1 /*-
2  * Copyright (c) 2007-2013 Bruce D. Evans
3  * All rights reserved.
4  *
5  * Redistribution and use in source and binary forms, with or without
6  * modification, are permitted provided that the following conditions
7  * are met:
8  * 1. Redistributions of source code must retain the above copyright
9  *    notice unmodified, this list of conditions, and the following
10  *    disclaimer.
11  * 2. Redistributions in binary form must reproduce the above copyright
12  *    notice, this list of conditions and the following disclaimer in the
13  *    documentation and/or other materials provided with the distribution.
14  *
15  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16  * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17  * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18  * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19  * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21  * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22  * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23  * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24  * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25  */
26 
27 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD$");
29 
30 /**
31  * Implementation of the natural logarithm of x for 128-bit format.
32  *
33  * First decompose x into its base 2 representation:
34  *
35  *    log(x) = log(X * 2**k), where X is in [1, 2)
36  *           = log(X) + k * log(2).
37  *
38  * Let X = X_i + e, where X_i is the center of one of the intervals
39  * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
40  * and X is in this interval.  Then
41  *
42  *    log(X) = log(X_i + e)
43  *           = log(X_i * (1 + e / X_i))
44  *           = log(X_i) + log(1 + e / X_i).
45  *
46  * The values log(X_i) are tabulated below.  Let d = e / X_i and use
47  *
48  *    log(1 + d) = p(d)
49  *
50  * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
51  * suitably high degree.
52  *
53  * To get sufficiently small roundoff errors, k * log(2), log(X_i), and
54  * sometimes (if |k| is not large) the first term in p(d) must be evaluated
55  * and added up in extra precision.  Extra precision is not needed for the
56  * rest of p(d).  In the worst case when k = 0 and log(X_i) is 0, the final
57  * error is controlled mainly by the error in the second term in p(d).  The
58  * error in this term itself is at most 0.5 ulps from the d*d operation in
59  * it.  The error in this term relative to the first term is thus at most
60  * 0.5 * |-0.5| * |d| < 1.0/1024 ulps.  We aim for an accumulated error of
61  * at most twice this at the point of the final rounding step.  Thus the
62  * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps.  Exhaustive
63  * testing of a float variant of this function showed a maximum final error
64  * of 0.5008 ulps.  Non-exhaustive testing of a double variant of this
65  * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
66  *
67  * We made the maximum of |d| (and thus the total relative error and the
68  * degree of p(d)) small by using a large number of intervals.  Using
69  * centers of intervals instead of endpoints reduces this maximum by a
70  * factor of 2 for a given number of intervals.  p(d) is special only
71  * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
72  * naturally.  The most accurate minimax polynomial of a given degree might
73  * be different, but then we wouldn't want it since we would have to do
74  * extra work to avoid roundoff error (especially for P0*d instead of d).
75  */
76 
77 #ifdef DEBUG
78 #include <assert.h>
79 #include <fenv.h>
80 #endif
81 
82 #include "fpmath.h"
83 #include "math.h"
84 #ifndef NO_STRUCT_RETURN
85 #define	STRUCT_RETURN
86 #endif
87 #include "math_private.h"
88 
89 #if !defined(NO_UTAB) && !defined(NO_UTABL)
90 #define	USE_UTAB
91 #endif
92 
93 /*
94  * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]:
95  * |log(1 + d)/d - p(d)| < 2**-122.7
96  */
97 static const long double
98 P2 = -0.5L,
99 P3 =  3.33333333333333333333333333333233795e-1L,	/*  0x15555555555555555555555554d42.0p-114L */
100 P4 = -2.49999999999999999999999999941139296e-1L,	/* -0x1ffffffffffffffffffffffdab14e.0p-115L */
101 P5 =  2.00000000000000000000000085468039943e-1L,	/*  0x19999999999999999999a6d3567f4.0p-115L */
102 P6 = -1.66666666666666666666696142372698408e-1L,	/* -0x15555555555555555567267a58e13.0p-115L */
103 P7 =  1.42857142857142857119522943477166120e-1L,	/*  0x1249249249249248ed79a0ae434de.0p-115L */
104 P8 = -1.24999999999999994863289015033581301e-1L;	/* -0x1fffffffffffffa13e91765e46140.0p-116L */
105 /* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */
106 static const double
107 P9 =  1.1111111111111401e-1,		/*  0x1c71c71c71c7ed.0p-56 */
108 P10 = -1.0000000000040135e-1,		/* -0x199999999a0a92.0p-56 */
109 P11 =  9.0909090728136258e-2,		/*  0x1745d173962111.0p-56 */
110 P12 = -8.3333318851855284e-2,		/* -0x1555551722c7a3.0p-56 */
111 P13 =  7.6928634666404178e-2,		/*  0x13b1985204a4ae.0p-56 */
112 P14 = -7.1626810078462499e-2;		/* -0x12562276cdc5d0.0p-56 */
113 
114 static volatile const double zero = 0;
115 
116 #define	INTERVALS	128
117 #define	LOG2_INTERVALS	7
118 #define	TSIZE		(INTERVALS + 1)
119 #define	G(i)		(T[(i)].G)
120 #define	F_hi(i)		(T[(i)].F_hi)
121 #define	F_lo(i)		(T[(i)].F_lo)
122 #define	ln2_hi		F_hi(TSIZE - 1)
123 #define	ln2_lo		F_lo(TSIZE - 1)
124 #define	E(i)		(U[(i)].E)
125 #define	H(i)		(U[(i)].H)
126 
127 static const struct {
128 	float	G;			/* 1/(1 + i/128) rounded to 8/9 bits */
129 	float	F_hi;			/* log(1 / G_i) rounded (see below) */
130 	/* The compiler will insert 8 bytes of padding here. */
131 	long double F_lo;		/* next 113 bits for log(1 / G_i) */
132 } T[TSIZE] = {
133 	/*
134 	 * ln2_hi and each F_hi(i) are rounded to a number of bits that
135 	 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
136 	 *
137 	 * The last entry (for X just below 2) is used to define ln2_hi
138 	 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
139 	 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
140 	 * This is needed for accuracy when x is just below 1.  (To avoid
141 	 * special cases, such x are "reduced" strangely to X just below
142 	 * 2 and dk = -1, and then the exact cancellation is needed
143 	 * because any the error from any non-exactness would be too
144 	 * large).
145 	 *
146 	 * The relevant range of dk is [-16445, 16383].  The maximum number
147 	 * of bits in F_hi(i) that works is very dependent on i but has
148 	 * a minimum of 93.  We only need about 12 bits in F_hi(i) for
149 	 * it to provide enough extra precision.
150 	 *
151 	 * We round F_hi(i) to 24 bits so that it can have type float,
152 	 * mainly to minimize the size of the table.  Using all 24 bits
153 	 * in a float for it automatically satisfies the above constraints.
154 	 */
155      0x800000.0p-23,  0,               0,
156      0xfe0000.0p-24,  0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L,
157      0xfc0000.0p-24,  0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L,
158      0xfa0000.0p-24,  0xc24929.0p-29,  0x1191957d173697cf302cc9476f561.0p-143L,
159      0xf80000.0p-24,  0x820aec.0p-28,  0x13ce8888e02e78eba9b1113bc1c18.0p-142L,
160      0xf60000.0p-24,  0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L,
161      0xf48000.0p-24,  0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L,
162      0xf30000.0p-24,  0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L,
163      0xf10000.0p-24,  0xf7518e.0p-28,  0x1ae1eec1b036c484993c549c4bf40.0p-151L,
164      0xef0000.0p-24,  0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L,
165      0xed8000.0p-24,  0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L,
166      0xec0000.0p-24,  0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L,
167      0xea0000.0p-24,  0xb80698.0p-27,  0x15d581c1e8da99ded322fb08b8462.0p-141L,
168      0xe80000.0p-24,  0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L,
169      0xe70000.0p-24,  0xd273b2.0p-27,  0x163786f5251aefe0ded34c8318f52.0p-145L,
170      0xe50000.0p-24,  0xe442c0.0p-27,  0x1bc4b2368e32d56699c1799a244d4.0p-144L,
171      0xe38000.0p-24,  0xf1b83f.0p-27,  0x1c6090f684e6766abceccab1d7174.0p-141L,
172      0xe20000.0p-24,  0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L,
173      0xe08000.0p-24,  0x8673f6.0p-26,  0x1b9985194b6affd511b534b72a28e.0p-140L,
174      0xdf0000.0p-24,  0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L,
175      0xdd8000.0p-24,  0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L,
176      0xdc0000.0p-24,  0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L,
177      0xda8000.0p-24,  0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L,
178      0xd90000.0p-24,  0xa93f2f.0p-26,  0x1286d633e8e5697dc6a402a56fce1.0p-141L,
179      0xd78000.0p-24,  0xb05988.0p-26,  0x16128eba9367707ebfa540e45350c.0p-144L,
180      0xd60000.0p-24,  0xb78094.0p-26,  0x16ead577390d31ef0f4c9d43f79b2.0p-140L,
181      0xd50000.0p-24,  0xbc4c6c.0p-26,  0x151131ccf7c7b75e7d900b521c48d.0p-141L,
182      0xd38000.0p-24,  0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L,
183      0xd20000.0p-24,  0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L,
184      0xd10000.0p-24,  0xcfb620.0p-26,  0x1c2259904d686581799fbce0b5f19.0p-141L,
185      0xcf8000.0p-24,  0xd71653.0p-26,  0x1ece57a8d5ae54f550444ecf8b995.0p-140L,
186      0xce0000.0p-24,  0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L,
187      0xcd0000.0p-24,  0xe37fde.0p-26,  0x1bc03dc271a74d3a85b5b43c0e727.0p-141L,
188      0xcb8000.0p-24,  0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L,
189      0xca0000.0p-24,  0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L,
190      0xc90000.0p-24,  0xf7ad6f.0p-26,  0x1373ff977baa6911c7bafcb4d84fb.0p-141L,
191      0xc80000.0p-24,  0xfcc8e3.0p-26,  0x196766f2fb328337cc050c6d83b22.0p-140L,
192      0xc68000.0p-24,  0x823f30.0p-25,  0x19bd076f7c434e5fcf1a212e2a91e.0p-139L,
193      0xc58000.0p-24,  0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L,
194      0xc40000.0p-24,  0x88bc74.0p-25,  0x113f23def19c5a0fe396f40f1dda9.0p-141L,
195      0xc30000.0p-24,  0x8b5ae6.0p-25,  0x1759f6e6b37de945a049a962e66c6.0p-139L,
196      0xc20000.0p-24,  0x8dfccb.0p-25,  0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L,
197      0xc10000.0p-24,  0x90a22b.0p-25,  0x1a1d71a87deba46bae9827221dc98.0p-139L,
198      0xbf8000.0p-24,  0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L,
199      0xbe8000.0p-24,  0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L,
200      0xbd8000.0p-24,  0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L,
201      0xbc8000.0p-24,  0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L,
202      0xbb0000.0p-24,  0xa0cda1.0p-25,  0x1eaf46390dbb2438273918db7df5c.0p-141L,
203      0xba0000.0p-24,  0xa38c6e.0p-25,  0x138e20d831f698298adddd7f32686.0p-141L,
204      0xb90000.0p-24,  0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L,
205      0xb80000.0p-24,  0xa91570.0p-25,  0x1ce28f5f3840b263acb4351104631.0p-140L,
206      0xb70000.0p-24,  0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L,
207      0xb60000.0p-24,  0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L,
208      0xb50000.0p-24,  0xb18018.0p-25,  0x16755892770633947ffe651e7352f.0p-139L,
209      0xb40000.0p-24,  0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L,
210      0xb30000.0p-24,  0xb73077.0p-25,  0x1abc65c8595f088b61a335f5b688c.0p-140L,
211      0xb20000.0p-24,  0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L,
212      0xb10000.0p-24,  0xbcf133.0p-25,  0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L,
213      0xb00000.0p-24,  0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L,
214      0xaf0000.0p-24,  0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L,
215      0xae8000.0p-24,  0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L,
216      0xad8000.0p-24,  0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L,
217      0xac8000.0p-24,  0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L,
218      0xab8000.0p-24,  0xcd1aae.0p-25,  0x19deb5ce6a6a8717d5626e16acc7d.0p-141L,
219      0xaa8000.0p-24,  0xd0192f.0p-25,  0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L,
220      0xaa0000.0p-24,  0xd19a20.0p-25,  0x1127d3c6457f9d79f51dcc73014c9.0p-141L,
221      0xa90000.0p-24,  0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L,
222      0xa80000.0p-24,  0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L,
223      0xa70000.0p-24,  0xdab7d0.0p-25,  0x1118a425494b610665377f15625b6.0p-140L,
224      0xa68000.0p-24,  0xdc40d5.0p-25,  0x1966f24d29d3a2d1b2176010478be.0p-140L,
225      0xa58000.0p-24,  0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L,
226      0xa48000.0p-24,  0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L,
227      0xa40000.0p-24,  0xe3ffce.0p-25,  0x1d155324911f56db28da4d629d00a.0p-140L,
228      0xa30000.0p-24,  0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L,
229      0xa20000.0p-24,  0xea4812.0p-25,  0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L,
230      0xa18000.0p-24,  0xebdd3d.0p-25,  0x1b3cfb3f7511dd73692609040ccc2.0p-139L,
231      0xa08000.0p-24,  0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L,
232      0xa00000.0p-24,  0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L,
233      0x9f0000.0p-24,  0xf3da16.0p-25,  0x1eed6b9aafac8d42f78d3e65d3727.0p-141L,
234      0x9e8000.0p-24,  0xf576e9.0p-25,  0x1d593218675af269647b783d88999.0p-139L,
235      0x9d8000.0p-24,  0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L,
236      0x9d0000.0p-24,  0xfa553f.0p-25,  0x1c063259bcade02951686d5373aec.0p-139L,
237      0x9c0000.0p-24,  0xfd9ac5.0p-25,  0x1ef491085fa3c1649349630531502.0p-139L,
238      0x9b8000.0p-24,  0xff3f8c.0p-25,  0x1d607a7c2b8c5320619fb9433d841.0p-139L,
239      0x9a8000.0p-24,  0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L,
240      0x9a0000.0p-24,  0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L,
241      0x990000.0p-24,  0x83c5f8.0p-24,  0x14cf15a048907b7d7f47ddb45c5a3.0p-139L,
242      0x988000.0p-24,  0x849c7d.0p-24,  0x1cbb1d35fb82873b04a9af1dd692c.0p-138L,
243      0x978000.0p-24,  0x864ba6.0p-24,  0x1128639b814f9b9770d8cb6573540.0p-138L,
244      0x970000.0p-24,  0x87244c.0p-24,  0x184733853300f002e836dfd47bd41.0p-139L,
245      0x968000.0p-24,  0x87fdaa.0p-24,  0x109d23aef77dd5cd7cc94306fb3ff.0p-140L,
246      0x958000.0p-24,  0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L,
247      0x950000.0p-24,  0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L,
248      0x948000.0p-24,  0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L,
249      0x938000.0p-24,  0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L,
250      0x930000.0p-24,  0x8e03c2.0p-24,  0x135cc00e566f76b87333891e0dec4.0p-138L,
251      0x928000.0p-24,  0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L,
252      0x918000.0p-24,  0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L,
253      0x910000.0p-24,  0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L,
254      0x908000.0p-24,  0x9267e7.0p-24,  0x1be03669a5268d21148c6002becd3.0p-139L,
255      0x8f8000.0p-24,  0x942f04.0p-24,  0x10b28e0e26c336af90e00533323ba.0p-139L,
256      0x8f0000.0p-24,  0x9513c3.0p-24,  0x1a1d820da57cf2f105a89060046aa.0p-138L,
257      0x8e8000.0p-24,  0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L,
258      0x8e0000.0p-24,  0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L,
259      0x8d0000.0p-24,  0x98aed2.0p-24,  0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L,
260      0x8c8000.0p-24,  0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L,
261      0x8c0000.0p-24,  0x9a8145.0p-24,  0x1b3b190b83f9527e6aba8f2d783c1.0p-138L,
262      0x8b8000.0p-24,  0x9b6bbf.0p-24,  0x13a69fad7e7abe7ba81c664c107e0.0p-138L,
263      0x8b0000.0p-24,  0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L,
264      0x8a8000.0p-24,  0x9d433b.0p-24,  0x1c95c444b807a246726b304ccae56.0p-139L,
265      0x898000.0p-24,  0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L,
266      0x890000.0p-24,  0xa00ce1.0p-24,  0x125ca93186cf0f38b4619a2483399.0p-141L,
267      0x888000.0p-24,  0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L,
268      0x880000.0p-24,  0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L,
269      0x878000.0p-24,  0xa2de62.0p-24,  0x193224e8516c008d3602a7b41c6e8.0p-139L,
270      0x870000.0p-24,  0xa3d0a9.0p-24,  0x1fa28b4d2541aca7d5844606b2421.0p-139L,
271      0x868000.0p-24,  0xa4c3d6.0p-24,  0x1c1b5760fb4571acbcfb03f16daf4.0p-138L,
272      0x858000.0p-24,  0xa6acea.0p-24,  0x1fed5d0f65949c0a345ad743ae1ae.0p-140L,
273      0x850000.0p-24,  0xa7a2d4.0p-24,  0x1ad270c9d749362382a7688479e24.0p-140L,
274      0x848000.0p-24,  0xa899ab.0p-24,  0x199ff15ce532661ea9643a3a2d378.0p-139L,
275      0x840000.0p-24,  0xa99171.0p-24,  0x1a19e15ccc45d257530a682b80490.0p-139L,
276      0x838000.0p-24,  0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L,
277      0x830000.0p-24,  0xab83d1.0p-24,  0x1aee319980bff3303dd481779df69.0p-139L,
278      0x828000.0p-24,  0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L,
279      0x820000.0p-24,  0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L,
280      0x818000.0p-24,  0xae768f.0p-24,  0x17c35c55a04a82ab19f77652d977a.0p-141L,
281      0x810000.0p-24,  0xaf7415.0p-24,  0x1448324047019b48d7b98c1cf7234.0p-138L,
282      0x808000.0p-24,  0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L,
283      0x800000.0p-24,  0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L,
284 };
285 
286 #ifdef USE_UTAB
287 static const struct {
288 	float	H;			/* 1 + i/INTERVALS (exact) */
289 	float	E;			/* H(i) * G(i) - 1 (exact) */
290 } U[TSIZE] = {
291 	 0x800000.0p-23,  0,
292 	 0x810000.0p-23, -0x800000.0p-37,
293 	 0x820000.0p-23, -0x800000.0p-35,
294 	 0x830000.0p-23, -0x900000.0p-34,
295 	 0x840000.0p-23, -0x800000.0p-33,
296 	 0x850000.0p-23, -0xc80000.0p-33,
297 	 0x860000.0p-23, -0xa00000.0p-36,
298 	 0x870000.0p-23,  0x940000.0p-33,
299 	 0x880000.0p-23,  0x800000.0p-35,
300 	 0x890000.0p-23, -0xc80000.0p-34,
301 	 0x8a0000.0p-23,  0xe00000.0p-36,
302 	 0x8b0000.0p-23,  0x900000.0p-33,
303 	 0x8c0000.0p-23, -0x800000.0p-35,
304 	 0x8d0000.0p-23, -0xe00000.0p-33,
305 	 0x8e0000.0p-23,  0x880000.0p-33,
306 	 0x8f0000.0p-23, -0xa80000.0p-34,
307 	 0x900000.0p-23, -0x800000.0p-35,
308 	 0x910000.0p-23,  0x800000.0p-37,
309 	 0x920000.0p-23,  0x900000.0p-35,
310 	 0x930000.0p-23,  0xd00000.0p-35,
311 	 0x940000.0p-23,  0xe00000.0p-35,
312 	 0x950000.0p-23,  0xc00000.0p-35,
313 	 0x960000.0p-23,  0xe00000.0p-36,
314 	 0x970000.0p-23, -0x800000.0p-38,
315 	 0x980000.0p-23, -0xc00000.0p-35,
316 	 0x990000.0p-23, -0xd00000.0p-34,
317 	 0x9a0000.0p-23,  0x880000.0p-33,
318 	 0x9b0000.0p-23,  0xe80000.0p-35,
319 	 0x9c0000.0p-23, -0x800000.0p-35,
320 	 0x9d0000.0p-23,  0xb40000.0p-33,
321 	 0x9e0000.0p-23,  0x880000.0p-34,
322 	 0x9f0000.0p-23, -0xe00000.0p-35,
323 	 0xa00000.0p-23,  0x800000.0p-33,
324 	 0xa10000.0p-23, -0x900000.0p-36,
325 	 0xa20000.0p-23, -0xb00000.0p-33,
326 	 0xa30000.0p-23, -0xa00000.0p-36,
327 	 0xa40000.0p-23,  0x800000.0p-33,
328 	 0xa50000.0p-23, -0xf80000.0p-35,
329 	 0xa60000.0p-23,  0x880000.0p-34,
330 	 0xa70000.0p-23, -0x900000.0p-33,
331 	 0xa80000.0p-23, -0x800000.0p-35,
332 	 0xa90000.0p-23,  0x900000.0p-34,
333 	 0xaa0000.0p-23,  0xa80000.0p-33,
334 	 0xab0000.0p-23, -0xac0000.0p-34,
335 	 0xac0000.0p-23, -0x800000.0p-37,
336 	 0xad0000.0p-23,  0xf80000.0p-35,
337 	 0xae0000.0p-23,  0xf80000.0p-34,
338 	 0xaf0000.0p-23, -0xac0000.0p-33,
339 	 0xb00000.0p-23, -0x800000.0p-33,
340 	 0xb10000.0p-23, -0xb80000.0p-34,
341 	 0xb20000.0p-23, -0x800000.0p-34,
342 	 0xb30000.0p-23, -0xb00000.0p-35,
343 	 0xb40000.0p-23, -0x800000.0p-35,
344 	 0xb50000.0p-23, -0xe00000.0p-36,
345 	 0xb60000.0p-23, -0x800000.0p-35,
346 	 0xb70000.0p-23, -0xb00000.0p-35,
347 	 0xb80000.0p-23, -0x800000.0p-34,
348 	 0xb90000.0p-23, -0xb80000.0p-34,
349 	 0xba0000.0p-23, -0x800000.0p-33,
350 	 0xbb0000.0p-23, -0xac0000.0p-33,
351 	 0xbc0000.0p-23,  0x980000.0p-33,
352 	 0xbd0000.0p-23,  0xbc0000.0p-34,
353 	 0xbe0000.0p-23,  0xe00000.0p-36,
354 	 0xbf0000.0p-23, -0xb80000.0p-35,
355 	 0xc00000.0p-23, -0x800000.0p-33,
356 	 0xc10000.0p-23,  0xa80000.0p-33,
357 	 0xc20000.0p-23,  0x900000.0p-34,
358 	 0xc30000.0p-23, -0x800000.0p-35,
359 	 0xc40000.0p-23, -0x900000.0p-33,
360 	 0xc50000.0p-23,  0x820000.0p-33,
361 	 0xc60000.0p-23,  0x800000.0p-38,
362 	 0xc70000.0p-23, -0x820000.0p-33,
363 	 0xc80000.0p-23,  0x800000.0p-33,
364 	 0xc90000.0p-23, -0xa00000.0p-36,
365 	 0xca0000.0p-23, -0xb00000.0p-33,
366 	 0xcb0000.0p-23,  0x840000.0p-34,
367 	 0xcc0000.0p-23, -0xd00000.0p-34,
368 	 0xcd0000.0p-23,  0x800000.0p-33,
369 	 0xce0000.0p-23, -0xe00000.0p-35,
370 	 0xcf0000.0p-23,  0xa60000.0p-33,
371 	 0xd00000.0p-23, -0x800000.0p-35,
372 	 0xd10000.0p-23,  0xb40000.0p-33,
373 	 0xd20000.0p-23, -0x800000.0p-35,
374 	 0xd30000.0p-23,  0xaa0000.0p-33,
375 	 0xd40000.0p-23, -0xe00000.0p-35,
376 	 0xd50000.0p-23,  0x880000.0p-33,
377 	 0xd60000.0p-23, -0xd00000.0p-34,
378 	 0xd70000.0p-23,  0x9c0000.0p-34,
379 	 0xd80000.0p-23, -0xb00000.0p-33,
380 	 0xd90000.0p-23, -0x800000.0p-38,
381 	 0xda0000.0p-23,  0xa40000.0p-33,
382 	 0xdb0000.0p-23, -0xdc0000.0p-34,
383 	 0xdc0000.0p-23,  0xc00000.0p-35,
384 	 0xdd0000.0p-23,  0xca0000.0p-33,
385 	 0xde0000.0p-23, -0xb80000.0p-34,
386 	 0xdf0000.0p-23,  0xd00000.0p-35,
387 	 0xe00000.0p-23,  0xc00000.0p-33,
388 	 0xe10000.0p-23, -0xf40000.0p-34,
389 	 0xe20000.0p-23,  0x800000.0p-37,
390 	 0xe30000.0p-23,  0x860000.0p-33,
391 	 0xe40000.0p-23, -0xc80000.0p-33,
392 	 0xe50000.0p-23, -0xa80000.0p-34,
393 	 0xe60000.0p-23,  0xe00000.0p-36,
394 	 0xe70000.0p-23,  0x880000.0p-33,
395 	 0xe80000.0p-23, -0xe00000.0p-33,
396 	 0xe90000.0p-23, -0xfc0000.0p-34,
397 	 0xea0000.0p-23, -0x800000.0p-35,
398 	 0xeb0000.0p-23,  0xe80000.0p-35,
399 	 0xec0000.0p-23,  0x900000.0p-33,
400 	 0xed0000.0p-23,  0xe20000.0p-33,
401 	 0xee0000.0p-23, -0xac0000.0p-33,
402 	 0xef0000.0p-23, -0xc80000.0p-34,
403 	 0xf00000.0p-23, -0x800000.0p-35,
404 	 0xf10000.0p-23,  0x800000.0p-35,
405 	 0xf20000.0p-23,  0xb80000.0p-34,
406 	 0xf30000.0p-23,  0x940000.0p-33,
407 	 0xf40000.0p-23,  0xc80000.0p-33,
408 	 0xf50000.0p-23, -0xf20000.0p-33,
409 	 0xf60000.0p-23, -0xc80000.0p-33,
410 	 0xf70000.0p-23, -0xa20000.0p-33,
411 	 0xf80000.0p-23, -0x800000.0p-33,
412 	 0xf90000.0p-23, -0xc40000.0p-34,
413 	 0xfa0000.0p-23, -0x900000.0p-34,
414 	 0xfb0000.0p-23, -0xc80000.0p-35,
415 	 0xfc0000.0p-23, -0x800000.0p-35,
416 	 0xfd0000.0p-23, -0x900000.0p-36,
417 	 0xfe0000.0p-23, -0x800000.0p-37,
418 	 0xff0000.0p-23, -0x800000.0p-39,
419 	 0x800000.0p-22,  0,
420 };
421 #endif /* USE_UTAB */
422 
423 #ifdef STRUCT_RETURN
424 #define	RETURN1(rp, v) do {	\
425 	(rp)->hi = (v);		\
426 	(rp)->lo_set = 0;	\
427 	return;			\
428 } while (0)
429 
430 #define	RETURN2(rp, h, l) do {	\
431 	(rp)->hi = (h);		\
432 	(rp)->lo = (l);		\
433 	(rp)->lo_set = 1;	\
434 	return;			\
435 } while (0)
436 
437 struct ld {
438 	long double hi;
439 	long double lo;
440 	int	lo_set;
441 };
442 #else
443 #define	RETURN1(rp, v)	RETURNF(v)
444 #define	RETURN2(rp, h, l)	RETURNI((h) + (l))
445 #endif
446 
447 #ifdef STRUCT_RETURN
448 static inline __always_inline void
k_logl(long double x,struct ld * rp)449 k_logl(long double x, struct ld *rp)
450 #else
451 long double
452 logl(long double x)
453 #endif
454 {
455 	long double d, val_hi, val_lo;
456 	double dd, dk;
457 	uint64_t lx, llx;
458 	int i, k;
459 	uint16_t hx;
460 
461 	EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
462 	k = -16383;
463 #if 0 /* Hard to do efficiently.  Don't do it until we support all modes. */
464 	if (x == 1)
465 		RETURN1(rp, 0);		/* log(1) = +0 in all rounding modes */
466 #endif
467 	if (hx == 0 || hx >= 0x8000) {	/* zero, negative or subnormal? */
468 		if (((hx & 0x7fff) | lx | llx) == 0)
469 			RETURN1(rp, -1 / zero);	/* log(+-0) = -Inf */
470 		if (hx != 0)
471 			/* log(neg or NaN) = qNaN: */
472 			RETURN1(rp, (x - x) / zero);
473 		x *= 0x1.0p113;		/* subnormal; scale up x */
474 		EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
475 		k = -16383 - 113;
476 	} else if (hx >= 0x7fff)
477 		RETURN1(rp, x + x);	/* log(Inf or NaN) = Inf or qNaN */
478 #ifndef STRUCT_RETURN
479 	ENTERI();
480 #endif
481 	k += hx;
482 	dk = k;
483 
484 	/* Scale x to be in [1, 2). */
485 	SET_LDBL_EXPSIGN(x, 0x3fff);
486 
487 	/* 0 <= i <= INTERVALS: */
488 #define	L2I	(49 - LOG2_INTERVALS)
489 	i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
490 
491 	/*
492 	 * -0.005280 < d < 0.004838.  In particular, the infinite-
493 	 * precision |d| is <= 2**-7.  Rounding of G(i) to 8 bits
494 	 * ensures that d is representable without extra precision for
495 	 * this bound on |d| (since when this calculation is expressed
496 	 * as x*G(i)-1, the multiplication needs as many extra bits as
497 	 * G(i) has and the subtraction cancels 8 bits).  But for
498 	 * most i (107 cases out of 129), the infinite-precision |d|
499 	 * is <= 2**-8.  G(i) is rounded to 9 bits for such i to give
500 	 * better accuracy (this works by improving the bound on |d|,
501 	 * which in turn allows rounding to 9 bits in more cases).
502 	 * This is only important when the original x is near 1 -- it
503 	 * lets us avoid using a special method to give the desired
504 	 * accuracy for such x.
505 	 */
506 	if (0)
507 		d = x * G(i) - 1;
508 	else {
509 #ifdef USE_UTAB
510 		d = (x - H(i)) * G(i) + E(i);
511 #else
512 		long double x_hi;
513 		double x_lo;
514 
515 		/*
516 		 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
517 		 * G(i) has at most 9 bits, so the splitting point is not
518 		 * critical.
519 		 */
520 		INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
521 		    llx & 0xffffffffff000000ULL);
522 		x_lo = x - x_hi;
523 		d = x_hi * G(i) - 1 + x_lo * G(i);
524 #endif
525 	}
526 
527 	/*
528 	 * Our algorithm depends on exact cancellation of F_lo(i) and
529 	 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
530 	 * at the end of the table.  This and other technical complications
531 	 * make it difficult to avoid the double scaling in (dk*ln2) *
532 	 * log(base) for base != e without losing more accuracy and/or
533 	 * efficiency than is gained.
534 	 */
535 	/*
536 	 * Use double precision operations wherever possible, since long
537 	 * double operations are emulated and are very slow on the only
538 	 * known machines that support ld128 (sparc64).  Also, don't try
539 	 * to improve parallelism by increasing the number of operations,
540 	 * since any parallelism on such machines is needed for the
541 	 * emulation.  Horner's method is good for this, and is also good
542 	 * for accuracy.  Horner's method doesn't handle the `lo' term
543 	 * well, either for efficiency or accuracy.  However, for accuracy
544 	 * we evaluate d * d * P2 separately to take advantage of
545 	 * by P2 being exact, and this gives a good place to sum the 'lo'
546 	 * term too.
547 	 */
548 	dd = (double)d;
549 	val_lo = d * d * d * (P3 +
550 	    d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
551 	    dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
552 	    dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2;
553 	val_hi = d;
554 #ifdef DEBUG
555 	if (fetestexcept(FE_UNDERFLOW))
556 		breakpoint();
557 #endif
558 
559 	_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
560 	RETURN2(rp, val_hi, val_lo);
561 }
562 
563 long double
log1pl(long double x)564 log1pl(long double x)
565 {
566 	long double d, d_hi, f_lo, val_hi, val_lo;
567 	long double f_hi, twopminusk;
568 	double d_lo, dd, dk;
569 	uint64_t lx, llx;
570 	int i, k;
571 	int16_t ax, hx;
572 
573 	DOPRINT_START(&x);
574 	EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
575 	if (hx < 0x3fff) {		/* x < 1, or x neg NaN */
576 		ax = hx & 0x7fff;
577 		if (ax >= 0x3fff) {	/* x <= -1, or x neg NaN */
578 			if (ax == 0x3fff && (lx | llx) == 0)
579 				RETURNP(-1 / zero);	/* log1p(-1) = -Inf */
580 			/* log1p(x < 1, or x NaN) = qNaN: */
581 			RETURNP((x - x) / (x - x));
582 		}
583 		if (ax <= 0x3f8d) {	/* |x| < 2**-113 */
584 			if ((int)x == 0)
585 				RETURNP(x);	/* x with inexact if x != 0 */
586 		}
587 		f_hi = 1;
588 		f_lo = x;
589 	} else if (hx >= 0x7fff) {	/* x +Inf or non-neg NaN */
590 		RETURNP(x + x);		/* log1p(Inf or NaN) = Inf or qNaN */
591 	} else if (hx < 0x40e1) {	/* 1 <= x < 2**226 */
592 		f_hi = x;
593 		f_lo = 1;
594 	} else {			/* 2**226 <= x < +Inf */
595 		f_hi = x;
596 		f_lo = 0;		/* avoid underflow of the P3 term */
597 	}
598 	ENTERI();
599 	x = f_hi + f_lo;
600 	f_lo = (f_hi - x) + f_lo;
601 
602 	EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
603 	k = -16383;
604 
605 	k += hx;
606 	dk = k;
607 
608 	SET_LDBL_EXPSIGN(x, 0x3fff);
609 	twopminusk = 1;
610 	SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
611 	f_lo *= twopminusk;
612 
613 	i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
614 
615 	/*
616 	 * x*G(i)-1 (with a reduced x) can be represented exactly, as
617 	 * above, but now we need to evaluate the polynomial on d =
618 	 * (x+f_lo)*G(i)-1 and extra precision is needed for that.
619 	 * Since x+x_lo is a hi+lo decomposition and subtracting 1
620 	 * doesn't lose too many bits, an inexact calculation for
621 	 * f_lo*G(i) is good enough.
622 	 */
623 	if (0)
624 		d_hi = x * G(i) - 1;
625 	else {
626 #ifdef USE_UTAB
627 		d_hi = (x - H(i)) * G(i) + E(i);
628 #else
629 		long double x_hi;
630 		double x_lo;
631 
632 		INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
633 		    llx & 0xffffffffff000000ULL);
634 		x_lo = x - x_hi;
635 		d_hi = x_hi * G(i) - 1 + x_lo * G(i);
636 #endif
637 	}
638 	d_lo = f_lo * G(i);
639 
640 	/*
641 	 * This is _2sumF(d_hi, d_lo) inlined.  The condition
642 	 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
643 	 * always satisifed, so it is not clear that this works, but
644 	 * it works in practice.  It works even if it gives a wrong
645 	 * normalized d_lo, since |d_lo| > |d_hi| implies that i is
646 	 * nonzero and d is tiny, so the F(i) term dominates d_lo.
647 	 * In float precision:
648 	 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
649 	 * And if d is only a little tinier than that, we would have
650 	 * another underflow problem for the P3 term; this is also ruled
651 	 * out by exhaustive testing.)
652 	 */
653 	d = d_hi + d_lo;
654 	d_lo = d_hi - d + d_lo;
655 	d_hi = d;
656 
657 	dd = (double)d;
658 	val_lo = d * d * d * (P3 +
659 	    d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
660 	    dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
661 	    dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2;
662 	val_hi = d_hi;
663 #ifdef DEBUG
664 	if (fetestexcept(FE_UNDERFLOW))
665 		breakpoint();
666 #endif
667 
668 	_3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
669 	RETURN2PI(val_hi, val_lo);
670 }
671 
672 #ifdef STRUCT_RETURN
673 
674 long double
logl(long double x)675 logl(long double x)
676 {
677 	struct ld r;
678 
679 	ENTERI();
680 	DOPRINT_START(&x);
681 	k_logl(x, &r);
682 	RETURNSPI(&r);
683 }
684 
685 /*
686  * 29+113 bit decompositions.  The bits are distributed so that the products
687  * of the hi terms are exact in double precision.  The types are chosen so
688  * that the products of the hi terms are done in at least double precision,
689  * without any explicit conversions.  More natural choices would require a
690  * slow long double precision multiplication.
691  */
692 static const double
693 invln10_hi =  4.3429448176175356e-1,		/*  0x1bcb7b15000000.0p-54 */
694 invln2_hi =  1.4426950402557850e0;		/*  0x17154765000000.0p-52 */
695 static const long double
696 invln10_lo =  1.41498268538580090791605082294397000e-10L,	/*  0x137287195355baaafad33dc323ee3.0p-145L */
697 invln2_lo =  6.33178418956604368501892137426645911e-10L;	/*  0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
698 
699 long double
log10l(long double x)700 log10l(long double x)
701 {
702 	struct ld r;
703 	long double lo;
704 	float hi;
705 
706 	ENTERI();
707 	DOPRINT_START(&x);
708 	k_logl(x, &r);
709 	if (!r.lo_set)
710 		RETURNPI(r.hi);
711 	_2sumF(r.hi, r.lo);
712 	hi = r.hi;
713 	lo = r.lo + (r.hi - hi);
714 	RETURN2PI(invln10_hi * hi,
715 	    (invln10_lo + invln10_hi) * lo + invln10_lo * hi);
716 }
717 
718 long double
log2l(long double x)719 log2l(long double x)
720 {
721 	struct ld r;
722 	long double lo;
723 	float hi;
724 
725 	ENTERI();
726 	DOPRINT_START(&x);
727 	k_logl(x, &r);
728 	if (!r.lo_set)
729 		RETURNPI(r.hi);
730 	_2sumF(r.hi, r.lo);
731 	hi = r.hi;
732 	lo = r.lo + (r.hi - hi);
733 	RETURN2PI(invln2_hi * hi,
734 	    (invln2_lo + invln2_hi) * lo + invln2_lo * hi);
735 }
736 
737 #endif /* STRUCT_RETURN */
738