From patchwork Mon Oct 25 11:57:52 2021 Content-Type: text/plain; charset="utf-8" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit X-Patchwork-Submitter: Adhemerval Zanella X-Patchwork-Id: 46604 Return-Path: X-Original-To: patchwork@sourceware.org Delivered-To: patchwork@sourceware.org Received: from server2.sourceware.org (localhost [IPv6:::1]) by sourceware.org (Postfix) with ESMTP id 7BE5B385843C for ; Mon, 25 Oct 2021 12:02:07 +0000 (GMT) DKIM-Filter: OpenDKIM Filter v2.11.0 sourceware.org 7BE5B385843C DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=sourceware.org; s=default; t=1635163327; bh=qMaM/yEcjFMGytPEZ6fLezjYarbULGSiYiE6vk1h7kI=; h=To:Subject:Date:In-Reply-To:References:List-Id:List-Unsubscribe: List-Archive:List-Post:List-Help:List-Subscribe:From:Reply-To: From; b=I2PY+tf6Yr4Ail8eZKqXPowxD83OJuA5kMrtydHzUoLI/HrkbjuhhjXB0Rw0SU2UF QjF4TdwZ6DPfEAYkHJM0WEruKXn0KuPsySE8StDtRRlvQYWsugc7gNtEys24SQ746z kCw78FcPSphl9PbvROHI6oO+O7y3Kotd5MaucHiU= X-Original-To: libc-alpha@sourceware.org Delivered-To: libc-alpha@sourceware.org Received: from mail-ua1-x92e.google.com (mail-ua1-x92e.google.com [IPv6:2607:f8b0:4864:20::92e]) by sourceware.org (Postfix) with ESMTPS id AA6523858413 for ; Mon, 25 Oct 2021 11:58:09 +0000 (GMT) DMARC-Filter: OpenDMARC Filter v1.4.1 sourceware.org AA6523858413 Received: by mail-ua1-x92e.google.com with SMTP id f24so2331853uav.12 for ; Mon, 25 Oct 2021 04:58:09 -0700 (PDT) X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20210112; h=x-gm-message-state:from:to:cc:subject:date:message-id:in-reply-to :references:mime-version:content-transfer-encoding; bh=qMaM/yEcjFMGytPEZ6fLezjYarbULGSiYiE6vk1h7kI=; b=W64CI04Fhit344fuLO2kUernlEiztr6pYfe6YXQMRbEUsK+NjTNZrk+PRFXGFT6YmY MboUc198Yb7ycpqJP39g/IrfTvMO3MB+Kaq1YiDbY5PqLORmosBN2N1Z3i5W4meWsdTp sculUKm5OxJYRjw/j7F2JEdxdBva+OgkkzsnRB33QzxtHD+l2gUN2VFKqF8Tdvd0CmsS BCguO4/tzBLGQd5xlmBNXTlwYpWCXXHZDISnXQu29baBq0vmKId8byQDKzZICat3fOen zMNx1y0y7d6EOSQEwP6UK94nIVWyS352dza51wX5a2X789MWYTRnEkKc2zenAuaAnd1M 5O8g== X-Gm-Message-State: AOAM5320Fg/JiXgYRBGcCEO6CRpLtPEPyiZ+5qBiHTL5cY8fQVq4vxMw D0jGPjmDPvq0r0sPn8zeRPNYLNO6ej0Bdg== X-Google-Smtp-Source: ABdhPJzyAaOby59HC0c83EBx1JQr8SssA1n6xpksrU53QT+Ry6xjFd2fXX7LExWOu1KXR28YAnFUjg== X-Received: by 2002:a9f:21b7:: with SMTP id 52mr14054932uac.9.1635163088938; Mon, 25 Oct 2021 04:58:08 -0700 (PDT) Received: from birita.. ([2804:431:c7ca:2654:4c60:ad20:95b6:1d6c]) by smtp.gmail.com with ESMTPSA id k11sm4230490vsh.3.2021.10.25.04.58.07 (version=TLS1_3 cipher=TLS_AES_256_GCM_SHA384 bits=256/256); Mon, 25 Oct 2021 04:58:08 -0700 (PDT) To: libc-alpha@sourceware.org Subject: [PATCH v2 5/9] math: Use an improved algorithm for hypotl (ldbl-96) Date: Mon, 25 Oct 2021 08:57:52 -0300 Message-Id: <20211025115756.11767-6-adhemerval.zanella@linaro.org> X-Mailer: git-send-email 2.32.0 In-Reply-To: <20211025115756.11767-1-adhemerval.zanella@linaro.org> References: <20211025115756.11767-1-adhemerval.zanella@linaro.org> MIME-Version: 1.0 X-Spam-Status: No, score=-12.2 required=5.0 tests=BAYES_00, DKIM_SIGNED, DKIM_VALID, DKIM_VALID_AU, DKIM_VALID_EF, GIT_PATCH_0, KAM_ASCII_DIVIDERS, KAM_SHORT, RCVD_IN_DNSWL_NONE, SPF_HELO_NONE, SPF_PASS, TXREP autolearn=ham autolearn_force=no version=3.4.4 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on server2.sourceware.org X-BeenThere: libc-alpha@sourceware.org X-Mailman-Version: 2.1.29 Precedence: list List-Id: Libc-alpha mailing list List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , X-Patchwork-Original-From: Adhemerval Zanella via Libc-alpha From: Adhemerval Zanella Reply-To: Adhemerval Zanella Errors-To: libc-alpha-bounces+patchwork=sourceware.org@sourceware.org Sender: "Libc-alpha" This implementation is based on 'An Improved Algorithm for hypot(a,b)' by Carlos F. Borges [1] using the MyHypot3 with the following changes: - Handle qNaN and sNaN. - Tune the 'widely varying operands' to avoid spurious underflow due the multiplication and fix the return value for upwards rounding mode. - Handle required underflow exception for subnormal results. The main advantage of the new algorithm is its precision. With a random 1e8 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc current implementation shows around 0.02% results with an error of 1 ulp (23158 results) while the new implementation only shows 0.0001% of total (111). [1] https://arxiv.org/pdf/1904.09481.pdf --- sysdeps/ieee754/ldbl-96/e_hypotl.c | 227 ++++++++++++----------------- 1 file changed, 97 insertions(+), 130 deletions(-) diff --git a/sysdeps/ieee754/ldbl-96/e_hypotl.c b/sysdeps/ieee754/ldbl-96/e_hypotl.c index 44e72353c0..85dcbc894b 100644 --- a/sysdeps/ieee754/ldbl-96/e_hypotl.c +++ b/sysdeps/ieee754/ldbl-96/e_hypotl.c @@ -1,142 +1,109 @@ -/* e_hypotl.c -- long double version of e_hypot.c. - */ +/* Euclidean distance function. Long Double/Binary96 version. + Copyright (C) 2021 Free Software Foundation, Inc. + This file is part of the GNU C Library. -/* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. - * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ + 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. -/* __ieee754_hypotl(x,y) - * - * Method : - * If (assume round-to-nearest) z=x*x+y*y - * has error less than sqrt(2)/2 ulp, than - * sqrt(z) has error less than 1 ulp (exercise). - * - * So, compute sqrt(x*x+y*y) with some care as - * follows to get the error below 1 ulp: - * - * Assume x>y>0; - * (if possible, set rounding to round-to-nearest) - * 1. if x > 2y use - * x1*x1+(y*y+(x2*(x+x1))) for x*x+y*y - * where x1 = x with lower 32 bits cleared, x2 = x-x1; else - * 2. if x <= 2y use - * t1*y1+((x-y)*(x-y)+(t1*y2+t2*y)) - * where t1 = 2x with lower 32 bits cleared, t2 = 2x-t1, - * y1= y with lower 32 bits chopped, y2 = y-y1. - * - * NOTE: scaling may be necessary if some argument is too - * large or too tiny - * - * Special cases: - * hypot(x,y) is INF if x or y is +INF or -INF; else - * hypot(x,y) is NAN if x or y is NAN. - * - * Accuracy: - * hypot(x,y) returns sqrt(x^2+y^2) with error less - * than 1 ulps (units in the last place) - */ + 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 + . */ + +/* This implementation is based on 'An Improved Algorithm for hypot(a,b)' by + Carlos F. Borges [1] using the MyHypot3 with the following changes: + + - Handle qNaN and sNaN. + - Tune the 'widely varying operands' to avoid spurious underflow + due the multiplication and fix the return value for upwards + rounding mode. + - Handle required underflow exception for subnormal results. + + [1] https://arxiv.org/pdf/1904.09481.pdf */ #include #include #include #include -long double __ieee754_hypotl(long double x, long double y) +/* sqrt (LDBL_EPSILON / 2.0) */ +#define SQRT_EPS_DIV_2 0x8p-35L +/* DBL_MIN / (sqrt (LDBL_EPSILON / 2.0)) */ +#define LDBL_MIN_THRESHOLD 0x8p-16353L +/* eps (long double) * sqrt (LDBL_MIN) */ +#define SCALE 0x8p-8257L +/* 1 / eps (sqrt (LDBL_MIN) */ +#define INV_SCALE 0x8p+8251L +/* sqrt (LDBL_MAX) */ +#define SQRT_LDBL_MAX 0xb.504f333f9de6484p+8188L +/* sqrt (LDBL_MIN) */ +#define SQRT_LDBL_MIN 0x8p-8194L + +long double +__ieee754_hypotl (long double x, long double y) { - long double a,b,t1,t2,y1,y2,w; - uint32_t j,k,ea,eb; + if ((isinf (x) || isinf (y)) + && !issignaling (x) && !issignaling (y)) + return INFINITY; + if (isnan (x) || isnan (y)) + return x + y; + + long double ax = fabsl (x); + long double ay = fabsl (y); + if (ay > ax) + { + long double tmp = ax; + ax = ay; + ay = tmp; + } + + /* Widely varying operands. The LDBL_MIN_THRESHOLD check is used to avoid + a spurious underflow from the multiplication. */ + if (ax >= LDBL_MIN_THRESHOLD && ay <= ax * SQRT_EPS_DIV_2) + return (ay == 0.0) ? ax : ax + LDBL_TRUE_MIN; + + long double scale = SCALE; + if (ax > SQRT_LDBL_MAX) + { + ax *= scale; + ay *= scale; + scale = INV_SCALE; + } + else if (ay < SQRT_LDBL_MIN) + { + ax /= scale; + ay /= scale; + } + else + scale = 1.0L; + + long double h = sqrtl (ax * ax + ay * ay); - GET_LDOUBLE_EXP(ea,x); - ea &= 0x7fff; - GET_LDOUBLE_EXP(eb,y); - eb &= 0x7fff; - if(eb > ea) {a=y;b=x;j=ea; ea=eb;eb=j;} else {a=x;b=y;} - SET_LDOUBLE_EXP(a,ea); /* a <- |a| */ - SET_LDOUBLE_EXP(b,eb); /* b <- |b| */ - if((ea-eb)>0x46) {return a+b;} /* x/y > 2**70 */ - k=0; - if(__builtin_expect(ea > 0x5f3f,0)) { /* a>2**8000 */ - if(ea == 0x7fff) { /* Inf or NaN */ - uint32_t exp __attribute__ ((unused)); - uint32_t high,low; - w = a+b; /* for sNaN */ - if (issignaling (a) || issignaling (b)) - return w; - GET_LDOUBLE_WORDS(exp,high,low,a); - if(((high&0x7fffffff)|low)==0) w = a; - GET_LDOUBLE_WORDS(exp,high,low,b); - if(((eb^0x7fff)|(high&0x7fffffff)|low)==0) w = b; - return w; - } - /* scale a and b by 2**-9600 */ - ea -= 0x2580; eb -= 0x2580; k += 9600; - SET_LDOUBLE_EXP(a,ea); - SET_LDOUBLE_EXP(b,eb); - } - if(__builtin_expect(eb < 0x20bf, 0)) { /* b < 2**-8000 */ - if(eb == 0) { /* subnormal b or 0 */ - uint32_t exp __attribute__ ((unused)); - uint32_t high,low; - GET_LDOUBLE_WORDS(exp,high,low,b); - if((high|low)==0) return a; - SET_LDOUBLE_WORDS(t1, 0x7ffd, 0x80000000, 0); /* t1=2^16382 */ - b *= t1; - a *= t1; - k -= 16382; - GET_LDOUBLE_EXP (ea, a); - GET_LDOUBLE_EXP (eb, b); - if (eb > ea) - { - t1 = a; - a = b; - b = t1; - j = ea; - ea = eb; - eb = j; - } - } else { /* scale a and b by 2^9600 */ - ea += 0x2580; /* a *= 2^9600 */ - eb += 0x2580; /* b *= 2^9600 */ - k -= 9600; - SET_LDOUBLE_EXP(a,ea); - SET_LDOUBLE_EXP(b,eb); - } - } - /* medium size a and b */ - w = a-b; - if (w>b) { - uint32_t high; - GET_LDOUBLE_MSW(high,a); - SET_LDOUBLE_WORDS(t1,ea,high,0); - t2 = a-t1; - w = sqrtl(t1*t1-(b*(-b)-t2*(a+t1))); - } else { - uint32_t high; - GET_LDOUBLE_MSW(high,b); - a = a+a; - SET_LDOUBLE_WORDS(y1,eb,high,0); - y2 = b - y1; - GET_LDOUBLE_MSW(high,a); - SET_LDOUBLE_WORDS(t1,ea+1,high,0); - t2 = a - t1; - w = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b))); - } - if(k!=0) { - uint32_t exp; - t1 = 1.0; - GET_LDOUBLE_EXP(exp,t1); - SET_LDOUBLE_EXP(t1,exp+k); - w *= t1; - math_check_force_underflow_nonneg (w); - return w; - } else return w; + long double t1; + long double t2; + if (h == 0.0L) + return h; + if (h <= 2.0L * ay) + { + long double delta = h - ay; + t1 = ax * (2.0L * delta - ax); + t2 = (delta - 2.0L * (ax - ay)) * delta; + } + else + { + long double delta = h - ax; + t1 = 2.0L * delta * (ax - 2.0L * ay); + t2 = (4.0L * delta - ay) * ay + delta * delta; + } + h -= (t1 + t2) / (2.0L * h); + h *= scale; + math_check_force_underflow_nonneg (h); + return h; } libm_alias_finite (__ieee754_hypotl, __hypotl)