1 /* Single-precision 10^x function.
2 Copyright (C) 2020-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include <math.h>
20 #include <math-narrow-eval.h>
21 #include <stdint.h>
22 #include <libm-alias-finite.h>
23 #include <libm-alias-float.h>
24 #include <shlib-compat.h>
25 #include <math-svid-compat.h>
26 #include "math_config.h"
27
28 /*
29 EXP2F_TABLE_BITS 5
30 EXP2F_POLY_ORDER 3
31
32 Max. ULP error: 0.502 (normal range, nearest rounding).
33 Max. relative error: 2^-33.240 (before rounding to float).
34 Wrong count: 169839.
35 Non-nearest ULP error: 1 (rounded ULP error).
36
37 Detailed error analysis (for EXP2F_TABLE_BITS=3 thus N=32):
38
39 - We first compute z = RN(InvLn10N * x) where
40
41 InvLn10N = N*log(10)/log(2) * (1 + theta1) with |theta1| < 2^-54.150
42
43 since z is rounded to nearest:
44
45 z = InvLn10N * x * (1 + theta2) with |theta2| < 2^-53
46
47 thus z = N*log(10)/log(2) * x * (1 + theta3) with |theta3| < 2^-52.463
48
49 - Since |x| < 38 in the main branch, we deduce:
50
51 z = N*log(10)/log(2) * x + theta4 with |theta4| < 2^-40.483
52
53 - We then write z = k + r where k is an integer and |r| <= 0.5 (exact)
54
55 - We thus have
56
57 x = z*log(2)/(N*log(10)) - theta4*log(2)/(N*log(10))
58 = z*log(2)/(N*log(10)) + theta5 with |theta5| < 2^-47.215
59
60 10^x = 2^(k/N) * 2^(r/N) * 10^theta5
61 = 2^(k/N) * 2^(r/N) * (1 + theta6) with |theta6| < 2^-46.011
62
63 - Then 2^(k/N) is approximated by table lookup, the maximal relative error
64 is for (k%N) = 5, with
65
66 s = 2^(i/N) * (1 + theta7) with |theta7| < 2^-53.249
67
68 - 2^(r/N) is approximated by a degree-3 polynomial, where the maximal
69 mathematical relative error is 2^-33.243.
70
71 - For C[0] * r + C[1], assuming no FMA is used, since |r| <= 0.5 and
72 |C[0]| < 1.694e-6, |C[0] * r| < 8.47e-7, and the absolute error on
73 C[0] * r is bounded by 1/2*ulp(8.47e-7) = 2^-74. Then we add C[1] with
74 |C[1]| < 0.000235, thus the absolute error on C[0] * r + C[1] is bounded
75 by 2^-65.994 (z is bounded by 0.000236).
76
77 - For r2 = r * r, since |r| <= 0.5, we have |r2| <= 0.25 and the absolute
78 error is bounded by 1/4*ulp(0.25) = 2^-56 (the factor 1/4 is because |r2|
79 cannot exceed 1/4, and on the left of 1/4 the distance between two
80 consecutive numbers is 1/4*ulp(1/4)).
81
82 - For y = C[2] * r + 1, assuming no FMA is used, since |r| <= 0.5 and
83 |C[2]| < 0.0217, the absolute error on C[2] * r is bounded by 2^-60,
84 and thus the absolute error on C[2] * r + 1 is bounded by 1/2*ulp(1)+2^60
85 < 2^-52.988, and |y| < 1.01085 (the error bound is better if a FMA is
86 used).
87
88 - for z * r2 + y: the absolute error on z is bounded by 2^-65.994, with
89 |z| < 0.000236, and the absolute error on r2 is bounded by 2^-56, with
90 r2 < 0.25, thus |z*r2| < 0.000059, and the absolute error on z * r2
91 (including the rounding error) is bounded by:
92
93 2^-65.994 * 0.25 + 0.000236 * 2^-56 + 1/2*ulp(0.000059) < 2^-66.429.
94
95 Now we add y, with |y| < 1.01085, and error on y bounded by 2^-52.988,
96 thus the absolute error is bounded by:
97
98 2^-66.429 + 2^-52.988 + 1/2*ulp(1.01085) < 2^-51.993
99
100 - Now we convert the error on y into relative error. Recall that y
101 approximates 2^(r/N), for |r| <= 0.5 and N=32. Thus 2^(-0.5/32) <= y,
102 and the relative error on y is bounded by
103
104 2^-51.993/2^(-0.5/32) < 2^-51.977
105
106 - Taking into account the mathematical relative error of 2^-33.243 we have:
107
108 y = 2^(r/N) * (1 + theta8) with |theta8| < 2^-33.242
109
110 - Since we had s = 2^(k/N) * (1 + theta7) with |theta7| < 2^-53.249,
111 after y = y * s we get y = 2^(k/N) * 2^(r/N) * (1 + theta9)
112 with |theta9| < 2^-33.241
113
114 - Finally, taking into account the error theta6 due to the rounding error on
115 InvLn10N, and when multiplying InvLn10N * x, we get:
116
117 y = 10^x * (1 + theta10) with |theta10| < 2^-33.240
118
119 - Converting into binary64 ulps, since y < 2^53*ulp(y), the error is at most
120 2^19.76 ulp(y)
121
122 - If the result is a binary32 value in the normal range (i.e., >= 2^-126),
123 then the error is at most 2^-9.24 ulps, i.e., less than 0.00166 (in the
124 subnormal range, the error in ulps might be larger).
125
126 Note that this bound is tight, since for x=-0x25.54ac0p0 the final value of
127 y (before conversion to float) differs from 879853 ulps from the correctly
128 rounded value, and 879853 ~ 2^19.7469. */
129
130 #define N (1 << EXP2F_TABLE_BITS)
131
132 #define InvLn10N (0x3.5269e12f346e2p0 * N) /* log(10)/log(2) to nearest */
133 #define T __exp2f_data.tab
134 #define C __exp2f_data.poly_scaled
135 #define SHIFT __exp2f_data.shift
136
137 static inline uint32_t
138 top13 (float x)
139 {
140 return asuint (x) >> 19;
141 }
142
143 float
144 __exp10f (float x)
145 {
146 uint32_t abstop;
147 uint64_t ki, t;
148 double kd, xd, z, r, r2, y, s;
149
150 xd = (double) x;
151 abstop = top13 (x) & 0xfff; /* Ignore sign. */
152 if (__glibc_unlikely (abstop >= top13 (38.0f)))
153 {
154 /* |x| >= 38 or x is nan. */
155 if (asuint (x) == asuint (-INFINITY))
156 return 0.0f;
157 if (abstop >= top13 (INFINITY))
158 return x + x;
159 /* 0x26.8826ap0 is the largest value such that 10^x < 2^128. */
160 if (x > 0x26.8826ap0f)
161 return __math_oflowf (0);
162 /* -0x2d.278d4p0 is the smallest value such that 10^x > 2^-150. */
163 if (x < -0x2d.278d4p0f)
164 return __math_uflowf (0);
165 #if WANT_ERRNO_UFLOW
166 if (x < -0x2c.da7cfp0)
167 return __math_may_uflowf (0);
168 #endif
169 /* the smallest value such that 10^x >= 2^-126 (normal range)
170 is x = -0x25.ee060p0 */
171 /* we go through here for 2014929 values out of 2060451840
172 (not counting NaN and infinities, i.e., about 0.1% */
173 }
174
175 /* x*N*Ln10/Ln2 = k + r with r in [-1/2, 1/2] and int k. */
176 z = InvLn10N * xd;
177 /* |xd| < 38 thus |z| < 1216 */
178 #if TOINT_INTRINSICS
179 kd = roundtoint (z);
180 ki = converttoint (z);
181 #else
182 # define SHIFT __exp2f_data.shift
183 kd = math_narrow_eval ((double) (z + SHIFT)); /* Needs to be double. */
184 ki = asuint64 (kd);
185 kd -= SHIFT;
186 #endif
187 r = z - kd;
188
189 /* 10^x = 10^(k/N) * 10^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
190 t = T[ki % N];
191 t += ki << (52 - EXP2F_TABLE_BITS);
192 s = asdouble (t);
193 z = C[0] * r + C[1];
194 r2 = r * r;
195 y = C[2] * r + 1;
196 y = z * r2 + y;
197 y = y * s;
198 return (float) y;
199 }
200 #ifndef __exp10f
201 strong_alias (__exp10f, __ieee754_exp10f)
202 libm_alias_finite (__ieee754_exp10f, __exp10f)
203 /* For architectures that already provided exp10f without SVID support, there
204 is no need to add a new version. */
205 #if !LIBM_SVID_COMPAT
206 # define EXP10F_VERSION GLIBC_2_26
207 #else
208 # define EXP10F_VERSION GLIBC_2_32
209 #endif
210 versioned_symbol (libm, __exp10f, exp10f, EXP10F_VERSION);
211 libm_alias_float_other (__exp10, exp10)
212 #endif