it computes the rounded (to nearest) value of log(x). */ /*********************************************************************/ double SECTION __ieee754_log (double x) { -#define M 4 - static const int pr[M] = { 8, 10, 18, 32 }; - int i, j, n, ux, dx, p; + int i, j, n, ux, dx; double dbl_n, u, p0, q, r0, w, nln2a, luai, lubi, lvaj, lvbj, - sij, ssij, ttij, A, B, B0, y, y1, y2, polI, polII, sa, sb, - t1, t2, t7, t8, t, ra, rb, ww, - a0, aa0, s1, s2, ss2, s3, ss3, a1, aa1, a, aa, b, bb, c; + sij, ssij, ttij, A, B, B0, polI, polII, t8, a, aa, b, bb, c; #ifndef DLA_FMS - double t3, t4, t5, t6; + double t1, t2, t3, t4, t5; #endif number num; - mp_no mpx, mpy, mpy1, mpy2, mperr; #include "ulog.tbl" #include "ulog.h" @@ -101,7 +92,7 @@ __ieee754_log (double x) if (w == 0.0) return 0.0; - /*--- Stage I, the case abs(x-1) < 0.03 */ + /*--- The case abs(x-1) < 0.03 */ t8 = MHALF * w; EMULV (t8, w, a, aa, t1, t2, t3, t4, t5); @@ -118,50 +109,9 @@ __ieee754_log (double x) polII *= w * w * w; c = (aa + bb) + polII; - /* End stage I, case abs(x-1) < 0.03 */ - if ((y = b + (c + b * E2)) == b + (c - b * E2)) - return y; - - /*--- Stage II, the case abs(x-1) < 0.03 */ - - a = d19.d + w * d20.d; - a = d18.d + w * a; - a = d17.d + w * a; - a = d16.d + w * a; - a = d15.d + w * a; - a = d14.d + w * a; - a = d13.d + w * a; - a = d12.d + w * a; - a = d11.d + w * a; - - EMULV (w, a, s2, ss2, t1, t2, t3, t4, t5); - ADD2 (d10.d, dd10.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d9.d, dd9.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d8.d, dd8.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d7.d, dd7.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d6.d, dd6.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d5.d, dd5.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d4.d, dd4.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d3.d, dd3.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (d2.d, dd2.d, s2, ss2, s3, ss3, t1, t2); - MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - MUL2 (w, 0, s2, ss2, s3, ss3, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (w, 0, s3, ss3, b, bb, t1, t2); + return b + (c + b * E2); - /* End stage II, case abs(x-1) < 0.03 */ - if ((y = b + (bb + b * E4)) == b + (bb - b * E4)) - return y; - goto stage_n; - - /*--- Stage I, the case abs(x-1) > 0.03 */ + /*--- The case abs(x-1) > 0.03 */ case_03: /* Find n,u such that x = u*2**n, 1/sqrt(2) < u < sqrt(2) */ @@ -203,58 +153,7 @@ case_03: B0 = (((lubi + lvbj) + ssij) + ttij) + dbl_n * LN2B; B = polI + B0; - /* End stage I, case abs(x-1) >= 0.03 */ - if ((y = A + (B + E1)) == A + (B - E1)) - return y; - - - /*--- Stage II, the case abs(x-1) > 0.03 */ - - /* Improve the accuracy of r0 */ - EMULV (p0, r0, sa, sb, t1, t2, t3, t4, t5); - t = r0 * ((1 - sa) - sb); - EADD (r0, t, ra, rb); - - /* Compute w */ - MUL2 (q, 0, ra, rb, w, ww, t1, t2, t3, t4, t5, t6, t7, t8); - - EADD (A, B0, a0, aa0); - - /* Evaluate polynomial III */ - s1 = (c3.d + (c4.d + c5.d * w) * w) * w; - EADD (c2.d, s1, s2, ss2); - MUL2 (s2, ss2, w, ww, s3, ss3, t1, t2, t3, t4, t5, t6, t7, t8); - MUL2 (s3, ss3, w, ww, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8); - ADD2 (s2, ss2, w, ww, s3, ss3, t1, t2); - ADD2 (s3, ss3, a0, aa0, a1, aa1, t1, t2); - - /* End stage II, case abs(x-1) >= 0.03 */ - if ((y = a1 + (aa1 + E3)) == a1 + (aa1 - E3)) - return y; - - - /* Final stages. Use multi-precision arithmetic. */ -stage_n: - - for (i = 0; i < M; i++) - { - p = pr[i]; - __dbl_mp (x, &mpx, p); - __dbl_mp (y, &mpy, p); - __mplog (&mpx, &mpy, p); - __dbl_mp (e[i].d, &mperr, p); - __add (&mpy, &mperr, &mpy1, p); - __sub (&mpy, &mperr, &mpy2, p); - __mp_dbl (&mpy1, &y1, p); - __mp_dbl (&mpy2, &y2, p); - if (y1 == y2) - { - LIBC_PROBE (slowlog, 3, &p, &x, &y1); - return y1; - } - } - LIBC_PROBE (slowlog_inexact, 3, &p, &x, &y1); - return y1; + return A + (B + E1); } #ifndef __ieee754_log diff --git a/sysdeps/ieee754/dbl-64/ulog.h b/sysdeps/ieee754/dbl-64/ulog.h index 36a31137b759f604fba611d68a60efc90dc8d20d..b9bb8ab046ed1b10bf1bf8a7b66c820d9d6a39cb 100644 --- a/sysdeps/ieee754/dbl-64/ulog.h +++ b/sysdeps/ieee754/dbl-64/ulog.h @@ -42,43 +42,6 @@ /**/ b6 = {{0x3fbc71c5, 0x25db58ac} }, /* 0.111... */ /**/ b7 = {{0xbfb9a4ac, 0x11a2a61c} }, /* -0.100... */ /**/ b8 = {{0x3fb75077, 0x0df2b591} }, /* 0.091... */ - /* polynomial III */ -#if 0 -/**/ c1 = {{0x3ff00000, 0x00000000} }, /* 1 */ -#endif -/**/ c2 = {{0xbfe00000, 0x00000000} }, /* -1/2 */ -/**/ c3 = {{0x3fd55555, 0x55555555} }, /* 1/3 */ -/**/ c4 = {{0xbfd00000, 0x00000000} }, /* -1/4 */ -/**/ c5 = {{0x3fc99999, 0x9999999a} }, /* 1/5 */ - /* polynomial IV */ -/**/ d2 = {{0xbfe00000, 0x00000000} }, /* -1/2 */ -/**/ dd2 = {{0x00000000, 0x00000000} }, /* -1/2-d2 */ -/**/ d3 = {{0x3fd55555, 0x55555555} }, /* 1/3 */ -/**/ dd3 = {{0x3c755555, 0x55555555} }, /* 1/3-d3 */ -/**/ d4 = {{0xbfd00000, 0x00000000} }, /* -1/4 */ -/**/ dd4 = {{0x00000000, 0x00000000} }, /* -1/4-d4 */ -/**/ d5 = {{0x3fc99999, 0x9999999a} }, /* 1/5 */ -/**/ dd5 = {{0xbc699999, 0x9999999a} }, /* 1/5-d5 */ -/**/ d6 = {{0xbfc55555, 0x55555555} }, /* -1/6 */ -/**/ dd6 = {{0xbc655555, 0x55555555} }, /* -1/6-d6 */ -/**/ d7 = {{0x3fc24924, 0x92492492} }, /* 1/7 */ -/**/ dd7 = {{0x3c624924, 0x92492492} }, /* 1/7-d7 */ -/**/ d8 = {{0xbfc00000, 0x00000000} }, /* -1/8 */ -/**/ dd8 = {{0x00000000, 0x00000000} }, /* -1/8-d8 */ -/**/ d9 = {{0x3fbc71c7, 0x1c71c71c} }, /* 1/9 */ -/**/ dd9 = {{0x3c5c71c7, 0x1c71c71c} }, /* 1/9-d9 */ -/**/ d10 = {{0xbfb99999, 0x9999999a} }, /* -1/10 */ -/**/ dd10 = {{0x3c599999, 0x9999999a} }, /* -1/10-d10 */ -/**/ d11 = {{0x3fb745d1, 0x745d1746} }, /* 1/11 */ -/**/ d12 = {{0xbfb55555, 0x55555555} }, /* -1/12 */ -/**/ d13 = {{0x3fb3b13b, 0x13b13b14} }, /* 1/13 */ -/**/ d14 = {{0xbfb24924, 0x92492492} }, /* -1/14 */ -/**/ d15 = {{0x3fb11111, 0x11111111} }, /* 1/15 */ -/**/ d16 = {{0xbfb00000, 0x00000000} }, /* -1/16 */ -/**/ d17 = {{0x3fae1e1e, 0x1e1e1e1e} }, /* 1/17 */ -/**/ d18 = {{0xbfac71c7, 0x1c71c71c} }, /* -1/18 */ -/**/ d19 = {{0x3faaf286, 0xbca1af28} }, /* 1/19 */ -/**/ d20 = {{0xbfa99999, 0x9999999a} }, /* -1/20 */ /* constants */ /**/ sqrt_2 = {{0x3ff6a09e, 0x667f3bcc} }, /* sqrt(2) */ /**/ h1 = {{0x3fd2e000, 0x00000000} }, /* 151/2**9 */ @@ -89,12 +52,6 @@ /**/ ln2b = {{0x3d2ef357, 0x93c76730} }, /* ln(2)-ln2a */ /**/ e1 = {{0x3bbcc868, 0x00000000} }, /* 6.095e-21 */ /**/ e2 = {{0x3c1138ce, 0x00000000} }, /* 2.334e-19 */ -/**/ e3 = {{0x3aa1565d, 0x00000000} }, /* 2.801e-26 */ -/**/ e4 = {{0x39809d88, 0x00000000} }, /* 1.024e-31 */ -/**/ e[M] ={{{0x37da223a, 0x00000000} }, /* 1.2e-39 */ -/**/ {{0x35c851c4, 0x00000000} }, /* 1.3e-49 */ -/**/ {{0x2ab85e51, 0x00000000} }, /* 6.8e-103 */ -/**/ {{0x17383827, 0x00000000} }},/* 8.1e-197 */ /**/ two54 = {{0x43500000, 0x00000000} }, /* 2**54 */ /**/ u03 = {{0x3f9eb851, 0xeb851eb8} }; /* 0.03 */ @@ -114,43 +71,6 @@ /**/ b6 = {{0x25db58ac, 0x3fbc71c5} }, /* 0.111... */ /**/ b7 = {{0x11a2a61c, 0xbfb9a4ac} }, /* -0.100... */ /**/ b8 = {{0x0df2b591, 0x3fb75077} }, /* 0.091... */ - /* polynomial III */ -#if 0 -/**/ c1 = {{0x00000000, 0x3ff00000} }, /* 1 */ -#endif -/**/ c2 = {{0x00000000, 0xbfe00000} }, /* -1/2 */ -/**/ c3 = {{0x55555555, 0x3fd55555} }, /* 1/3 */ -/**/ c4 = {{0x00000000, 0xbfd00000} }, /* -1/4 */ -/**/ c5 = {{0x9999999a, 0x3fc99999} }, /* 1/5 */ - /* polynomial IV */ -/**/ d2 = {{0x00000000, 0xbfe00000} }, /* -1/2 */ -/**/ dd2 = {{0x00000000, 0x00000000} }, /* -1/2-d2 */ -/**/ d3 = {{0x55555555, 0x3fd55555} }, /* 1/3 */ -/**/ dd3 = {{0x55555555, 0x3c755555} }, /* 1/3-d3 */ -/**/ d4 = {{0x00000000, 0xbfd00000} }, /* -1/4 */ -/**/ dd4 = {{0x00000000, 0x00000000} }, /* -1/4-d4 */ -/**/ d5 = {{0x9999999a, 0x3fc99999} }, /* 1/5 */ -/**/ dd5 = {{0x9999999a, 0xbc699999} }, /* 1/5-d5 */ -/**/ d6 = {{0x55555555, 0xbfc55555} }, /* -1/6 */ -/**/ dd6 = {{0x55555555, 0xbc655555} }, /* -1/6-d6 */ -/**/ d7 = {{0x92492492, 0x3fc24924} }, /* 1/7 */ -/**/ dd7 = {{0x92492492, 0x3c624924} }, /* 1/7-d7 */ -/**/ d8 = {{0x00000000, 0xbfc00000} }, /* -1/8 */ -/**/ dd8 = {{0x00000000, 0x00000000} }, /* -1/8-d8 */ -/**/ d9 = {{0x1c71c71c, 0x3fbc71c7} }, /* 1/9 */ -/**/ dd9 = {{0x1c71c71c, 0x3c5c71c7} }, /* 1/9-d9 */ -/**/ d10 = {{0x9999999a, 0xbfb99999} }, /* -1/10 */ -/**/ dd10 = {{0x9999999a, 0x3c599999} }, /* -1/10-d10 */ -/**/ d11 = {{0x745d1746, 0x3fb745d1} }, /* 1/11 */ -/**/ d12 = {{0x55555555, 0xbfb55555} }, /* -1/12 */ -/**/ d13 = {{0x13b13b14, 0x3fb3b13b} }, /* 1/13 */ -/**/ d14 = {{0x92492492, 0xbfb24924} }, /* -1/14 */ -/**/ d15 = {{0x11111111, 0x3fb11111} }, /* 1/15 */ -/**/ d16 = {{0x00000000, 0xbfb00000} }, /* -1/16 */ -/**/ d17 = {{0x1e1e1e1e, 0x3fae1e1e} }, /* 1/17 */ -/**/ d18 = {{0x1c71c71c, 0xbfac71c7} }, /* -1/18 */ -/**/ d19 = {{0xbca1af28, 0x3faaf286} }, /* 1/19 */ -/**/ d20 = {{0x9999999a, 0xbfa99999} }, /* -1/20 */ /* constants */ /**/ sqrt_2 = {{0x667f3bcc, 0x3ff6a09e} }, /* sqrt(2) */ /**/ h1 = {{0x00000000, 0x3fd2e000} }, /* 151/2**9 */ @@ -161,12 +81,6 @@ /**/ ln2b = {{0x93c76730, 0x3d2ef357} }, /* ln(2)-ln2a */ /**/ e1 = {{0x00000000, 0x3bbcc868} }, /* 6.095e-21 */ /**/ e2 = {{0x00000000, 0x3c1138ce} }, /* 2.334e-19 */ -/**/ e3 = {{0x00000000, 0x3aa1565d} }, /* 2.801e-26 */ -/**/ e4 = {{0x00000000, 0x39809d88} }, /* 1.024e-31 */ -/**/ e[M] ={{{0x00000000, 0x37da223a} }, /* 1.2e-39 */ -/**/ {{0x00000000, 0x35c851c4} }, /* 1.3e-49 */ -/**/ {{0x00000000, 0x2ab85e51} }, /* 6.8e-103 */ -/**/ {{0x00000000, 0x17383827} }},/* 8.1e-197 */ /**/ two54 = {{0x00000000, 0x43500000} }, /* 2**54 */ /**/ u03 = {{0xeb851eb8, 0x3f9eb851} }; /* 0.03 */