[libc] Implement correctly rounded log2f based on RLIBM library.

Implement log2f based on RLIBM library correctly rounded for all rounding modes. Reviewed By: sivachandra, michaelrj, santoshn, jpl169, zimmermann6 Differential Revision: https://reviews.llvm.org/D115828
llvm · Jan 14, 2022 · 63d2df0 · 63d2df0
1 parent 2a11fbb
commit 63d2df0
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Showing 21 changed files with 528 additions and 47 deletions.
diff --git a/libc/config/linux/aarch64/entrypoints.txt b/libc/config/linux/aarch64/entrypoints.txt
@@ -145,6 +145,7 @@ set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.ldexp
     libc.src.math.ldexpf
     libc.src.math.ldexpl
+    libc.src.math.log2f
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf

diff --git a/libc/config/linux/x86_64/entrypoints.txt b/libc/config/linux/x86_64/entrypoints.txt
@@ -144,6 +144,7 @@ set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.llround
     libc.src.math.llroundf
     libc.src.math.llroundl
+    libc.src.math.log2f
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf

diff --git a/libc/config/windows/entrypoints.txt b/libc/config/windows/entrypoints.txt
@@ -118,6 +118,7 @@ set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.llround
     libc.src.math.llroundf
     libc.src.math.llroundl
+    libc.src.math.log2f
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf

diff --git a/libc/spec/stdc.td b/libc/spec/stdc.td
@@ -396,6 +396,8 @@ def StdC : StandardSpec<"stdc"> {
           FunctionSpec<"ldexpf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<IntType>]>,
           FunctionSpec<"ldexpl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<IntType>]>,
 
+          FunctionSpec<"log2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
+
           FunctionSpec<"logf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"logb", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,

diff --git a/libc/src/__support/FPUtil/PolyEval.h b/libc/src/__support/FPUtil/PolyEval.h
@@ -37,7 +37,9 @@ INLINE_FMA static inline T polyeval(T x, T a0, Ts... a) {
 
 #ifdef LLVM_LIBC_ARCH_X86_64
 
-#include "x86_64/PolyEval.h"
+// [DISABLED] There is a regression with using vectorized version for polyeval
+// compared to the naive Horner's scheme with fma.  Need further investigation
+// #include "x86_64/PolyEval.h"
 
 #endif // LLVM_LIBC_ARCH_X86_64
 

diff --git a/libc/src/math/CMakeLists.txt b/libc/src/math/CMakeLists.txt
@@ -116,6 +116,8 @@ add_math_entrypoint_object(ldexp)
 add_math_entrypoint_object(ldexpf)
 add_math_entrypoint_object(ldexpl)
 
+add_math_entrypoint_object(log2f)
+
 add_math_entrypoint_object(logf)
 
 add_math_entrypoint_object(logb)

diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt
@@ -645,13 +645,38 @@ add_entrypoint_object(
     -O2
 )
 
+add_object_library(
+  common_constants
+  HDRS
+    common_constants.h
+  SRCS
+    common_constants.cpp
+  COMPILE_OPTIONS
+    -Wno-c++17-extensions
+)
+
+add_entrypoint_object(
+  log2f
+  SRCS
+    log2f.cpp
+  HDRS
+    ../log2f.h
+  DEPENDS
+    .common_constants
+    libc.src.__support.FPUtil.fputil
+  COMPILE_OPTIONS
+    -O3
+    -Wno-c++17-extensions
+)
+
 add_entrypoint_object(
   logf
   SRCS
     logf.cpp
   HDRS
     ../logf.h
   DEPENDS
+    .common_constants
     libc.src.__support.FPUtil.fputil
   COMPILE_OPTIONS
     -O3

diff --git a/libc/src/math/generic/common_constants.cpp b/libc/src/math/generic/common_constants.cpp
@@ -0,0 +1,59 @@
+//===-- Common constants for math functions ---------------------*- 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
+//
+//===----------------------------------------------------------------------===//
+
+#include "common_constants.h"
+
+namespace __llvm_libc {
+
+// Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127.
+const double ONE_OVER_F[128] = {
+    0x1.0000000000000p+0, 0x1.fc07f01fc07f0p-1, 0x1.f81f81f81f820p-1,
+    0x1.f44659e4a4271p-1, 0x1.f07c1f07c1f08p-1, 0x1.ecc07b301ecc0p-1,
+    0x1.e9131abf0b767p-1, 0x1.e573ac901e574p-1, 0x1.e1e1e1e1e1e1ep-1,
+    0x1.de5d6e3f8868ap-1, 0x1.dae6076b981dbp-1, 0x1.d77b654b82c34p-1,
+    0x1.d41d41d41d41dp-1, 0x1.d0cb58f6ec074p-1, 0x1.cd85689039b0bp-1,
+    0x1.ca4b3055ee191p-1, 0x1.c71c71c71c71cp-1, 0x1.c3f8f01c3f8f0p-1,
+    0x1.c0e070381c0e0p-1, 0x1.bdd2b899406f7p-1, 0x1.bacf914c1bad0p-1,
+    0x1.b7d6c3dda338bp-1, 0x1.b4e81b4e81b4fp-1, 0x1.b2036406c80d9p-1,
+    0x1.af286bca1af28p-1, 0x1.ac5701ac5701bp-1, 0x1.a98ef606a63bep-1,
+    0x1.a6d01a6d01a6dp-1, 0x1.a41a41a41a41ap-1, 0x1.a16d3f97a4b02p-1,
+    0x1.9ec8e951033d9p-1, 0x1.9c2d14ee4a102p-1, 0x1.999999999999ap-1,
+    0x1.970e4f80cb872p-1, 0x1.948b0fcd6e9e0p-1, 0x1.920fb49d0e229p-1,
+    0x1.8f9c18f9c18fap-1, 0x1.8d3018d3018d3p-1, 0x1.8acb90f6bf3aap-1,
+    0x1.886e5f0abb04ap-1, 0x1.8618618618618p-1, 0x1.83c977ab2beddp-1,
+    0x1.8181818181818p-1, 0x1.7f405fd017f40p-1, 0x1.7d05f417d05f4p-1,
+    0x1.7ad2208e0ecc3p-1, 0x1.78a4c8178a4c8p-1, 0x1.767dce434a9b1p-1,
+    0x1.745d1745d1746p-1, 0x1.724287f46debcp-1, 0x1.702e05c0b8170p-1,
+    0x1.6e1f76b4337c7p-1, 0x1.6c16c16c16c17p-1, 0x1.6a13cd1537290p-1,
+    0x1.6816816816817p-1, 0x1.661ec6a5122f9p-1, 0x1.642c8590b2164p-1,
+    0x1.623fa77016240p-1, 0x1.6058160581606p-1, 0x1.5e75bb8d015e7p-1,
+    0x1.5c9882b931057p-1, 0x1.5ac056b015ac0p-1, 0x1.58ed2308158edp-1,
+    0x1.571ed3c506b3ap-1, 0x1.5555555555555p-1, 0x1.5390948f40febp-1,
+    0x1.51d07eae2f815p-1, 0x1.5015015015015p-1, 0x1.4e5e0a72f0539p-1,
+    0x1.4cab88725af6ep-1, 0x1.4afd6a052bf5bp-1, 0x1.49539e3b2d067p-1,
+    0x1.47ae147ae147bp-1, 0x1.460cbc7f5cf9ap-1, 0x1.446f86562d9fbp-1,
+    0x1.42d6625d51f87p-1, 0x1.4141414141414p-1, 0x1.3fb013fb013fbp-1,
+    0x1.3e22cbce4a902p-1, 0x1.3c995a47babe7p-1, 0x1.3b13b13b13b14p-1,
+    0x1.3991c2c187f63p-1, 0x1.3813813813814p-1, 0x1.3698df3de0748p-1,
+    0x1.3521cfb2b78c1p-1, 0x1.33ae45b57bcb2p-1, 0x1.323e34a2b10bfp-1,
+    0x1.30d190130d190p-1, 0x1.2f684bda12f68p-1, 0x1.2e025c04b8097p-1,
+    0x1.2c9fb4d812ca0p-1, 0x1.2b404ad012b40p-1, 0x1.29e4129e4129ep-1,
+    0x1.288b01288b013p-1, 0x1.27350b8812735p-1, 0x1.25e22708092f1p-1,
+    0x1.2492492492492p-1, 0x1.23456789abcdfp-1, 0x1.21fb78121fb78p-1,
+    0x1.20b470c67c0d9p-1, 0x1.1f7047dc11f70p-1, 0x1.1e2ef3b3fb874p-1,
+    0x1.1cf06ada2811dp-1, 0x1.1bb4a4046ed29p-1, 0x1.1a7b9611a7b96p-1,
+    0x1.19453808ca29cp-1, 0x1.1811811811812p-1, 0x1.16e0689427379p-1,
+    0x1.15b1e5f75270dp-1, 0x1.1485f0e0acd3bp-1, 0x1.135c81135c811p-1,
+    0x1.12358e75d3033p-1, 0x1.1111111111111p-1, 0x1.0fef010fef011p-1,
+    0x1.0ecf56be69c90p-1, 0x1.0db20a88f4696p-1, 0x1.0c9714fbcda3bp-1,
+    0x1.0b7e6ec259dc8p-1, 0x1.0a6810a6810a7p-1, 0x1.0953f39010954p-1,
+    0x1.0842108421084p-1, 0x1.073260a47f7c6p-1, 0x1.0624dd2f1a9fcp-1,
+    0x1.05197f7d73404p-1, 0x1.0410410410410p-1, 0x1.03091b51f5e1ap-1,
+    0x1.0204081020408p-1, 0x1.0101010101010p-1};
+
+} // namespace __llvm_libc
diff --git a/libc/src/math/generic/common_constants.h b/libc/src/math/generic/common_constants.h
@@ -0,0 +1,19 @@
+//===-- Common constants for math functions ---------------------*- 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_MATH_GENERIC_COMMON_CONSTANTS_H
+#define LLVM_LIBC_SRC_MATH_GENERIC_COMMON_CONSTANTS_H
+
+namespace __llvm_libc {
+
+// Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127.
+extern const double ONE_OVER_F[128];
+
+} // namespace __llvm_libc
+
+#endif // LLVM_LIBC_SRC_MATH_GENERIC_COMMON_CONSTANTS_H
diff --git a/libc/src/math/generic/log2f.cpp b/libc/src/math/generic/log2f.cpp
@@ -0,0 +1,154 @@
+//===-- Single-precision log2(x) 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/math/log2f.h"
+#include "common_constants.h" // Lookup table for (1/f)
+#include "src/__support/FPUtil/BasicOperations.h"
+#include "src/__support/FPUtil/FEnvUtils.h"
+#include "src/__support/FPUtil/FMA.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/PolyEval.h"
+#include "src/__support/common.h"
+
+// This is a correctly-rounded algorithm for log2(x) in single precision with
+// round-to-nearest, tie-to-even mode from the RLIBM project at:
+// https://people.cs.rutgers.edu/~sn349/rlibm
+
+// Step 1 - Range reduction:
+//   For x = 2^m * 1.mant, log2(x) = m + log2(1.m)
+//   If x is denormal, we normalize it by multiplying x by 2^23 and subtracting
+//   m by 23.
+
+// Step 2 - Another range reduction:
+//   To compute log(1.mant), let f be the highest 8 bits including the hidden
+// bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the
+// mantissa. Then we have the following approximation formula:
+//   log2(1.mant) = log2(f) + log2(1.mant / f)
+//                = log2(f) + log2(1 + d/f)
+//                ~ log2(f) + P(d/f)
+// since d/f is sufficiently small.
+//   log2(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables.
+
+// Step 3 - Polynomial approximation:
+//   To compute P(d/f), we use a single degree-5 polynomial in double precision
+// which provides correct rounding for all but few exception values.
+//   For more detail about how this polynomial is obtained, please refer to the
+// papers:
+//   Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce
+// Correctly Rounded Results of an Elementary Function for Multiple
+// Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN
+// Symposium on Principles of Programming Languages (POPL-2022), Philadelphia,
+// USA, Jan. 16-22, 2022.
+// https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf
+//   Aanjaneya, M., Lim, J., and Nagarakatte, S., "RLibm-Prog: Progressive
+// Polynomial Approximations for Fast Correctly Rounded Math Libraries",
+// Dept. of Comp. Sci., Rutgets U., Technical Report DCS-TR-758, Nov. 2021.
+// https://arxiv.org/pdf/2111.12852.pdf.
+
+namespace __llvm_libc {
+
+// Lookup table for log2(f) = log2(1 + n*2^(-7)) where n = 0..127.
+static constexpr double LOG2_F[128] = {
+    0x0.0000000000000p+0, 0x1.6fe50b6ef0851p-7, 0x1.6e79685c2d22ap-6,
+    0x1.11cd1d5133413p-5, 0x1.6bad3758efd87p-5, 0x1.c4dfab90aab5fp-5,
+    0x1.0eb389fa29f9bp-4, 0x1.3aa2fdd27f1c3p-4, 0x1.663f6fac91316p-4,
+    0x1.918a16e46335bp-4, 0x1.bc84240adabbap-4, 0x1.e72ec117fa5b2p-4,
+    0x1.08c588cda79e4p-3, 0x1.1dcd197552b7bp-3, 0x1.32ae9e278ae1ap-3,
+    0x1.476a9f983f74dp-3, 0x1.5c01a39fbd688p-3, 0x1.70742d4ef027fp-3,
+    0x1.84c2bd02f03b3p-3, 0x1.98edd077e70dfp-3, 0x1.acf5e2db4ec94p-3,
+    0x1.c0db6cdd94deep-3, 0x1.d49ee4c325970p-3, 0x1.e840be74e6a4dp-3,
+    0x1.fbc16b902680ap-3, 0x1.0790adbb03009p-2, 0x1.11307dad30b76p-2,
+    0x1.1ac05b291f070p-2, 0x1.24407ab0e073ap-2, 0x1.2db10fc4d9aafp-2,
+    0x1.37124cea4cdedp-2, 0x1.406463b1b0449p-2, 0x1.49a784bcd1b8bp-2,
+    0x1.52dbdfc4c96b3p-2, 0x1.5c01a39fbd688p-2, 0x1.6518fe4677ba7p-2,
+    0x1.6e221cd9d0cdep-2, 0x1.771d2ba7efb3cp-2, 0x1.800a563161c54p-2,
+    0x1.88e9c72e0b226p-2, 0x1.91bba891f1709p-2, 0x1.9a802391e232fp-2,
+    0x1.a33760a7f6051p-2, 0x1.abe18797f1f49p-2, 0x1.b47ebf73882a1p-2,
+    0x1.bd0f2e9e79031p-2, 0x1.c592fad295b56p-2, 0x1.ce0a4923a587dp-2,
+    0x1.d6753e032ea0fp-2, 0x1.ded3fd442364cp-2, 0x1.e726aa1e754d2p-2,
+    0x1.ef6d67328e220p-2, 0x1.f7a8568cb06cfp-2, 0x1.ffd799a83ff9bp-2,
+    0x1.03fda8b97997fp-1, 0x1.0809cf27f703dp-1, 0x1.0c10500d63aa6p-1,
+    0x1.10113b153c8eap-1, 0x1.140c9faa1e544p-1, 0x1.18028cf72976ap-1,
+    0x1.1bf311e95d00ep-1, 0x1.1fde3d30e8126p-1, 0x1.23c41d42727c8p-1,
+    0x1.27a4c0585cbf8p-1, 0x1.2b803473f7ad1p-1, 0x1.2f56875eb3f26p-1,
+    0x1.3327c6ab49ca7p-1, 0x1.36f3ffb6d9162p-1, 0x1.3abb3faa02167p-1,
+    0x1.3e7d9379f7016p-1, 0x1.423b07e986aa9p-1, 0x1.45f3a98a20739p-1,
+    0x1.49a784bcd1b8bp-1, 0x1.4d56a5b33cec4p-1, 0x1.510118708a8f9p-1,
+    0x1.54a6e8ca5438ep-1, 0x1.5848226989d34p-1, 0x1.5be4d0cb51435p-1,
+    0x1.5f7cff41e09afp-1, 0x1.6310b8f553048p-1, 0x1.66a008e4788ccp-1,
+    0x1.6a2af9e5a0f0ap-1, 0x1.6db196a76194ap-1, 0x1.7133e9b156c7cp-1,
+    0x1.74b1fd64e0754p-1, 0x1.782bdbfdda657p-1, 0x1.7ba18f93502e4p-1,
+    0x1.7f1322182cf16p-1, 0x1.82809d5be7073p-1, 0x1.85ea0b0b27b26p-1,
+    0x1.894f74b06ef8bp-1, 0x1.8cb0e3b4b3bbep-1, 0x1.900e6160002cdp-1,
+    0x1.9367f6da0ab2fp-1, 0x1.96bdad2acb5f6p-1, 0x1.9a0f8d3b0e050p-1,
+    0x1.9d5d9fd5010b3p-1, 0x1.a0a7eda4c112dp-1, 0x1.a3ee7f38e181fp-1,
+    0x1.a7315d02f20c8p-1, 0x1.aa708f58014d3p-1, 0x1.adac1e711c833p-1,
+    0x1.b0e4126bcc86cp-1, 0x1.b418734a9008cp-1, 0x1.b74948f5532dap-1,
+    0x1.ba769b39e4964p-1, 0x1.bda071cc67e6ep-1, 0x1.c0c6d447c5dd3p-1,
+    0x1.c3e9ca2e1a055p-1, 0x1.c7095ae91e1c7p-1, 0x1.ca258dca93316p-1,
+    0x1.cd3e6a0ca8907p-1, 0x1.d053f6d260896p-1, 0x1.d3663b27f31d5p-1,
+    0x1.d6753e032ea0fp-1, 0x1.d9810643d6615p-1, 0x1.dc899ab3ff56cp-1,
+    0x1.df8f02086af2cp-1, 0x1.e29142e0e0140p-1, 0x1.e59063c8822cep-1,
+    0x1.e88c6b3626a73p-1, 0x1.eb855f8ca88fbp-1, 0x1.ee7b471b3a950p-1,
+    0x1.f16e281db7630p-1, 0x1.f45e08bcf0655p-1, 0x1.f74aef0efafaep-1,
+    0x1.fa34e1177c233p-1, 0x1.fd1be4c7f2af9p-1};
+
+INLINE_FMA
+LLVM_LIBC_FUNCTION(float, log2f, (float x)) {
+  using FPBits = typename fputil::FPBits<float>;
+  FPBits xbits(x);
+  int m = 0;
+
+  // Hard to round value(s).
+  if (FPBits(x).uintval() == 0x3f81d0b5U) {
+    int rounding_mode = fputil::get_round();
+    if (rounding_mode == FE_DOWNWARD || rounding_mode == FE_TOWARDZERO) {
+      return 0x1.4cdc4cp-6f;
+    }
+  }
+
+  // Exceptional inputs.
+  if (xbits.uintval() < FPBits::MIN_NORMAL ||
+      xbits.uintval() > FPBits::MAX_NORMAL) {
+    if (xbits.is_zero()) {
+      return static_cast<float>(FPBits::neg_inf());
+    }
+    if (xbits.get_sign() && !xbits.is_nan()) {
+      return FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
+    }
+    if (xbits.is_inf_or_nan()) {
+      return x;
+    }
+    // Normalize denormal inputs.
+    xbits.val *= 0x1.0p23f;
+    m = -23;
+  }
+
+  m += xbits.get_exponent();
+  // Set bits to 1.m
+  xbits.set_unbiased_exponent(0x7F);
+  // Get the 8 highest bits, use 7 bits (excluding the implicit hidden bit) for
+  // lookup tables.
+  int f_index = xbits.get_mantissa() >> 16;
+
+  FPBits f(xbits.val);
+  // Clear the lowest 16 bits.
+  f.bits &= ~0x0000'FFFF;
+
+  double d = static_cast<float>(xbits) - static_cast<float>(f);
+  d *= ONE_OVER_F[f_index];
+
+  double extra_factor = static_cast<double>(m) + LOG2_F[f_index];
+  double r = __llvm_libc::fputil::polyeval(
+      d, extra_factor, 0x1.71547652bd4fp+0, -0x1.7154769b978c7p-1,
+      0x1.ec71a99e349c8p-2, -0x1.720d90e6aac6cp-2, 0x1.5132da3583dap-2);
+
+  return static_cast<float>(r);
+}
+
+} // namespace __llvm_libc
diff --git a/libc/src/math/generic/logf.cpp b/libc/src/math/generic/logf.cpp
@@ -7,6 +7,7 @@
 //===----------------------------------------------------------------------===//
 
 #include "src/math/logf.h"
+#include "common_constants.h" // Lookup table for (1/f)
 #include "src/__support/FPUtil/BasicOperations.h"
 #include "src/__support/FPUtil/FMA.h"
 #include "src/__support/FPUtil/FPBits.h"
@@ -95,52 +96,6 @@ static constexpr double LOG_F[128] = {
     0x1.58cadb5cd7989p-1, 0x1.5ad404c359f2cp-1, 0x1.5cdb1dc6c1764p-1,
     0x1.5ee02a9241675p-1, 0x1.60e32f44788d8p-1};
 
-// Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127.
-static constexpr double ONE_OVER_F[128] = {
-    0x1.0000000000000p+0, 0x1.fc07f01fc07f0p-1, 0x1.f81f81f81f820p-1,
-    0x1.f44659e4a4271p-1, 0x1.f07c1f07c1f08p-1, 0x1.ecc07b301ecc0p-1,
-    0x1.e9131abf0b767p-1, 0x1.e573ac901e574p-1, 0x1.e1e1e1e1e1e1ep-1,
-    0x1.de5d6e3f8868ap-1, 0x1.dae6076b981dbp-1, 0x1.d77b654b82c34p-1,
-    0x1.d41d41d41d41dp-1, 0x1.d0cb58f6ec074p-1, 0x1.cd85689039b0bp-1,
-    0x1.ca4b3055ee191p-1, 0x1.c71c71c71c71cp-1, 0x1.c3f8f01c3f8f0p-1,
-    0x1.c0e070381c0e0p-1, 0x1.bdd2b899406f7p-1, 0x1.bacf914c1bad0p-1,
-    0x1.b7d6c3dda338bp-1, 0x1.b4e81b4e81b4fp-1, 0x1.b2036406c80d9p-1,
-    0x1.af286bca1af28p-1, 0x1.ac5701ac5701bp-1, 0x1.a98ef606a63bep-1,
-    0x1.a6d01a6d01a6dp-1, 0x1.a41a41a41a41ap-1, 0x1.a16d3f97a4b02p-1,
-    0x1.9ec8e951033d9p-1, 0x1.9c2d14ee4a102p-1, 0x1.999999999999ap-1,
-    0x1.970e4f80cb872p-1, 0x1.948b0fcd6e9e0p-1, 0x1.920fb49d0e229p-1,
-    0x1.8f9c18f9c18fap-1, 0x1.8d3018d3018d3p-1, 0x1.8acb90f6bf3aap-1,
-    0x1.886e5f0abb04ap-1, 0x1.8618618618618p-1, 0x1.83c977ab2beddp-1,
-    0x1.8181818181818p-1, 0x1.7f405fd017f40p-1, 0x1.7d05f417d05f4p-1,
-    0x1.7ad2208e0ecc3p-1, 0x1.78a4c8178a4c8p-1, 0x1.767dce434a9b1p-1,
-    0x1.745d1745d1746p-1, 0x1.724287f46debcp-1, 0x1.702e05c0b8170p-1,
-    0x1.6e1f76b4337c7p-1, 0x1.6c16c16c16c17p-1, 0x1.6a13cd1537290p-1,
-    0x1.6816816816817p-1, 0x1.661ec6a5122f9p-1, 0x1.642c8590b2164p-1,
-    0x1.623fa77016240p-1, 0x1.6058160581606p-1, 0x1.5e75bb8d015e7p-1,
-    0x1.5c9882b931057p-1, 0x1.5ac056b015ac0p-1, 0x1.58ed2308158edp-1,
-    0x1.571ed3c506b3ap-1, 0x1.5555555555555p-1, 0x1.5390948f40febp-1,
-    0x1.51d07eae2f815p-1, 0x1.5015015015015p-1, 0x1.4e5e0a72f0539p-1,
-    0x1.4cab88725af6ep-1, 0x1.4afd6a052bf5bp-1, 0x1.49539e3b2d067p-1,
-    0x1.47ae147ae147bp-1, 0x1.460cbc7f5cf9ap-1, 0x1.446f86562d9fbp-1,
-    0x1.42d6625d51f87p-1, 0x1.4141414141414p-1, 0x1.3fb013fb013fbp-1,
-    0x1.3e22cbce4a902p-1, 0x1.3c995a47babe7p-1, 0x1.3b13b13b13b14p-1,
-    0x1.3991c2c187f63p-1, 0x1.3813813813814p-1, 0x1.3698df3de0748p-1,
-    0x1.3521cfb2b78c1p-1, 0x1.33ae45b57bcb2p-1, 0x1.323e34a2b10bfp-1,
-    0x1.30d190130d190p-1, 0x1.2f684bda12f68p-1, 0x1.2e025c04b8097p-1,
-    0x1.2c9fb4d812ca0p-1, 0x1.2b404ad012b40p-1, 0x1.29e4129e4129ep-1,
-    0x1.288b01288b013p-1, 0x1.27350b8812735p-1, 0x1.25e22708092f1p-1,
-    0x1.2492492492492p-1, 0x1.23456789abcdfp-1, 0x1.21fb78121fb78p-1,
-    0x1.20b470c67c0d9p-1, 0x1.1f7047dc11f70p-1, 0x1.1e2ef3b3fb874p-1,
-    0x1.1cf06ada2811dp-1, 0x1.1bb4a4046ed29p-1, 0x1.1a7b9611a7b96p-1,
-    0x1.19453808ca29cp-1, 0x1.1811811811812p-1, 0x1.16e0689427379p-1,
-    0x1.15b1e5f75270dp-1, 0x1.1485f0e0acd3bp-1, 0x1.135c81135c811p-1,
-    0x1.12358e75d3033p-1, 0x1.1111111111111p-1, 0x1.0fef010fef011p-1,
-    0x1.0ecf56be69c90p-1, 0x1.0db20a88f4696p-1, 0x1.0c9714fbcda3bp-1,
-    0x1.0b7e6ec259dc8p-1, 0x1.0a6810a6810a7p-1, 0x1.0953f39010954p-1,
-    0x1.0842108421084p-1, 0x1.073260a47f7c6p-1, 0x1.0624dd2f1a9fcp-1,
-    0x1.05197f7d73404p-1, 0x1.0410410410410p-1, 0x1.03091b51f5e1ap-1,
-    0x1.0204081020408p-1, 0x1.0101010101010p-1};
-
 INLINE_FMA
 LLVM_LIBC_FUNCTION(float, logf, (float x)) {
   constexpr double LOG_2 = 0x1.62e42fefa39efp-1;