3 * Relative error logarithm
4 * Natural logarithm of 1+x, 128-bit long double precision
10 * long double x, y, log1pl();
18 * Returns the base e (2.718...) logarithm of 1+x.
20 * The argument 1+x is separated into its exponent and fractional
21 * parts. If the exponent is between -1 and +1, the logarithm
22 * of the fraction is approximated by
24 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
26 * Otherwise, setting z = 2(w-1)/(w+1),
28 * log(w) = z + z^3 P(z)/Q(z).
35 * arithmetic domain # trials peak rms
36 * IEEE -1, 8 100000 1.9e-34 4.3e-35
39 /* Copyright 2001 by Stephen L. Moshier
41 This library is free software; you can redistribute it and/or
42 modify it under the terms of the GNU Lesser General Public
43 License as published by the Free Software Foundation; either
44 version 2.1 of the License, or (at your option) any later version.
46 This library is distributed in the hope that it will be useful,
47 but WITHOUT ANY WARRANTY; without even the implied warranty of
48 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
49 Lesser General Public License for more details.
51 You should have received a copy of the GNU Lesser General Public
52 License along with this library; if not, see
53 <http://www.gnu.org/licenses/>. */
58 #include <math_private.h>
60 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
61 * 1/sqrt(2) <= 1+x < sqrt(2)
62 * Theoretical peak relative error = 5.3e-37,
63 * relative peak error spread = 2.3e-14
65 static const _Float128
66 P12
= L(1.538612243596254322971797716843006400388E-6),
67 P11
= L(4.998469661968096229986658302195402690910E-1),
68 P10
= L(2.321125933898420063925789532045674660756E1
),
69 P9
= L(4.114517881637811823002128927449878962058E2
),
70 P8
= L(3.824952356185897735160588078446136783779E3
),
71 P7
= L(2.128857716871515081352991964243375186031E4
),
72 P6
= L(7.594356839258970405033155585486712125861E4
),
73 P5
= L(1.797628303815655343403735250238293741397E5
),
74 P4
= L(2.854829159639697837788887080758954924001E5
),
75 P3
= L(3.007007295140399532324943111654767187848E5
),
76 P2
= L(2.014652742082537582487669938141683759923E5
),
77 P1
= L(7.771154681358524243729929227226708890930E4
),
78 P0
= L(1.313572404063446165910279910527789794488E4
),
79 /* Q12 = 1.000000000000000000000000000000000000000E0L, */
80 Q11
= L(4.839208193348159620282142911143429644326E1
),
81 Q10
= L(9.104928120962988414618126155557301584078E2
),
82 Q9
= L(9.147150349299596453976674231612674085381E3
),
83 Q8
= L(5.605842085972455027590989944010492125825E4
),
84 Q7
= L(2.248234257620569139969141618556349415120E5
),
85 Q6
= L(6.132189329546557743179177159925690841200E5
),
86 Q5
= L(1.158019977462989115839826904108208787040E6
),
87 Q4
= L(1.514882452993549494932585972882995548426E6
),
88 Q3
= L(1.347518538384329112529391120390701166528E6
),
89 Q2
= L(7.777690340007566932935753241556479363645E5
),
90 Q1
= L(2.626900195321832660448791748036714883242E5
),
91 Q0
= L(3.940717212190338497730839731583397586124E4
);
93 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
94 * where z = 2(x-1)/(x+1)
95 * 1/sqrt(2) <= x < sqrt(2)
96 * Theoretical peak relative error = 1.1e-35,
97 * relative peak error spread 1.1e-9
99 static const _Float128
100 R5
= L(-8.828896441624934385266096344596648080902E-1),
101 R4
= L(8.057002716646055371965756206836056074715E1
),
102 R3
= L(-2.024301798136027039250415126250455056397E3
),
103 R2
= L(2.048819892795278657810231591630928516206E4
),
104 R1
= L(-8.977257995689735303686582344659576526998E4
),
105 R0
= L(1.418134209872192732479751274970992665513E5
),
106 /* S6 = 1.000000000000000000000000000000000000000E0L, */
107 S5
= L(-1.186359407982897997337150403816839480438E2
),
108 S4
= L(3.998526750980007367835804959888064681098E3
),
109 S3
= L(-5.748542087379434595104154610899551484314E4
),
110 S2
= L(4.001557694070773974936904547424676279307E5
),
111 S1
= L(-1.332535117259762928288745111081235577029E6
),
112 S0
= L(1.701761051846631278975701529965589676574E6
);
115 static const _Float128 C1
= L(6.93145751953125E-1);
116 static const _Float128 C2
= L(1.428606820309417232121458176568075500134E-6);
118 static const _Float128 sqrth
= L(0.7071067811865475244008443621048490392848);
119 /* ln (2^16384 * (1 - 2^-113)) */
120 static const _Float128 zero
= 0;
123 __log1pl (_Float128 xm1
)
125 _Float128 x
, y
, z
, r
, s
;
126 ieee854_long_double_shape_type u
;
130 /* Test for NaN or infinity input. */
133 if ((hx
& 0x7fffffff) >= 0x7fff0000)
134 return xm1
+ fabsl (xm1
);
136 /* log1p(+- 0) = +- 0. */
137 if (((hx
& 0x7fffffff) == 0)
138 && (u
.parts32
.w1
| u
.parts32
.w2
| u
.parts32
.w3
) == 0)
141 if ((hx
& 0x7fffffff) < 0x3f8e0000)
143 math_check_force_underflow (xm1
);
148 if (xm1
>= L(0x1p
113))
153 /* log1p(-1) = -inf */
157 return (-1 / zero
); /* log1p(-1) = -inf */
159 return (zero
/ (x
- x
));
162 /* Separate mantissa from exponent. */
164 /* Use frexp used so that denormal numbers will be handled properly. */
165 x
= __frexpl (x
, &e
);
167 /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
168 where z = 2(x-1)/x+1). */
169 if ((e
> 2) || (e
< -2))
172 { /* 2( 2x-1 )/( 2x+1 ) */
175 y
= L(0.5) * z
+ L(0.5);
178 { /* 2 (x-1)/(x+1) */
181 y
= L(0.5) * x
+ L(0.5);
206 /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
212 x
= 2 * x
- 1; /* 2x - 1 */
224 r
= (((((((((((P12
* x