1 /*
2 * Double-precision vector log(1+x) function.
3 *
4 * Copyright (c) 2022-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "v_math.h"
9 #include "estrin.h"
10 #include "pl_sig.h"
11 #include "pl_test.h"
12
13 #if V_SUPPORTED
14
15 #define Ln2Hi v_f64 (0x1.62e42fefa3800p-1)
16 #define Ln2Lo v_f64 (0x1.ef35793c76730p-45)
17 #define HfRt2Top 0x3fe6a09e00000000 /* top32(asuint64(sqrt(2)/2)) << 32. */
18 #define OneMHfRt2Top \
19 0x00095f6200000000 /* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) \
20 << 32. */
21 #define OneTop12 0x3ff
22 #define BottomMask 0xffffffff
23 #define AbsMask 0x7fffffffffffffff
24 #define C(i) v_f64 (__log1p_data.coeffs[i])
25
26 static inline v_f64_t
eval_poly(v_f64_t f)27 eval_poly (v_f64_t f)
28 {
29 v_f64_t f2 = f * f;
30 v_f64_t f4 = f2 * f2;
31 v_f64_t f8 = f4 * f4;
32 return ESTRIN_18 (f, f2, f4, f8, f8 * f8, C);
33 }
34
35 VPCS_ATTR
36 NOINLINE static v_f64_t
specialcase(v_f64_t x,v_f64_t y,v_u64_t special)37 specialcase (v_f64_t x, v_f64_t y, v_u64_t special)
38 {
39 return v_call_f64 (log1p, x, y, special);
40 }
41
42 /* Vector log1p approximation using polynomial on reduced interval. Routine is a
43 modification of the algorithm used in scalar log1p, with no shortcut for k=0
44 and no narrowing for f and k. Maximum observed error is 2.46 ULP:
45 __v_log1p(0x1.654a1307242a4p+11) got 0x1.fd5565fb590f4p+2
46 want 0x1.fd5565fb590f6p+2 . */
V_NAME(log1p)47 VPCS_ATTR v_f64_t V_NAME (log1p) (v_f64_t x)
48 {
49 v_u64_t ix = v_as_u64_f64 (x);
50 v_u64_t ia = ix & AbsMask;
51 v_u64_t special
52 = v_cond_u64 ((ia >= v_u64 (0x7ff0000000000000))
53 | (ix >= 0xbff0000000000000) | (ix == 0x8000000000000000));
54
55 #if WANT_SIMD_EXCEPT
56 if (unlikely (v_any_u64 (special)))
57 x = v_sel_f64 (special, v_f64 (0), x);
58 #endif
59
60 /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
61 is in [sqrt(2)/2, sqrt(2)]):
62 log1p(x) = k*log(2) + log1p(f).
63
64 f may not be representable exactly, so we need a correction term:
65 let m = round(1 + x), c = (1 + x) - m.
66 c << m: at very small x, log1p(x) ~ x, hence:
67 log(1+x) - log(m) ~ c/m.
68
69 We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */
70
71 /* Obtain correctly scaled k by manipulation in the exponent.
72 The scalar algorithm casts down to 32-bit at this point to calculate k and
73 u_red. We stay in double-width to obtain f and k, using the same constants
74 as the scalar algorithm but shifted left by 32. */
75 v_f64_t m = x + 1;
76 v_u64_t mi = v_as_u64_f64 (m);
77 v_u64_t u = mi + OneMHfRt2Top;
78
79 v_s64_t ki = v_as_s64_u64 (u >> 52) - OneTop12;
80 v_f64_t k = v_to_f64_s64 (ki);
81
82 /* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */
83 v_u64_t utop = (u & 0x000fffff00000000) + HfRt2Top;
84 v_u64_t u_red = utop | (mi & BottomMask);
85 v_f64_t f = v_as_f64_u64 (u_red) - 1;
86
87 /* Correction term c/m. */
88 v_f64_t cm = (x - (m - 1)) / m;
89
90 /* Approximate log1p(x) on the reduced input using a polynomial. Because
91 log1p(0)=0 we choose an approximation of the form:
92 x + C0*x^2 + C1*x^3 + C2x^4 + ...
93 Hence approximation has the form f + f^2 * P(f)
94 where P(x) = C0 + C1*x + C2x^2 + ...
95 Assembling this all correctly is dealt with at the final step. */
96 v_f64_t p = eval_poly (f);
97
98 v_f64_t ylo = v_fma_f64 (k, Ln2Lo, cm);
99 v_f64_t yhi = v_fma_f64 (k, Ln2Hi, f);
100 v_f64_t y = v_fma_f64 (f * f, p, ylo + yhi);
101
102 if (unlikely (v_any_u64 (special)))
103 return specialcase (v_as_f64_u64 (ix), y, special);
104
105 return y;
106 }
107 VPCS_ALIAS
108
109 PL_SIG (V, D, 1, log1p, -0.9, 10.0)
110 PL_TEST_ULP (V_NAME (log1p), 1.97)
111 PL_TEST_EXPECT_FENV (V_NAME (log1p), WANT_SIMD_EXCEPT)
112 PL_TEST_INTERVAL (V_NAME (log1p), -10.0, 10.0, 10000)
113 PL_TEST_INTERVAL (V_NAME (log1p), 0.0, 0x1p-23, 50000)
114 PL_TEST_INTERVAL (V_NAME (log1p), 0x1p-23, 0.001, 50000)
115 PL_TEST_INTERVAL (V_NAME (log1p), 0.001, 1.0, 50000)
116 PL_TEST_INTERVAL (V_NAME (log1p), 0.0, -0x1p-23, 50000)
117 PL_TEST_INTERVAL (V_NAME (log1p), -0x1p-23, -0.001, 50000)
118 PL_TEST_INTERVAL (V_NAME (log1p), -0.001, -1.0, 50000)
119 PL_TEST_INTERVAL (V_NAME (log1p), -1.0, inf, 5000)
120 #endif
121