[libc][math] Refactor exp implementation to header-only in src/__support/math folder. #148761
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bassiounix
commented
Jul 15, 2025
bassiounix
commented
- [libc][math] Refactor exp implementation to header-only in src/__support/math folder.
- Reapply "[libc][math] Refactor exp implementation to header-only in src/__support/math folder." (Revert "[libc][math] Refactor exp implementation to header-only in src/__support/math folder." #148668)
bassiounix
requested review from
rupprecht,
keith and
aaronmondal
as code owners
llvmbot
added
backend:MIPS
backend:RISC-V
libc
debuginfo
bazel
|
@llvm/pr-subscribers-backend-risc-v @llvm/pr-subscribers-backend-mips Author: Muhammad Bassiouni (bassiounix) Changes
Patch is 109.58 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/148761.diff 27 Files Affected:
diff --git a/libc/fuzzing/math/CMakeLists.txt b/libc/fuzzing/math/CMakeLists.txt
index e3c29651917fc..bf2be5c0b3cea 100644
--- a/libc/fuzzing/math/CMakeLists.txt
+++ b/libc/fuzzing/math/CMakeLists.txt
@@ -62,6 +62,42 @@ add_libc_fuzzer(
libc.src.math.nextafterl
)
+add_libc_fuzzer(
+ exp_fuzz
+ NEED_MPFR
+ SRCS
+ exp_fuzz.cpp
+ DEPENDS
+ libc.src.math.exp
+)
+
+add_libc_fuzzer(
+ exp10_fuzz
+ NEED_MPFR
+ SRCS
+ exp10_fuzz.cpp
+ DEPENDS
+ libc.src.math.exp10
+)
+
+add_libc_fuzzer(
+ exp2_fuzz
+ NEED_MPFR
+ SRCS
+ exp2_fuzz.cpp
+ DEPENDS
+ libc.src.math.exp2
+)
+
+add_libc_fuzzer(
+ expm1_fuzz
+ NEED_MPFR
+ SRCS
+ expm1_fuzz.cpp
+ DEPENDS
+ libc.src.math.expm1
+)
+
add_libc_fuzzer(
asin_fuzz
NEED_MPFR
diff --git a/libc/fuzzing/math/exp10_fuzz.cpp b/libc/fuzzing/math/exp10_fuzz.cpp
new file mode 100644
index 0000000000000..2baef03a264a4
--- /dev/null
+++ b/libc/fuzzing/math/exp10_fuzz.cpp
@@ -0,0 +1,38 @@
+//===-- exp10_fuzz.cpp ----------------------------------------------------===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+///
+/// Fuzzing test for llvm-libc exp10 implementation.
+///
+//===----------------------------------------------------------------------===//
+
+#include "src/math/exp10.h"
+#include "utils/MPFRWrapper/mpfr_inc.h"
+#include <math.h>
+
+extern "C" int LLVMFuzzerTestOneInput(double x) {
+ // remove NaN and inf
+ if (isnan(x) || isinf(x))
+ return 0;
+ // signed zeros already tested in unit tests
+ if (signbit(x) && x == 0.0)
+ return 0;
+ mpfr_t input;
+ mpfr_init2(input, 53);
+ mpfr_set_d(input, x, MPFR_RNDN);
+ int output = mpfr_exp10(input, input, MPFR_RNDN);
+ mpfr_subnormalize(input, output, MPFR_RNDN);
+ double to_compare = mpfr_get_d(input, MPFR_RNDN);
+
+ double result = LIBC_NAMESPACE::exp10(x);
+
+ if (result != to_compare)
+ __builtin_trap();
+
+ mpfr_clear(input);
+ return 0;
+}
diff --git a/libc/fuzzing/math/exp2_fuzz.cpp b/libc/fuzzing/math/exp2_fuzz.cpp
new file mode 100644
index 0000000000000..8a2959047a6ca
--- /dev/null
+++ b/libc/fuzzing/math/exp2_fuzz.cpp
@@ -0,0 +1,38 @@
+//===-- exp2_fuzz.cpp -----------------------------------------------------===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+///
+/// Fuzzing test for llvm-libc exp2 implementation.
+///
+//===----------------------------------------------------------------------===//
+
+#include "src/math/exp2.h"
+#include "utils/MPFRWrapper/mpfr_inc.h"
+#include <math.h>
+
+extern "C" int LLVMFuzzerTestOneInput(double x) {
+ // remove NaN and inf
+ if (isnan(x) || isinf(x))
+ return 0;
+ // signed zeros already tested in unit tests
+ if (signbit(x) && x == 0.0)
+ return 0;
+ mpfr_t input;
+ mpfr_init2(input, 53);
+ mpfr_set_d(input, x, MPFR_RNDN);
+ int output = mpfr_exp2(input, input, MPFR_RNDN);
+ mpfr_subnormalize(input, output, MPFR_RNDN);
+ double to_compare = mpfr_get_d(input, MPFR_RNDN);
+
+ double result = LIBC_NAMESPACE::exp2(x);
+
+ if (result != to_compare)
+ __builtin_trap();
+
+ mpfr_clear(input);
+ return 0;
+}
diff --git a/libc/fuzzing/math/exp_fuzz.cpp b/libc/fuzzing/math/exp_fuzz.cpp
new file mode 100644
index 0000000000000..97bc12dfa64c9
--- /dev/null
+++ b/libc/fuzzing/math/exp_fuzz.cpp
@@ -0,0 +1,38 @@
+//===-- exp_fuzz.cpp ------------------------------------------------------===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+///
+/// Fuzzing test for llvm-libc exp implementation.
+///
+//===----------------------------------------------------------------------===//
+
+#include "src/math/exp.h"
+#include "utils/MPFRWrapper/mpfr_inc.h"
+#include <math.h>
+
+extern "C" int LLVMFuzzerTestOneInput(double x) {
+ // remove NaN and inf
+ if (isnan(x) || isinf(x))
+ return 0;
+ // signed zeros already tested in unit tests
+ if (signbit(x) && x == 0.0)
+ return 0;
+ mpfr_t input;
+ mpfr_init2(input, 53);
+ mpfr_set_d(input, x, MPFR_RNDN);
+ int output = mpfr_exp(input, input, MPFR_RNDN);
+ mpfr_subnormalize(input, output, MPFR_RNDN);
+ double to_compare = mpfr_get_d(input, MPFR_RNDN);
+
+ double result = LIBC_NAMESPACE::exp(x);
+
+ if (result != to_compare)
+ __builtin_trap();
+
+ mpfr_clear(input);
+ return 0;
+}
diff --git a/libc/fuzzing/math/expm1_fuzz.cpp b/libc/fuzzing/math/expm1_fuzz.cpp
new file mode 100644
index 0000000000000..db507bb02b1d7
--- /dev/null
+++ b/libc/fuzzing/math/expm1_fuzz.cpp
@@ -0,0 +1,38 @@
+//===-- expm1_fuzz.cpp ----------------------------------------------------===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+///
+/// Fuzzing test for llvm-libc expm1 implementation.
+///
+//===----------------------------------------------------------------------===//
+
+#include "src/math/expm1.h"
+#include "utils/MPFRWrapper/mpfr_inc.h"
+#include <math.h>
+
+extern "C" int LLVMFuzzerTestOneInput(double x) {
+ // remove NaN and inf
+ if (isnan(x) || isinf(x))
+ return 0;
+ // signed zeros already tested in unit tests
+ if (signbit(x) && x == 0.0)
+ return 0;
+ mpfr_t input;
+ mpfr_init2(input, 53);
+ mpfr_set_d(input, x, MPFR_RNDN);
+ int output = mpfr_expm1(input, input, MPFR_RNDN);
+ mpfr_subnormalize(input, output, MPFR_RNDN);
+ double to_compare = mpfr_get_d(input, MPFR_RNDN);
+
+ double result = LIBC_NAMESPACE::expm1(x);
+
+ if (result != to_compare)
+ __builtin_trap();
+
+ mpfr_clear(input);
+ return 0;
+}
diff --git a/libc/shared/math.h b/libc/shared/math.h
index b2f1a03e0940d..3012cbb938816 100644
--- a/libc/shared/math.h
+++ b/libc/shared/math.h
@@ -11,6 +11,7 @@
#include "libc_common.h"
+#include "math/exp.h"
#include "math/expf.h"
#include "math/expf16.h"
#include "math/frexpf.h"
diff --git a/libc/shared/math/exp.h b/libc/shared/math/exp.h
new file mode 100644
index 0000000000000..7cdd6331e613a
--- /dev/null
+++ b/libc/shared/math/exp.h
@@ -0,0 +1,23 @@
+//===-- Shared exp 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_EXP_H
+#define LLVM_LIBC_SHARED_MATH_EXP_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/exp.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::exp;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXP_H
diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index 900c0ab04d3a3..f7ef9e7694fe6 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -110,3 +110,42 @@ add_header_library(
DEPENDS
libc.src.__support.FPUtil.manipulation_functions
)
+
+add_header_library(
+ exp_constants
+ HDRS
+ exp_constants.h
+ DEPENDS
+ libc.src.__support.FPUtil.triple_double
+)
+
+add_header_library(
+ exp_utils
+ HDRS
+ exp_utils.h
+ DEPENDS
+ libc.src.__support.CPP.optional
+ libc.src.__support.CPP.bit
+ libc.src.__support.FPUtil.fp_bits
+)
+
+add_header_library(
+ exp
+ HDRS
+ exp.h
+ DEPENDS
+ .exp_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
+)
diff --git a/libc/src/__support/math/exp.h b/libc/src/__support/math/exp.h
new file mode 100644
index 0000000000000..5c8ea089635ef
--- /dev/null
+++ b/libc/src/__support/math/exp.h
@@ -0,0 +1,448 @@
+//===-- Implementation header for exp ---------------------------*- 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_EXP_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_EXP_H
+
+#include "exp_constants.h"
+#include "exp_utils.h"
+#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 {
+
+using fputil::DoubleDouble;
+using fputil::TripleDouble;
+using Float128 = typename fputil::DyadicFloat<128>;
+
+using LIBC_NAMESPACE::operator""_u128;
+
+// log2(e)
+static constexpr double LOG2_E = 0x1.71547652b82fep+0;
+
+// Error bounds:
+// Errors when using double precision.
+static constexpr double ERR_D = 0x1.8p-63;
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+// Errors when using double-double precision.
+static constexpr double ERR_DD = 0x1.0p-99;
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+// -2^-12 * log(2)
+// > a = -2^-12 * log(2);
+// > b = round(a, 30, RN);
+// > c = round(a - b, 30, RN);
+// > d = round(a - b - c, D, RN);
+// Errors < 1.5 * 2^-133
+static constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
+static constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+static constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
+static constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+namespace {
+
+// Polynomial approximations with double precision:
+// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
+// For |dx| < 2^-13 + 2^-30:
+// | output - expm1(dx) / dx | < 2^-51.
+static double poly_approx_d(double dx) {
+ // dx^2
+ double dx2 = dx * dx;
+ // c0 = 1 + dx / 2
+ double c0 = fputil::multiply_add(dx, 0.5, 1.0);
+ // c1 = 1/6 + dx / 24
+ double c1 =
+ fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3);
+ // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
+ double p = fputil::multiply_add(dx2, c1, c0);
+ return p;
+}
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+// Polynomial approximation with double-double precision:
+// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
+// For |dx| < 2^-13 + 2^-30:
+// | output - exp(dx) | < 2^-101
+static DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
+ // Taylor polynomial.
+ constexpr DoubleDouble COEFFS[] = {
+ {0, 0x1p0}, // 1
+ {0, 0x1p0}, // 1
+ {0, 0x1p-1}, // 1/2
+ {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6
+ {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24
+ {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120
+ {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720
+ };
+
+ 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 + dx + dx^2 / 2 + ... + dx^7 / 5040
+// For |dx| < 2^-13 + 2^-30:
+// | output - exp(dx) | < 2^-126.
+static Float128 poly_approx_f128(const Float128 &dx) {
+ constexpr Float128 COEFFS_128[]{
+ {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
+ {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
+ {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5
+ {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6
+ {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24
+ {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120
+ {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720
+ {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040
+ };
+
+ 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 exp(x) using 128-bit precision.
+// TODO(lntue): investigate triple-double precision implementation for this
+// step.
+static Float128 exp_f128(double x, double kd, int idx1, int idx2) {
+ // Recalculate dx:
+
+ double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+ double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
+ double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133
+
+ Float128 dx = fputil::quick_add(
+ Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
+
+ // 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 += static_cast<int>(kd) >> 12;
+
+ return r;
+}
+
+// Compute exp(x) with double-double precision.
+static DoubleDouble exp_double_double(double x, double kd,
+ const DoubleDouble &exp_mid) {
+ // Recalculate dx:
+ // dx = x - k * 2^-12 * log(2)
+ double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+ double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
+ double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130
+
+ DoubleDouble dx = fputil::exact_add(t1, t2);
+ dx.lo += t3;
+
+ // Degree-6 Taylor polynomial approximation in double-double precision.
+ // | p - exp(x) | < 2^-100.
+ DoubleDouble p = poly_approx_dd(dx);
+
+ // Error bounds: 2^-99.
+ DoubleDouble r = fputil::quick_mult(exp_mid, p);
+
+ return r;
+}
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+// Check for exceptional cases when
+// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
+static 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| <= 2^-53
+ if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
+ // exp(x) ~ 1 + x
+ return 1 + x;
+ }
+
+ // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
+
+ // x <= log(2^-1075) or -inf/nan
+ if (x_u >= 0xc087'4910'd52d'3052ULL) {
+ // 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;
+ }
+
+ // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 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
+
+namespace math {
+
+static double exp(double x) {
+ using FPBits = typename fputil::FPBits<double>;
+ FPBits xbits(x);
+
+ uint64_t x_u = xbits.uintval();
+
+ // Upper bound: max normal number = 2^1023 * (2 - 2^-52)
+ // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
+ // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
+ // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
+ // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
+
+ // Lower bound: min denormal number / 2 = 2^-1075
+ // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
+
+ // Another lower bound: min normal number = 2^-1022
+ // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
+
+ // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
+ if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
+ (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
+ x_u < 0x3ca0000000000000)) {
+ return set_exceptional(x);
+ }
+
+ // Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
+
+ // Range reduction:
+ // Let x = log(2) * (hi + mid1 + mid2) + lo
+ // in which:
+ // hi is an integer
+ // mid1 * 2^6 is an integer
+ // mid2 * 2^12 is an integer
+ // then:
+ // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(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.
+ // - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
+ //
+ // They can be defined by:
+ // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
+ // If we store L2E = round(log2(e), D, RN), then:
+ // log2(e) - L2E ~ 1.5 * 2^(-56)
+ // So the errors when computing in double precision is:
+ // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
+ // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
+ // + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
+ // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN
+ // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
+ // So if:
+ // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
+ // in double precision, the reduced argument:
+ // lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
+ // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
+ // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
+ // < 2^-13 + 2^-41
+ //
+
+ // The following trick computes the round(x * L2E) more efficiently
+ // than using the rounding instructions, with the tradeoff for less accuracy,
+ // and hence a slightly larger range for the reduced argument `lo`.
+ //
+ // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
+ // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
+ // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t.
+ // Thus, the goal is to be able to use an additional addition and fixed width
+ // sh...
[truncated]
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✅ With the latest revision this PR passed the C/C++ code formatter. |
lntue
reviewed
Jul 15, 2025
lntue
approved these changes
Jul 15, 2025
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