1 /*
2 * Double-precision log(1+x) function.
3 *
4 * Copyright (c) 2022-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "poly_scalar_f64.h"
9 #include "math_config.h"
10 #include "test_sig.h"
11 #include "test_defs.h"
12
13 #define Ln2Hi 0x1.62e42fefa3800p-1
14 #define Ln2Lo 0x1.ef35793c76730p-45
15 #define HfRt2Top 0x3fe6a09e /* top32(asuint64(sqrt(2)/2)). */
16 #define OneMHfRt2Top \
17 0x00095f62 /* top32(asuint64(1)) - top32(asuint64(sqrt(2)/2)). */
18 #define OneTop12 0x3ff
19 #define BottomMask 0xffffffff
20 #define OneMHfRt2 0x3fd2bec333018866
21 #define Rt2MOne 0x3fda827999fcef32
22 #define AbsMask 0x7fffffffffffffff
23 #define ExpM63 0x3c00
24
25 static inline double
eval_poly(double f)26 eval_poly (double f)
27 {
28 double f2 = f * f;
29 double f4 = f2 * f2;
30 double f8 = f4 * f4;
31 return estrin_18_f64 (f, f2, f4, f8, f8 * f8, __log1p_data.coeffs);
32 }
33
34 /* log1p approximation using polynomial on reduced interval. Largest
35 observed errors are near the lower boundary of the region where k
36 is 0.
37 Maximum measured error: 1.75ULP.
38 log1p(-0x1.2e1aea97b3e5cp-2) got -0x1.65fb8659a2f9p-2
39 want -0x1.65fb8659a2f92p-2. */
40 double
log1p(double x)41 log1p (double x)
42 {
43 uint64_t ix = asuint64 (x);
44 uint64_t ia = ix & AbsMask;
45 uint32_t ia16 = ia >> 48;
46
47 /* Handle special cases first. */
48 if (unlikely (ia16 >= 0x7ff0 || ix >= 0xbff0000000000000
49 || ix == 0x8000000000000000))
50 {
51 if (ix == 0x8000000000000000 || ix == 0x7ff0000000000000)
52 {
53 /* x == -0 => log1p(x) = -0.
54 x == Inf => log1p(x) = Inf. */
55 return x;
56 }
57 if (ix == 0xbff0000000000000)
58 {
59 /* x == -1 => log1p(x) = -Inf. */
60 return __math_divzero (-1);
61 ;
62 }
63 if (ia16 >= 0x7ff0)
64 {
65 /* x == +/-NaN => log1p(x) = NaN. */
66 return __math_invalid (asdouble (ia));
67 }
68 /* x < -1 => log1p(x) = NaN.
69 x == -Inf => log1p(x) = NaN. */
70 return __math_invalid (x);
71 }
72
73 /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
74 is in [sqrt(2)/2, sqrt(2)]):
75 log1p(x) = k*log(2) + log1p(f).
76
77 f may not be representable exactly, so we need a correction term:
78 let m = round(1 + x), c = (1 + x) - m.
79 c << m: at very small x, log1p(x) ~ x, hence:
80 log(1+x) - log(m) ~ c/m.
81
82 We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */
83
84 uint64_t sign = ix & ~AbsMask;
85 if (ia <= OneMHfRt2 || (!sign && ia <= Rt2MOne))
86 {
87 if (unlikely (ia16 <= ExpM63))
88 {
89 /* If exponent of x <= -63 then shortcut the polynomial and avoid
90 underflow by just returning x, which is exactly rounded in this
91 region. */
92 return x;
93 }
94 /* If x is in [sqrt(2)/2 - 1, sqrt(2) - 1] then we can shortcut all the
95 logic below, as k = 0 and f = x and therefore representable exactly.
96 All we need is to return the polynomial. */
97 return fma (x, eval_poly (x) * x, x);
98 }
99
100 /* Obtain correctly scaled k by manipulation in the exponent. */
101 double m = x + 1;
102 uint64_t mi = asuint64 (m);
103 uint32_t u = (mi >> 32) + OneMHfRt2Top;
104 int32_t k = (int32_t) (u >> 20) - OneTop12;
105
106 /* Correction term c/m. */
107 double cm = (x - (m - 1)) / m;
108
109 /* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */
110 uint32_t utop = (u & 0x000fffff) + HfRt2Top;
111 uint64_t u_red = ((uint64_t) utop << 32) | (mi & BottomMask);
112 double f = asdouble (u_red) - 1;
113
114 /* Approximate log1p(x) on the reduced input using a polynomial. Because
115 log1p(0)=0 we choose an approximation of the form:
116 x + C0*x^2 + C1*x^3 + C2x^4 + ...
117 Hence approximation has the form f + f^2 * P(f)
118 where P(x) = C0 + C1*x + C2x^2 + ... */
119 double p = fma (f, eval_poly (f) * f, f);
120
121 double kd = k;
122 double y = fma (Ln2Lo, kd, cm);
123 return y + fma (Ln2Hi, kd, p);
124 }
125
126 TEST_SIG (S, D, 1, log1p, -0.9, 10.0)
127 TEST_ULP (log1p, 1.26)
128 TEST_SYM_INTERVAL (log1p, 0.0, 0x1p-23, 50000)
129 TEST_SYM_INTERVAL (log1p, 0x1p-23, 0.001, 50000)
130 TEST_SYM_INTERVAL (log1p, 0.001, 1.0, 50000)
131 TEST_SYM_INTERVAL (log1p, 1.0, inf, 5000)
132