[libc][math] Refactor exp2 implementation to header-only in src/__support/math folder. #161297
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@llvm/pr-subscribers-libc Author: Muhammad Bassiouni (bassiounix) ChangesPart of #147386 in preparation for: https://discourse.llvm.org/t/rfc-make-clang-builtin-math-functions-constexpr-with-llvm-libc-to-support-c-23-constexpr-math-functions/86450 Patch is 69.02 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/161297.diff 24 Files Affected:
diff --git a/libc/shared/math.h b/libc/shared/math.h
index 4b2a0d8c712ad..924d0cb4f3bfa 100644
--- a/libc/shared/math.h
+++ b/libc/shared/math.h
@@ -47,6 +47,7 @@
#include "math/exp10f16.h"
#include "math/exp10m1f.h"
#include "math/exp10m1f16.h"
+#include "math/exp2.h"
#include "math/expf.h"
#include "math/expf16.h"
#include "math/frexpf.h"
diff --git a/libc/shared/math/exp2.h b/libc/shared/math/exp2.h
new file mode 100644
index 0000000000000..6f1e143376a8a
--- /dev/null
+++ b/libc/shared/math/exp2.h
@@ -0,0 +1,23 @@
+//===-- Shared exp2 function ------------------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_EXP2_H
+#define LLVM_LIBC_SHARED_MATH_EXP2_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp2.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp2;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP2_H
diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 98f9bb42f91f4..4130fdf02078b 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -373,6 +373,15 @@ add_header_library(
libc.src.__support.macros.optimization
)
+add_header_library(
+ common_constants
+ HDRS
+ common_constants.h
+ DEPENDS
+ libc.src.__support.macros.config
+ libc.src.__support.number_pair
+)
+
add_header_library(
cos
HDRS
@@ -704,6 +713,28 @@ add_header_library(
libc.src.__support.macros.optimization
)
+add_header_library(
+ exp2
+ HDRS
+ exp2.h
+ DEPENDS
+ .common_constants
+ .exp_utils
+ libc.src.__support.CPP.bit
+ libc.src.__support.CPP.optional
+ libc.src.__support.FPUtil.dyadic_float
+ libc.src.__support.FPUtil.fenv_impl
+ libc.src.__support.FPUtil.fp_bits
+ libc.src.__support.FPUtil.multiply_add
+ libc.src.__support.FPUtil.nearest_integer
+ libc.src.__support.FPUtil.polyeval
+ libc.src.__support.FPUtil.rounding_mode
+ libc.src.__support.FPUtil.triple_double
+ libc.src.__support.integer_literals
+ libc.src.__support.macros.optimization
+ libc.src.errno.errno
+)
+
add_header_library(
exp10
HDRS
diff --git a/libc/src/math/generic/common_constants.cpp b/libc/src/__support/math/common_constants.h
similarity index 95%
rename from libc/src/math/generic/common_constants.cpp
rename to libc/src/__support/math/common_constants.h
index 2a15df243b5a4..53abbfeef3412 100644
--- a/libc/src/math/generic/common_constants.cpp
+++ b/libc/src/__support/math/common_constants.h
@@ -6,12 +6,29 @@
//
//===----------------------------------------------------------------------===//
-#include "common_constants.h"
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_COMMON_CONSTANTS_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_COMMON_CONSTANTS_H
+
#include "src/__support/macros/config.h"
#include "src/__support/number_pair.h"
namespace LIBC_NAMESPACE_DECL {
+namespace common_constants_internal {
+
+// log(2) generated by Sollya with:
+// > a = 2^-43 * nearestint(2^43*log(2));
+// LSB = 2^-43 is chosen so that e_x * LOG_2_HI is exact for -1075 < e_x < 1024.
+static constexpr double LOG_2_HI = 0x1.62e42fefa38p-1; // LSB = 2^-43
+// > b = round(log10(2) - a, D, RN);
+static constexpr double LOG_2_LO = 0x1.ef35793c7673p-45; // LSB = 2^-97
+
+// Minimax polynomial for (log(1 + x) - x)/x^2, generated by sollya with:
+// > P = fpminimax((log(1 + x) - x)/x^2, 5, [|D...|], [-2^-8, 2^-7]);
+constexpr double LOG_COEFFS[6] = {-0x1.fffffffffffffp-2, 0x1.5555555554a9bp-2,
+ -0x1.0000000094567p-2, 0x1.99999dcc9823cp-3,
+ -0x1.55550ac2e537ap-3, 0x1.21a02c4e624d7p-3};
+
// Range reduction constants for logarithms.
// r(0) = 1, r(127) = 0.5
// r(k) = 2^-8 * ceil(2^8 * (1 - 2^-8) / (1 + k*2^-7))
@@ -19,7 +36,7 @@ namespace LIBC_NAMESPACE_DECL {
// precision, and -2^-8 <= v < 2^-7.
// TODO(lntue): Add reference to how the constants are derived after the
// resulting paper is ready.
-alignas(8) const float R[128] = {
+alignas(8) static constexpr float R[128] = {
0x1p0, 0x1.fcp-1, 0x1.f8p-1, 0x1.f4p-1, 0x1.fp-1, 0x1.ecp-1, 0x1.e8p-1,
0x1.e4p-1, 0x1.ep-1, 0x1.dep-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d4p-1, 0x1.dp-1,
0x1.ccp-1, 0x1.cap-1, 0x1.c6p-1, 0x1.c4p-1, 0x1.cp-1, 0x1.bep-1, 0x1.bap-1,
@@ -40,7 +57,7 @@ alignas(8) const float R[128] = {
0x1.0ap-1, 0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1,
0x1.02p-1, 0x1.0p-1};
-const double RD[128] = {
+static constexpr double RD[128] = {
0x1p0, 0x1.fcp-1, 0x1.f8p-1, 0x1.f4p-1, 0x1.fp-1, 0x1.ecp-1, 0x1.e8p-1,
0x1.e4p-1, 0x1.ep-1, 0x1.dep-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d4p-1, 0x1.dp-1,
0x1.ccp-1, 0x1.cap-1, 0x1.c6p-1, 0x1.c4p-1, 0x1.cp-1, 0x1.bep-1, 0x1.bap-1,
@@ -65,7 +82,7 @@ const double RD[128] = {
// available.
// Generated by Sollya with the formula: CD[i] = RD[i]*(1 + i*2^-7) - 1
// for RD[i] defined on the table above.
-const double CD[128] = {
+static constexpr double CD[128] = {
0.0, -0x1p-14, -0x1p-12, -0x1.2p-11, -0x1p-10, -0x1.9p-10,
-0x1.2p-9, -0x1.88p-9, -0x1p-8, -0x1.9p-11, -0x1.fp-10, -0x1.9cp-9,
-0x1p-12, -0x1.cp-10, -0x1.bp-9, -0x1.5p-11, -0x1.4p-9, 0x1p-14,
@@ -90,7 +107,7 @@ const double CD[128] = {
-0x1p-14, -0x1p-8,
};
-const double LOG_R[128] = {
+static constexpr double LOG_R[128] = {
0x0.0000000000000p0, 0x1.010157588de71p-7, 0x1.0205658935847p-6,
0x1.8492528c8cabfp-6, 0x1.0415d89e74444p-5, 0x1.466aed42de3eap-5,
0x1.894aa149fb343p-5, 0x1.ccb73cdddb2ccp-5, 0x1.08598b59e3a07p-4,
@@ -135,7 +152,7 @@ const double LOG_R[128] = {
0x1.5707a26bb8c66p-1, 0x1.5af405c3649ep-1, 0x1.5af405c3649ep-1,
0x1.5ee82aa24192p-1, 0x0.000000000000p0};
-const double LOG2_R[128] = {
+static constexpr double LOG2_R[128] = {
0x0.0000000000000p+0, 0x1.72c7ba20f7327p-7, 0x1.743ee861f3556p-6,
0x1.184b8e4c56af8p-5, 0x1.77394c9d958d5p-5, 0x1.d6ebd1f1febfep-5,
0x1.1bb32a600549dp-4, 0x1.4c560fe68af88p-4, 0x1.7d60496cfbb4cp-4,
@@ -188,7 +205,7 @@ const double LOG2_R[128] = {
// print("{", -c, ",", -b, "},");
// };
// We replace LOG_R[0] with log10(1.0) == 0.0
-alignas(16) const NumberPair<double> LOG_R_DD[128] = {
+alignas(16) static constexpr NumberPair<double> LOG_R_DD[128] = {
{0.0, 0.0},
{-0x1.0c76b999d2be8p-46, 0x1.010157589p-7},
{-0x1.3dc5b06e2f7d2p-45, 0x1.0205658938p-6},
@@ -324,7 +341,7 @@ alignas(16) const NumberPair<double> LOG_R_DD[128] = {
// Output range:
// [-0x1.3ffcp-15, 0x1.3e3dp-15]
// We store S2[i] = 2^16 (r(i - 2^6) - 1).
-alignas(8) const int S2[193] = {
+alignas(8) static constexpr int S2[193] = {
0x101, 0xfd, 0xf9, 0xf5, 0xf1, 0xed, 0xe9, 0xe5, 0xe1,
0xdd, 0xd9, 0xd5, 0xd1, 0xcd, 0xc9, 0xc5, 0xc1, 0xbd,
0xb9, 0xb4, 0xb0, 0xac, 0xa8, 0xa4, 0xa0, 0x9c, 0x98,
@@ -348,7 +365,7 @@ alignas(8) const int S2[193] = {
-0x1cd, -0x1d1, -0x1d5, -0x1d9, -0x1dd, -0x1e0, -0x1e4, -0x1e8, -0x1ec,
-0x1f0, -0x1f4, -0x1f8, -0x1fc};
-const double R2[193] = {
+static constexpr double R2[193] = {
0x1.0101p0, 0x1.00fdp0, 0x1.00f9p0, 0x1.00f5p0, 0x1.00f1p0,
0x1.00edp0, 0x1.00e9p0, 0x1.00e5p0, 0x1.00e1p0, 0x1.00ddp0,
0x1.00d9p0, 0x1.00d5p0, 0x1.00d1p0, 0x1.00cdp0, 0x1.00c9p0,
@@ -395,7 +412,7 @@ const double R2[193] = {
// Output range:
// [-0x1.01928p-22 , 0x1p-22]
// We store S[i] = 2^21 (r(i - 80) - 1).
-alignas(8) const int S3[161] = {
+alignas(8) static constexpr int S3[161] = {
0x50, 0x4f, 0x4e, 0x4d, 0x4c, 0x4b, 0x4a, 0x49, 0x48, 0x47, 0x46,
0x45, 0x44, 0x43, 0x42, 0x41, 0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b,
0x3a, 0x39, 0x38, 0x37, 0x36, 0x35, 0x34, 0x33, 0x32, 0x31, 0x30,
@@ -418,7 +435,7 @@ alignas(8) const int S3[161] = {
// Output range:
// [-0x1.0002143p-29 , 0x1p-29]
// We store S[i] = 2^28 (r(i - 65) - 1).
-alignas(8) const int S4[130] = {
+alignas(8) static constexpr int S4[130] = {
0x41, 0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b, 0x3a, 0x39, 0x38, 0x37,
0x36, 0x35, 0x34, 0x33, 0x32, 0x31, 0x30, 0x2f, 0x2e, 0x2d, 0x2c,
0x2b, 0x2a, 0x29, 0x28, 0x27, 0x26, 0x25, 0x24, 0x23, 0x22, 0x21,
@@ -439,7 +456,7 @@ alignas(8) const int S4[130] = {
// Table is generated with Sollya as follow:
// > display = hexadecimal;
// > for i from -104 to 89 do { D(exp(i)); };
-const double EXP_M1[195] = {
+static constexpr double EXP_M1[195] = {
0x1.f1e6b68529e33p-151, 0x1.525be4e4e601dp-149, 0x1.cbe0a45f75eb1p-148,
0x1.3884e838aea68p-146, 0x1.a8c1f14e2af5dp-145, 0x1.20a717e64a9bdp-143,
0x1.8851d84118908p-142, 0x1.0a9bdfb02d240p-140, 0x1.6a5bea046b42ep-139,
@@ -511,7 +528,7 @@ const double EXP_M1[195] = {
// Table is generated with Sollya as follow:
// > display = hexadecimal;
// > for i from 0 to 127 do { D(exp(i / 128)); };
-const double EXP_M2[128] = {
+static constexpr double EXP_M2[128] = {
0x1.0000000000000p0, 0x1.0202015600446p0, 0x1.04080ab55de39p0,
0x1.06122436410ddp0, 0x1.08205601127edp0, 0x1.0a32a84e9c1f6p0,
0x1.0c49236829e8cp0, 0x1.0e63cfa7ab09dp0, 0x1.1082b577d34edp0,
@@ -557,4 +574,8 @@ const double EXP_M2[128] = {
0x1.568bb722dd593p1, 0x1.593b7d72305bbp1,
};
+} // namespace common_constants_internal
+
} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SRC___SUPPORT_MATH_COMMON_CONSTANTS_H
diff --git a/libc/src/__support/math/exp2.h b/libc/src/__support/math/exp2.h
new file mode 100644
index 0000000000000..7eaa465f93841
--- /dev/null
+++ b/libc/src/__support/math/exp2.h
@@ -0,0 +1,425 @@
+//===-- Implementation header for exp2 --------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_EXP2_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP2_H
+
+#include "common_constants.h" // Lookup tables EXP2_MID1 and EXP_M2.
+#include "exp_constants.h"
+#include "exp_utils.h" // ziv_test_denorm.
+#include "src/__support/CPP/bit.h"
+#include "src/__support/CPP/optional.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/PolyEval.h"
+#include "src/__support/FPUtil/double_double.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/FPUtil/rounding_mode.h"
+#include "src/__support/FPUtil/triple_double.h"
+#include "src/__support/common.h"
+#include "src/__support/integer_literals.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace math {
+
+namespace exp2_internal {
+
+using namespace common_constants_internal;
+
+using fputil::DoubleDouble;
+using fputil::TripleDouble;
+using Float128 = typename fputil::DyadicFloat<128>;
+
+using LIBC_NAMESPACE::operator""_u128;
+
+// Error bounds:
+// Errors when using double precision.
+#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+constexpr double ERR_D = 0x1.0p-63;
+#else
+constexpr double ERR_D = 0x1.8p-63;
+#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+// Errors when using double-double precision.
+constexpr double ERR_DD = 0x1.0p-100;
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+// Polynomial approximations with double precision. Generated by Sollya with:
+// > P = fpminimax((2^x - 1)/x, 3, [|D...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);
+// > P;
+// Error bounds:
+// | output - (2^dx - 1) / dx | < 1.5 * 2^-52.
+LIBC_INLINE static double poly_approx_d(double dx) {
+ // dx^2
+ double dx2 = dx * dx;
+ double c0 =
+ fputil::multiply_add(dx, 0x1.ebfbdff82c58ep-3, 0x1.62e42fefa39efp-1);
+ double c1 =
+ fputil::multiply_add(dx, 0x1.3b2aba7a95a89p-7, 0x1.c6b08e8fc0c0ep-5);
+ double p = fputil::multiply_add(dx2, c1, c0);
+ return p;
+}
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+// Polynomial approximation with double-double precision. Generated by Solya
+// with:
+// > P = fpminimax((2^x - 1)/x, 5, [|DD...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);
+// Error bounds:
+// | output - 2^(dx) | < 2^-101
+LIBC_INLINE static constexpr DoubleDouble
+poly_approx_dd(const DoubleDouble &dx) {
+ // Taylor polynomial.
+ constexpr DoubleDouble COEFFS[] = {
+ {0, 0x1p0},
+ {0x1.abc9e3b39824p-56, 0x1.62e42fefa39efp-1},
+ {-0x1.5e43a53e4527bp-57, 0x1.ebfbdff82c58fp-3},
+ {-0x1.d37963a9444eep-59, 0x1.c6b08d704a0cp-5},
+ {0x1.4eda1a81133dap-62, 0x1.3b2ab6fba4e77p-7},
+ {-0x1.c53fd1ba85d14p-64, 0x1.5d87fe7a265a5p-10},
+ {0x1.d89250b013eb8p-70, 0x1.430912f86cb8ep-13},
+ };
+
+ DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
+ COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
+ return p;
+}
+
+// Polynomial approximation with 128-bit precision:
+// Return exp(dx) ~ 1 + a0 * dx + a1 * dx^2 + ... + a6 * dx^7
+// For |dx| < 2^-13 + 2^-30:
+// | output - exp(dx) | < 2^-126.
+LIBC_INLINE static constexpr Float128 poly_approx_f128(const Float128 &dx) {
+ constexpr Float128 COEFFS_128[]{
+ {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
+ {Sign::POS, -128, 0xb17217f7'd1cf79ab'c9e3b398'03f2f6af_u128},
+ {Sign::POS, -128, 0x3d7f7bff'058b1d50'de2d60dd'9c9a1d9f_u128},
+ {Sign::POS, -132, 0xe35846b8'2505fc59'9d3b15d9'e7fb6897_u128},
+ {Sign::POS, -134, 0x9d955b7d'd273b94e'184462f6'bcd2b9e7_u128},
+ {Sign::POS, -137, 0xaec3ff3c'53398883'39ea1bb9'64c51a89_u128},
+ {Sign::POS, -138, 0x2861225f'345c396a'842c5341'8fa8ae61_u128},
+ {Sign::POS, -144, 0xffe5fe2d'109a319d'7abeb5ab'd5ad2079_u128},
+ };
+
+ Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
+ COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
+ COEFFS_128[6], COEFFS_128[7]);
+ return p;
+}
+
+// Compute 2^(x) using 128-bit precision.
+// TODO(lntue): investigate triple-double precision implementation for this
+// step.
+LIBC_INLINE static constexpr Float128 exp2_f128(double x, int hi, int idx1,
+ int idx2) {
+ Float128 dx = Float128(x);
+
+ // TODO: Skip recalculating exp_mid1 and exp_mid2.
+ Float128 exp_mid1 =
+ fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
+ fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
+ Float128(EXP2_MID1[idx1].lo)));
+
+ Float128 exp_mid2 =
+ fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
+ fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
+ Float128(EXP2_MID2[idx2].lo)));
+
+ Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
+
+ Float128 p = poly_approx_f128(dx);
+
+ Float128 r = fputil::quick_mul(exp_mid, p);
+
+ r.exponent += hi;
+
+ return r;
+}
+
+// Compute 2^x with double-double precision.
+LIBC_INLINE static DoubleDouble
+exp2_double_double(double x, const DoubleDouble &exp_mid) {
+ DoubleDouble dx({0, x});
+
+ // Degree-6 polynomial approximation in double-double precision.
+ // | p - 2^x | < 2^-103.
+ DoubleDouble p = poly_approx_dd(dx);
+
+ // Error bounds: 2^-102.
+ DoubleDouble r = fputil::quick_mult(exp_mid, p);
+
+ return r;
+}
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+// When output is denormal.
+LIBC_INLINE static double exp2_denorm(double x) {
+ // Range reduction.
+ int k =
+ static_cast<int>(cpp::bit_cast<uint64_t>(x + 0x1.8000'0000'4p21) >> 19);
+ double kd = static_cast<double>(k);
+
+ uint32_t idx1 = (k >> 6) & 0x3f;
+ uint32_t idx2 = k & 0x3f;
+
+ int hi = k >> 12;
+
+ DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
+ DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
+ DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
+
+ // |dx| < 2^-13 + 2^-30.
+ double dx = fputil::multiply_add(kd, -0x1.0p-12, x); // exact
+
+ double mid_lo = dx * exp_mid.hi;
+
+ // Approximate (2^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
+ double p = poly_approx_d(dx);
+
+ double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
+
+#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ return ziv_test_denorm</*SKIP_ZIV_TEST=*/true>(hi, exp_mid.hi, lo, ERR_D)
+ .value();
+#else
+ if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
+ LIBC_LIKELY(r.has_value()))
+ return r.value();
+
+ // Use double-double
+ DoubleDouble r_dd = exp2_double_double(dx, exp_mid);
+
+ if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
+ LIBC_LIKELY(r.has_value()))
+ return r.value();
+
+ // Use 128-bit precision
+ Float128 r_f128 = exp2_f128(dx, hi, idx1, idx2);
+
+ return static_cast<double>(r_f128);
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+}
+
+// Check for exceptional cases when:
+// * log2(1 - 2^-54) < x < log2(1 + 2^-53)
+// * x >= 1024
+// * x <= -1022
+// * x is inf or nan
+LIBC_INLINE static constexpr double set_exceptional(double x) {
+ using FPBits = typename fputil::FPBits<double>;
+ FPBits xbits(x);
+
+ uint64_t x_u = xbits.uintval();
+ uint64_t x_abs = xbits.abs().uintval();
+
+ // |x| < log2(1 + 2^-53)
+ if (x_abs <= 0x3ca71547652b82fd) {
+ // 2^(x) ~ 1 + x/2
+ return fputil::multiply_add(x, 0.5, 1.0);
+ }
+
+ // x <= -1022 || x >= 1024 or inf/nan.
+ if (x_u > 0xc08ff00000000000) {
+ // x <= -1075 or -inf/nan
+ if (x_u >= 0xc090cc0000000000) {
+ // exp(-Inf) = 0
+ if (xbits.is_inf())
+ return 0.0;
+
+ // exp(nan) = nan
+ if (xbits.is_nan())
+ return x;
+
+ if (fputil::quick_get_round() == FE_UPWARD)
+ return FPBits::min_subnormal().get_val();
+ fputil::set_errno_if_required(ERANGE);
+ fputil::raise_except_if_required(FE_UNDERFLOW);
+ return 0.0;
+ }
+
+ return exp2_denorm(x);
+ }
+
+ // x >= 1024 or +inf/nan
+ // x is finite
+ if (x_u < 0x7ff0'0000'0000'0000ULL) {
+ int rounding = fputil::quick_get_round();
+ if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
+ return FPBits::max_normal().get_val();
+
+ fputil::set_errno_if_required(ERANGE);
+ fputil::raise_except_if_required(FE_OVERFLOW);
+ }
+ // x is +inf or nan
+ return x + FPBits::inf().get_val();
+}
+
+} // namespace exp2_internal
+
+LIBC_INLINE static constexpr double exp2(double x) {
+ using namespace exp2_internal;
+ using FPBits = typename fputil::FPBits<double>;
+ FPBits xbits(x);
+
+ uint64_t x_u = xbits.uintval();
+
+ // x < -1022 or x >= 1024 or log2(1 - 2^-54) < x < log2(1 + 2^-53).
+ if (LIBC_UNLIKELY(x_u > 0xc08ff00000000000 ||
+ (x_u <= 0xbc971547652b82fe && x_u >= 0x4090000000000000) ||
+ x_u <= 0x3ca71547652b82fd)) {
+ return set_exceptional(x);
+ }
+
+ // Now -1075 < x <= log2(1 - 2^-54) or log2(1 + 2^-53) < x < 1024
+
+ // Range reduction:
+ // Let x = (hi + mid1 + mid2) + lo
+ // in which:
+ // hi is an integer
+ // mid1 * 2^6 is an integer
+ // mid2 * 2^12 is an integer
+ // then:
+ // 2^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 2^(lo).
+ // With this formula:
+ // - multiplying by 2^hi is exact and cheap, simply by adding the exponent
+ // field.
+ // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
+ // - 2^(lo) ~ 1 + a0*lo + a1 * lo^2 + ...
+ //
+ // We compute (hi + mid1 + mid2) together by perform the rounding on x * 2^12.
+ // Since |x| < |-1075)| < 2^11,
+ // |x * 2^12| < 2^11 * 2^12 < 2^23,
+ // So we can fit the rounded result round(x * 2^12) in int32_t.
+ // Thus, the goal is to be able to use an additional addition and fixed width
+ // shift to get an int32_t representing round(x * 2^12).
+ //...
[truncated]
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Part of #147386
in preparation for: https://discourse.llvm.org/t/rfc-make-clang-builtin-math-functions-constexpr-with-llvm-libc-to-support-c-23-constexpr-math-functions/86450