1 /*
2 * Double-precision SVE 10^x function.
3 *
4 * Copyright (c) 2023-2025, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "sv_math.h"
9 #include "test_sig.h"
10 #include "test_defs.h"
11
12 #define SpecialBound 307.0 /* floor (log10 (2^1023)). */
13
14 static const struct data
15 {
16 double c1, c3, c2, c4, c0;
17 double shift, log10_2, log2_10_hi, log2_10_lo, scale_thres, special_bound;
18 } data = {
19 /* Coefficients generated using Remez algorithm.
20 rel error: 0x1.9fcb9b3p-60
21 abs error: 0x1.a20d9598p-60 in [ -log10(2)/128, log10(2)/128 ]
22 max ulp err 0.52 +0.5. */
23 .c0 = 0x1.26bb1bbb55516p1,
24 .c1 = 0x1.53524c73cd32ap1,
25 .c2 = 0x1.0470591daeafbp1,
26 .c3 = 0x1.2bd77b1361ef6p0,
27 .c4 = 0x1.142b5d54e9621p-1,
28 /* 1.5*2^46+1023. This value is further explained below. */
29 .shift = 0x1.800000000ffc0p+46,
30 .log10_2 = 0x1.a934f0979a371p1, /* 1/log2(10). */
31 .log2_10_hi = 0x1.34413509f79ffp-2, /* log2(10). */
32 .log2_10_lo = -0x1.9dc1da994fd21p-59,
33 .scale_thres = 1280.0,
34 .special_bound = SpecialBound,
35 };
36
37 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */
38 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */
39 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */
40 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */
41
42 /* Update of both special and non-special cases, if any special case is
43 detected. */
44 static inline svfloat64_t
special_case(svbool_t pg,svfloat64_t s,svfloat64_t y,svfloat64_t n,const struct data * d)45 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n,
46 const struct data *d)
47 {
48 /* s=2^n may overflow, break it up into s=s1*s2,
49 such that exp = s + s*y can be computed as s1*(s2+s2*y)
50 and s1*s1 overflows only if n>0. */
51
52 /* If n<=0 then set b to 0x6, 0 otherwise. */
53 svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */
54 svuint64_t b = svdup_u64_z (p_sign, SpecialOffset);
55
56 /* Set s1 to generate overflow depending on sign of exponent n. */
57 svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1));
58 /* Offset s to avoid overflow in final result if n is below threshold. */
59 svfloat64_t s2 = svreinterpret_f64 (
60 svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
61
62 /* |n| > 1280 => 2^(n) overflows. */
63 svbool_t p_cmp = svacgt (pg, n, d->scale_thres);
64
65 svfloat64_t r1 = svmul_x (svptrue_b64 (), s1, s1);
66 svfloat64_t r2 = svmla_x (pg, s2, s2, y);
67 svfloat64_t r0 = svmul_x (svptrue_b64 (), r2, s1);
68
69 return svsel (p_cmp, r1, r0);
70 }
71
72 /* Fast vector implementation of exp10 using FEXPA instruction.
73 Maximum measured error is 1.02 ulp.
74 SV_NAME_D1 (exp10)(-0x1.2862fec805e58p+2) got 0x1.885a89551d782p-16
75 want 0x1.885a89551d781p-16. */
SV_NAME_D1(exp10)76 svfloat64_t SV_NAME_D1 (exp10) (svfloat64_t x, svbool_t pg)
77 {
78 const struct data *d = ptr_barrier (&data);
79 svbool_t no_big_scale = svacle (pg, x, d->special_bound);
80 svbool_t special = svnot_z (pg, no_big_scale);
81
82 /* n = round(x/(log10(2)/N)). */
83 svfloat64_t shift = sv_f64 (d->shift);
84 svfloat64_t z = svmla_x (pg, shift, x, d->log10_2);
85 svfloat64_t n = svsub_x (pg, z, shift);
86
87 /* r = x - n*log10(2)/N. */
88 svfloat64_t log2_10 = svld1rq (svptrue_b64 (), &d->log2_10_hi);
89 svfloat64_t r = x;
90 r = svmls_lane (r, n, log2_10, 0);
91 r = svmls_lane (r, n, log2_10, 1);
92
93 /* scale = 2^(n/N), computed using FEXPA. FEXPA does not propagate NaNs, so
94 for consistent NaN handling we have to manually propagate them. This
95 comes at significant performance cost. */
96 svuint64_t u = svreinterpret_u64 (z);
97 svfloat64_t scale = svexpa (u);
98 svfloat64_t c24 = svld1rq (svptrue_b64 (), &d->c2);
99 /* Approximate exp10(r) using polynomial. */
100 svfloat64_t r2 = svmul_x (svptrue_b64 (), r, r);
101 svfloat64_t p12 = svmla_lane (sv_f64 (d->c1), r, c24, 0);
102 svfloat64_t p34 = svmla_lane (sv_f64 (d->c3), r, c24, 1);
103 svfloat64_t p14 = svmla_x (pg, p12, p34, r2);
104
105 svfloat64_t y = svmla_x (pg, svmul_x (svptrue_b64 (), r, d->c0), r2, p14);
106
107 /* Assemble result as exp10(x) = 2^n * exp10(r). If |x| > SpecialBound
108 multiplication may overflow, so use special case routine. */
109 if (unlikely (svptest_any (pg, special)))
110 {
111 /* FEXPA zeroes the sign bit, however the sign is meaningful to the
112 special case function so needs to be copied.
113 e = sign bit of u << 46. */
114 svuint64_t e = svand_x (pg, svlsl_x (pg, u, 46), 0x8000000000000000);
115 /* Copy sign to scale. */
116 scale = svreinterpret_f64 (svadd_x (pg, e, svreinterpret_u64 (scale)));
117 return special_case (pg, scale, y, n, d);
118 }
119
120 /* No special case. */
121 return svmla_x (pg, scale, scale, y);
122 }
123
124 #if WANT_EXP10_TESTS
125 TEST_SIG (SV, D, 1, exp10, -9.9, 9.9)
126 TEST_ULP (SV_NAME_D1 (exp10), 0.52)
127 TEST_DISABLE_FENV (SV_NAME_D1 (exp10))
128 TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), 0, SpecialBound, 10000)
129 TEST_SYM_INTERVAL (SV_NAME_D1 (exp10), SpecialBound, inf, 1000)
130 #endif
131 CLOSE_SVE_ATTR
132