[libc-commits] [libc] [llvm] [libc][math] Refactor pow to Header Only. (PR #176529)
via libc-commits
libc-commits at lists.llvm.org
Fri Jan 16 17:39:44 PST 2026
llvmbot wrote:
<!--LLVM PR SUMMARY COMMENT-->
@llvm/pr-subscribers-libc
Author: Anonmiraj (AnonMiraj)
<details>
<summary>Changes</summary>
@<!-- -->bassiounix
closes : #<!-- -->176516
---
Patch is 47.90 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/176529.diff
9 Files Affected:
- (modified) libc/shared/math.h (+1)
- (added) libc/shared/math/pow.h (+22)
- (modified) libc/src/__support/math/CMakeLists.txt (+19)
- (added) libc/src/__support/math/pow.h (+544)
- (modified) libc/src/math/generic/CMakeLists.txt (+1-12)
- (modified) libc/src/math/generic/pow.cpp (+2-521)
- (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 (+21-5)
``````````diff
diff --git a/libc/shared/math.h b/libc/shared/math.h
index 017c94f8ad54a..5ebadfb530682 100644
--- a/libc/shared/math.h
+++ b/libc/shared/math.h
@@ -74,6 +74,7 @@
#include "math/logbf.h"
#include "math/logbf128.h"
#include "math/logbf16.h"
+#include "math/pow.h"
#include "math/rsqrtf.h"
#include "math/rsqrtf16.h"
#include "math/sin.h"
diff --git a/libc/shared/math/pow.h b/libc/shared/math/pow.h
new file mode 100644
index 0000000000000..a5a31d70ed0ca
--- /dev/null
+++ b/libc/shared/math/pow.h
@@ -0,0 +1,22 @@
+//===-- Shared pow 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_POW_H
+#define LLVM_LIBC_SHARED_MATH_POW_H
+#include "shared/libc_common.h"
+#include "src/__support/math/pow.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::pow;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_POW_H
diff --git a/libc/src/__support/math/CMakeLists.txt b/libc/src/__support/math/CMakeLists.txt
index cb03b18678145..05d0ee2c4561c 100644
--- a/libc/src/__support/math/CMakeLists.txt
+++ b/libc/src/__support/math/CMakeLists.txt
@@ -1139,6 +1139,25 @@ add_header_library(
libc.src.__support.uint128
)
+add_header_library(
+ pow
+ HDRS
+ pow.h
+ DEPENDS
+ libc.src.__support.CPP.bit
+ 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.sqrt
+ libc.src.__support.FPUtil.triple_double
+ libc.src.__support.macros.optimization
+ libc.src.__support.math.common_constants
+ libc.src.__support.math.exp10f
+ libc.src.__support.math.exp2f
+)
+
add_header_library(
sin
HDRS
diff --git a/libc/src/__support/math/pow.h b/libc/src/__support/math/pow.h
new file mode 100644
index 0000000000000..8e4507cf0242f
--- /dev/null
+++ b/libc/src/__support/math/pow.h
@@ -0,0 +1,544 @@
+//===-- Implementation header for pow ------------------------*- 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_POW_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_POW_H
+
+#include "hdr/errno_macros.h"
+#include "hdr/fenv_macros.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/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/FPUtil/sqrt.h" // Speedup for pow(x, 1/2) = sqrt(x)
+#include "src/__support/common.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#include "src/__support/math/common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
+#include "src/__support/math/exp_constants.h" // Lookup tables EXP_M1 and EXP_M2.
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace math {
+
+using fputil::DoubleDouble;
+
+namespace {
+
+using namespace common_constants_internal;
+
+// Constants for log2(x) range reduction, generated by Sollya with:
+// > for i from 0 to 127 do {
+// r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^-7) );
+// b = nearestint(log2(r) * 2^41) * 2^-41;
+// c = round(log2(r) - b, D, RN);
+// print("{", -c, ",", -b, "},");
+// };
+// This is the same as -log2(RD[i]), with the least significant bits of the
+// high part set to be 2^-41, so that the sum of high parts + e_x is exact in
+// double precision.
+// We also replace the first and the last ones to be 0.
+constexpr DoubleDouble LOG2_R_DD[128] = {
+ {0.0, 0.0},
+ {-0x1.19b14945cf6bap-44, 0x1.72c7ba21p-7},
+ {-0x1.95539356f93dcp-43, 0x1.743ee862p-6},
+ {0x1.abe0a48f83604p-43, 0x1.184b8e4c5p-5},
+ {0x1.635577970e04p-43, 0x1.77394c9d9p-5},
+ {-0x1.401fbaaa67e3cp-45, 0x1.d6ebd1f2p-5},
+ {-0x1.5b1799ceaeb51p-43, 0x1.1bb32a6008p-4},
+ {0x1.7c407050799bfp-43, 0x1.4c560fe688p-4},
+ {0x1.da6339da288fcp-43, 0x1.7d60496cf8p-4},
+ {0x1.be4f6f22dbbadp-43, 0x1.960caf9ab8p-4},
+ {-0x1.c760bc9b188c4p-45, 0x1.c7b528b71p-4},
+ {0x1.164e932b2d51cp-44, 0x1.f9c95dc1dp-4},
+ {0x1.924ae921f7ecap-45, 0x1.097e38ce6p-3},
+ {-0x1.6d25a5b8a19b2p-44, 0x1.22dadc2ab4p-3},
+ {0x1.e50a1644ac794p-43, 0x1.3c6fb650ccp-3},
+ {0x1.f34baa74a7942p-43, 0x1.494f863b8cp-3},
+ {-0x1.8f7aac147fdc1p-46, 0x1.633a8bf438p-3},
+ {0x1.f84be19cb9578p-43, 0x1.7046031c78p-3},
+ {-0x1.66cccab240e9p-46, 0x1.8a8980abfcp-3},
+ {-0x1.3f7a55cd2af4cp-47, 0x1.97c1cb13c8p-3},
+ {0x1.3458cde69308cp-43, 0x1.b2602497d4p-3},
+ {-0x1.667f21fa8423fp-44, 0x1.bfc67a8p-3},
+ {0x1.d2fe4574e09b9p-47, 0x1.dac22d3e44p-3},
+ {0x1.367bde40c5e6dp-43, 0x1.e857d3d36p-3},
+ {0x1.d45da26510033p-46, 0x1.01d9bbcfa6p-2},
+ {-0x1.7204f55bbf90dp-44, 0x1.08bce0d96p-2},
+ {-0x1.d4f1b95e0ff45p-43, 0x1.169c05364p-2},
+ {0x1.c20d74c0211bfp-44, 0x1.1d982c9d52p-2},
+ {0x1.ad89a083e072ap-43, 0x1.249cd2b13cp-2},
+ {0x1.cd0cb4492f1bcp-43, 0x1.32bfee370ep-2},
+ {-0x1.2101a9685c779p-47, 0x1.39de8e155ap-2},
+ {0x1.9451cd394fe8dp-43, 0x1.4106017c3ep-2},
+ {0x1.661e393a16b95p-44, 0x1.4f6fbb2cecp-2},
+ {-0x1.c6d8d86531d56p-44, 0x1.56b22e6b58p-2},
+ {0x1.c1c885adb21d3p-43, 0x1.5dfdcf1eeap-2},
+ {0x1.3bb5921006679p-45, 0x1.6552b49986p-2},
+ {0x1.1d406db502403p-43, 0x1.6cb0f6865cp-2},
+ {0x1.55a63e278bad5p-43, 0x1.7b89f02cf2p-2},
+ {-0x1.66ae2a7ada553p-49, 0x1.8304d90c12p-2},
+ {-0x1.66cccab240e9p-45, 0x1.8a8980abfcp-2},
+ {-0x1.62404772a151dp-45, 0x1.921800924ep-2},
+ {0x1.ac9bca36fd02ep-44, 0x1.99b072a96cp-2},
+ {0x1.4bc302ffa76fbp-43, 0x1.a8ff97181p-2},
+ {0x1.01fea1ec47c71p-43, 0x1.b0b67f4f46p-2},
+ {-0x1.f20203b3186a6p-43, 0x1.b877c57b1cp-2},
+ {-0x1.2642415d47384p-45, 0x1.c043859e3p-2},
+ {-0x1.bc76a2753b99bp-50, 0x1.c819dc2d46p-2},
+ {-0x1.da93ae3a5f451p-43, 0x1.cffae611aep-2},
+ {-0x1.50e785694a8c6p-43, 0x1.d7e6c0abc4p-2},
+ {0x1.c56138c894641p-43, 0x1.dfdd89d586p-2},
+ {0x1.5669df6a2b592p-43, 0x1.e7df5fe538p-2},
+ {-0x1.ea92d9e0e8ac2p-48, 0x1.efec61b012p-2},
+ {0x1.a0331af2e6feap-43, 0x1.f804ae8d0cp-2},
+ {0x1.9518ce032f41dp-48, 0x1.0014332bep-1},
+ {-0x1.b3b3864c60011p-44, 0x1.042bd4b9a8p-1},
+ {-0x1.103e8f00d41c8p-45, 0x1.08494c66b9p-1},
+ {0x1.65be75cc3da17p-43, 0x1.0c6caaf0c5p-1},
+ {0x1.3676289cd3dd4p-43, 0x1.1096015deep-1},
+ {-0x1.41dfc7d7c3321p-43, 0x1.14c560fe69p-1},
+ {0x1.e0cda8bd74461p-44, 0x1.18fadb6e2dp-1},
+ {0x1.2a606046ad444p-44, 0x1.1d368296b5p-1},
+ {0x1.f9ea977a639cp-43, 0x1.217868b0c3p-1},
+ {-0x1.50520a377c7ecp-45, 0x1.25c0a0463cp-1},
+ {0x1.6e3cb71b554e7p-47, 0x1.2a0f3c3407p-1},
+ {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
+ {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
+ {-0x1.979a5db68721dp-45, 0x1.32bfee370fp-1},
+ {0x1.1ee969a95f529p-43, 0x1.37222bb707p-1},
+ {0x1.bb4b69336b66ep-43, 0x1.3b8b1c68fap-1},
+ {0x1.d5e6a8a4fb059p-45, 0x1.3ffad4e74fp-1},
+ {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
+ {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
+ {-0x1.9bcaf1aa4168ap-43, 0x1.48eef19318p-1},
+ {0x1.1646b761c48dep-44, 0x1.4d7380dcc4p-1},
+ {0x1.2f0c0bfe9dbecp-43, 0x1.51ff2e3021p-1},
+ {0x1.29904613e33cp-43, 0x1.5692101d9bp-1},
+ {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
+ {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
+ {-0x1.125d6cbcd1095p-44, 0x1.5fcdce2728p-1},
+ {-0x1.bd9b32266d92cp-43, 0x1.6476d98adap-1},
+ {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
+ {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
+ {-0x1.ce60916e52e91p-44, 0x1.6ddfc2a79p-1},
+ {0x1.f1f5ae718f241p-43, 0x1.729fd26b7p-1},
+ {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
+ {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
+ {0x1.fed21f9cb2cc5p-43, 0x1.7c37a9227ep-1},
+ {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
+ {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
+ {0x1.5b338360c2ae2p-43, 0x1.85efd062c6p-1},
+ {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
+ {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
+ {-0x1.bdc81c4db3134p-44, 0x1.8fc924c89bp-1},
+ {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
+ {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
+ {0x1.e41fa0a62e6aep-44, 0x1.99c48be206p-1},
+ {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
+ {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
+ {-0x1.3f94e00e7d6bcp-46, 0x1.a3e2f4ac44p-1},
+ {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
+ {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
+ {0x1.1659d8e2d7d38p-44, 0x1.ae255819fp-1},
+ {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
+ {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
+ {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
+ {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
+ {0x1.871a7610e40bdp-45, 0x1.bdce9dcc96p-1},
+ {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
+ {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
+ {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
+ {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
+ {-0x1.9ad57391924a7p-43, 0x1.cdcebd2374p-1},
+ {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
+ {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
+ {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
+ {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
+ {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
+ {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
+ {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
+ {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
+ {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
+ {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
+ {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
+ {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
+ {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
+ {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
+ {0x1.ef5d00e390ap-44, 0x1.fa406bd244p-1},
+ {0.0, 1.0},
+};
+
+LIBC_INLINE static constexpr bool is_odd_integer(double x) {
+ using FPBits = fputil::FPBits<double>;
+ FPBits xbits(x);
+ uint64_t x_u = xbits.uintval();
+ unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
+ unsigned lsb =
+ static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
+ constexpr unsigned UNIT_EXPONENT =
+ static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
+ return (x_e + lsb == UNIT_EXPONENT);
+}
+
+LIBC_INLINE static constexpr bool is_integer(double x) {
+ using FPBits = fputil::FPBits<double>;
+ FPBits xbits(x);
+ uint64_t x_u = xbits.uintval();
+ unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
+ unsigned lsb =
+ static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
+ constexpr unsigned UNIT_EXPONENT =
+ static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
+ return (x_e + lsb >= UNIT_EXPONENT);
+}
+
+} // namespace
+
+LIBC_INLINE static constexpr double pow(double x, double y) {
+ using FPBits = fputil::FPBits<double>;
+
+ FPBits xbits(x), ybits(y);
+
+ bool x_sign = xbits.sign() == Sign::NEG;
+ bool y_sign = ybits.sign() == Sign::NEG;
+
+ FPBits x_abs = xbits.abs();
+ FPBits y_abs = ybits.abs();
+
+ uint64_t x_mant = xbits.get_mantissa();
+ uint64_t y_mant = ybits.get_mantissa();
+ uint64_t x_u = xbits.uintval();
+ uint64_t x_a = x_abs.uintval();
+ uint64_t y_a = y_abs.uintval();
+
+ double e_x = static_cast<double>(xbits.get_exponent());
+ uint64_t sign = 0;
+
+ ///////// BEGIN - Check exceptional cases ////////////////////////////////////
+ // If x or y is signaling NaN
+ if (x_abs.is_signaling_nan() || y_abs.is_signaling_nan()) {
+ fputil::raise_except_if_required(FE_INVALID);
+ return FPBits::quiet_nan().get_val();
+ }
+
+ // The double precision number that is closest to 1 is (1 - 2^-53), which has
+ // log2(1 - 2^-53) ~ -1.715...p-53.
+ // So if |y| > |1075 / log2(1 - 2^-53)|, and x is finite:
+ // |y * log2(x)| = 0 or > 1075.
+ // Hence x^y will either overflow or underflow if x is not zero.
+ if (LIBC_UNLIKELY(y_mant == 0 || y_a > 0x43d7'4910'd52d'3052 ||
+ x_u == FPBits::one().uintval() ||
+ x_u >= FPBits::inf().uintval() ||
+ x_u < FPBits::min_normal().uintval())) {
+ // Exceptional exponents.
+ if (y == 0.0)
+ return 1.0;
+
+ switch (y_a) {
+ case 0x3fe0'0000'0000'0000: { // y = +-0.5
+ // TODO: speed up x^(-1/2) with rsqrt(x) when available.
+ if (LIBC_UNLIKELY(
+ (x == 0.0 || x_u == FPBits::inf(Sign::NEG).uintval()))) {
+ // pow(-0, 1/2) = +0
+ // pow(-inf, 1/2) = +inf
+ // Make sure it works correctly for FTZ/DAZ.
+ return y_sign ? 1.0 / (x * x) : (x * x);
+ }
+ return y_sign ? (1.0 / fputil::sqrt<double>(x)) : fputil::sqrt<double>(x);
+ }
+ case 0x3ff0'0000'0000'0000: // y = +-1.0
+ return y_sign ? (1.0 / x) : x;
+ case 0x4000'0000'0000'0000: // y = +-2.0;
+ return y_sign ? (1.0 / (x * x)) : (x * x);
+ }
+
+ // |y| > |1075 / log2(1 - 2^-53)|.
+ if (y_a > 0x43d7'4910'd52d'3052) {
+ if (y_a >= 0x7ff0'0000'0000'0000) {
+ // y is inf or nan
+ if (y_mant != 0) {
+ // y is NaN
+ // pow(1, NaN) = 1
+ // pow(x, NaN) = NaN
+ return (x_u == FPBits::one().uintval()) ? 1.0 : y;
+ }
+
+ // Now y is +-Inf
+ if (x_abs.is_nan()) {
+ // pow(NaN, +-Inf) = NaN
+ return x;
+ }
+
+ if (x_a == 0x3ff0'0000'0000'0000) {
+ // pow(+-1, +-Inf) = 1.0
+ return 1.0;
+ }
+
+ if (x == 0.0 && y_sign) {
+ // pow(+-0, -Inf) = +inf and raise FE_DIVBYZERO
+ fputil::set_errno_if_required(EDOM);
+ fputil::raise_except_if_required(FE_DIVBYZERO);
+ return FPBits::inf().get_val();
+ }
+ // pow (|x| < 1, -inf) = +inf
+ // pow (|x| < 1, +inf) = 0.0
+ // pow (|x| > 1, -inf) = 0.0
+ // pow (|x| > 1, +inf) = +inf
+ return ((x_a < FPBits::one().uintval()) == y_sign)
+ ? FPBits::inf().get_val()
+ : 0.0;
+ }
+ // x^y will overflow / underflow in double precision. Set y to a
+ // large enough exponent but not too large, so that the computations
+ // won't overflow in double precision.
+ y = y_sign ? -0x1.0p100 : 0x1.0p100;
+ }
+
+ // y is finite and non-zero.
+
+ if (x_u == FPBits::one().uintval()) {
+ // pow(1, y) = 1
+ return 1.0;
+ }
+
+ // TODO: Speed things up with pow(2, y) = exp2(y) and pow(10, y) = exp10(y).
+
+ if (x == 0.0) {
+ bool out_is_neg = x_sign && is_odd_integer(y);
+ if (y_sign) {
+ // pow(0, negative number) = inf
+ fputil::set_errno_if_required(EDOM);
+ fputil::raise_except_if_required(FE_DIVBYZERO);
+ return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).get_val();
+ }
+ // pow(0, positive number) = 0
+ return out_is_neg ? -0.0 : 0.0;
+ }
+
+ if (x_a == FPBits::inf().uintval()) {
+ bool out_is_neg = x_sign && is_odd_integer(y);
+ if (y_sign)
+ return out_is_neg ? -0.0 : 0.0;
+ return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).get_val();
+ }
+
+ if (x_a > FPBits::inf().uintval()) {
+ // x is NaN.
+ // pow (aNaN, 0) is already taken care above.
+ return x;
+ }
+
+ // Normalize denormal inputs.
+ if (x_a < FPBits::min_normal().uintval()) {
+ e_x -= 64.0;
+ x_mant = FPBits(x * 0x1.0p64).get_mantissa();
+ }
+
+ // x is finite and negative, and y is a finite integer.
+ if (x_sign) {
+ if (is_integer(y)) {
+ x = -x;
+ if (is_odd_integer(y))
+ // sign = -1.0;
+ sign = 0x8000'0000'0000'0000;
+ } else {
+ // pow( negative, non-integer ) = NaN
+ fputil::set_errno_if_required(EDOM);
+ fputil::raise_except_if_required(FE_INVALID);
+ return FPBits::quiet_nan().get_val();
+ }
+ }
+ }
+
+ ///////// END - Check exceptional cases //////////////////////////////////////
+
+ // x^y = 2^( y * log2(x) )
+ // = 2^( y * ( e_x + log2(m_x) ) )
+ // First we compute log2(x) = e_x + log2(m_x)
+
+ // Extract exponent field of x.
+
+ // Use the highest 7 fractional bits of m_x as the index for look up tables.
+ unsigned idx_x = static_cast<unsigned>(x_mant >> (FPBits::FRACTION_LEN - 7));
+ // Add the hidden bit to the mantissa.
+ // 1 <= m_x < 2
+ FPBits m_x = FPBits(x_mant | 0x3ff0'0000'0000'0000);
+
+ // Reduced argument for log2(m_x):
+ // dx = r * m_x - 1.
+ // The computation is exact, and -2^-8 <= dx < 2^-7.
+ // Then m_x = (1 + dx) / r, and
+ // log2(m_x) = log2( (1 + dx) / r )
+ // = log2(1 + dx) - log2(r).
+
+ // In order for the overall computations x^y = 2^(y * log2(x)) to have the
+ // relative errors < 2^-52 (1ULP), we will need to evaluate the exponent part
+ // y * log2(x) with absolute errors < 2^-52 (or better, 2^-53). Since the
+ // whole exponent range for double precision is bounded by
+ // |y * log2(x)| < 1076 ~ 2^10, we need to evaluate log2(x) with absolute
+ // errors < 2^-53 * 2^-10 = 2^-63.
+
+ // With that requirement, we use the following degree-6 polynomial
+ // approximation:
+ // P(dx) ~ log2(1 + dx) / dx
+ // Generated by Sollya with:
+ // > P = fpminimax(log2(1 + x)/x, 6, [|D...|], [-2^-8, 2^-7]); P;
+ // > dirtyinfnorm(log2(1 + x) - x*P, [-2^-8, 2^-7]);
+ // 0x1.d03cc...p-66
+ constexpr double COEFFS[] = {0x1.71547652b82fep0, -0x1.71547652b82e7p-1,
+ 0x1.ec709dc3b1fd5p-2, -0x1.7154766124215p-2,
+ 0x1.2776bd90259d8p-2, -0x1.ec586c6f3d311p-3,
+ 0x1.9c4775eccf524p-3};
+ // Error: ulp(dx^2) <= (2^-7)^2 * 2^-52 = 2^-66
+ // Extra errors from various computations and rounding directions, the overall
+ // errors we can be bounded by 2^-65.
+
+ DoubleDouble dx_c0;
+
+ // Perform exact range reduction and exact product dx * c0.
+#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+ double dx = fputil::multiply_add(RD[idx_x], m_x.get_val(), -1.0); // Exact
+ dx_c0 = fputil::exact_mult(COEFFS[0], dx);
+#else
+ double c = FPBits(m_x.uintval() & 0x3fff'e000'0000'0000).get_val();
+ double dx =
+ fputil::multiply_add(RD[idx_x], m_x.get_val() - c, CD[idx_x]); // Exact
+ dx_c0 = fputil::exact_mult<double, 28>(dx, COEFFS[0]); // Exact
+#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+
+ double dx2 = dx * dx;
+ double c0 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
+ double c1 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
+ double c2 = fputil::multiply_add(dx, COEFFS[6], COEFFS[5]);
+
+ double p = fputil::polyeval(dx2, c0, c1, c2);
+
+ // s = e_x - log2(r) + dx * P(dx)
+ // Absolute error bound:
+ // |log2(x) - log2_x.hi - log2_x.lo| < 2^-65.
+
+ // Notice that e_x - log2(r).hi is exact, so we perform an exact sum of
+ // e_x - log2(r).hi and the high part of the product dx * c0:
+ // log2_x_hi.hi + log2_x_hi.lo = e_x - log2(r).hi + (dx * c0).hi
+ DoubleDouble log2_x_hi =
+ fputil::exact_add(e_x + LOG2_R_DD[idx_x].hi, dx_c0.hi);
+ // The low part is dx^2 * p + low part of (dx * c0) + low part of -log2(r).
+ double log2_x_lo =
+ fputil::multiply_add(dx2, p, dx_c0.lo + LOG2_R_DD[idx_x].lo);
+ // Perform accurate sums.
+ DoubleDouble log2_x = fputil::exact_add(log2_x_hi.hi, log2_x_lo);
+ log2_x.lo += log2_x_hi.lo;
+
+ // To compute 2^(y * log2(x)), we break the exponent into 3 parts:
+ // y * log(2) = hi + mid + lo, where
+ // hi is an integer
+ // mid * 2^6 is an integer
+ // |lo| <= 2^-7
+ // Then:
+ // x^y = 2^(y * log2(x)) = 2^hi * 2^mid * 2^lo,
+ // In which 2^mid is obtained from a look-up table of size 2^6 = 64 elements,
+ // and 2^lo ~ 1 + lo * P(lo).
+ // Thus, we have:
+ /...
[truncated]
``````````
</details>
https://github.com/llvm/llvm-project/pull/176529
More information about the libc-commits
mailing list