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/*
* Double-precision vector exp(x) - 1 function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "v_math.h"
#include "pl_sig.h"
#include "pl_test.h"
#if V_SUPPORTED
#define InvLn2 v_f64 (0x1.71547652b82fep0)
#define MLn2hi v_f64 (-0x1.62e42fefa39efp-1)
#define MLn2lo v_f64 (-0x1.abc9e3b39803fp-56)
#define Shift v_f64 (0x1.8p52)
#define TinyBound \
0x3cc0000000000000 /* 0x1p-51, below which expm1(x) is within 2 ULP of x. */
#define SpecialBound \
0x40862b7d369a5aa9 /* 0x1.62b7d369a5aa9p+9. For |x| > SpecialBound, the \
final stage of the algorithm overflows so fall back to \
scalar. */
#define AbsMask 0x7fffffffffffffff
#define One 0x3ff0000000000000
#define C(i) v_f64 (__expm1_poly[i])
static inline v_f64_t
eval_poly (v_f64_t f, v_f64_t f2)
{
/* Evaluate custom polynomial using Estrin scheme. */
v_f64_t p_01 = v_fma_f64 (f, C (1), C (0));
v_f64_t p_23 = v_fma_f64 (f, C (3), C (2));
v_f64_t p_45 = v_fma_f64 (f, C (5), C (4));
v_f64_t p_67 = v_fma_f64 (f, C (7), C (6));
v_f64_t p_89 = v_fma_f64 (f, C (9), C (8));
v_f64_t p_03 = v_fma_f64 (f2, p_23, p_01);
v_f64_t p_47 = v_fma_f64 (f2, p_67, p_45);
v_f64_t p_8a = v_fma_f64 (f2, C (10), p_89);
v_f64_t f4 = f2 * f2;
v_f64_t p_07 = v_fma_f64 (f4, p_47, p_03);
return v_fma_f64 (f4 * f4, p_8a, p_07);
}
/* Double-precision vector exp(x) - 1 function.
The maximum error observed error is 2.18 ULP:
__v_expm1(0x1.634ba0c237d7bp-2) got 0x1.a8b9ea8d66e22p-2
want 0x1.a8b9ea8d66e2p-2. */
VPCS_ATTR
v_f64_t V_NAME (expm1) (v_f64_t x)
{
v_u64_t ix = v_as_u64_f64 (x);
v_u64_t ax = ix & AbsMask;
#if WANT_SIMD_EXCEPT
/* If fp exceptions are to be triggered correctly, fall back to the scalar
variant for all lanes if any of them should trigger an exception. */
v_u64_t special = v_cond_u64 ((ax >= SpecialBound) | (ax <= TinyBound));
if (unlikely (v_any_u64 (special)))
return v_call_f64 (expm1, x, x, v_u64 (-1));
#else
/* Large input, NaNs and Infs. */
v_u64_t special
= v_cond_u64 ((ax >= SpecialBound) | (ix == 0x8000000000000000));
#endif
/* Reduce argument to smaller range:
Let i = round(x / ln2)
and f = x - i * ln2, then f is in [-ln2/2, ln2/2].
exp(x) - 1 = 2^i * (expm1(f) + 1) - 1
where 2^i is exact because i is an integer. */
v_f64_t j = v_fma_f64 (InvLn2, x, Shift) - Shift;
v_s64_t i = v_to_s64_f64 (j);
v_f64_t f = v_fma_f64 (j, MLn2hi, x);
f = v_fma_f64 (j, MLn2lo, f);
/* Approximate expm1(f) using polynomial.
Taylor expansion for expm1(x) has the form:
x + ax^2 + bx^3 + cx^4 ....
So we calculate the polynomial P(f) = a + bf + cf^2 + ...
and assemble the approximation expm1(f) ~= f + f^2 * P(f). */
v_f64_t f2 = f * f;
v_f64_t p = v_fma_f64 (f2, eval_poly (f, f2), f);
/* Assemble the result.
expm1(x) ~= 2^i * (p + 1) - 1
Let t = 2^i. */
v_f64_t t = v_as_f64_u64 (v_as_u64_s64 (i << 52) + One);
/* expm1(x) ~= p * t + (t - 1). */
v_f64_t y = v_fma_f64 (p, t, t - 1);
#if !WANT_SIMD_EXCEPT
if (unlikely (v_any_u64 (special)))
return v_call_f64 (expm1, x, y, special);
#endif
return y;
}
VPCS_ALIAS
PL_SIG (V, D, 1, expm1, -9.9, 9.9)
PL_TEST_ULP (V_NAME (expm1), 1.68)
PL_TEST_EXPECT_FENV (V_NAME (expm1), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME (expm1), 0, 0x1p-51, 1000)
PL_TEST_INTERVAL (V_NAME (expm1), -0, -0x1p-51, 1000)
PL_TEST_INTERVAL (V_NAME (expm1), 0x1p-51, 0x1.63108c75a1937p+9, 100000)
PL_TEST_INTERVAL (V_NAME (expm1), -0x1p-51, -0x1.740bf7c0d927dp+9, 100000)
PL_TEST_INTERVAL (V_NAME (expm1), 0x1.63108c75a1937p+9, inf, 100)
PL_TEST_INTERVAL (V_NAME (expm1), -0x1.740bf7c0d927dp+9, -inf, 100)
#endif
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