1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Double-precision atanh(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2022-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner
8*f3087befSAndrew Turner #include "math_config.h"
9*f3087befSAndrew Turner #include "poly_scalar_f64.h"
10*f3087befSAndrew Turner #include "test_sig.h"
11*f3087befSAndrew Turner #include "test_defs.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner #define AbsMask 0x7fffffffffffffff
14*f3087befSAndrew Turner #define Half 0x3fe0000000000000
15*f3087befSAndrew Turner #define One 0x3ff0000000000000
16*f3087befSAndrew Turner #define Ln2Hi 0x1.62e42fefa3800p-1
17*f3087befSAndrew Turner #define Ln2Lo 0x1.ef35793c76730p-45
18*f3087befSAndrew Turner #define OneMHfRt2Top \
19*f3087befSAndrew Turner 0x00095f62 /* top32(asuint64(1)) - top32(asuint64(sqrt(2)/2)). */
20*f3087befSAndrew Turner #define OneTop12 0x3ff
21*f3087befSAndrew Turner #define HfRt2Top 0x3fe6a09e /* top32(asuint64(sqrt(2)/2)). */
22*f3087befSAndrew Turner #define BottomMask 0xffffffff
23*f3087befSAndrew Turner
24*f3087befSAndrew Turner static inline double
log1p_inline(double x)25*f3087befSAndrew Turner log1p_inline (double x)
26*f3087befSAndrew Turner {
27*f3087befSAndrew Turner /* Helper for calculating log(1 + x) using order-18 polynomial on a reduced
28*f3087befSAndrew Turner interval. Copied from log1p_2u.c, with no special-case handling. See that
29*f3087befSAndrew Turner file for details of the algorithm. */
30*f3087befSAndrew Turner double m = x + 1;
31*f3087befSAndrew Turner uint64_t mi = asuint64 (m);
32*f3087befSAndrew Turner
33*f3087befSAndrew Turner /* Decompose x + 1 into (f + 1) * 2^k, with k chosen such that f is in
34*f3087befSAndrew Turner [sqrt(2)/2, sqrt(2)]. */
35*f3087befSAndrew Turner uint32_t u = (mi >> 32) + OneMHfRt2Top;
36*f3087befSAndrew Turner int32_t k = (int32_t) (u >> 20) - OneTop12;
37*f3087befSAndrew Turner uint32_t utop = (u & 0x000fffff) + HfRt2Top;
38*f3087befSAndrew Turner uint64_t u_red = ((uint64_t) utop << 32) | (mi & BottomMask);
39*f3087befSAndrew Turner double f = asdouble (u_red) - 1;
40*f3087befSAndrew Turner
41*f3087befSAndrew Turner /* Correction term for round-off in f. */
42*f3087befSAndrew Turner double cm = (x - (m - 1)) / m;
43*f3087befSAndrew Turner
44*f3087befSAndrew Turner /* Approximate log1p(f) with polynomial. */
45*f3087befSAndrew Turner double f2 = f * f;
46*f3087befSAndrew Turner double f4 = f2 * f2;
47*f3087befSAndrew Turner double f8 = f4 * f4;
48*f3087befSAndrew Turner double p = fma (
49*f3087befSAndrew Turner f, estrin_18_f64 (f, f2, f4, f8, f8 * f8, __log1p_data.coeffs) * f, f);
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner /* Recombine log1p(x) = k*log2 + log1p(f) + c/m. */
52*f3087befSAndrew Turner double kd = k;
53*f3087befSAndrew Turner double y = fma (Ln2Lo, kd, cm);
54*f3087befSAndrew Turner return y + fma (Ln2Hi, kd, p);
55*f3087befSAndrew Turner }
56*f3087befSAndrew Turner
57*f3087befSAndrew Turner /* Approximation for double-precision inverse tanh(x), using a simplified
58*f3087befSAndrew Turner version of log1p. Greatest observed error is 3.00 ULP:
59*f3087befSAndrew Turner atanh(0x1.e58f3c108d714p-4) got 0x1.e7da77672a647p-4
60*f3087befSAndrew Turner want 0x1.e7da77672a64ap-4. */
61*f3087befSAndrew Turner double
atanh(double x)62*f3087befSAndrew Turner atanh (double x)
63*f3087befSAndrew Turner {
64*f3087befSAndrew Turner uint64_t ix = asuint64 (x);
65*f3087befSAndrew Turner uint64_t sign = ix & ~AbsMask;
66*f3087befSAndrew Turner uint64_t ia = ix & AbsMask;
67*f3087befSAndrew Turner
68*f3087befSAndrew Turner if (unlikely (ia == One))
69*f3087befSAndrew Turner return __math_divzero (sign >> 32);
70*f3087befSAndrew Turner
71*f3087befSAndrew Turner if (unlikely (ia > One))
72*f3087befSAndrew Turner return __math_invalid (x);
73*f3087befSAndrew Turner
74*f3087befSAndrew Turner double halfsign = asdouble (Half | sign);
75*f3087befSAndrew Turner double ax = asdouble (ia);
76*f3087befSAndrew Turner return halfsign * log1p_inline ((2 * ax) / (1 - ax));
77*f3087befSAndrew Turner }
78*f3087befSAndrew Turner
79*f3087befSAndrew Turner TEST_SIG (S, D, 1, atanh, -1.0, 1.0)
80*f3087befSAndrew Turner TEST_ULP (atanh, 3.00)
81*f3087befSAndrew Turner TEST_SYM_INTERVAL (atanh, 0, 0x1p-23, 10000)
82*f3087befSAndrew Turner TEST_SYM_INTERVAL (atanh, 0x1p-23, 1, 90000)
83*f3087befSAndrew Turner TEST_SYM_INTERVAL (atanh, 1, inf, 100)
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