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/*
* Single-precision scalar atan2(x) function.
*
* Copyright (c) 2021-2022, Arm Limited.
* SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
*/
#include <stdbool.h>
#include "atanf_common.h"
#include "math_config.h"
#include "pl_sig.h"
#include "pl_test.h"
#define Pi (0x1.921fb6p+1f)
#define PiOver2 (0x1.921fb6p+0f)
#define PiOver4 (0x1.921fb6p-1f)
#define SignMask (0x80000000)
/* We calculate atan2f by P(n/d), where n and d are similar to the input
arguments, and P is a polynomial. The polynomial may underflow.
POLY_UFLOW_BOUND is the lower bound of the difference in exponents of n and d
for which P underflows, and is used to special-case such inputs. */
#define POLY_UFLOW_BOUND 24
static inline int32_t
biased_exponent (float f)
{
uint32_t fi = asuint (f);
int32_t ex = (int32_t) ((fi & 0x7f800000) >> 23);
if (unlikely (ex == 0))
{
/* Subnormal case - we still need to get the exponent right for subnormal
numbers as division may take us back inside the normal range. */
return ex - __builtin_clz (fi << 9);
}
return ex;
}
/* Fast implementation of scalar atan2f. Largest observed error is
2.88ulps in [99.0, 101.0] x [99.0, 101.0]:
atan2f(0x1.9332d8p+6, 0x1.8cb6c4p+6) got 0x1.964646p-1
want 0x1.964640p-1. */
float
atan2f (float y, float x)
{
uint32_t ix = asuint (x);
uint32_t iy = asuint (y);
uint32_t sign_x = ix & SignMask;
uint32_t sign_y = iy & SignMask;
uint32_t iax = ix & ~SignMask;
uint32_t iay = iy & ~SignMask;
/* x or y is NaN. */
if ((iax > 0x7f800000) || (iay > 0x7f800000))
return x + y;
/* m = 2 * sign(x) + sign(y). */
uint32_t m = ((iy >> 31) & 1) | ((ix >> 30) & 2);
/* The following follows glibc ieee754 implementation, except
that we do not use +-tiny shifts (non-nearest rounding mode). */
int32_t exp_diff = biased_exponent (x) - biased_exponent (y);
/* Special case for (x, y) either on or very close to the x axis. Either y =
0, or y is tiny and x is huge (difference in exponents >=
POLY_UFLOW_BOUND). In the second case, we only want to use this special
case when x is negative (i.e. quadrants 2 or 3). */
if (unlikely (iay == 0 || (exp_diff >= POLY_UFLOW_BOUND && m >= 2)))
{
switch (m)
{
case 0:
case 1:
return y; /* atan(+-0,+anything)=+-0. */
case 2:
return Pi; /* atan(+0,-anything) = pi. */
case 3:
return -Pi; /* atan(-0,-anything) =-pi. */
}
}
/* Special case for (x, y) either on or very close to the y axis. Either x =
0, or x is tiny and y is huge (difference in exponents >=
POLY_UFLOW_BOUND). */
if (unlikely (iax == 0 || exp_diff <= -POLY_UFLOW_BOUND))
return sign_y ? -PiOver2 : PiOver2;
/* x is INF. */
if (iax == 0x7f800000)
{
if (iay == 0x7f800000)
{
switch (m)
{
case 0:
return PiOver4; /* atan(+INF,+INF). */
case 1:
return -PiOver4; /* atan(-INF,+INF). */
case 2:
return 3.0f * PiOver4; /* atan(+INF,-INF). */
case 3:
return -3.0f * PiOver4; /* atan(-INF,-INF). */
}
}
else
{
switch (m)
{
case 0:
return 0.0f; /* atan(+...,+INF). */
case 1:
return -0.0f; /* atan(-...,+INF). */
case 2:
return Pi; /* atan(+...,-INF). */
case 3:
return -Pi; /* atan(-...,-INF). */
}
}
}
/* y is INF. */
if (iay == 0x7f800000)
return sign_y ? -PiOver2 : PiOver2;
uint32_t sign_xy = sign_x ^ sign_y;
float ax = asfloat (iax);
float ay = asfloat (iay);
bool pred_aygtax = (ay > ax);
/* Set up z for call to atanf. */
float n = pred_aygtax ? -ax : ay;
float d = pred_aygtax ? ay : ax;
float z = n / d;
float ret;
if (unlikely (m < 2 && exp_diff >= POLY_UFLOW_BOUND))
{
/* If (x, y) is very close to x axis and x is positive, the polynomial
will underflow and evaluate to z. */
ret = z;
}
else
{
/* Work out the correct shift. */
float shift = sign_x ? -2.0f : 0.0f;
shift = pred_aygtax ? shift + 1.0f : shift;
shift *= PiOver2;
ret = eval_poly (z, z, shift);
}
/* Account for the sign of x and y. */
return asfloat (asuint (ret) ^ sign_xy);
}
/* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
PL_SIG (S, F, 2, atan2)
PL_TEST_ULP (atan2f, 2.4)
PL_TEST_INTERVAL (atan2f, -10.0, 10.0, 50000)
PL_TEST_INTERVAL (atan2f, -1.0, 1.0, 40000)
PL_TEST_INTERVAL (atan2f, 0.0, 1.0, 40000)
PL_TEST_INTERVAL (atan2f, 1.0, 100.0, 40000)
PL_TEST_INTERVAL (atan2f, 1e6, 1e32, 40000)
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