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/*
* Double-precision vector log(1+x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "v_math.h"
#include "estrin.h"
#include "pl_sig.h"
#include "pl_test.h"
#if V_SUPPORTED
#define Ln2Hi v_f64 (0x1.62e42fefa3800p-1)
#define Ln2Lo v_f64 (0x1.ef35793c76730p-45)
#define HfRt2Top 0x3fe6a09e00000000 /* top32(asuint64(sqrt(2)/2)) << 32. */
#define OneMHfRt2Top \
0x00095f6200000000 /* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) \
<< 32. */
#define OneTop12 0x3ff
#define BottomMask 0xffffffff
#define AbsMask 0x7fffffffffffffff
#define C(i) v_f64 (__log1p_data.coeffs[i])
static inline v_f64_t
eval_poly (v_f64_t f)
{
v_f64_t f2 = f * f;
v_f64_t f4 = f2 * f2;
v_f64_t f8 = f4 * f4;
return ESTRIN_18 (f, f2, f4, f8, f8 * f8, C);
}
VPCS_ATTR
NOINLINE static v_f64_t
specialcase (v_f64_t x, v_f64_t y, v_u64_t special)
{
return v_call_f64 (log1p, x, y, special);
}
/* Vector log1p approximation using polynomial on reduced interval. Routine is a
modification of the algorithm used in scalar log1p, with no shortcut for k=0
and no narrowing for f and k. Maximum observed error is 2.46 ULP:
__v_log1p(0x1.654a1307242a4p+11) got 0x1.fd5565fb590f4p+2
want 0x1.fd5565fb590f6p+2 . */
VPCS_ATTR v_f64_t V_NAME (log1p) (v_f64_t x)
{
v_u64_t ix = v_as_u64_f64 (x);
v_u64_t ia = ix & AbsMask;
v_u64_t special
= v_cond_u64 ((ia >= v_u64 (0x7ff0000000000000))
| (ix >= 0xbff0000000000000) | (ix == 0x8000000000000000));
#if WANT_SIMD_EXCEPT
if (unlikely (v_any_u64 (special)))
x = v_sel_f64 (special, v_f64 (0), x);
#endif
/* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
is in [sqrt(2)/2, sqrt(2)]):
log1p(x) = k*log(2) + log1p(f).
f may not be representable exactly, so we need a correction term:
let m = round(1 + x), c = (1 + x) - m.
c << m: at very small x, log1p(x) ~ x, hence:
log(1+x) - log(m) ~ c/m.
We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */
/* Obtain correctly scaled k by manipulation in the exponent.
The scalar algorithm casts down to 32-bit at this point to calculate k and
u_red. We stay in double-width to obtain f and k, using the same constants
as the scalar algorithm but shifted left by 32. */
v_f64_t m = x + 1;
v_u64_t mi = v_as_u64_f64 (m);
v_u64_t u = mi + OneMHfRt2Top;
v_s64_t ki = v_as_s64_u64 (u >> 52) - OneTop12;
v_f64_t k = v_to_f64_s64 (ki);
/* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */
v_u64_t utop = (u & 0x000fffff00000000) + HfRt2Top;
v_u64_t u_red = utop | (mi & BottomMask);
v_f64_t f = v_as_f64_u64 (u_red) - 1;
/* Correction term c/m. */
v_f64_t cm = (x - (m - 1)) / m;
/* Approximate log1p(x) on the reduced input using a polynomial. Because
log1p(0)=0 we choose an approximation of the form:
x + C0*x^2 + C1*x^3 + C2x^4 + ...
Hence approximation has the form f + f^2 * P(f)
where P(x) = C0 + C1*x + C2x^2 + ...
Assembling this all correctly is dealt with at the final step. */
v_f64_t p = eval_poly (f);
v_f64_t ylo = v_fma_f64 (k, Ln2Lo, cm);
v_f64_t yhi = v_fma_f64 (k, Ln2Hi, f);
v_f64_t y = v_fma_f64 (f * f, p, ylo + yhi);
if (unlikely (v_any_u64 (special)))
return specialcase (v_as_f64_u64 (ix), y, special);
return y;
}
VPCS_ALIAS
PL_SIG (V, D, 1, log1p, -0.9, 10.0)
PL_TEST_ULP (V_NAME (log1p), 1.97)
PL_TEST_EXPECT_FENV (V_NAME (log1p), WANT_SIMD_EXCEPT)
PL_TEST_INTERVAL (V_NAME (log1p), -10.0, 10.0, 10000)
PL_TEST_INTERVAL (V_NAME (log1p), 0.0, 0x1p-23, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), 0x1p-23, 0.001, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), 0.001, 1.0, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), 0.0, -0x1p-23, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), -0x1p-23, -0.001, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), -0.001, -1.0, 50000)
PL_TEST_INTERVAL (V_NAME (log1p), -1.0, inf, 5000)
#endif
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