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/*
* Double-precision atanh(x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "math_config.h"
#include "estrin.h"
#include "pl_sig.h"
#include "pl_test.h"
#define AbsMask 0x7fffffffffffffff
#define Half 0x3fe0000000000000
#define One 0x3ff0000000000000
#define Ln2Hi 0x1.62e42fefa3800p-1
#define Ln2Lo 0x1.ef35793c76730p-45
#define OneMHfRt2Top \
0x00095f62 /* top32(asuint64(1)) - top32(asuint64(sqrt(2)/2)). */
#define OneTop12 0x3ff
#define HfRt2Top 0x3fe6a09e /* top32(asuint64(sqrt(2)/2)). */
#define BottomMask 0xffffffff
#define C(i) __log1p_data.coeffs[i]
static inline double
log1p_inline (double x)
{
/* Helper for calculating log(1 + x) using order-18 polynomial on a reduced
interval. Copied from log1p_2u.c, with no special-case handling. See that
file for details of the algorithm. */
double m = x + 1;
uint64_t mi = asuint64 (m);
/* Decompose x + 1 into (f + 1) * 2^k, with k chosen such that f is in
[sqrt(2)/2, sqrt(2)]. */
uint32_t u = (mi >> 32) + OneMHfRt2Top;
int32_t k = (int32_t) (u >> 20) - OneTop12;
uint32_t utop = (u & 0x000fffff) + HfRt2Top;
uint64_t u_red = ((uint64_t) utop << 32) | (mi & BottomMask);
double f = asdouble (u_red) - 1;
/* Correction term for round-off in f. */
double cm = (x - (m - 1)) / m;
/* Approximate log1p(f) with polynomial. */
double f2 = f * f;
double f4 = f2 * f2;
double f8 = f4 * f4;
double p = fma (f, ESTRIN_18 (f, f2, f4, f8, f8 * f8, C) * f, f);
/* Recombine log1p(x) = k*log2 + log1p(f) + c/m. */
double kd = k;
double y = fma (Ln2Lo, kd, cm);
return y + fma (Ln2Hi, kd, p);
}
/* Approximation for double-precision inverse tanh(x), using a simplified
version of log1p. Greatest observed error is 3.00 ULP:
atanh(0x1.e58f3c108d714p-4) got 0x1.e7da77672a647p-4
want 0x1.e7da77672a64ap-4. */
double
atanh (double x)
{
uint64_t ix = asuint64 (x);
uint64_t sign = ix & ~AbsMask;
uint64_t ia = ix & AbsMask;
if (unlikely (ia == One))
return __math_divzero (sign >> 32);
if (unlikely (ia > One))
return __math_invalid (x);
double halfsign = asdouble (Half | sign);
double ax = asdouble (ia);
return halfsign * log1p_inline ((2 * ax) / (1 - ax));
}
PL_SIG (S, D, 1, atanh, -1.0, 1.0)
PL_TEST_ULP (atanh, 3.00)
PL_TEST_INTERVAL (atanh, 0, 0x1p-23, 10000)
PL_TEST_INTERVAL (atanh, -0, -0x1p-23, 10000)
PL_TEST_INTERVAL (atanh, 0x1p-23, 1, 90000)
PL_TEST_INTERVAL (atanh, -0x1p-23, -1, 90000)
PL_TEST_INTERVAL (atanh, 1, inf, 100)
PL_TEST_INTERVAL (atanh, -1, -inf, 100)
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