1 /*
2  * Double-precision asinh(x) function
3  *
4  * Copyright (c) 2022-2023, Arm Limited.
5  * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6  */
7 #include "poly_scalar_f64.h"
8 #include "math_config.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11 
12 #define AbsMask 0x7fffffffffffffff
13 #define ExpM26 0x3e50000000000000 /* asuint64(0x1.0p-26).  */
14 #define One 0x3ff0000000000000	  /* asuint64(1.0).  */
15 #define Exp511 0x5fe0000000000000 /* asuint64(0x1.0p511).  */
16 #define Ln2 0x1.62e42fefa39efp-1
17 
18 double
19 optr_aor_log_f64 (double);
20 
21 /* Scalar double-precision asinh implementation. This routine uses different
22    approaches on different intervals:
23 
24    |x| < 2^-26: Return x. Function is exact in this region.
25 
26    |x| < 1: Use custom order-17 polynomial. This is least accurate close to 1.
27      The largest observed error in this region is 1.47 ULPs:
28      asinh(0x1.fdfcd00cc1e6ap-1) got 0x1.c1d6bf874019bp-1
29 				want 0x1.c1d6bf874019cp-1.
30 
31    |x| < 2^511: Upper bound of this region is close to sqrt(DBL_MAX). Calculate
32      the result directly using the definition asinh(x) = ln(x + sqrt(x*x + 1)).
33      The largest observed error in this region is 2.03 ULPs:
34      asinh(-0x1.00094e0f39574p+0) got -0x1.c3508eb6a681ep-1
35 				 want -0x1.c3508eb6a682p-1.
36 
37    |x| >= 2^511: We cannot square x without overflow at a low
38      cost. At very large x, asinh(x) ~= ln(2x). At huge x we cannot
39      even double x without overflow, so calculate this as ln(x) +
40      ln(2). The largest observed error in this region is 0.98 ULPs at many
41      values, for instance:
42      asinh(0x1.5255a4cf10319p+975) got 0x1.52652f4cb26cbp+9
43 				  want 0x1.52652f4cb26ccp+9.  */
44 double
45 asinh (double x)
46 {
47   uint64_t ix = asuint64 (x);
48   uint64_t ia = ix & AbsMask;
49   double ax = asdouble (ia);
50   uint64_t sign = ix & ~AbsMask;
51 
52   if (ia < ExpM26)
53     {
54       return x;
55     }
56 
57   if (ia < One)
58     {
59       double x2 = x * x;
60       double z2 = x2 * x2;
61       double z4 = z2 * z2;
62       double z8 = z4 * z4;
63       double p = estrin_17_f64 (x2, z2, z4, z8, z8 * z8, __asinh_data.poly);
64       double y = fma (p, x2 * ax, ax);
65       return asdouble (asuint64 (y) | sign);
66     }
67 
68   if (unlikely (ia >= Exp511))
69     {
70       return asdouble (asuint64 (optr_aor_log_f64 (ax) + Ln2) | sign);
71     }
72 
73   return asdouble (asuint64 (optr_aor_log_f64 (ax + sqrt (ax * ax + 1)))
74 		   | sign);
75 }
76 
77 PL_SIG (S, D, 1, asinh, -10.0, 10.0)
78 PL_TEST_ULP (asinh, 1.54)
79 PL_TEST_INTERVAL (asinh, -0x1p-26, 0x1p-26, 50000)
80 PL_TEST_INTERVAL (asinh, 0x1p-26, 1.0, 40000)
81 PL_TEST_INTERVAL (asinh, -0x1p-26, -1.0, 10000)
82 PL_TEST_INTERVAL (asinh, 1.0, 100.0, 40000)
83 PL_TEST_INTERVAL (asinh, -1.0, -100.0, 10000)
84 PL_TEST_INTERVAL (asinh, 100.0, inf, 50000)
85 PL_TEST_INTERVAL (asinh, -100.0, -inf, 10000)
86