[llvm-branch-commits] [libc] [llvm] [libc][math] Refactor expm1 implementation to header-only in src/__support/math folder. (PR #162127)
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llvm-branch-commits at lists.llvm.org
Mon Oct 6 10:45:28 PDT 2025
llvmbot wrote:
<!--LLVM PR SUMMARY COMMENT-->
@llvm/pr-subscribers-libc
Author: Muhammad Bassiouni (bassiounix)
<details>
<summary>Changes</summary>
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
---
Patch is 45.06 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/162127.diff
9 Files Affected:
- (modified) libc/shared/math.h (+1)
- (added) libc/shared/math/expm1.h (+23)
- (modified) libc/src/__support/math/CMakeLists.txt (+20)
- (added) libc/src/__support/math/expm1.h (+518)
- (modified) libc/src/math/generic/CMakeLists.txt (+1-12)
- (modified) libc/src/math/generic/expm1.cpp (+2-490)
- (modified) libc/test/shared/CMakeLists.txt (+1)
- (modified) libc/test/shared/shared_math_test.cpp (+1)
- (modified) utils/bazel/llvm-project-overlay/libc/BUILD.bazel (+19-9)
``````````diff
diff --git a/libc/shared/math.h b/libc/shared/math.h
index bcddb39021a9c..a5c9de4fb7029 100644
--- a/libc/shared/math.h
+++ b/libc/shared/math.h
@@ -54,6 +54,7 @@
#include "math/exp2m1f16.h"
#include "math/expf.h"
#include "math/expf16.h"
+#include "math/expm1.h"
#include "math/frexpf.h"
#include "math/frexpf128.h"
#include "math/frexpf16.h"
diff --git a/libc/shared/math/expm1.h b/libc/shared/math/expm1.h
new file mode 100644
index 0000000000000..4c8dbdc013a11
--- /dev/null
+++ b/libc/shared/math/expm1.h
@@ -0,0 +1,23 @@
+//===-- Shared expm1 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_EXPM1_H
+#define LLVM_LIBC_SHARED_MATH_EXPM1_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/expm1.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::expm1;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_EXPM1_H
diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index aaace44d04d3b..74f17b9fd8099 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -886,6 +886,26 @@ add_header_library(
libc.src.__support.macros.properties.cpu_features
)
+add_header_library(
+ expm1
+ HDRS
+ expm1.h
+ DEPENDS
+ .common_constants
+ .exp_constants
+ libc.src.__support.CPP.bit
+ 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.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(
range_reduction_double
HDRS
diff --git a/libc/src/__support/math/expm1.h b/libc/src/__support/math/expm1.h
new file mode 100644
index 0000000000000..2d8b3eacae4f6
--- /dev/null
+++ b/libc/src/__support/math/expm1.h
@@ -0,0 +1,518 @@
+//===-- Implementation header for expm1 -------------------------*- 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_EXPM1_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_EXPM1_H
+
+#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
+#include "exp_constants.h"
+#include "src/__support/CPP/bit.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/except_value_utils.h"
+#include "src/__support/FPUtil/multiply_add.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 expm1_internal {
+
+#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0)
+#define LIBC_MATH_EXPM1_SKIP_ACCURATE_PASS
+#endif
+
+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.
+// 0x1.8p-63;
+static constexpr uint64_t ERR_D = 0x3c08000000000000;
+// Errors when using double-double precision.
+// 0x1.0p-99
+[[maybe_unused]] static constexpr uint64_t ERR_DD = 0x39c0000000000000;
+
+// -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;
+static constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
+static constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
+
+using namespace common_constants_internal;
+
+// 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.
+LIBC_INLINE static constexpr 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;
+}
+
+// Polynomial approximation with double-double precision:
+// Return expm1(dx) / dx ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040
+// For |dx| < 2^-13 + 2^-30:
+// | output - expm1(dx) | < 2^-101
+LIBC_INLINE static constexpr DoubleDouble
+poly_approx_dd(const DoubleDouble &dx) {
+ // Taylor polynomial.
+ constexpr DoubleDouble COEFFS[] = {
+ {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
+ {0x1.a01a01a01a01ap-73, 0x1.a01a01a01a01ap-13}, // 1/5040
+ };
+
+ 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 ~ 1 + dx / 2 + dx^2 / 6 + ... + dx^6 / 5040
+// For |dx| < 2^-13 + 2^-30:
+// | output - exp(dx) | < 2^-126.
+[[maybe_unused]] 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, 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]);
+ return p;
+}
+
+#ifdef DEBUGDEBUG
+std::ostream &operator<<(std::ostream &OS, const Float128 &r) {
+ OS << (r.sign == Sign::NEG ? "-(" : "(") << r.mantissa.val[0] << " + "
+ << r.mantissa.val[1] << " * 2^64) * 2^" << r.exponent << "\n";
+ return OS;
+}
+
+std::ostream &operator<<(std::ostream &OS, const DoubleDouble &r) {
+ OS << std::hexfloat << "(" << r.hi << " + " << r.lo << ")"
+ << std::defaultfloat << "\n";
+ return OS;
+}
+#endif
+
+// Compute exp(x) - 1 using 128-bit precision.
+// TODO(lntue): investigate triple-double precision implementation for this
+// step.
+[[maybe_unused]] LIBC_INLINE static constexpr Float128
+expm1_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);
+
+ int hi = static_cast<int>(kd) >> 12;
+ Float128 minus_one{Sign::NEG, -127 - hi,
+ 0x80000000'00000000'00000000'00000000_u128};
+
+ Float128 exp_mid_m1 = fputil::quick_add(exp_mid, minus_one);
+
+ Float128 p = poly_approx_f128(dx);
+
+ // r = exp_mid * (1 + dx * P) - 1
+ // = (exp_mid - 1) + (dx * exp_mid) * P
+ Float128 r =
+ fputil::multiply_add(fputil::quick_mul(exp_mid, dx), p, exp_mid_m1);
+
+ r.exponent += hi;
+
+#ifdef DEBUGDEBUG
+ std::cout << "=== VERY SLOW PASS ===\n"
+ << " kd: " << kd << "\n"
+ << " hi: " << hi << "\n"
+ << " minus_one: " << minus_one << " dx: " << dx
+ << "exp_mid_m1: " << exp_mid_m1 << " exp_mid: " << exp_mid
+ << " p: " << p << " r: " << r << std::endl;
+#endif
+
+ return r;
+}
+
+// Compute exp(x) - 1 with double-double precision.
+LIBC_INLINE static constexpr DoubleDouble
+exp_double_double(double x, double kd, const DoubleDouble &exp_mid,
+ const DoubleDouble &hi_part) {
+ // 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::multiply_add(fputil::quick_mult(exp_mid, dx), p, hi_part);
+
+#ifdef DEBUGDEBUG
+ std::cout << "=== SLOW PASS ===\n"
+ << " dx: " << dx << " p: " << p << " r: " << r << std::endl;
+#endif
+
+ return r;
+}
+
+// Check for exceptional cases when
+// |x| <= 2^-53 or x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9
+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| <= 2^-53.
+ if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
+ // expm1(x) ~ x.
+
+ if (LIBC_UNLIKELY(x_abs <= 0x0370'0000'0000'0000ULL)) {
+ if (LIBC_UNLIKELY(x_abs == 0))
+ return x;
+ // |x| <= 2^-968, need to scale up a bit before rounding, then scale it
+ // back down.
+ return 0x1.0p-200 * fputil::multiply_add(x, 0x1.0p+200, 0x1.0p-1022);
+ }
+
+ // 2^-968 < |x| <= 2^-53.
+ return fputil::round_result_slightly_up(x);
+ }
+
+ // x < log(2^-54) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
+
+ // x < log(2^-54) or -inf/nan
+ if (x_u >= 0xc042'b708'8723'20e2ULL) {
+ // expm1(-Inf) = -1
+ if (xbits.is_inf())
+ return -1.0;
+
+ // exp(nan) = nan
+ if (xbits.is_nan())
+ return x;
+
+ return fputil::round_result_slightly_up(-1.0);
+ }
+
+ // 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 expm1_internal
+
+LIBC_INLINE static constexpr double expm1(double x) {
+ using namespace expm1_internal;
+
+ using FPBits = typename fputil::FPBits<double>;
+
+ FPBits xbits(x);
+
+ bool x_is_neg = xbits.is_neg();
+ 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: log(2^-54) = -0x1.2b708872320e2p5
+ // > round(log(2^-54), D, RN) = -0x1.2b708872320e2p5
+
+ // x < log(2^-54) or x >= 0x1.6232bdd7abcd3p+9 or |x| <= 2^-53.
+
+ if (LIBC_UNLIKELY(x_u >= 0xc042b708872320e2 ||
+ (x_u <= 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
+ x_u <= 0x3ca0000000000000)) {
+ return set_exceptional(x);
+ }
+
+ // Now log(2^-54) <= 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
+ // shift to get an int32_t representing round(x * 2^12 * L2E).
+ //
+ // Assuming int32_t using 2-complement representation, since the mantissa part
+ // of a double precision is unsigned with the leading bit hidden, if we add an
+ // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
+ // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
+ // considered as a proper 2-complement representations of x*2^12*L2E.
+ //
+ // One small problem with this approach is that the sum (x*2^12*L2E + C) in
+ // double precision is rounded to the least significant bit of the dorminant
+ // factor C. In order to minimize the rounding errors from this addition, we
+ // want to minimize e1. Another constraint that we want is that after
+ // shifting the mantissa so that the least significant bit of int32_t
+ // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
+ // any adjustment. So combining these 2 requirements, we can choose
+ // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
+ // after right shifting the mantissa, the resulting int32_t has correct sign.
+ // With this choice of C, the number of mantissa bits we need to shift to the
+ // right is: 52 - 33 = 19.
+ //
+ // Moreover, since the integer right shifts are equivalent to rounding down,
+ // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
+ // +infinity. So in particular, we can compute:
+ // hmm = x * 2^12 * L2E + C,
+ // where C = 2^33 + 2^32 + 2^-1, then if
+ // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
+ // the reduced argument:
+ // lo = x - log(2) * 2^-12 * k is bounded by:
+ // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
+ // = 2^-13 + 2^-31 + 2^-41.
+ //
+ // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the
+ // exponent 2^12 is not needed. So we can simply define
+ // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
+ // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19).
+
+ // Rounding errors <= 2^-31 + 2^-41.
+ double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21);
+ int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 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);
+
+ // -2^(-hi)
+ double one_scaled =
+ FPBits::create_value(Sign::NEG, FPBits::EXP_BIAS - hi, 0).get_val();
+
+ // 2^(mid1 + mid2) - 2^(-hi)
+ DoubleDouble hi_part = x_is_neg ? fputil::exact_add(one_scaled, exp_mid.hi)
+ : fputil::exact_add(exp_mid.hi, one_scaled);
+
+ hi_part.lo += exp_mid.lo;
+
+ // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
+ // = 2^11 * 2^-13 * 2^-52
+ // = 2^-54.
+ // |dx| < 2^-13 + 2^-30.
+ double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
+ double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h);
+
+ // We use the degree-4 Taylor polynomial to approximate exp(lo):
+ // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
+ // So that the errors are bounded by:
+ // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
+ // Let P_ be an evaluation of P where all intermediate computations are in
+ // double precision. Using either Horner's or Estrin's schemes, the evaluated
+ // errors can be bounded by:
+ // |P_(dx) - P(dx)| < 2^-51
+ // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
+ // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.
+ // Since we approximate
+ // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
+ // We use the expression:
+ // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
+ // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
+ // with errors bounded by 1.5 * 2^-63.
+
+ // Finally, we have the following approximation formula:
+ // expm1(x) = 2^hi * 2^(mid1 + mid2) * exp(lo) - 1
+ // = 2^hi * ( 2^(mid1 + mid2) * exp(lo) - 2^(-hi) )
+ // ~ 2^hi * ( (exp_mid.hi - 2^-hi) +
+ // + (exp_mid.hi * dx * P_(dx) + exp_mid.lo))
+
+ double mid_lo = dx * exp_mid.hi;
+
+ // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
+ double p = poly_approx_d(dx);
+
+ double lo = fputil::multiply_add(p, mid_lo, hi_part.lo);
+
+ // TODO: The following line leaks encoding abstraction. Use FPBits methods
+ // instead.
+ uint64_t err = x_is_neg ? (static_cast<uint64_t>(-hi) << 52) : 0;
+
+ double err_d = cpp::bit_cast<double>(ERR_D + err);
+
+ double upper = hi_part.hi + (lo + err_d);
+ double lower = hi_part.hi + (lo - e...
[truncated]
``````````
</details>
https://github.com/llvm/llvm-project/pull/162127
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