[libc-commits] [libc] [libc][math] Add float-only option for atan2f. (PR #122979)
via libc-commits
libc-commits at lists.llvm.org
Tue Feb 11 13:07:59 PST 2025
https://github.com/lntue updated https://github.com/llvm/llvm-project/pull/122979
>From 420bcff640b59b14c0e2600e8a316ac8be492941 Mon Sep 17 00:00:00 2001
From: Tue Ly <lntue.h at gmail.com>
Date: Tue, 11 Feb 2025 18:50:56 +0000
Subject: [PATCH 1/2] [libc][math] Add float-only option for atan2f.
---
libc/src/__support/FPUtil/double_double.h | 97 +++----
libc/src/__support/macros/optimization.h | 5 +
libc/src/math/generic/CMakeLists.txt | 2 +
libc/src/math/generic/atan2f.cpp | 10 +
libc/src/math/generic/atan2f_float.h | 239 ++++++++++++++++++
.../math/generic/range_reduction_double_fma.h | 12 +-
.../generic/range_reduction_double_nofma.h | 12 +-
7 files changed, 322 insertions(+), 55 deletions(-)
create mode 100644 libc/src/math/generic/atan2f_float.h
diff --git a/libc/src/__support/FPUtil/double_double.h b/libc/src/__support/FPUtil/double_double.h
index db3c2c8a3d7a6..825038b22290a 100644
--- a/libc/src/__support/FPUtil/double_double.h
+++ b/libc/src/__support/FPUtil/double_double.h
@@ -20,41 +20,52 @@ namespace fputil {
#define DEFAULT_DOUBLE_SPLIT 27
-using DoubleDouble = LIBC_NAMESPACE::NumberPair<double>;
+template <typename T> struct DefaultSplit;
+template <> struct DefaultSplit<float> {
+ static constexpr size_t VALUE = 12;
+};
+template <> struct DefaultSplit<double> {
+ static constexpr size_t VALUE = 27;
+};
+
+using DoubleDouble = NumberPair<double>;
+using FloatFloat = NumberPair<float>;
// The output of Dekker's FastTwoSum algorithm is correct, i.e.:
// r.hi + r.lo = a + b exactly
// and |r.lo| < eps(r.lo)
// Assumption: |a| >= |b|, or a = 0.
-template <bool FAST2SUM = true>
-LIBC_INLINE constexpr DoubleDouble exact_add(double a, double b) {
- DoubleDouble r{0.0, 0.0};
+template <bool FAST2SUM = true, typename T = double>
+LIBC_INLINE constexpr NumberPair<T> exact_add(T a, T b) {
+ NumberPair<T> r{0.0, 0.0};
if constexpr (FAST2SUM) {
r.hi = a + b;
- double t = r.hi - a;
+ T t = r.hi - a;
r.lo = b - t;
} else {
r.hi = a + b;
- double t1 = r.hi - a;
- double t2 = r.hi - t1;
- double t3 = b - t1;
- double t4 = a - t2;
+ T t1 = r.hi - a;
+ T t2 = r.hi - t1;
+ T t3 = b - t1;
+ T t4 = a - t2;
r.lo = t3 + t4;
}
return r;
}
// Assumption: |a.hi| >= |b.hi|
-LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a,
- const DoubleDouble &b) {
- DoubleDouble r = exact_add(a.hi, b.hi);
- double lo = a.lo + b.lo;
+template <typename T>
+LIBC_INLINE constexpr NumberPair<T> add(const NumberPair<T> &a,
+ const NumberPair<T> &b) {
+ NumberPair<T> r = exact_add(a.hi, b.hi);
+ T lo = a.lo + b.lo;
return exact_add(r.hi, r.lo + lo);
}
// Assumption: |a.hi| >= |b|
-LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a, double b) {
- DoubleDouble r = exact_add<false>(a.hi, b);
+template <typename T>
+LIBC_INLINE constexpr NumberPair<T> add(const NumberPair<T> &a, T b) {
+ NumberPair<T> r = exact_add<false>(a.hi, b);
return exact_add(r.hi, r.lo + a.lo);
}
@@ -63,12 +74,12 @@ LIBC_INLINE constexpr DoubleDouble add(const DoubleDouble &a, double b) {
// Zimmermann, P., "Note on the Veltkamp/Dekker Algorithms with Directed
// Roundings," https://inria.hal.science/hal-04480440.
// Default splitting constant = 2^ceil(prec(double)/2) + 1 = 2^27 + 1.
-template <size_t N = DEFAULT_DOUBLE_SPLIT>
-LIBC_INLINE constexpr DoubleDouble split(double a) {
- DoubleDouble r{0.0, 0.0};
+template <typename T = double, size_t N = DefaultSplit<T>::VALUE>
+LIBC_INLINE constexpr NumberPair<T> split(T a) {
+ NumberPair<T> r{0.0, 0.0};
// CN = 2^N.
- constexpr double CN = static_cast<double>(1 << N);
- constexpr double C = CN + 1.0;
+ constexpr T CN = static_cast<T>(1 << N);
+ constexpr T C = CN + 1.0;
double t1 = C * a;
double t2 = a - t1;
r.hi = t1 + t2;
@@ -77,16 +88,15 @@ LIBC_INLINE constexpr DoubleDouble split(double a) {
}
// Helper for non-fma exact mult where the first number is already split.
-template <size_t SPLIT_B = DEFAULT_DOUBLE_SPLIT>
-LIBC_INLINE DoubleDouble exact_mult(const DoubleDouble &as, double a,
- double b) {
- DoubleDouble bs = split<SPLIT_B>(b);
- DoubleDouble r{0.0, 0.0};
+template <typename T = double, size_t SPLIT_B = DefaultSplit<T>::VALUE>
+LIBC_INLINE NumberPair<T> exact_mult(const NumberPair<T> &as, T a, T b) {
+ NumberPair<T> bs = split<T, SPLIT_B>(b);
+ NumberPair<T> r{0.0, 0.0};
r.hi = a * b;
- double t1 = as.hi * bs.hi - r.hi;
- double t2 = as.hi * bs.lo + t1;
- double t3 = as.lo * bs.hi + t2;
+ T t1 = as.hi * bs.hi - r.hi;
+ T t2 = as.hi * bs.lo + t1;
+ T t3 = as.lo * bs.hi + t2;
r.lo = as.lo * bs.lo + t3;
return r;
@@ -99,18 +109,18 @@ LIBC_INLINE DoubleDouble exact_mult(const DoubleDouble &as, double a,
// Using Theorem 1 in the paper above, without FMA instruction, if we restrict
// the generated constants to precision <= 51, and splitting it by 2^28 + 1,
// then a * b = r.hi + r.lo is exact for all rounding modes.
-template <size_t SPLIT_B = 27>
-LIBC_INLINE DoubleDouble exact_mult(double a, double b) {
- DoubleDouble r{0.0, 0.0};
+template <typename T = double, size_t SPLIT_B = DefaultSplit<T>::VALUE>
+LIBC_INLINE NumberPair<T> exact_mult(T a, T b) {
+ NumberPair<T> r{0.0, 0.0};
#ifdef LIBC_TARGET_CPU_HAS_FMA
r.hi = a * b;
r.lo = fputil::multiply_add(a, b, -r.hi);
#else
// Dekker's Product.
- DoubleDouble as = split(a);
+ NumberPair<T> as = split(a);
- r = exact_mult<SPLIT_B>(as, a, b);
+ r = exact_mult<T, SPLIT_B>(as, a, b);
#endif // LIBC_TARGET_CPU_HAS_FMA
return r;
@@ -125,7 +135,7 @@ LIBC_INLINE DoubleDouble quick_mult(double a, const DoubleDouble &b) {
template <size_t SPLIT_B = 27>
LIBC_INLINE DoubleDouble quick_mult(const DoubleDouble &a,
const DoubleDouble &b) {
- DoubleDouble r = exact_mult<SPLIT_B>(a.hi, b.hi);
+ DoubleDouble r = exact_mult<double, SPLIT_B>(a.hi, b.hi);
double t1 = multiply_add(a.hi, b.lo, r.lo);
double t2 = multiply_add(a.lo, b.hi, t1);
r.lo = t2;
@@ -157,19 +167,20 @@ LIBC_INLINE DoubleDouble multiply_add<DoubleDouble>(const DoubleDouble &a,
// rl = q * (ah - bh * rh) + q * (al - bl * rh)
// as accurate as possible, then the error is bounded by:
// |(ah + al) / (bh + bl) - (rh + rl)| < O(bl/bh) * (2^-52 + al/ah + bl/bh)
-LIBC_INLINE DoubleDouble div(const DoubleDouble &a, const DoubleDouble &b) {
- DoubleDouble r;
- double q = 1.0 / b.hi;
+template <typename T>
+LIBC_INLINE NumberPair<T> div(const NumberPair<T> &a, const NumberPair<T> &b) {
+ NumberPair<T> r;
+ T q = T(1) / b.hi;
r.hi = a.hi * q;
#ifdef LIBC_TARGET_CPU_HAS_FMA
- double e_hi = fputil::multiply_add(b.hi, -r.hi, a.hi);
- double e_lo = fputil::multiply_add(b.lo, -r.hi, a.lo);
+ T e_hi = fputil::multiply_add(b.hi, -r.hi, a.hi);
+ T e_lo = fputil::multiply_add(b.lo, -r.hi, a.lo);
#else
- DoubleDouble b_hi_r_hi = fputil::exact_mult(b.hi, -r.hi);
- DoubleDouble b_lo_r_hi = fputil::exact_mult(b.lo, -r.hi);
- double e_hi = (a.hi + b_hi_r_hi.hi) + b_hi_r_hi.lo;
- double e_lo = (a.lo + b_lo_r_hi.hi) + b_lo_r_hi.lo;
+ NumberPair<T> b_hi_r_hi = fputil::exact_mult(b.hi, -r.hi);
+ NumberPair<T> b_lo_r_hi = fputil::exact_mult(b.lo, -r.hi);
+ T e_hi = (a.hi + b_hi_r_hi.hi) + b_hi_r_hi.lo;
+ T e_lo = (a.lo + b_lo_r_hi.hi) + b_lo_r_hi.lo;
#endif // LIBC_TARGET_CPU_HAS_FMA
r.lo = q * (e_hi + e_lo);
diff --git a/libc/src/__support/macros/optimization.h b/libc/src/__support/macros/optimization.h
index a2634950d431b..253843e5e37aa 100644
--- a/libc/src/__support/macros/optimization.h
+++ b/libc/src/__support/macros/optimization.h
@@ -45,6 +45,7 @@ LIBC_INLINE constexpr bool expects_bool_condition(T value, T expected) {
#define LIBC_MATH_FAST \
(LIBC_MATH_SKIP_ACCURATE_PASS | LIBC_MATH_SMALL_TABLES | \
LIBC_MATH_NO_ERRNO | LIBC_MATH_NO_EXCEPT)
+#define LIBC_MATH_INTERMEDIATE_COMP_IN_FLOAT 0x10
#ifndef LIBC_MATH
#define LIBC_MATH 0
@@ -58,4 +59,8 @@ LIBC_INLINE constexpr bool expects_bool_condition(T value, T expected) {
#define LIBC_MATH_HAS_SMALL_TABLES
#endif
+#if (LIBC_MATH & LIBC_MATH_INTERMEDIATE_COMP_IN_FLOAT)
+#define LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT
+#endif
+
#endif // LLVM_LIBC_SRC___SUPPORT_MACROS_OPTIMIZATION_H
diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt
index 9faf46d491426..2bda741b453f5 100644
--- a/libc/src/math/generic/CMakeLists.txt
+++ b/libc/src/math/generic/CMakeLists.txt
@@ -4052,8 +4052,10 @@ add_entrypoint_object(
atan2f.cpp
HDRS
../atan2f.h
+ atan2f_float.h
DEPENDS
.inv_trigf_utils
+ libc.src.__support.FPUtil.double_double
libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.multiply_add
libc.src.__support.FPUtil.nearest_integer
diff --git a/libc/src/math/generic/atan2f.cpp b/libc/src/math/generic/atan2f.cpp
index db7639396cdd7..5ac2b29438ea9 100644
--- a/libc/src/math/generic/atan2f.cpp
+++ b/libc/src/math/generic/atan2f.cpp
@@ -17,6 +17,14 @@
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
+ defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
+
+// We use float-float implementation to reduce size.
+#include "src/math/generic/atan2f_float.h"
+
+#else
+
namespace LIBC_NAMESPACE_DECL {
namespace {
@@ -324,3 +332,5 @@ LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
}
} // namespace LIBC_NAMESPACE_DECL
+
+#endif
diff --git a/libc/src/math/generic/atan2f_float.h b/libc/src/math/generic/atan2f_float.h
new file mode 100644
index 0000000000000..1819a3c3fb0a0
--- /dev/null
+++ b/libc/src/math/generic/atan2f_float.h
@@ -0,0 +1,239 @@
+//===-- Single-precision atan2f 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/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/double_double.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/FPUtil/rounding_mode.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#include "src/math/atan2f.h"
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace {
+
+using FloatFloat = fputil::FloatFloat;
+
+// atan(i/64) with i = 0..16, generated by Sollya with:
+// > for i from 0 to 16 do {
+// a = round(atan(i/16), SG, RN);
+// b = round(atan(i/16) - a, SG, RN);
+// print("{", b, ",", a, "},");
+// };
+constexpr FloatFloat ATAN_I[17] = {
+ {0.0f, 0.0f},
+ {-0x1.1a6042p-30f, 0x1.ff55bcp-5f},
+ {-0x1.54f424p-30f, 0x1.fd5baap-4f},
+ {0x1.79cb6p-28f, 0x1.7b97b4p-3f},
+ {-0x1.b4dfc8p-29f, 0x1.f5b76p-3f},
+ {-0x1.1f0286p-27f, 0x1.362774p-2f},
+ {0x1.e4defp-30f, 0x1.6f6194p-2f},
+ {0x1.e611fep-29f, 0x1.a64eecp-2f},
+ {0x1.586ed4p-28f, 0x1.dac67p-2f},
+ {-0x1.6499e6p-26f, 0x1.0657eap-1f},
+ {0x1.7bdfd6p-26f, 0x1.1e00bap-1f},
+ {-0x1.98e422p-28f, 0x1.345f02p-1f},
+ {0x1.934f7p-28f, 0x1.4978fap-1f},
+ {0x1.c5a6c6p-27f, 0x1.5d5898p-1f},
+ {0x1.5e118cp-27f, 0x1.700a7cp-1f},
+ {-0x1.1d4eb6p-26f, 0x1.819d0cp-1f},
+ {-0x1.777a5cp-26f, 0x1.921fb6p-1f},
+};
+
+// Approximate atan(x) for |x| <= 2^-5.
+// Using degree-3 Taylor polynomial:
+// P = x - x^3/3
+// Then the absolute error is bounded by:
+// |atan(x) - P(x)| < |x|^5/5 < 2^(-5*5) / 5 < 2^-27.
+// And the relative error is bounded by:
+// |(atan(x) - P(x))/atan(x)| < |x|^4 / 4 < 2^-22.
+// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
+// ulp(x_hi^3 / 3) gives us:
+// P(x) ~ x_hi - x_hi^3/3 + x_lo * (1 - x_hi^2)
+FloatFloat atan_eval(const FloatFloat &x) {
+ FloatFloat p;
+ p.hi = x.hi;
+ float x_hi_sq = x.hi * x.hi;
+ // c0 ~ - x_hi^2 / 3
+ float c0 = -0x1.555556p-2f * x_hi_sq;
+ // c1 ~ x_lo * (1 - x_hi^2)
+ float c1 = fputil::multiply_add(x_hi_sq, -x.lo, x.lo);
+ // p.lo ~ - x_hi^3 / 3 + x_lo * (1 - x_hi*2)
+ p.lo = fputil::multiply_add(x.hi, c0, c1);
+ return p;
+}
+
+} // 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/16 | <= 1/32.
+// In particular,
+// idx := 2^-4 * round(2^4 * n/d)
+// Then for the fast pass, we find a polynomial approximation for:
+// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
+// with Q(x) = x - x^3/3 be the cubic Taylor polynomial of atan(x).
+// It's error in float-float precision is estimated in Sollya to be:
+// > P = x - x^3/3;
+// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
+// 0x1.995...p-28.
+
+LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
+ using FPBits = typename fputil::FPBits<float>;
+ constexpr float IS_NEG[2] = {1.0f, -1.0f};
+ constexpr FloatFloat ZERO = {0.0f, 0.0f};
+ constexpr FloatFloat MZERO = {-0.0f, -0.0f};
+ constexpr FloatFloat PI = {-0x1.777a5cp-24f, 0x1.921fb6p1f};
+ constexpr FloatFloat MPI = {0x1.777a5cp-24f, -0x1.921fb6p1f};
+ constexpr FloatFloat PI_OVER_4 = {-0x1.777a5cp-26f, 0x1.921fb6p-1f};
+ constexpr FloatFloat PI_OVER_2 = {-0x1.777a5cp-25f, 0x1.921fb6p0f};
+ constexpr FloatFloat MPI_OVER_2 = {-0x1.777a5cp-25f, 0x1.921fb6p0f};
+ constexpr FloatFloat THREE_PI_OVER_4 = {-0x1.99bc5cp-28f, 0x1.2d97c8p1f};
+ // Adjustment for constant term:
+ // CONST_ADJ[x_sign][y_sign][recip]
+ constexpr FloatFloat CONST_ADJ[2][2][2] = {
+ {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
+ {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
+
+ 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();
+ uint32_t x_abs = x_bits.uintval();
+ uint32_t y_abs = y_bits.uintval();
+ bool recip = x_abs < y_abs;
+ uint32_t min_abs = recip ? x_abs : y_abs;
+ uint32_t 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);
+
+ float num = FPBits(min_abs).get_val();
+ float 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 > 0xffU - 64U || min_exp < 64U)) {
+ if (x_bits.is_nan() || y_bits.is_nan())
+ return FPBits::quiet_nan().get_val();
+ unsigned x_except = x == 0.0f ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
+ unsigned y_except = y == 0.0f ? 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 FloatFloat 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)) {
+ FloatFloat r = EXCEPTS[y_except][x_except][x_sign];
+ return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);
+ }
+ bool scale_up = min_exp < 64U;
+ bool scale_down = max_exp > 0xffU - 64U;
+ // At least one input is denormal, multiply both numerator and denominator
+ // by some large enough power of 2 to normalize denormal inputs.
+ if (scale_up) {
+ num *= 0x1.0p32f;
+ if (!scale_down)
+ den *= 0x1.0p32f;
+ } else if (scale_down) {
+ den *= 0x1.0p-32f;
+ if (!scale_up)
+ num *= 0x1.0p-32f;
+ }
+
+ min_abs = FPBits(num).uintval();
+ max_abs = FPBits(den).uintval();
+ min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
+ max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
+ }
+
+ float final_sign = IS_NEG[(x_sign != y_sign) != recip];
+ FloatFloat const_term = CONST_ADJ[x_sign][y_sign][recip];
+ unsigned exp_diff = max_exp - min_exp;
+ // We have the following bound for normalized n and d:
+ // 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
+ if (LIBC_UNLIKELY(exp_diff > 25)) {
+ return fputil::multiply_add(final_sign, const_term.hi,
+ final_sign * (const_term.lo + num / den));
+ }
+
+ float k = fputil::nearest_integer(16.0f * num / den);
+ unsigned idx = static_cast<unsigned>(k);
+ // k = idx / 16
+ k *= 0x1.0p-4f;
+
+ // Range reduction:
+ // atan(n/d) - atan(k/64) = atan((n/d - k/16) / (1 + (n/d) * (k/16)))
+ // = atan((n - d * k/16)) / (d + n * k/16))
+ FloatFloat num_k = fputil::exact_mult(num, k);
+ FloatFloat den_k = fputil::exact_mult(den, k);
+
+ // num_dd = n - d * k
+ FloatFloat num_ff = fputil::exact_add(num - den_k.hi, -den_k.lo);
+ // den_dd = d + n * k
+ FloatFloat den_ff = fputil::exact_add(den, num_k.hi);
+ den_ff.lo += num_k.lo;
+
+ // q = (n - d * k) / (d + n * k)
+ FloatFloat q = fputil::div(num_ff, den_ff);
+ // p ~ atan(q)
+ FloatFloat p = atan_eval(q);
+
+ FloatFloat r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));
+ return final_sign * r.hi;
+}
+
+} // namespace LIBC_NAMESPACE_DECL
diff --git a/libc/src/math/generic/range_reduction_double_fma.h b/libc/src/math/generic/range_reduction_double_fma.h
index cab031c28baa1..8e0bc3a42462c 100644
--- a/libc/src/math/generic/range_reduction_double_fma.h
+++ b/libc/src/math/generic/range_reduction_double_fma.h
@@ -33,14 +33,14 @@ LIBC_INLINE unsigned LargeRangeReduction::fast(double x, DoubleDouble &u) {
// 2^62 <= |x_reduced| < 2^(62 + 16) = 2^78
x_reduced = xbits.get_val();
// x * c_hi = ph.hi + ph.lo exactly.
- DoubleDouble ph =
- fputil::exact_mult<SPLIT>(x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][0]);
+ DoubleDouble ph = fputil::exact_mult<double, SPLIT>(
+ x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][0]);
// x * c_mid = pm.hi + pm.lo exactly.
- DoubleDouble pm =
- fputil::exact_mult<SPLIT>(x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][1]);
+ DoubleDouble pm = fputil::exact_mult<double, SPLIT>(
+ x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][1]);
// x * c_lo = pl.hi + pl.lo exactly.
- DoubleDouble pl =
- fputil::exact_mult<SPLIT>(x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][2]);
+ DoubleDouble pl = fputil::exact_mult<double, SPLIT>(
+ x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][2]);
// Extract integral parts and fractional parts of (ph.lo + pm.hi).
double sum_hi = ph.lo + pm.hi;
double kd = fputil::nearest_integer(sum_hi);
diff --git a/libc/src/math/generic/range_reduction_double_nofma.h b/libc/src/math/generic/range_reduction_double_nofma.h
index 5640732947798..606c3f8185d61 100644
--- a/libc/src/math/generic/range_reduction_double_nofma.h
+++ b/libc/src/math/generic/range_reduction_double_nofma.h
@@ -34,14 +34,14 @@ LIBC_INLINE unsigned LargeRangeReduction::fast(double x, DoubleDouble &u) {
x_reduced = xbits.get_val();
// x * c_hi = ph.hi + ph.lo exactly.
DoubleDouble x_split = fputil::split(x_reduced);
- DoubleDouble ph = fputil::exact_mult<SPLIT>(x_split, x_reduced,
- ONE_TWENTY_EIGHT_OVER_PI[idx][0]);
+ DoubleDouble ph = fputil::exact_mult<double, SPLIT>(
+ x_split, x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][0]);
// x * c_mid = pm.hi + pm.lo exactly.
- DoubleDouble pm = fputil::exact_mult<SPLIT>(x_split, x_reduced,
- ONE_TWENTY_EIGHT_OVER_PI[idx][1]);
+ DoubleDouble pm = fputil::exact_mult<double, SPLIT>(
+ x_split, x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][1]);
// x * c_lo = pl.hi + pl.lo exactly.
- DoubleDouble pl = fputil::exact_mult<SPLIT>(x_split, x_reduced,
- ONE_TWENTY_EIGHT_OVER_PI[idx][2]);
+ DoubleDouble pl = fputil::exact_mult<double, SPLIT>(
+ x_split, x_reduced, ONE_TWENTY_EIGHT_OVER_PI[idx][2]);
// Extract integral parts and fractional parts of (ph.lo + pm.hi).
double sum_hi = ph.lo + pm.hi;
double kd = fputil::nearest_integer(sum_hi);
>From 8c051a70e562b785c31dd20b7c7abc65e402c307 Mon Sep 17 00:00:00 2001
From: Tue Ly <lntue.h at gmail.com>
Date: Tue, 11 Feb 2025 21:07:40 +0000
Subject: [PATCH 2/2] Address comments.
---
libc/src/__support/FPUtil/double_double.h | 2 +-
libc/src/math/generic/atan2f_float.h | 7 +++----
libc/src/math/generic/pow.cpp | 2 +-
3 files changed, 5 insertions(+), 6 deletions(-)
diff --git a/libc/src/__support/FPUtil/double_double.h b/libc/src/__support/FPUtil/double_double.h
index 825038b22290a..8c318cfbed363 100644
--- a/libc/src/__support/FPUtil/double_double.h
+++ b/libc/src/__support/FPUtil/double_double.h
@@ -25,7 +25,7 @@ template <> struct DefaultSplit<float> {
static constexpr size_t VALUE = 12;
};
template <> struct DefaultSplit<double> {
- static constexpr size_t VALUE = 27;
+ static constexpr size_t VALUE = DEFAULT_DOUBLE_SPLIT;
};
using DoubleDouble = NumberPair<double>;
diff --git a/libc/src/math/generic/atan2f_float.h b/libc/src/math/generic/atan2f_float.h
index 1819a3c3fb0a0..f03d65a1d99b3 100644
--- a/libc/src/math/generic/atan2f_float.h
+++ b/libc/src/math/generic/atan2f_float.h
@@ -148,8 +148,8 @@ LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
bool recip = x_abs < y_abs;
uint32_t min_abs = recip ? x_abs : y_abs;
uint32_t 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);
+ auto min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
+ auto max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
float num = FPBits(min_abs).get_val();
float den = FPBits(max_abs).get_val();
@@ -205,10 +205,9 @@ LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
unsigned exp_diff = max_exp - min_exp;
// We have the following bound for normalized n and d:
// 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
- if (LIBC_UNLIKELY(exp_diff > 25)) {
+ if (LIBC_UNLIKELY(exp_diff > 25))
return fputil::multiply_add(final_sign, const_term.hi,
final_sign * (const_term.lo + num / den));
- }
float k = fputil::nearest_integer(16.0f * num / den);
unsigned idx = static_cast<unsigned>(k);
diff --git a/libc/src/math/generic/pow.cpp b/libc/src/math/generic/pow.cpp
index 213dbd959039c..a2a0bb698f81a 100644
--- a/libc/src/math/generic/pow.cpp
+++ b/libc/src/math/generic/pow.cpp
@@ -400,7 +400,7 @@ LLVM_LIBC_FUNCTION(double, pow, (double x, double y)) {
#else
double c = FPBits(m_x.uintval() & 0x3fff'e000'0000'0000).get_val();
dx = fputil::multiply_add(RD[idx_x], m_x.get_val() - c, CD[idx_x]); // Exact
- dx_c0 = fputil::exact_mult<28>(dx, COEFFS[0]); // Exact
+ dx_c0 = fputil::exact_mult<double, 28>(dx, COEFFS[0]); // Exact
#endif // LIBC_TARGET_CPU_HAS_FMA
double dx2 = dx * dx;
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