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[libc][math] Refactor atan2 implementation to header-only in src/__support/math folder.
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9 files changed

+266
-198
lines changed

9 files changed

+266
-198
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‎libc/shared/math.h

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@@ -23,6 +23,7 @@
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#include "math/asinhf.h"
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#include "math/asinhf16.h"
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#include "math/atan.h"
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#include "math/atan2.h"
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#include "math/atanf.h"
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#include "math/atanf16.h"
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#include "math/erff.h"

‎libc/shared/math/atan2.h

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//===-- Shared atan2 function -----------------------------------*- C++ -*-===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#ifndef LLVM_LIBC_SHARED_MATH_ATAN2_H
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#define LLVM_LIBC_SHARED_MATH_ATAN2_H
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#include "shared/libc_common.h"
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#include "src/__support/math/atan2.h"
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namespace LIBC_NAMESPACE_DECL {
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namespace shared {
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using math::atan2;
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} // namespace shared
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} // namespace LIBC_NAMESPACE_DECL
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#endif // LLVM_LIBC_SHARED_MATH_ATAN2_H

‎libc/src/__support/math/CMakeLists.txt

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@@ -158,7 +158,7 @@ add_header_library(
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asinhf16
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HDRS
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asinhf16.h
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DEPENDS
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DEPENDS
162162
.acoshf_utils
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libc.src.__support.FPUtil.fenv_impl
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libc.src.__support.FPUtil.fp_bits
@@ -176,7 +176,7 @@ add_header_library(
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atan_utils
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HDRS
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atan_utils.h
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DEPENDS
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DEPENDS
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libc.src.__support.integer_literals
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libc.src.__support.FPUtil.double_double
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libc.src.__support.FPUtil.dyadic_float
@@ -189,7 +189,21 @@ add_header_library(
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atan
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HDRS
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atan.h
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DEPENDS
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DEPENDS
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.atan_utils
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libc.src.__support.FPUtil.double_double
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libc.src.__support.FPUtil.fenv_impl
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libc.src.__support.FPUtil.fp_bits
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libc.src.__support.FPUtil.multiply_add
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libc.src.__support.FPUtil.nearest_integer
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libc.src.__support.macros.optimization
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)
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add_header_library(
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atan2
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HDRS
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atan2.h
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DEPENDS
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.atan_utils
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libc.src.__support.FPUtil.double_double
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libc.src.__support.FPUtil.fenv_impl

‎libc/src/__support/math/atan2.h

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//===-- Implementation header for atan2 -------------------------*- C++ -*-===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H
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#define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H
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#include "atan_utils.h"
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#include "src/__support/FPUtil/FEnvImpl.h"
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#include "src/__support/FPUtil/FPBits.h"
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#include "src/__support/FPUtil/double_double.h"
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#include "src/__support/FPUtil/multiply_add.h"
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#include "src/__support/FPUtil/nearest_integer.h"
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#include "src/__support/macros/config.h"
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#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
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namespace LIBC_NAMESPACE_DECL {
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namespace math {
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// There are several range reduction steps we can take for atan2(y, x) as
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// follow:
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// * Range reduction 1: signness
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// atan2(y, x) will return a number between -PI and PI representing the angle
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// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
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// In particular, we have that:
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// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
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// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
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// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
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// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
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// Since atan function is odd, we can use the formula:
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// atan(-u) = -atan(u)
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// to adjust the above conditions a bit further:
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// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
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// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
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// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
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// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
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// Which can be simplified to:
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// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
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// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
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// * Range reduction 2: reciprocal
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// Now that the argument inside atan is positive, we can use the formula:
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// atan(1/x) = pi/2 - atan(x)
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// to make the argument inside atan <= 1 as follow:
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// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
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// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
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// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
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// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
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// * Range reduction 3: look up table.
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// After the previous two range reduction steps, we reduce the problem to
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// compute atan(u) with 0 <= u <= 1, or to be precise:
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// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
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// An accurate polynomial approximation for the whole [0, 1] input range will
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// require a very large degree. To make it more efficient, we reduce the input
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// range further by finding an integer idx such that:
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// | n/d - idx/64 | <= 1/128.
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// In particular,
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// idx := round(2^6 * n/d)
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// Then for the fast pass, we find a polynomial approximation for:
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// atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
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// For the accurate pass, we use the addition formula:
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// atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
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// = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
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// And for the fast pass, we use degree-9 Taylor polynomial to compute the RHS:
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// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9
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// with absolute errors bounded by:
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// |atan(u) - P(u)| < |u|^11 / 11 < 2^-80
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// and relative errors bounded by:
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// |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73.
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LIBC_INLINE static constexpr double atan2(double y, double x) {
79+
using namespace atan_internal;
80+
using FPBits = fputil::FPBits<double>;
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constexpr double IS_NEG[2] = {1.0, -1.0};
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constexpr DoubleDouble ZERO = {0.0, 0.0};
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constexpr DoubleDouble MZERO = {-0.0, -0.0};
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constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1};
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constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1};
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constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54,
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0x1.921fb54442d18p0};
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constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54,
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-0x1.921fb54442d18p0};
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constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55,
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0x1.921fb54442d18p-1};
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constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54,
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0x1.2d97c7f3321d2p+1};
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// Adjustment for constant term:
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// CONST_ADJ[x_sign][y_sign][recip]
97+
constexpr DoubleDouble CONST_ADJ[2][2][2] = {
98+
{{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
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{{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
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FPBits x_bits(x), y_bits(y);
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bool x_sign = x_bits.sign().is_neg();
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bool y_sign = y_bits.sign().is_neg();
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x_bits = x_bits.abs();
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y_bits = y_bits.abs();
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uint64_t x_abs = x_bits.uintval();
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uint64_t y_abs = y_bits.uintval();
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bool recip = x_abs < y_abs;
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uint64_t min_abs = recip ? x_abs : y_abs;
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uint64_t max_abs = !recip ? x_abs : y_abs;
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unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
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unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
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double num = FPBits(min_abs).get_val();
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double den = FPBits(max_abs).get_val();
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// Check for exceptional cases, whether inputs are 0, inf, nan, or close to
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// overflow, or close to underflow.
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if (LIBC_UNLIKELY(max_exp > 0x7ffU - 128U || min_exp < 128U)) {
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if (x_bits.is_nan() || y_bits.is_nan()) {
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if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
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fputil::raise_except_if_required(FE_INVALID);
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return FPBits::quiet_nan().get_val();
124+
}
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unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
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unsigned y_except = y == 0.0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);
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// Exceptional cases:
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// EXCEPT[y_except][x_except][x_is_neg]
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// with x_except & y_except:
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// 0: zero
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// 1: finite, non-zero
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// 2: infinity
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constexpr DoubleDouble EXCEPTS[3][3][2] = {
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{{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
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{{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
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{{PI_OVER_2, PI_OVER_2},
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{PI_OVER_2, PI_OVER_2},
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{PI_OVER_4, THREE_PI_OVER_4}},
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};
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if ((x_except != 1) || (y_except != 1)) {
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DoubleDouble r = EXCEPTS[y_except][x_except][x_sign];
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return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);
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}
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bool scale_up = min_exp < 128U;
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bool scale_down = max_exp > 0x7ffU - 128U;
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// At least one input is denormal, multiply both numerator and denominator
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// by some large enough power of 2 to normalize denormal inputs.
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if (scale_up) {
151+
num *= 0x1.0p64;
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if (!scale_down)
153+
den *= 0x1.0p64;
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} else if (scale_down) {
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den *= 0x1.0p-64;
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if (!scale_up)
157+
num *= 0x1.0p-64;
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}
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min_abs = FPBits(num).uintval();
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max_abs = FPBits(den).uintval();
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min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
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max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
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}
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double final_sign = IS_NEG[(x_sign != y_sign) != recip];
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DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
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unsigned exp_diff = max_exp - min_exp;
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// We have the following bound for normalized n and d:
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// 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
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if (LIBC_UNLIKELY(exp_diff > 54)) {
172+
return fputil::multiply_add(final_sign, const_term.hi,
173+
final_sign * (const_term.lo + num / den));
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}
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double k = fputil::nearest_integer(64.0 * num / den);
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unsigned idx = static_cast<unsigned>(k);
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// k = idx / 64
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k *= 0x1.0p-6;
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// Range reduction:
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// atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
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// = atan((n - d * k/64)) / (d + n * k/64))
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DoubleDouble num_k = fputil::exact_mult(num, k);
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DoubleDouble den_k = fputil::exact_mult(den, k);
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// num_dd = n - d * k
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DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo);
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// den_dd = d + n * k
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DoubleDouble den_dd = fputil::exact_add(den, num_k.hi);
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den_dd.lo += num_k.lo;
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// q = (n - d * k) / (d + n * k)
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DoubleDouble q = fputil::div(num_dd, den_dd);
195+
// p ~ atan(q)
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DoubleDouble p = atan_eval(q);
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DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));
199+
r.hi *= final_sign;
200+
r.lo *= final_sign;
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return r.hi + r.lo;
203+
}
204+
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} // namespace math
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} // namespace LIBC_NAMESPACE_DECL
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#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H

‎libc/src/math/generic/CMakeLists.txt

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Original file line numberDiff line numberDiff line change
@@ -4079,13 +4079,7 @@ add_entrypoint_object(
40794079
HDRS
40804080
../atan2.h
40814081
DEPENDS
4082-
libc.src.__support.math.atan_utils
4083-
libc.src.__support.FPUtil.double_double
4084-
libc.src.__support.FPUtil.fenv_impl
4085-
libc.src.__support.FPUtil.fp_bits
4086-
libc.src.__support.FPUtil.multiply_add
4087-
libc.src.__support.FPUtil.nearest_integer
4088-
libc.src.__support.macros.optimization
4082+
40894083
)
40904084

40914085
add_entrypoint_object(

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