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+/*-
+ * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
+ *
+ * Copyright (c) 2007-2013 Bruce D. Evans
+ * 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.
+ */
+
+#include <sys/cdefs.h>
+__FBSDID("$FreeBSD$");
+
+/**
+ * Implementation of the natural logarithm of x for Intel 80-bit format.
+ *
+ * First decompose x into its base 2 representation:
+ *
+ * log(x) = log(X * 2**k), where X is in [1, 2)
+ * = log(X) + k * log(2).
+ *
+ * Let X = X_i + e, where X_i is the center of one of the intervals
+ * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
+ * and X is in this interval. Then
+ *
+ * log(X) = log(X_i + e)
+ * = log(X_i * (1 + e / X_i))
+ * = log(X_i) + log(1 + e / X_i).
+ *
+ * The values log(X_i) are tabulated below. Let d = e / X_i and use
+ *
+ * log(1 + d) = p(d)
+ *
+ * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
+ * suitably high degree.
+ *
+ * To get sufficiently small roundoff errors, k * log(2), log(X_i), and
+ * sometimes (if |k| is not large) the first term in p(d) must be evaluated
+ * and added up in extra precision. Extra precision is not needed for the
+ * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
+ * error is controlled mainly by the error in the second term in p(d). The
+ * error in this term itself is at most 0.5 ulps from the d*d operation in
+ * it. The error in this term relative to the first term is thus at most
+ * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
+ * at most twice this at the point of the final rounding step. Thus the
+ * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
+ * testing of a float variant of this function showed a maximum final error
+ * of 0.5008 ulps. Non-exhaustive testing of a double variant of this
+ * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
+ *
+ * We made the maximum of |d| (and thus the total relative error and the
+ * degree of p(d)) small by using a large number of intervals. Using
+ * centers of intervals instead of endpoints reduces this maximum by a
+ * factor of 2 for a given number of intervals. p(d) is special only
+ * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
+ * naturally. The most accurate minimax polynomial of a given degree might
+ * be different, but then we wouldn't want it since we would have to do
+ * extra work to avoid roundoff error (especially for P0*d instead of d).
+ */
+
+#ifdef DEBUG
+#include <assert.h>
+#include <fenv.h>
+#endif
+
+#ifdef __i386__
+#include <ieeefp.h>
+#endif
+
+#include "fpmath.h"
+#include "math.h"
+#define i386_SSE_GOOD
+#ifndef NO_STRUCT_RETURN
+#define STRUCT_RETURN
+#endif
+#include "math_private.h"
+
+#if !defined(NO_UTAB) && !defined(NO_UTABL)
+#define USE_UTAB
+#endif
+
+/*
+ * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]:
+ * |log(1 + d)/d - p(d)| < 2**-70.7
+ */
+static const double
+P2 = -0.5,
+P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */
+P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */
+P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */
+P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */
+P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */
+P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */
+
+static volatile const double zero = 0;
+
+#define INTERVALS 128
+#define LOG2_INTERVALS 7
+#define TSIZE (INTERVALS + 1)
+#define G(i) (T[(i)].G)
+#define F_hi(i) (T[(i)].F_hi)
+#define F_lo(i) (T[(i)].F_lo)
+#define ln2_hi F_hi(TSIZE - 1)
+#define ln2_lo F_lo(TSIZE - 1)
+#define E(i) (U[(i)].E)
+#define H(i) (U[(i)].H)
+
+static const struct {
+ float G; /* 1/(1 + i/128) rounded to 8/9 bits */
+ float F_hi; /* log(1 / G_i) rounded (see below) */
+ double F_lo; /* next 53 bits for log(1 / G_i) */
+} T[TSIZE] = {
+ /*
+ * ln2_hi and each F_hi(i) are rounded to a number of bits that
+ * makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
+ *
+ * The last entry (for X just below 2) is used to define ln2_hi
+ * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
+ * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
+ * This is needed for accuracy when x is just below 1. (To avoid
+ * special cases, such x are "reduced" strangely to X just below
+ * 2 and dk = -1, and then the exact cancellation is needed
+ * because any the error from any non-exactness would be too
+ * large).
+ *
+ * We want to share this table between double precision and ld80,
+ * so the relevant range of dk is the larger one of ld80
+ * ([-16445, 16383]) and the relevant exactness requirement is
+ * the stricter one of double precision. The maximum number of
+ * bits in F_hi(i) that works is very dependent on i but has
+ * a minimum of 33. We only need about 12 bits in F_hi(i) for
+ * it to provide enough extra precision in double precision (11
+ * more than that are required for ld80).
+ *
+ * We round F_hi(i) to 24 bits so that it can have type float,
+ * mainly to minimize the size of the table. Using all 24 bits
+ * in a float for it automatically satisfies the above constraints.
+ */
+ { 0x800000.0p-23, 0, 0 },
+ { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84 },
+ { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84 },
+ { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83 },
+ { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82 },
+ { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82 },
+ { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83 },
+ { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82 },
+ { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91 },
+ { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81 },
+ { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82 },
+ { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83 },
+ { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81 },
+ { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83 },
+ { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85 },
+ { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84 },
+ { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81 },
+ { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82 },
+ { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80 },
+ { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83 },
+ { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82 },
+ { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80 },
+ { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82 },
+ { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81 },
+ { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84 },
+ { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80 },
+ { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81 },
+ { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80 },
+ { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82 },
+ { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81 },
+ { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80 },
+ { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81 },
+ { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81 },
+ { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85 },
+ { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87 },
+ { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81 },
+ { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80 },
+ { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79 },
+ { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79 },
+ { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81 },
+ { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79 },
+ { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81 },
+ { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79 },
+ { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80 },
+ { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81 },
+ { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79 },
+ { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79 },
+ { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81 },
+ { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81 },
+ { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82 },
+ { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80 },
+ { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79 },
+ { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83 },
+ { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79 },
+ { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82 },
+ { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80 },
+ { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79 },
+ { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79 },
+ { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80 },
+ { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79 },
+ { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80 },
+ { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80 },
+ { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79 },
+ { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81 },
+ { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79 },
+ { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81 },
+ { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81 },
+ { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79 },
+ { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80 },
+ { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80 },
+ { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82 },
+ { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80 },
+ { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80 },
+ { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80 },
+ { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80 },
+ { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79 },
+ { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79 },
+ { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80 },
+ { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81 },
+ { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79 },
+ { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84 },
+ { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79 },
+ { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79 },
+ { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79 },
+ { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78 },
+ { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81 },
+ { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 },
+ { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 },
+ { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 },
+ { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 },
+ { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 },
+ { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 },
+ { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 },
+ { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 },
+ { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 },
+ { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 },
+ { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 },
+ { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 },
+ { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 },
+ { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 },
+ { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 },
+ { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 },
+ { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 },
+ { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 },
+ { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 },
+ { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 },
+ { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 },
+ { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 },
+ { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 },
+ { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 },
+ { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 },
+ { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 },
+ { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 },
+ { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 },
+ { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 },
+ { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 },
+ { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 },
+ { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 },
+ { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 },
+ { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 },
+ { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 },
+ { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 },
+ { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 },
+ { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 },
+ { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 },
+ { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 },
+ { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 },
+ { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 },
+ { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 },
+};
+
+#ifdef USE_UTAB
+static const struct {
+ float H; /* 1 + i/INTERVALS (exact) */
+ float E; /* H(i) * G(i) - 1 (exact) */
+} U[TSIZE] = {
+ { 0x800000.0p-23, 0 },
+ { 0x810000.0p-23, -0x800000.0p-37 },
+ { 0x820000.0p-23, -0x800000.0p-35 },
+ { 0x830000.0p-23, -0x900000.0p-34 },
+ { 0x840000.0p-23, -0x800000.0p-33 },
+ { 0x850000.0p-23, -0xc80000.0p-33 },
+ { 0x860000.0p-23, -0xa00000.0p-36 },
+ { 0x870000.0p-23, 0x940000.0p-33 },
+ { 0x880000.0p-23, 0x800000.0p-35 },
+ { 0x890000.0p-23, -0xc80000.0p-34 },
+ { 0x8a0000.0p-23, 0xe00000.0p-36 },
+ { 0x8b0000.0p-23, 0x900000.0p-33 },
+ { 0x8c0000.0p-23, -0x800000.0p-35 },
+ { 0x8d0000.0p-23, -0xe00000.0p-33 },
+ { 0x8e0000.0p-23, 0x880000.0p-33 },
+ { 0x8f0000.0p-23, -0xa80000.0p-34 },
+ { 0x900000.0p-23, -0x800000.0p-35 },
+ { 0x910000.0p-23, 0x800000.0p-37 },
+ { 0x920000.0p-23, 0x900000.0p-35 },
+ { 0x930000.0p-23, 0xd00000.0p-35 },
+ { 0x940000.0p-23, 0xe00000.0p-35 },
+ { 0x950000.0p-23, 0xc00000.0p-35 },
+ { 0x960000.0p-23, 0xe00000.0p-36 },
+ { 0x970000.0p-23, -0x800000.0p-38 },
+ { 0x980000.0p-23, -0xc00000.0p-35 },
+ { 0x990000.0p-23, -0xd00000.0p-34 },
+ { 0x9a0000.0p-23, 0x880000.0p-33 },
+ { 0x9b0000.0p-23, 0xe80000.0p-35 },
+ { 0x9c0000.0p-23, -0x800000.0p-35 },
+ { 0x9d0000.0p-23, 0xb40000.0p-33 },
+ { 0x9e0000.0p-23, 0x880000.0p-34 },
+ { 0x9f0000.0p-23, -0xe00000.0p-35 },
+ { 0xa00000.0p-23, 0x800000.0p-33 },
+ { 0xa10000.0p-23, -0x900000.0p-36 },
+ { 0xa20000.0p-23, -0xb00000.0p-33 },
+ { 0xa30000.0p-23, -0xa00000.0p-36 },
+ { 0xa40000.0p-23, 0x800000.0p-33 },
+ { 0xa50000.0p-23, -0xf80000.0p-35 },
+ { 0xa60000.0p-23, 0x880000.0p-34 },
+ { 0xa70000.0p-23, -0x900000.0p-33 },
+ { 0xa80000.0p-23, -0x800000.0p-35 },
+ { 0xa90000.0p-23, 0x900000.0p-34 },
+ { 0xaa0000.0p-23, 0xa80000.0p-33 },
+ { 0xab0000.0p-23, -0xac0000.0p-34 },
+ { 0xac0000.0p-23, -0x800000.0p-37 },
+ { 0xad0000.0p-23, 0xf80000.0p-35 },
+ { 0xae0000.0p-23, 0xf80000.0p-34 },
+ { 0xaf0000.0p-23, -0xac0000.0p-33 },
+ { 0xb00000.0p-23, -0x800000.0p-33 },
+ { 0xb10000.0p-23, -0xb80000.0p-34 },
+ { 0xb20000.0p-23, -0x800000.0p-34 },
+ { 0xb30000.0p-23, -0xb00000.0p-35 },
+ { 0xb40000.0p-23, -0x800000.0p-35 },
+ { 0xb50000.0p-23, -0xe00000.0p-36 },
+ { 0xb60000.0p-23, -0x800000.0p-35 },
+ { 0xb70000.0p-23, -0xb00000.0p-35 },
+ { 0xb80000.0p-23, -0x800000.0p-34 },
+ { 0xb90000.0p-23, -0xb80000.0p-34 },
+ { 0xba0000.0p-23, -0x800000.0p-33 },
+ { 0xbb0000.0p-23, -0xac0000.0p-33 },
+ { 0xbc0000.0p-23, 0x980000.0p-33 },
+ { 0xbd0000.0p-23, 0xbc0000.0p-34 },
+ { 0xbe0000.0p-23, 0xe00000.0p-36 },
+ { 0xbf0000.0p-23, -0xb80000.0p-35 },
+ { 0xc00000.0p-23, -0x800000.0p-33 },
+ { 0xc10000.0p-23, 0xa80000.0p-33 },
+ { 0xc20000.0p-23, 0x900000.0p-34 },
+ { 0xc30000.0p-23, -0x800000.0p-35 },
+ { 0xc40000.0p-23, -0x900000.0p-33 },
+ { 0xc50000.0p-23, 0x820000.0p-33 },
+ { 0xc60000.0p-23, 0x800000.0p-38 },
+ { 0xc70000.0p-23, -0x820000.0p-33 },
+ { 0xc80000.0p-23, 0x800000.0p-33 },
+ { 0xc90000.0p-23, -0xa00000.0p-36 },
+ { 0xca0000.0p-23, -0xb00000.0p-33 },
+ { 0xcb0000.0p-23, 0x840000.0p-34 },
+ { 0xcc0000.0p-23, -0xd00000.0p-34 },
+ { 0xcd0000.0p-23, 0x800000.0p-33 },
+ { 0xce0000.0p-23, -0xe00000.0p-35 },
+ { 0xcf0000.0p-23, 0xa60000.0p-33 },
+ { 0xd00000.0p-23, -0x800000.0p-35 },
+ { 0xd10000.0p-23, 0xb40000.0p-33 },
+ { 0xd20000.0p-23, -0x800000.0p-35 },
+ { 0xd30000.0p-23, 0xaa0000.0p-33 },
+ { 0xd40000.0p-23, -0xe00000.0p-35 },
+ { 0xd50000.0p-23, 0x880000.0p-33 },
+ { 0xd60000.0p-23, -0xd00000.0p-34 },
+ { 0xd70000.0p-23, 0x9c0000.0p-34 },
+ { 0xd80000.0p-23, -0xb00000.0p-33 },
+ { 0xd90000.0p-23, -0x800000.0p-38 },
+ { 0xda0000.0p-23, 0xa40000.0p-33 },
+ { 0xdb0000.0p-23, -0xdc0000.0p-34 },
+ { 0xdc0000.0p-23, 0xc00000.0p-35 },
+ { 0xdd0000.0p-23, 0xca0000.0p-33 },
+ { 0xde0000.0p-23, -0xb80000.0p-34 },
+ { 0xdf0000.0p-23, 0xd00000.0p-35 },
+ { 0xe00000.0p-23, 0xc00000.0p-33 },
+ { 0xe10000.0p-23, -0xf40000.0p-34 },
+ { 0xe20000.0p-23, 0x800000.0p-37 },
+ { 0xe30000.0p-23, 0x860000.0p-33 },
+ { 0xe40000.0p-23, -0xc80000.0p-33 },
+ { 0xe50000.0p-23, -0xa80000.0p-34 },
+ { 0xe60000.0p-23, 0xe00000.0p-36 },
+ { 0xe70000.0p-23, 0x880000.0p-33 },
+ { 0xe80000.0p-23, -0xe00000.0p-33 },
+ { 0xe90000.0p-23, -0xfc0000.0p-34 },
+ { 0xea0000.0p-23, -0x800000.0p-35 },
+ { 0xeb0000.0p-23, 0xe80000.0p-35 },
+ { 0xec0000.0p-23, 0x900000.0p-33 },
+ { 0xed0000.0p-23, 0xe20000.0p-33 },
+ { 0xee0000.0p-23, -0xac0000.0p-33 },
+ { 0xef0000.0p-23, -0xc80000.0p-34 },
+ { 0xf00000.0p-23, -0x800000.0p-35 },
+ { 0xf10000.0p-23, 0x800000.0p-35 },
+ { 0xf20000.0p-23, 0xb80000.0p-34 },
+ { 0xf30000.0p-23, 0x940000.0p-33 },
+ { 0xf40000.0p-23, 0xc80000.0p-33 },
+ { 0xf50000.0p-23, -0xf20000.0p-33 },
+ { 0xf60000.0p-23, -0xc80000.0p-33 },
+ { 0xf70000.0p-23, -0xa20000.0p-33 },
+ { 0xf80000.0p-23, -0x800000.0p-33 },
+ { 0xf90000.0p-23, -0xc40000.0p-34 },
+ { 0xfa0000.0p-23, -0x900000.0p-34 },
+ { 0xfb0000.0p-23, -0xc80000.0p-35 },
+ { 0xfc0000.0p-23, -0x800000.0p-35 },
+ { 0xfd0000.0p-23, -0x900000.0p-36 },
+ { 0xfe0000.0p-23, -0x800000.0p-37 },
+ { 0xff0000.0p-23, -0x800000.0p-39 },
+ { 0x800000.0p-22, 0 },
+};
+#endif /* USE_UTAB */
+
+#ifdef STRUCT_RETURN
+#define RETURN1(rp, v) do { \
+ (rp)->hi = (v); \
+ (rp)->lo_set = 0; \
+ return; \
+} while (0)
+
+#define RETURN2(rp, h, l) do { \
+ (rp)->hi = (h); \
+ (rp)->lo = (l); \
+ (rp)->lo_set = 1; \
+ return; \
+} while (0)
+
+struct ld {
+ long double hi;
+ long double lo;
+ int lo_set;
+};
+#else
+#define RETURN1(rp, v) RETURNF(v)
+#define RETURN2(rp, h, l) RETURNI((h) + (l))
+#endif
+
+#ifdef STRUCT_RETURN
+static inline __always_inline void
+k_logl(long double x, struct ld *rp)
+#else
+long double
+logl(long double x)
+#endif
+{
+ long double d, dk, val_hi, val_lo, z;
+ uint64_t ix, lx;
+ int i, k;
+ uint16_t hx;
+
+ EXTRACT_LDBL80_WORDS(hx, lx, x);
+ k = -16383;
+#if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
+ if (x == 1)
+ RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
+#endif
+ if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
+ if (((hx & 0x7fff) | lx) == 0)
+ RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
+ if (hx != 0)
+ /* log(neg or [pseudo-]NaN) = qNaN: */
+ RETURN1(rp, (x - x) / zero);
+ x *= 0x1.0p65; /* subnormal; scale up x */
+ /* including pseudo-subnormals */
+ EXTRACT_LDBL80_WORDS(hx, lx, x);
+ k = -16383 - 65;
+ } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0)
+ RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
+ /* log(pseudo-Inf) = qNaN */
+ /* log(pseudo-NaN) = qNaN */
+ /* log(unnormal) = qNaN */
+#ifndef STRUCT_RETURN
+ ENTERI();
+#endif
+ k += hx;
+ ix = lx & 0x7fffffffffffffffULL;
+ dk = k;
+
+ /* Scale x to be in [1, 2). */
+ SET_LDBL_EXPSIGN(x, 0x3fff);
+
+ /* 0 <= i <= INTERVALS: */
+#define L2I (64 - LOG2_INTERVALS)
+ i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
+
+ /*
+ * -0.005280 < d < 0.004838. In particular, the infinite-
+ * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
+ * ensures that d is representable without extra precision for
+ * this bound on |d| (since when this calculation is expressed
+ * as x*G(i)-1, the multiplication needs as many extra bits as
+ * G(i) has and the subtraction cancels 8 bits). But for
+ * most i (107 cases out of 129), the infinite-precision |d|
+ * is <= 2**-8. G(i) is rounded to 9 bits for such i to give
+ * better accuracy (this works by improving the bound on |d|,
+ * which in turn allows rounding to 9 bits in more cases).
+ * This is only important when the original x is near 1 -- it
+ * lets us avoid using a special method to give the desired
+ * accuracy for such x.
+ */
+ if (0)
+ d = x * G(i) - 1;
+ else {
+#ifdef USE_UTAB
+ d = (x - H(i)) * G(i) + E(i);
+#else
+ long double x_hi, x_lo;
+ float fx_hi;
+
+ /*
+ * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
+ * G(i) has at most 9 bits, so the splitting point is not
+ * critical.
+ */
+ SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
+ x_hi = fx_hi;
+ x_lo = x - x_hi;
+ d = x_hi * G(i) - 1 + x_lo * G(i);
+#endif
+ }
+
+ /*
+ * Our algorithm depends on exact cancellation of F_lo(i) and
+ * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
+ * at the end of the table. This and other technical complications
+ * make it difficult to avoid the double scaling in (dk*ln2) *
+ * log(base) for base != e without losing more accuracy and/or
+ * efficiency than is gained.
+ */
+ z = d * d;
+ val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
+ (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2;
+ val_hi = d;
+#ifdef DEBUG
+ if (fetestexcept(FE_UNDERFLOW))
+ breakpoint();
+#endif
+
+ _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
+ RETURN2(rp, val_hi, val_lo);
+}
+
+long double
+log1pl(long double x)
+{
+ long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z;
+ long double f_hi, twopminusk;
+ uint64_t ix, lx;
+ int i, k;
+ int16_t ax, hx;
+
+ DOPRINT_START(&x);
+ EXTRACT_LDBL80_WORDS(hx, lx, x);
+ if (hx < 0x3fff) { /* x < 1, or x neg NaN */
+ ax = hx & 0x7fff;
+ if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
+ if (ax == 0x3fff && lx == 0x8000000000000000ULL)
+ RETURNP(-1 / zero); /* log1p(-1) = -Inf */
+ /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */
+ RETURNP((x - x) / (x - x));
+ }
+ if (ax <= 0x3fbe) { /* |x| < 2**-64 */
+ if ((int)x == 0)
+ RETURNP(x); /* x with inexact if x != 0 */
+ }
+ f_hi = 1;
+ f_lo = x;
+ } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
+ RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
+ /* log1p(pseudo-Inf) = qNaN */
+ /* log1p(pseudo-NaN) = qNaN */
+ /* log1p(unnormal) = qNaN */
+ } else if (hx < 0x407f) { /* 1 <= x < 2**128 */
+ f_hi = x;
+ f_lo = 1;
+ } else { /* 2**128 <= x < +Inf */
+ f_hi = x;
+ f_lo = 0; /* avoid underflow of the P5 term */
+ }
+ ENTERI();
+ x = f_hi + f_lo;
+ f_lo = (f_hi - x) + f_lo;
+
+ EXTRACT_LDBL80_WORDS(hx, lx, x);
+ k = -16383;
+
+ k += hx;
+ ix = lx & 0x7fffffffffffffffULL;
+ dk = k;
+
+ SET_LDBL_EXPSIGN(x, 0x3fff);
+ twopminusk = 1;
+ SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
+ f_lo *= twopminusk;
+
+ i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
+
+ /*
+ * x*G(i)-1 (with a reduced x) can be represented exactly, as
+ * above, but now we need to evaluate the polynomial on d =
+ * (x+f_lo)*G(i)-1 and extra precision is needed for that.
+ * Since x+x_lo is a hi+lo decomposition and subtracting 1
+ * doesn't lose too many bits, an inexact calculation for
+ * f_lo*G(i) is good enough.
+ */
+ if (0)
+ d_hi = x * G(i) - 1;
+ else {
+#ifdef USE_UTAB
+ d_hi = (x - H(i)) * G(i) + E(i);
+#else
+ long double x_hi, x_lo;
+ float fx_hi;
+
+ SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
+ x_hi = fx_hi;
+ x_lo = x - x_hi;
+ d_hi = x_hi * G(i) - 1 + x_lo * G(i);
+#endif
+ }
+ d_lo = f_lo * G(i);
+
+ /*
+ * This is _2sumF(d_hi, d_lo) inlined. The condition
+ * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
+ * always satisifed, so it is not clear that this works, but
+ * it works in practice. It works even if it gives a wrong
+ * normalized d_lo, since |d_lo| > |d_hi| implies that i is
+ * nonzero and d is tiny, so the F(i) term dominates d_lo.
+ * In float precision:
+ * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
+ * And if d is only a little tinier than that, we would have
+ * another underflow problem for the P3 term; this is also ruled
+ * out by exhaustive testing.)
+ */
+ d = d_hi + d_lo;
+ d_lo = d_hi - d + d_lo;
+ d_hi = d;
+
+ z = d * d;
+ val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
+ (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2;
+ val_hi = d_hi;
+#ifdef DEBUG
+ if (fetestexcept(FE_UNDERFLOW))
+ breakpoint();
+#endif
+
+ _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
+ RETURN2PI(val_hi, val_lo);
+}
+
+#ifdef STRUCT_RETURN
+
+long double
+logl(long double x)
+{
+ struct ld r;
+
+ ENTERI();
+ DOPRINT_START(&x);
+ k_logl(x, &r);
+ RETURNSPI(&r);
+}
+
+/* Use macros since GCC < 8 rejects static const expressions in initializers. */
+#define invln10_hi 4.3429448190317999e-1 /* 0x1bcb7b1526e000.0p-54 */
+#define invln10_lo 7.1842412889749798e-14 /* 0x1438ca9aadd558.0p-96 */
+#define invln2_hi 1.4426950408887933e0 /* 0x171547652b8000.0p-52 */
+#define invln2_lo 1.7010652264631490e-13 /* 0x17f0bbbe87fed0.0p-95 */
+/* Let the compiler pre-calculate this sum to avoid FE_INEXACT at run time. */
+static const double invln10_lo_plus_hi = invln10_lo + invln10_hi;
+static const double invln2_lo_plus_hi = invln2_lo + invln2_hi;
+
+long double
+log10l(long double x)
+{
+ struct ld r;
+ long double hi, lo;
+
+ ENTERI();
+ DOPRINT_START(&x);
+ k_logl(x, &r);
+ if (!r.lo_set)
+ RETURNPI(r.hi);
+ _2sumF(r.hi, r.lo);
+ hi = (float)r.hi;
+ lo = r.lo + (r.hi - hi);
+ RETURN2PI(invln10_hi * hi,
+ invln10_lo_plus_hi * lo + invln10_lo * hi);
+}
+
+long double
+log2l(long double x)
+{
+ struct ld r;
+ long double hi, lo;
+
+ ENTERI();
+ DOPRINT_START(&x);
+ k_logl(x, &r);
+ if (!r.lo_set)
+ RETURNPI(r.hi);
+ _2sumF(r.hi, r.lo);
+ hi = (float)r.hi;
+ lo = r.lo + (r.hi - hi);
+ RETURN2PI(invln2_hi * hi,
+ invln2_lo_plus_hi * lo + invln2_lo * hi);
+}
+
+#endif /* STRUCT_RETURN */
generated by cgit v1.1 at 2024-09-09 14:14:12 +0000