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/*
* Double-precision atan(x) function.
*
* Copyright (c) 2022-2023, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include "pl_sig.h"
#include "pl_test.h"
#include "atan_common.h"
#define AbsMask 0x7fffffffffffffff
#define PiOver2 0x1.921fb54442d18p+0
#define TinyBound 0x3e1 /* top12(asuint64(0x1p-30)). */
#define BigBound 0x434 /* top12(asuint64(0x1p53)). */
#define OneTop 0x3ff
/* Fast implementation of double-precision atan.
Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
atan(0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
want 0x1.9225645bdd7c3p-1. */
double
atan (double x)
{
uint64_t ix = asuint64 (x);
uint64_t sign = ix & ~AbsMask;
uint64_t ia = ix & AbsMask;
uint32_t ia12 = ia >> 52;
if (unlikely (ia12 >= BigBound || ia12 < TinyBound))
{
if (ia12 < TinyBound)
/* Avoid underflow by returning x. */
return x;
if (ia > 0x7ff0000000000000)
/* Propagate NaN. */
return __math_invalid (x);
/* atan(x) rounds to PiOver2 for large x. */
return asdouble (asuint64 (PiOver2) ^ sign);
}
double z, az, shift;
if (ia12 >= OneTop)
{
/* For x > 1, use atan(x) = pi / 2 + atan(-1 / x). */
z = -1.0 / x;
shift = PiOver2;
/* Use absolute value only when needed (odd powers of z). */
az = -fabs (z);
}
else
{
/* For x < 1, approximate atan(x) directly. */
z = x;
shift = 0;
az = asdouble (ia);
}
/* Calculate polynomial, shift + z + z^3 * P(z^2). */
double y = eval_poly (z, az, shift);
/* Copy sign. */
return asdouble (asuint64 (y) ^ sign);
}
PL_SIG (S, D, 1, atan, -10.0, 10.0)
PL_TEST_ULP (atan, 1.78)
PL_TEST_INTERVAL (atan, 0, 0x1p-30, 10000)
PL_TEST_INTERVAL (atan, -0, -0x1p-30, 1000)
PL_TEST_INTERVAL (atan, 0x1p-30, 0x1p53, 900000)
PL_TEST_INTERVAL (atan, -0x1p-30, -0x1p53, 90000)
PL_TEST_INTERVAL (atan, 0x1p53, inf, 10000)
PL_TEST_INTERVAL (atan, -0x1p53, -inf, 1000)
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