From patchwork Mon Nov 1 20:20:56 2021 Content-Type: text/plain; charset="utf-8" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit X-Patchwork-Submitter: Adhemerval Zanella X-Patchwork-Id: 46920 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 8BC7F385802B for ; Mon, 1 Nov 2021 20:24:23 +0000 (GMT) DKIM-Filter: OpenDKIM Filter v2.11.0 sourceware.org 8BC7F385802B DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=sourceware.org; s=default; t=1635798263; bh=JV9NVd9pYhvrmIpR52JbY4sZFWLAXpo0DtuUOU5hVDs=; h=To:Subject:Date:In-Reply-To:References:List-Id:List-Unsubscribe: List-Archive:List-Post:List-Help:List-Subscribe:From:Reply-To:Cc: From; b=EjIxMo5eEPi//JG8fFPMzR22reZqrCG6jMlCIyT4TEeXCyqk3oEZR9NXXPkXxxH+M iwhqONIYZ+t6BCRnvph5dDgbVTRLkVBwoI9OSgBdwpNfCHGwQc5h8likv4MrMibdIi ZDFHMVQ/+4QNuyntKXnic/cDTp457RMqpBmK14S0= X-Original-To: libc-alpha@sourceware.org Delivered-To: libc-alpha@sourceware.org Received: from mail-qk1-x731.google.com (mail-qk1-x731.google.com [IPv6:2607:f8b0:4864:20::731]) by sourceware.org (Postfix) with ESMTPS id A48303858001 for ; Mon, 1 Nov 2021 20:21:10 +0000 (GMT) DMARC-Filter: OpenDMARC Filter v1.4.1 sourceware.org A48303858001 Received: by mail-qk1-x731.google.com with SMTP id bl12so3461791qkb.13 for ; Mon, 01 Nov 2021 13:21:10 -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=JV9NVd9pYhvrmIpR52JbY4sZFWLAXpo0DtuUOU5hVDs=; b=aiF6BtWQ9iriMv7dko0Wn3Hyidkyz+elS8BGJew7YzbDbNuHBk1zdi/VI8BxYuhybE 0nUuBIkK3vqhA8y44Nt/wIm2BUiA91cu/AFDfXbDvPcEuIwIXA3fCND/QXSh7s8kqLa9 nje3ZtJJURCTWWWNsrH4LZF4meg34pwTuQiIpaWy42zRI5r7wIbJ3gIo++YZSBzjysXN v42dnxIckOZg3MZNZGr5BJZc0VZXMlJ651paQZ6Ec6nCYhRbqVhe1tsstmAveQRszQek sUme6pn97mS/xOnyW6TTkDbXmEgZebbDokVv2Ay55mqkER42jgiOYfR01sXAbQ6uYFo2 HnNw== X-Gm-Message-State: AOAM531zPf0g+AxV4NLw8rbP2DKIoG6ne3JETaOinlRU6YgqqDs1nGHj UfxBgoFQqCHJc3JMx/bkubNDqos3r5ZXjg== X-Google-Smtp-Source: ABdhPJxstKPg0DjnsDkSyUslB/PVGEyRvj6yEWnKyj0aYQn/OkzL3LffnsMJ8anOsFG7VAWqLFKsvw== X-Received: by 2002:a05:620a:1910:: with SMTP id bj16mr12383514qkb.34.1635798070113; Mon, 01 Nov 2021 13:21:10 -0700 (PDT) Received: from birita.. ([2804:431:c7cb:b64f:7c54:165f:8728:a193]) by smtp.gmail.com with ESMTPSA id d11sm10023965qtx.81.2021.11.01.13.21.08 (version=TLS1_3 cipher=TLS_AES_256_GCM_SHA384 bits=256/256); Mon, 01 Nov 2021 13:21:09 -0700 (PDT) To: libc-alpha@sourceware.org Subject: [PATCH v3 4/7] math: Use an improved algorithm for hypotl (ldbl-128) Date: Mon, 1 Nov 2021 17:20:56 -0300 Message-Id: <20211101202059.1026032-5-adhemerval.zanella@linaro.org> X-Mailer: git-send-email 2.32.0 In-Reply-To: <20211101202059.1026032-1-adhemerval.zanella@linaro.org> References: <20211101202059.1026032-1-adhemerval.zanella@linaro.org> MIME-Version: 1.0 X-Spam-Status: No, score=-12.0 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 Cc: Wilco Dijkstra 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 1e9 input pairs in the range of [LDBL_MIN, LDBL_MAX], glibc current implementation shows around 0.05% results with an error of 1 ulp (453266 results) while the new implementation only shows 0.0001% of total (1280). Checked on aarch64-linux-gnu and x86_64-linux-gnu. [1] https://arxiv.org/pdf/1904.09481.pdf --- sysdeps/ieee754/ldbl-128/e_hypotl.c | 224 ++++++++++++---------------- 1 file changed, 97 insertions(+), 127 deletions(-) diff --git a/sysdeps/ieee754/ldbl-128/e_hypotl.c b/sysdeps/ieee754/ldbl-128/e_hypotl.c index cd4fdbc4a6..c5068e330a 100644 --- a/sysdeps/ieee754/ldbl-128/e_hypotl.c +++ b/sysdeps/ieee754/ldbl-128/e_hypotl.c @@ -1,141 +1,111 @@ -/* e_hypotl.c -- long double version of e_hypot.c. - */ +/* Euclidean distance function. Long Double/Binary128 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 sqrtl(2)/2 ulp, than - * sqrtl(z) has error less than 1 ulp (exercise). - * - * So, compute sqrtl(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 64 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 64 bits cleared, t2 = 2x-t1, - * y1= y with lower 64 bits chopped, y2 = y-y1. - * - * NOTE: scaling may be necessary if some argument is too - * large or too tiny - * - * Special cases: - * hypotl(x,y) is INF if x or y is +INF or -INF; else - * hypotl(x,y) is NAN if x or y is NAN. - * - * Accuracy: - * hypotl(x,y) returns sqrtl(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 +/* sqrt (LDBL_EPSILON / 2.0) */ +#define SQRT_EPS_DIV_2 L(0x1.6a09e667f3bcc908b2fb1366ea95p-57) +/* DBL_MIN / (sqrt (LDBL_EPSILON / 2.0)) */ +#define LDBL_MIN_THRESHOLD L(0x1.6a09e667f3bcc908b2fb1366ea96p-16326) +/* eps (long double) *(sqrt (LDBL_MIN) */ +#define SCALE L(0x1p-8303) +/* 1 / eps (sqrt (LDBL_MIN) */ +#define INV_SCALE L(0x1p+8303) +/* sqrt (LDBL_MAX) */ +#define SQRT_LDBL_MAX L(0x1.6a09e667f3bcc908b2fb1366ea95p+8191) +/* sqrt (LDBL_MIN) */ +#define SQRT_LDBL_MIN L(0x1p-8191) + _Float128 __ieee754_hypotl(_Float128 x, _Float128 y) { - _Float128 a,b,t1,t2,y1,y2,w; - int64_t j,k,ha,hb; + if (!isfinite(x) || !isfinite(y)) + { + if ((isinf (x) || isinf (y)) + && !issignaling (x) && !issignaling (y)) + return INFINITY; + return x + y; + } + + _Float128 ax = fabsl (x); + _Float128 ay = fabsl (y); + if (ay > ax) + { + _Float128 tmp = ax; + ax = ay; + ay = tmp; + } + + /* Widely varying operands. The DBL_MIN_THRESHOLD check is used to avoid + an 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; + + _Float128 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 = 1.0; + + _Float128 h = sqrtl (ax * ax + ay * ay); - GET_LDOUBLE_MSW64(ha,x); - ha &= 0x7fffffffffffffffLL; - GET_LDOUBLE_MSW64(hb,y); - hb &= 0x7fffffffffffffffLL; - if(hb > ha) {a=y;b=x;j=ha; ha=hb;hb=j;} else {a=x;b=y;} - SET_LDOUBLE_MSW64(a,ha); /* a <- |a| */ - SET_LDOUBLE_MSW64(b,hb); /* b <- |b| */ - if((ha-hb)>0x78000000000000LL) {return a+b;} /* x/y > 2**120 */ - k=0; - if(ha > 0x5f3f000000000000LL) { /* a>2**8000 */ - if(ha >= 0x7fff000000000000LL) { /* Inf or NaN */ - uint64_t low; - w = a+b; /* for sNaN */ - if (issignaling (a) || issignaling (b)) - return w; - GET_LDOUBLE_LSW64(low,a); - if(((ha&0xffffffffffffLL)|low)==0) w = a; - GET_LDOUBLE_LSW64(low,b); - if(((hb^0x7fff000000000000LL)|low)==0) w = b; - return w; - } - /* scale a and b by 2**-9600 */ - ha -= 0x2580000000000000LL; - hb -= 0x2580000000000000LL; k += 9600; - SET_LDOUBLE_MSW64(a,ha); - SET_LDOUBLE_MSW64(b,hb); - } - if(hb < 0x20bf000000000000LL) { /* b < 2**-8000 */ - if(hb <= 0x0000ffffffffffffLL) { /* subnormal b or 0 */ - uint64_t low; - GET_LDOUBLE_LSW64(low,b); - if((hb|low)==0) return a; - t1=0; - SET_LDOUBLE_MSW64(t1,0x7ffd000000000000LL); /* t1=2^16382 */ - b *= t1; - a *= t1; - k -= 16382; - GET_LDOUBLE_MSW64 (ha, a); - GET_LDOUBLE_MSW64 (hb, b); - if (hb > ha) - { - t1 = a; - a = b; - b = t1; - j = ha; - ha = hb; - hb = j; - } - } else { /* scale a and b by 2^9600 */ - ha += 0x2580000000000000LL; /* a *= 2^9600 */ - hb += 0x2580000000000000LL; /* b *= 2^9600 */ - k -= 9600; - SET_LDOUBLE_MSW64(a,ha); - SET_LDOUBLE_MSW64(b,hb); - } - } - /* medium size a and b */ - w = a-b; - if (w>b) { - t1 = 0; - SET_LDOUBLE_MSW64(t1,ha); - t2 = a-t1; - w = sqrtl(t1*t1-(b*(-b)-t2*(a+t1))); - } else { - a = a+a; - y1 = 0; - SET_LDOUBLE_MSW64(y1,hb); - y2 = b - y1; - t1 = 0; - SET_LDOUBLE_MSW64(t1,ha+0x0001000000000000LL); - t2 = a - t1; - w = sqrtl(t1*y1-(w*(-w)-(t1*y2+t2*b))); - } - if(k!=0) { - uint64_t high; - t1 = 1; - GET_LDOUBLE_MSW64(high,t1); - SET_LDOUBLE_MSW64(t1,high+(k<<48)); - w *= t1; - math_check_force_underflow_nonneg (w); - return w; - } else return w; + _Float128 t1; + _Float128 t2; + if (h == 0.0) + return h; + if (h <= 2.0 * ay) + { + _Float128 delta = h - ay; + t1 = ax * (2.0 * delta - ax); + t2 = (delta - 2.0 * (ax - ay)) * delta; + } + else + { + _Float128 delta = h - ax; + t1 = 2.0 * delta * (ax - 2 * ay); + t2 = (4.0 * delta - ay) * ay + delta * delta; + } + h -= (t1 + t2) / (2.0 * h); + h *= scale; + math_check_force_underflow_nonneg (h); + return h; } libm_alias_finite (__ieee754_hypotl, __hypotl)