[libc-commits] [libc] [libc][math] Implement a fast pass for atan2f128 with 1ULP error using DyadicFloat<128>. (PR #133150)
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Mon Mar 31 07:27:17 PDT 2025
================
@@ -0,0 +1,203 @@
+//===-- Quad-precision atan2 function -------------------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "src/math/atan2f128.h"
+#include "atan_utils.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/dyadic_float.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/integer_literals.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#include "src/__support/macros/properties/types.h"
+#include "src/__support/uint128.h"
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace {
+
+using Float128 = fputil::DyadicFloat<128>;
+
+static constexpr Float128 ZERO = {Sign::POS, 0, 0_u128};
+static constexpr Float128 MZERO = {Sign::NEG, 0, 0_u128};
+static constexpr Float128 PI = {Sign::POS, -126,
+ 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 MPI = {Sign::NEG, -126,
+ 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 PI_OVER_2 = {
+ Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 MPI_OVER_2 = {
+ Sign::NEG, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 PI_OVER_4 = {
+ Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 THREE_PI_OVER_4 = {
+ Sign::POS, -128, 0x96cbe3f9'990e91a7'9394c9e8'a0a5159d_u128};
+
+// Adjustment for constant term:
+// CONST_ADJ[x_sign][y_sign][recip]
+static constexpr Float128 CONST_ADJ[2][2][2] = {
+ {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
+ {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
+
+} // anonymous namespace
+
+// There are several range reduction steps we can take for atan2(y, x) as
+// follow:
+
+// * Range reduction 1: signness
+// atan2(y, x) will return a number between -PI and PI representing the angle
+// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
+// In particular, we have that:
+// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
+// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
+// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
+// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
+// Since atan function is odd, we can use the formula:
+// atan(-u) = -atan(u)
+// to adjust the above conditions a bit further:
+// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
+// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
+// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
+// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
+// Which can be simplified to:
+// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
+// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
+
+// * Range reduction 2: reciprocal
+// Now that the argument inside atan is positive, we can use the formula:
+// atan(1/x) = pi/2 - atan(x)
+// to make the argument inside atan <= 1 as follow:
+// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
+// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
+// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
+// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
+
+// * Range reduction 3: look up table.
+// After the previous two range reduction steps, we reduce the problem to
+// compute atan(u) with 0 <= u <= 1, or to be precise:
+// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
+// An accurate polynomial approximation for the whole [0, 1] input range will
+// require a very large degree. To make it more efficient, we reduce the input
+// range further by finding an integer idx such that:
+// | n/d - idx/64 | <= 1/128.
+// In particular,
+// idx := round(2^6 * n/d)
+// Then for the fast pass, we find a polynomial approximation for:
+// atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
+// For the accurate pass, we use the addition formula:
+// atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
+// = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
+// And for the fast pass, we use degree-13 minimax polynomial to compute the
+// RHS:
+// atan(u) ~ P(u) = u - c_3 * u^3 + c_5 * u^5 - c_7 * u^7 + c_9 *u^9 -
+// - c_11 * u^11 + c_13 * u^13
+// with absolute errors bounded by:
+// |atan(u) - P(u)| < 2^-121
+// and relative errors bounded by:
+// |(atan(u) - P(u)) / P(u)| < 2^-114.
+
+LLVM_LIBC_FUNCTION(float128, atan2f128, (float128 y, float128 x)) {
+ using FPBits = fputil::FPBits<float128>;
+ using Float128 = fputil::DyadicFloat<128>;
+
+ FPBits x_bits(x), y_bits(y);
+ bool x_sign = x_bits.sign().is_neg();
+ bool y_sign = y_bits.sign().is_neg();
+ x_bits = x_bits.abs();
+ y_bits = y_bits.abs();
+ UInt128 x_abs = x_bits.uintval();
+ UInt128 y_abs = y_bits.uintval();
+ bool recip = x_abs < y_abs;
+ UInt128 min_abs = recip ? x_abs : y_abs;
+ UInt128 max_abs = !recip ? x_abs : y_abs;
+ unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
+ unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
+
+ Float128 num(FPBits(min_abs).get_val());
+ Float128 den(FPBits(max_abs).get_val());
+
+ // Check for exceptional cases, whether inputs are 0, inf, nan, or close to
+ // overflow, or close to underflow.
+ if (LIBC_UNLIKELY(max_exp >= 0x7fffU || min_exp == 0U)) {
+ if (x_bits.is_nan() || y_bits.is_nan())
+ return FPBits::quiet_nan().get_val();
+ unsigned x_except = x == 0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
+ unsigned y_except = y == 0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);
+
+ // Exceptional cases:
+ // EXCEPT[y_except][x_except][x_is_neg]
+ // with x_except & y_except:
+ // 0: zero
+ // 1: finite, non-zero
+ // 2: infinity
+ constexpr Float128 EXCEPTS[3][3][2] = {
+ {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
+ {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
+ {{PI_OVER_2, PI_OVER_2},
+ {PI_OVER_2, PI_OVER_2},
+ {PI_OVER_4, THREE_PI_OVER_4}},
+ };
+
+ if ((x_except != 1) || (y_except != 1)) {
+ Float128 r = EXCEPTS[y_except][x_except][x_sign];
+ if (y_sign)
+ r.sign = (r.sign == Sign::POS) ? Sign::NEG : Sign::POS;
----------------
overmighty wrote:
```suggestion
r.sign = r.sign.negate();
```
https://github.com/llvm/llvm-project/pull/133150
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