From owner-freebsd-standards@FreeBSD.ORG Sun Dec 9 21:39:17 2007 Return-Path: Delivered-To: freebsd-standards@FreeBSD.ORG Received: from mx1.freebsd.org (mx1.freebsd.org [IPv6:2001:4f8:fff6::34]) by hub.freebsd.org (Postfix) with ESMTP id 2287416A46E for ; Sun, 9 Dec 2007 21:39:17 +0000 (UTC) (envelope-from sgk@troutmask.apl.washington.edu) Received: from troutmask.apl.washington.edu (troutmask.apl.washington.edu [128.208.78.105]) by mx1.freebsd.org (Postfix) with ESMTP id EABE713C4D1 for ; Sun, 9 Dec 2007 21:39:16 +0000 (UTC) (envelope-from sgk@troutmask.apl.washington.edu) Received: from troutmask.apl.washington.edu (localhost.apl.washington.edu [127.0.0.1]) by troutmask.apl.washington.edu (8.14.1/8.14.1) with ESMTP id lB9LYoZ4096090 for ; Sun, 9 Dec 2007 13:34:50 -0800 (PST) (envelope-from sgk@troutmask.apl.washington.edu) Received: (from sgk@localhost) by troutmask.apl.washington.edu (8.14.1/8.14.1/Submit) id lB9LYo1F096089 for freebsd-standards@FreeBSD.ORG; Sun, 9 Dec 2007 13:34:50 -0800 (PST) (envelope-from sgk) Date: Sun, 9 Dec 2007 13:34:50 -0800 From: Steve Kargl To: freebsd-standards@FreeBSD.ORG Message-ID: <20071209213450.GA95726@troutmask.apl.washington.edu> References: <20071103001305.GA82775@troutmask.apl.washington.edu> <20071209212505.GA9698@VARK.MIT.EDU> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: <20071209212505.GA9698@VARK.MIT.EDU> User-Agent: Mutt/1.4.2.3i Cc: Subject: Re: Implementation of expl() X-BeenThere: freebsd-standards@freebsd.org X-Mailman-Version: 2.1.5 Precedence: list List-Id: Standards compliance List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , X-List-Received-Date: Sun, 09 Dec 2007 21:39:17 -0000 On Sun, Dec 09, 2007 at 04:25:05PM -0500, David Schultz wrote: > On Fri, Nov 02, 2007, Steve Kargl wrote: > > With all the success of developing several other missing > > C99 math routines, I decided to tackle expl() (which I > > need to tackle tanhl()). > > Hmm, great, but where's the patch? :) Maybe the mailing list > software ate it. This is the current version. I need to revise how I computed the ploynomial coefficient. In short, I mapped r in [-ln(2)/2:ln(2)/2] into the range x in [-1,1] for the Chebyshev interpolation, but I never scaled x back into r. This is the reason why the lines "r = r * TOLN2;" exists. I don't remember if bde sent me comments on this code. I sure he has plenty. :) steve /*- * Copyright (c) 2007 Steven G. Kargl * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice unmodified, this list of conditions, and the following * disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ /* * Use argument reduction to compute exp(x). The reduction writes * x = k * ln(2) + r with r in the range [0, ln(2)]. This then * gives exp(x) = 2**k * exp(r), and exp(r) is evaluated via a nearly * minimax polynomial approximation such that r is mapped into [-1:1]. */ #include "math.h" #include "math_private.h" #include "fpmath.h" /* ln(LDBL_MAX) = 11356.523406294144 */ #define XMAX 0x2.c5c85fdf473de6ap12L /* ln(LDBL_MIN) = -11355.137111933024 */ #define XMIN -0x2.c5b2319c4843accp12L /* | ln(smallest subnormal) | = 11399.498531488861 */ #define GRAD 0x2.c877f9fc278aeaap12L #define LN2 0xb.17217f7d1cf79acp-4L /* ln(2) */ #define LN2HI 0xb.17217f7d2000000p-4L #define LN2LO -0x3.08654361c4c67fcp-44L #define TOLN2 0x2.e2a8eca5705fc2fp0L /* 2/ln(2) */ #define ILN2 0x1.71547652b82fe17p0L /* 1/ln(2) */ #define ILN2HI 0x1.71547652b800000p0L #define ILN2LO 0x2.fe1777d0ffda0d2p-44L #define LN2O2 0x5.8b90bfbe8e7bcd6p-4L /* ln(2)/2 */ #define ZERO 0.L /* * This set of coefficients is used in the polynomial approximation * for exp(r) where r is in [-ln(2)/2:ln(2)/2], and the r comes from * the argument reduction of x. */ #define C0 0x1.000000000000000p0L #define C1 0x5.8b90bfbe8e7bcd6p-4L #define C2 0xf.5fdeffc162c7543p-8L #define C3 0x1.c6b08d704a0bf8bp-8L #define C4 0x2.76556df749cee54p-12L #define C5 0x2.bb0ffcf14ce6221p-16L #define C6 0x2.861225f0d8f0edfp-20L #define C7 0x1.ffcbfc588b0c687p-24L #define C8 0x1.62c0223a5c823fdp-28L #define C9 0xd.a929e9caf3e1ed2p-36L #define C10 0x7.933d4562e3b2cd7p-40L #define C11 0x3.d1958e6a3764b64p-44L #define C12 0x1.c3bd650fc1e343ap-48L #define C13 0xc.0b0c98b3649ff26p-56L #define C14 0x4.c525936609b02cfp-60L #define C15 0x1.c36e84400493e74p-64L long double expl(long double x) { union IEEEl2bits z; int k, s; long double r; z.e = x; s = z.bits.sign; z.bits.sign = 0; /* x is either 0 or a subnormal number. */ if (z.bits.exp == 0) { if ((z.bits.manl | z.bits.manh) == 0) return (1); else return (1 + x); } if (XMIN <= x && x <= XMAX) { /* Argument reduction. */ k = (int) (z.e * ILN2HI + z.e * ILN2LO); r = z.e - k * LN2HI - k * LN2LO; if (r > LN2O2) { r -= LN2; k++; } /* Compute exp(r) via the polynomial approximation. */ r = r * TOLN2; z.e = C0 + r * (C1 + r * (C2 + r * (C3 + r * (C4 + r * (C5 + r * (C6 + r * (C7 + r * (C8 + r * (C9 + r * (C10 + r * (C11 + r * (C12 + r * (C13 + r * (C14 + r * C15)))))))))))))); if (s) { z.e = 1 / z.e; z.bits.exp -= k; } else z.bits.exp += k; return (z.e); } /* * If x = +Inf, then exp(x) = Inf. * If x = -Inf, then exp(x) = 0. * If x = NaN, then exp(x) = NaN. */ if (z.bits.exp == 32767) { mask_nbit_l(z); if (!(z.bits.manh | z.bits.manl)) return (s ? ZERO : 1.L / ZERO); return ((x - x) / (x - x)); } /* If x > 0 then, overflow to +Inf. */ if (!s) return (1.L / ZERO); /* For x < 0, check if gradual underflow is needed. */ if (z.e > GRAD) return (ZERO); /* Argument reduction. */ k = (int) (z.e * ILN2); r = z.e - k * LN2HI - k * LN2LO; if (r > LN2O2) { r -= LN2; k++; } /* Compute exp(r) via the polynomial approximation. */ r *= TOLN2; z.e = C0 + r * (C1 + r * (C2 + r * (C3 + r * (C4 + r * (C5 + r * (C6 + r * (C7 + r * (C8 + r * (C9 + r * (C10 + r * (C11 + r * (C12 + r * (C13 + r * (C14 + r * C15)))))))))))))); z.e = 1 / z.e; /* * FIXME: There has to be a better way to handle gradual underflow * because the relative absolute error is fairly large for numerous * vlaues of x as exp(x) goes to zero. */ z.e = scalbnl(z.e, -k); return (z.e); }