[libc][math] Make log10 correctly rounded for non-FMA targets and improve itsperformance.

[lntue]

Summary

Make log10 correctly rounded for non-FMA targets and improve its
performance.

Implemented fast pass and accurate pass:

Fast Pass:

• Range reduction step 0: Extract exponent and mantissa

x = 2^(e_x) * m_x

• Range reduction step 1: Use lookup tables of size 2^7 = 128 to reduce the argument to:

 -2^-8 <= v = r * m_x - 1 < 2^-7
where r = 2^-8 * ceil( 2^8 * (1 - 2^-8) / (1 + k * 2^-7) )
and k = trunc( (m_x - 1) * 2^7 )

• Polynomial approximation: approximate log(1 + v) by a degree-7 polynomial generated by Sollya with:

> P = fpminimax((log(1 + x) - x)/x^2, 5, [|D...|], [-2^-8, 2^-7]);

• Combine the results:

log10(x) ~ ( e_x * log(2) - log(r) + v + v^2 * P(v) ) * log10(e)

• Perform additive Ziv's test with errors bounded by P_ERR * v^2. Return the result if Ziv's test passed.

Accurate Pass:

• Take e_x, v, and the lookup table index from the range reduction step of fast pass.
• Perform 3 more range reduction steps:
  • Range reduction step 2: Use look-up tables of size 193 to reduce the argument to [-0x1.3ffcp-15, 0x1.3e3dp-15]

 v2 = r2 * (1 + v) - 1 = (1 + s2) * (1 + v) - 1 = s2 + v + s2 * v
where r2 = 2^-16 * round ( 2^16 / (1 + k * 2^-14) )
and k = trunc( v * 2^14 + 0.5 ).

• Range reduction step 3: Use look-up tables of size 161 to reduce the argument to [-0x1.01928p-22 , 0x1p-22]

 v3 = r3 * (1 + v2) - 1 = (1 + s3) * (1 + v2) - 1 = s3 + v2 + s3 * v2
where r3 = 2^-21 * round ( 2^21 / (1 + k * 2^-21) )
and k = trunc( v * 2^21 + 0.5 ).

• Range reduction step 4: Use look-up tables of size 130 to reduce the argument to [-0x1.0002143p-29 , 0x1p-29]

 v4 = r4 * (1 + v3) - 1 = (1 + s4) * (1 + v3) - 1 = s4 + v3 + s4 * v3
where r4 = 2^-28 * round ( 2^28 / (1 + k * 2^-28) )
and k = trunc( v * 2^28 + 0.5 ).

• Polynomial approximation: approximate log10(1 + v4) by a degree-4 minimax polynomial generated by Sollya with:

> P = fpminimax(log10(1 + x)/x, 3, [|128...|], [-0x1.0002143p-29 , 0x1p-29]);

• Combine the results:

log10(x) ~ e_x * log10(2) - log10(r) - log10(r2) - log10(r3) - log10(r4) + v * P(v)

• The combined results are computed using floating points of 128-bit precision.

Performance

• For 0.5 <= x <= 2, the fast pass hitting rate is about 99.92%.
• Reciprocal throughput from CORE-MATH's perf tool on Ryzen 5900X:

$ ./perf.sh log10
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 20.402 + 0.589 clc/call; Median-Min = 0.277 clc/call; Max = 22.752 clc/call;

-- CORE-MATH reciprocal throughput -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 75.797 + 3.317 clc/call; Median-Min = 3.407 clc/call; Max = 79.371 clc/call;

-- System LIBC reciprocal throughput --
[####################] 100 %
Ntrial = 20 ; Min = 22.668 + 0.184 clc/call; Median-Min = 0.181 clc/call; Max = 23.205 clc/call;

-- LIBC reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 25.977 + 0.183 clc/call; Median-Min = 0.138 clc/call; Max = 26.283 clc/call;

-- LIBC reciprocal throughput -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 22.140 + 0.980 clc/call; Median-Min = 0.853 clc/call; Max = 23.790 clc/call;

• Latency from CORE-MATH's perf tool on Ryzen 5900X:

$ ./perf.sh log10 --latency
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 54.613 + 0.357 clc/call; Median-Min = 0.287 clc/call; Max = 55.701 clc/call;

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 79.681 + 0.482 clc/call; Median-Min = 0.294 clc/call; Max = 81.604 clc/call;

-- System LIBC latency --
[####################] 100 %
Ntrial = 20 ; Min = 61.532 + 0.208 clc/call; Median-Min = 0.199 clc/call; Max = 62.256 clc/call;

-- LIBC latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 41.510 + 0.205 clc/call; Median-Min = 0.244 clc/call; Max = 41.867 clc/call;

-- LIBC latency -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 55.669 + 0.240 clc/call; Median-Min = 0.280 clc/call; Max = 56.056 clc/call;

• Accurate pass latency:

$ ./perf.sh log10 --latency --simple_stat
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH latency -- with FMA
640.688

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
667.354

-- LIBC latency -- with FMA
495.593

-- LIBC latency -- without FMA
504.143

[lntue]

Fix log10(1.0) = +0.0 for all rounding mode.

[lntue]

Update error bounds of the fast pass and sync to HEAD.

[lntue]

Sync to HEAD.

[zimmermann6]

all worst cases from core-math do pass. However I get a larger reciprocal throughput on a AMD EPYC 7282:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 21.270 + 0.322 clc/call; Median-Min = 0.302 clc/call; Max = 23.839 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 26.172 + 0.346 clc/call; Median-Min = 0.310 clc/call; Max = 26.798 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 49.688 + 0.306 clc/call; Median-Min = 0.308 clc/call; Max = 50.288 clc/call;

For the latency I get similar values:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 65.915 + 0.316 clc/call; Median-Min = 0.282 clc/call; Max = 66.518 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 73.612 + 0.389 clc/call; Median-Min = 0.326 clc/call; Max = 74.422 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 66.387 + 0.276 clc/call; Median-Min = 0.303 clc/call; Max = 67.140 clc/call;

libc/src/__support/FPUtil/double_double.h

 //===-- Utilities for double-double data type. ------------------*- 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_FPUTIL_DOUBLEDOUBLE_H
 #define LLVM_LIBC_SRC_SUPPORT_FPUTIL_DOUBLEDOUBLE_H
 
 #include "multiply_add.h"
 #include "src/__support/common.h"
+#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
 #include "src/__support/number_pair.h"
 
 namespace __llvm_libc::fputil {
 
 using DoubleDouble = __llvm_libc::NumberPair<double>;
 
-// Assumption: |a| >= |b|
+// The output of Dekker's FastTwoSum algorithm is correct, i.e.:
+//   r.hi + r.lo = a + b exactly
+//   and |r.lo| < eps(r.lo)
+// if ssumption: |a| >= |b|, or a = 0.
 LIBC_INLINE constexpr DoubleDouble exact_add(double a, double b) {
   DoubleDouble r{0.0, 0.0};
   r.hi = a + b;
   double t = r.hi - a;
   r.lo = b - t;
   return r;
 }
 
 // Assumption: |a.hi| >= |b.hi|
 LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, DoubleDouble b) {
   DoubleDouble r = exact_add(a.hi, b.hi);
   double lo = a.lo + b.lo;
   return exact_add(r.hi, r.lo + lo);
 }
 
 // Assumption: |a.hi| >= |b|
 LIBC_INLINE constexpr DoubleDouble add(DoubleDouble a, double b) {
   DoubleDouble r = exact_add(a.hi, b);
   return exact_add(r.hi, r.lo + a.lo);
 }
 
-// TODO(lntue): add a correct multiplication when FMA instructions are not
-// available.
+// Velkamp's Splitting for double precision.
+LIBC_INLINE constexpr DoubleDouble split(double a) {
+  DoubleDouble r{0.0, 0.0};
+  // Splitting constant = 2^ceil(prec(double)/2) + 1 = 2^27 + 1.
+  constexpr double C = 0x1.0p27 + 1.0;
+  double t1 = C * a;
+  double t2 = a - t1;
+  r.hi = t1 + t2;
+  r.lo = a - r.hi;
+  return r;
+}
+
 LIBC_INLINE DoubleDouble exact_mult(double a, double b) {
   DoubleDouble r{0.0, 0.0};
+
+#ifdef LIBC_TARGET_CPU_HAS_FMA
   r.hi = a * b;
   r.lo = fputil::multiply_add(a, b, -r.hi);
+#else
+  // Dekker's Product.
+  DoubleDouble as = split(a);
+  DoubleDouble bs = split(b);
+  r.hi = a * b;
+  double t1 = as.hi * bs.hi - r.hi;
+  double t2 = as.hi * bs.lo + t1;
+  double t3 = as.lo * bs.hi + t2;
+  r.lo = as.lo * bs.lo + t3;
+#endif // LIBC_TARGET_CPU_HAS_FMA
+
+  return r;
+}
+
+LIBC_INLINE DoubleDouble quick_mult(DoubleDouble a, DoubleDouble b) {
+  DoubleDouble r = exact_mult(a.hi, b.hi);
+  double t1 = fputil::multiply_add(a.hi, b.lo, r.lo);
+  double t2 = fputil::multiply_add(a.lo, b.hi, t1);
+  r.lo = t2;
   return r;
 }
 
 } // namespace __llvm_libc::fputil
 
 #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_DOUBLEDOUBLE_H

libc/src/math/generic/CMakeLists.txt

 )
 
 add_object_library(
   common_constants
   HDRS
     common_constants.h
   SRCS
     common_constants.cpp
+  DEPENDS
+    libc.src.__support.number_pair
 )
 
-# TODO(lntue): Make log10 correctly rounded for non-FMA targets.
 add_entrypoint_object(
   log10
   SRCS
     log10.cpp
   HDRS
     ../log10.h
   DEPENDS
+    .common_constants
     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.double_double
     libc.src.__support.FPUtil.dyadic_float
     libc.src.__support.macros.optimization
   COMPILE_OPTIONS
     -O3
 )
 
 add_entrypoint_object(

libc/src/math/generic/common_constants.h

 //===-- 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
 
+#include "src/__support/number_pair.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];
 
 // Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127.
 extern const double LOG_F[128];
 
 // Lookup table for range reduction constants r for logarithms.
 extern const float R[128];
 
 // Lookup table for range reduction constants r for logarithms.
 extern const double RD[128];
 
+// Lookup table for compensated constants for exact range reduction when FMA
+// instructions are not available.
+extern const double CD[128];
+
 // Lookup table for -log(r)
 extern const double LOG_R[128];
+extern const NumberPair<double> LOG_R_DD[128];
+
+// 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};
+
+// 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.
+constexpr double LOG_2_HI = 0x1.62e42fefa38p-1; // LSB = 2^-43
+// > b = round(log10(2) - a, D, RN);
+constexpr double LOG_2_LO = 0x1.ef35793c7673p-45; // LSB = 2^-97
 
 // Lookup table for exp(m) with m = -104, ..., 89.
 //   -104 = floor(log(single precision's min denormal))
 //     89 = ceil(log(single precision's max normal))
 // Table is generated with Sollya as follow:
 // > display = hexadecimal;
 // > for i from -104 to 89 do { D(exp(i)); };
 extern const double EXP_M1[195];

libc/src/math/generic/common_constants.cpp

 //===-- 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"
+#include "src/__support/number_pair.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,
alignas(64) const double RD[128] = {
     0x1.34p-1, 0x1.34p-1, 0x1.32p-1, 0x1.3p-1,  0x1.3p-1,  0x1.2ep-1, 0x1.2cp-1,
     0x1.2cp-1, 0x1.2ap-1, 0x1.28p-1, 0x1.28p-1, 0x1.26p-1, 0x1.24p-1, 0x1.24p-1,
     0x1.22p-1, 0x1.2p-1,  0x1.2p-1,  0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1,
     0x1.1ap-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.14p-1, 0x1.12p-1,
     0x1.1p-1,  0x1.1p-1,  0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0cp-1, 0x1.0ap-1,
     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};
 
+// Compensated constants for exact logarithm range reduction when FMA is not
+// available.
+// Generated by Sollya with the formula: CD[i] = RD[i]*(1 + i*2^-7) - 1
+// for RD[i] defined on the table above.
+alignas(64) const 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,
+    -0x1p-9,    0x1.ap-12,  -0x1.ep-10, 0x1.8p-12,   -0x1.1p-9,  -0x1p-15,
+    -0x1.6p-9,  -0x1.ap-11, -0x1.ep-9,  -0x1.f8p-10, -0x1p-12,   -0x1.cp-9,
+    -0x1.fp-10, -0x1.cp-12, -0x1p-8,    -0x1.54p-9,  -0x1.6p-10, -0x1.4p-13,
+    0x1p-10,    -0x1.88p-9, -0x1.08p-9, -0x1.2p-10,  -0x1p-12,   0x1.2p-11,
+    -0x1p-8,    -0x1.acp-9, -0x1.6p-9,  -0x1.1cp-9,  -0x1.cp-10, -0x1.58p-10,
+    -0x1p-10,   -0x1.7p-11, -0x1p-11,   -0x1.6p-12,  -0x1p-12,   -0x1.cp-13,
+    -0x1p-12,   -0x1.6p-12, -0x1p-11,   -0x1.7p-11,  -0x1p-10,   -0x1.58p-10,
+    -0x1.cp-10, -0x1.1cp-9, -0x1.6p-9,  -0x1.acp-9,  -0x1p-8,    0x1.5p-10,
+    0x1.2p-11,  -0x1p-12,   -0x1.2p-10, -0x1.08p-9,  -0x1.88p-9, 0x1.0cp-9,
+    0x1p-10,    -0x1.4p-13, -0x1.6p-10, -0x1.54p-9,  -0x1p-8,    0x1p-10,
+    -0x1.cp-12, -0x1.fp-10, -0x1.cp-9,  0x1.68p-10,  -0x1p-12,   -0x1.f8p-10,
+    -0x1.ep-9,  0x1.1p-10,  -0x1.ap-11, -0x1.6p-9,   0x1p-9,     -0x1p-15,
+    -0x1.1p-9,  0x1.48p-9,  0x1.8p-12,  -0x1.ep-10,  0x1.6p-9,   0x1.ap-12,
+    -0x1p-9,    0x1.48p-9,  0x1p-14,    -0x1.4p-9,   0x1p-9,     -0x1.5p-11,
+    -0x1.bp-9,  0x1.1p-10,  -0x1.cp-10, 0x1.54p-9,   -0x1p-12,   -0x1.9cp-9,
+    0x1.2p-10,  -0x1.fp-10, 0x1.3p-9,   -0x1.9p-11,  -0x1p-8,    0x1p-12,
+    -0x1.88p-9, 0x1.28p-10, -0x1.2p-9,  0x1.fp-10,   -0x1.9p-10, 0x1.4cp-9,
+    -0x1p-10,   0x1.9p-9,   -0x1.2p-11, 0x1.c4p-9,   -0x1p-12,   0x1.e8p-9,
+    -0x1p-14,   -0x1p-8,
+};
+
 alignas(64) const 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,
     0x1.1973bd1465567p-4, 0x1.3bdf5a7d1ee64p-4, 0x1.5e95a4d9791cbp-4,
     0x1.700d30aeac0e1p-4, 0x1.9335e5d594989p-4, 0x1.b6ac88dad5b1cp-4,
     0x1.c885801bc4b23p-4, 0x1.ec739830a112p-4,  0x1.fe89139dbd566p-4,
     0x1.1178e8227e47cp-3, 0x1.1aa2b7e23f72ap-3, 0x1.2d1610c86813ap-3,
alignas(64) const double LOG_R[128] = {
     0x1.38ae2171976e7p-1, 0x1.3c6080c36bfb5p-1, 0x1.3c6080c36bfb5p-1,
     0x1.4019c2125ca93p-1, 0x1.43d9ff2f923c5p-1, 0x1.43d9ff2f923c5p-1,
     0x1.47a1527e8a2d3p-1, 0x1.47a1527e8a2d3p-1, 0x1.4b6fd6f970c1fp-1,
     0x1.4b6fd6f970c1fp-1, 0x1.4f45a835a4e19p-1, 0x1.4f45a835a4e19p-1,
     0x1.5322e26867857p-1, 0x1.5322e26867857p-1, 0x1.5707a26bb8c66p-1,
     0x1.5707a26bb8c66p-1, 0x1.5af405c3649ep-1,  0x1.5af405c3649ep-1,
     0x1.5ee82aa24192p-1,  0x0.000000000000p0};
 
+// 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(log(r)*2^43) * 2^-43;
+//     c = round(log(r) - b, D, RN);
+//     print("{", -c, ",", -b, "},");
+//   };
+// We replace LOG_R[0] with log10(1.0) == 0.0
+alignas(64) const NumberPair<double> LOG_R_DD[128] = {
+    {0.0, 0.0},
+    {-0x1.0c76b999d2be8p-46, 0x1.010157589p-7},
+    {-0x1.3dc5b06e2f7d2p-45, 0x1.0205658938p-6},
+    {-0x1.aa0ba325a0c34p-45, 0x1.8492528c9p-6},
+    {0x1.111c05cf1d753p-47, 0x1.0415d89e74p-5},
+    {-0x1.c167375bdfd28p-45, 0x1.466aed42ep-5},
+    {-0x1.97995d05a267dp-46, 0x1.894aa149fcp-5},
+    {-0x1.a68f247d82807p-46, 0x1.ccb73cdddcp-5},
+    {-0x1.7e5dd7009902cp-46, 0x1.08598b59e4p-4},
+    {-0x1.5325d560d9e9bp-45, 0x1.1973bd1466p-4},
+    {0x1.cc85ea5db4ed7p-45, 0x1.3bdf5a7d1ep-4},
+    {-0x1.c69063c5d1d1ep-45, 0x1.5e95a4d97ap-4},
+    {0x1.c1e8da99ded32p-49, 0x1.700d30aeacp-4},
+    {0x1.3115c3abd47dap-45, 0x1.9335e5d594p-4},
+    {-0x1.390802bf768e5p-46, 0x1.b6ac88dad6p-4},
+    {0x1.646d1c65aacd3p-45, 0x1.c885801bc4p-4},
+    {-0x1.dc068afe645ep-45, 0x1.ec739830a2p-4},
+    {-0x1.534d64fa10afdp-45, 0x1.fe89139dbep-4},
+    {0x1.1ef78ce2d07f2p-45, 0x1.1178e8227ep-3},
+    {0x1.ca78e44389934p-45, 0x1.1aa2b7e23fp-3},
+    {0x1.39d6ccb81b4a1p-47, 0x1.2d1610c868p-3},
+    {0x1.62fa8234b7289p-51, 0x1.365fcb0159p-3},
+    {0x1.5837954fdb678p-45, 0x1.4913d8333bp-3},
+    {0x1.633e8e5697dc7p-45, 0x1.527e5e4a1bp-3},
+    {0x1.9cf8b2c3c2e78p-46, 0x1.6574ebe8c1p-3},
+    {-0x1.5118de59c21e1p-45, 0x1.6f0128b757p-3},
+    {0x1.e0ddb9a631e83p-46, 0x1.823c16551ap-3},
+    {-0x1.73d54aae92cd1p-47, 0x1.8beafeb39p-3},
+    {0x1.7f22858a0ff6fp-47, 0x1.95a5adcf7p-3},
+    {-0x1.8724350562169p-45, 0x1.a93ed3c8aep-3},
+    {-0x1.c358d4eace1aap-47, 0x1.b31d8575bdp-3},
+    {-0x1.d4bc4595412b6p-45, 0x1.bd087383bep-3},
+    {-0x1.84a7e75b6f6e4p-47, 0x1.d1037f2656p-3},
+    {-0x1.aff2af715b035p-45, 0x1.db13db0d49p-3},
+    {0x1.212276041f43p-51, 0x1.e530effe71p-3},
+    {-0x1.a211565bb8e11p-51, 0x1.ef5ade4ddp-3},
+    {0x1.bcbecca0cdf3p-46, 0x1.f991c6cb3bp-3},
+    {0x1.89cdb16ed4e91p-48, 0x1.07138604d58p-2},
+    {0x1.7188b163ceae9p-45, 0x1.0c42d67616p-2},
+    {-0x1.c210e63a5f01cp-45, 0x1.1178e8227e8p-2},
+    {0x1.b9acdf7a51681p-45, 0x1.16b5ccbacf8p-2},
+    {0x1.ca6ed5147bdb7p-45, 0x1.1bf99635a68p-2},
+    {0x1.c93c1df5bb3b6p-45, 0x1.269621134d8p-2},
+    {0x1.a9cfa4a5004f4p-45, 0x1.2bef07cdc9p-2},
+    {-0x1.8e27ad3213cb8p-45, 0x1.314f1e1d36p-2},
+    {0x1.16ecdb0f177c8p-46, 0x1.36b6776be1p-2},
+    {0x1.83b54b606bd5cp-46, 0x1.3c25277333p-2},
+    {0x1.8e436ec90e09dp-47, 0x1.419b423d5e8p-2},
+    {-0x1.f27ce0967d675p-45, 0x1.4718dc271c8p-2},
+    {-0x1.e20891b0ad8a4p-45, 0x1.4c9e09e173p-2},
+    {0x1.ebe708164c759p-45, 0x1.522ae0738ap-2},
+    {0x1.fadedee5d40efp-46, 0x1.57bf753c8dp-2},
+    {-0x1.a0b2a08a465dcp-47, 0x1.5d5bddf596p-2},
+    {-0x1.db623e731aep-45, 0x1.630030b3abp-2},
+    {0x1.0a0d32756ebap-45, 0x1.68ac83e9c68p-2},
+    {0x1.721657c222d87p-46, 0x1.6e60ee6af18p-2},
+    {0x1.d8b0949dc60b3p-45, 0x1.741d876c678p-2},
+    {0x1.9ec7d2efd1778p-45, 0x1.79e26687cf8p-2},
+    {-0x1.72090c812566ap-45, 0x1.7fafa3bd818p-2},
+    {0x1.fd56f3333778ap-45, 0x1.85855776dc8p-2},
+    {-0x1.05ae1e5e7047p-45, 0x1.8b639a88b3p-2},
+    {-0x1.766b52ee6307dp-46, 0x1.914a8635bf8p-2},
+    {-0x1.52313a502d9fp-46, 0x1.973a3431358p-2},
+    {-0x1.6279e10d0c0bp-45, 0x1.9d32bea15fp-2},
+    {0x1.3c6457f9d79f5p-45, 0x1.a33440224f8p-2},
+    {0x1.3c6457f9d79f5p-45, 0x1.a33440224f8p-2},
+    {0x1.e36f2bea77a5dp-46, 0x1.a93ed3c8ad8p-2},
+    {-0x1.17cc552774458p-45, 0x1.af5295248dp-2},
+    {0x1.095252d841995p-46, 0x1.b56fa044628p-2},
+    {0x1.7d85bf40a666dp-45, 0x1.bb9611b80ep-2},
+    {0x1.cec807fe8e18p-45, 0x1.c1c60693fap-2},
+    {0x1.cec807fe8e18p-45, 0x1.c1c60693fap-2},
+    {-0x1.9b6ddc15249aep-45, 0x1.c7ff9c74558p-2},
+    {-0x1.797c33ec7a6bp-47, 0x1.ce42f180648p-2},
+    {0x1.35bafe9a767a8p-45, 0x1.d490246def8p-2},
+    {-0x1.ea42d60dc616ap-46, 0x1.dae75484c98p-2},
+    {-0x1.326b207322938p-46, 0x1.e148a1a2728p-2},
+    {-0x1.326b207322938p-46, 0x1.e148a1a2728p-2},
+    {-0x1.465505372bd08p-45, 0x1.e7b42c3ddbp-2},
+    {0x1.f27f45a470251p-45, 0x1.ee2a156b41p-2},
+    {0x1.2cde56f014a8bp-46, 0x1.f4aa7ee0318p-2},
+    {0x1.2cde56f014a8bp-46, 0x1.f4aa7ee0318p-2},
+    {0x1.085fa3c164935p-47, 0x1.fb358af7a48p-2},
+    {-0x1.53ba3b1727b1cp-47, 0x1.00e5ae5b208p-1},
+    {-0x1.4c45fe79539ep-47, 0x1.04360be7604p-1},
+    {-0x1.4c45fe79539ep-47, 0x1.04360be7604p-1},
+    {0x1.6812241edf5fdp-45, 0x1.078bf0533c4p-1},
+    {0x1.f486b887e7e27p-46, 0x1.0ae76e2d054p-1},
+    {0x1.f486b887e7e27p-46, 0x1.0ae76e2d054p-1},
+    {0x1.c299807801742p-46, 0x1.0e4898611ccp-1},
+    {-0x1.58647bb9ddcb2p-45, 0x1.11af823c75cp-1},
+    {-0x1.58647bb9ddcb2p-45, 0x1.11af823c75cp-1},
+    {-0x1.edd97a293ae49p-45, 0x1.151c3f6f298p-1},
+    {0x1.4cc4ef8ab465p-46, 0x1.188ee40f23cp-1},
+    {0x1.4cc4ef8ab465p-46, 0x1.188ee40f23cp-1},
+    {0x1.cacdeed70e667p-51, 0x1.1c07849ae6p-1},
+    {-0x1.a7242c9fe81d3p-45, 0x1.1f8635fc618p-1},
+    {-0x1.a7242c9fe81d3p-45, 0x1.1f8635fc618p-1},
+    {0x1.2fc066e48667bp-46, 0x1.230b0d8bebcp-1},
+    {-0x1.b61f10522625p-47, 0x1.269621134dcp-1},
+    {-0x1.b61f10522625p-47, 0x1.269621134dcp-1},
+    {0x1.06d2be797882dp-45, 0x1.2a2786d0ecp-1},
+    {-0x1.7a6e507b9dc11p-46, 0x1.2dbf557b0ep-1},
+    {-0x1.7a6e507b9dc11p-46, 0x1.2dbf557b0ep-1},
+    {-0x1.74e93c5a0ed9cp-45, 0x1.315da443408p-1},
+    {-0x1.74e93c5a0ed9cp-45, 0x1.315da443408p-1},
+    {0x1.0b83f9527e6acp-46, 0x1.35028ad9d8cp-1},
+    {-0x1.18b7abb5569a4p-45, 0x1.38ae2171978p-1},
+    {-0x1.18b7abb5569a4p-45, 0x1.38ae2171978p-1},
+    {-0x1.2b7367cfe13c2p-47, 0x1.3c6080c36cp-1},
+    {-0x1.2b7367cfe13c2p-47, 0x1.3c6080c36cp-1},
+    {-0x1.6ce7930f0c74cp-45, 0x1.4019c2125ccp-1},
+    {-0x1.d984f481051f7p-48, 0x1.43d9ff2f924p-1},
+    {-0x1.d984f481051f7p-48, 0x1.43d9ff2f924p-1},
+    {-0x1.2cb6af94d60aap-45, 0x1.47a1527e8a4p-1},
+    {-0x1.2cb6af94d60aap-45, 0x1.47a1527e8a4p-1},
+    {0x1.f7115ed4c541cp-49, 0x1.4b6fd6f970cp-1},
+    {0x1.f7115ed4c541cp-49, 0x1.4b6fd6f970cp-1},
+    {-0x1.e6c516d93b8fbp-45, 0x1.4f45a835a5p-1},
+    {-0x1.e6c516d93b8fbp-45, 0x1.4f45a835a5p-1},
+    {0x1.5ccc45d257531p-47, 0x1.5322e268678p-1},
+    {0x1.5ccc45d257531p-47, 0x1.5322e268678p-1},
+    {0x1.9980bff3303ddp-47, 0x1.5707a26bb8cp-1},
+    {0x1.9980bff3303ddp-47, 0x1.5707a26bb8cp-1},
+    {0x1.dfa63ac10c9fbp-45, 0x1.5af405c3648p-1},
+    {0x1.dfa63ac10c9fbp-45, 0x1.5af405c3648p-1},
+    {0x1.202380cda46bep-45, 0x1.5ee82aa2418p-1},
+    {0.0, 0.0},
+};
+
 // Lookup table for exp(m) with m = -104, ..., 89.
 //   -104 = floor(log(single precision's min denormal))
 //     89 = ceil(log(single precision's max normal))
 // Table is generated with Sollya as follow:
 // > display = hexadecimal;
 // > for i from -104 to 89 do { D(exp(i)); };
 const double EXP_M1[195] = {
     0x1.f1e6b68529e33p-151, 0x1.525be4e4e601dp-149, 0x1.cbe0a45f75eb1p-148,

libc/src/math/generic/log10.cpp

 //===-- Double-precision log10(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/log10.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/common.h"
 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
 
+#include "common_constants.h"
+
 namespace __llvm_libc {
 
-// 192-bit precision dyadic floating point numbers.
-using Float192 = typename fputil::DyadicFloat<192>;
-using MType = typename Float192::MantissaType;
+// 128-bit precision dyadic floating point numbers.
+using Float128 = typename fputil::DyadicFloat<128>;
+using MType = typename Float128::MantissaType;
 
 namespace {
 
-// log10(2) generated by Sollya with:
-// > a = round(log10(2), 43, RN);
-constexpr double LOG10_2_HI = 0x1.34413509f78p-2; // LSB = 2^-43
-// > b = round(log10(2) - a, D, RN);
-constexpr double LOG10_2_LO = 0x1.fef311f12b358p-46; // LSB = 2^-98
-constexpr double LOG10_2_ULP[2] = {0x1.0p-97, 0.0};
-
-// Generated by Sollya with:
-// > for i from 0 to 127 do {
-//     r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
-//                            ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
-//     print(r, ",");
-//   };
-// To improve the accuracy with inputs close to 1, we replace R[0] with 1.0.
-constexpr double R[128] = {
-    1.0,       0x1.fap-1, 0x1.f6p-1, 0x1.f2p-1, 0x1.eep-1, 0x1.eap-1, 0x1.e8p-1,
-    0x1.e4p-1, 0x1.ep-1,  0x1.dcp-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d2p-1, 0x1.dp-1,
-    0x1.ccp-1, 0x1.c8p-1, 0x1.c6p-1, 0x1.c2p-1, 0x1.cp-1,  0x1.bcp-1, 0x1.bap-1,
-    0x1.b6p-1, 0x1.b4p-1, 0x1.bp-1,  0x1.aep-1, 0x1.aap-1, 0x1.a8p-1, 0x1.a6p-1,
-    0x1.a2p-1, 0x1.ap-1,  0x1.9ep-1, 0x1.9ap-1, 0x1.98p-1, 0x1.96p-1, 0x1.94p-1,
-    0x1.9p-1,  0x1.8ep-1, 0x1.8cp-1, 0x1.8ap-1, 0x1.88p-1, 0x1.84p-1, 0x1.82p-1,
-    0x1.8p-1,  0x1.7ep-1, 0x1.7cp-1, 0x1.7ap-1, 0x1.78p-1, 0x1.76p-1, 0x1.74p-1,
-    0x1.72p-1, 0x1.7p-1,  0x1.6ep-1, 0x1.6cp-1, 0x1.6ap-1, 0x1.68p-1, 0x1.66p-1,
-    0x1.64p-1, 0x1.62p-1, 0x1.6p-1,  0x1.5ep-1, 0x1.5cp-1, 0x1.5ap-1, 0x1.58p-1,
-    0x1.56p-1, 0x1.54p-1, 0x1.52p-1, 0x1.5p-1,  0x1.5p-1,  0x1.4ep-1, 0x1.4cp-1,
-    0x1.4ap-1, 0x1.48p-1, 0x1.46p-1, 0x1.46p-1, 0x1.44p-1, 0x1.42p-1, 0x1.4p-1,
-    0x1.3ep-1, 0x1.3ep-1, 0x1.3cp-1, 0x1.3ap-1, 0x1.38p-1, 0x1.38p-1, 0x1.36p-1,
-    0x1.34p-1, 0x1.32p-1, 0x1.32p-1, 0x1.3p-1,  0x1.2ep-1, 0x1.2ep-1, 0x1.2cp-1,
-    0x1.2ap-1, 0x1.2ap-1, 0x1.28p-1, 0x1.26p-1, 0x1.26p-1, 0x1.24p-1, 0x1.22p-1,
-    0x1.22p-1, 0x1.2p-1,  0x1.1ep-1, 0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1,
-    0x1.18p-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.12p-1, 0x1.12p-1,
-    0x1.1p-1,  0x1.1p-1,  0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0ap-1, 0x1.0ap-1,
-    0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1, 0x1.02p-1,
-    0x1.02p-1, 0x1p-1,
-};
+constexpr fputil::DoubleDouble LOG10_E = {0x1.95355baaafad3p-57,
+                                          0x1.bcb7b1526e50ep-2};
 
-// Generated by Sollya with:
-// for i from 0 to 127 do {
-//     r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
-//                           ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
-//     b = nearestint(log10(r)*2^43) * 2^-43;
-//     c = round(log10(r) - b, D, RN);
-//     print("{", -c, ",", -b, "},");
-//   };
-// We replace LOG10_R[0] with log10(1.0) == 0.0
-constexpr fputil::DoubleDouble LOG10_R[128] = {
-    {0.0, 0.0},
-    {-0x1.dfa6d47e47379p-45, 0x1.4f8205236p-8},
-    {-0x1.11bc6f2b9a3acp-45, 0x1.18b2dc8d3p-7},
-    {-0x1.685fc114e61bfp-46, 0x1.8a8c06bb2p-7},
-    {0x1.3c757d5b7376ap-45, 0x1.fd503c39p-7},
-    {0x1.f3cbd9c5111b1p-47, 0x1.3881a7b818p-6},
-    {-0x1.e22fab794a816p-45, 0x1.559bd2407p-6},
-    {0x1.421bab5f034a4p-45, 0x1.902c31d628p-6},
-    {0x1.fedb4b594a31bp-45, 0x1.cb38fccd88p-6},
-    {-0x1.c4e824b246c7fp-47, 0x1.0362241e64p-5},
-    {-0x1.df23ed2ebe477p-45, 0x1.125d0432ecp-5},
-    {-0x1.00c12f7a1b586p-47, 0x1.30838cdc3p-5},
-    {0x1.e5ff3439d368dp-46, 0x1.4eec0e2458p-5},
-    {0x1.2951bb9cd2fb7p-45, 0x1.5e3966b7e8p-5},
-    {0x1.fae96a708581ep-46, 0x1.7d070145f4p-5},
-    {0x1.badcf3d6e4566p-46, 0x1.9c197abfp-5},
-    {-0x1.b0197d2cb982ep-48, 0x1.abbcebd85p-5},
-    {-0x1.24b4a6b5ce4d4p-52, 0x1.cb38fccd8cp-5},
-    {-0x1.40bcd23c3e44cp-45, 0x1.db11ed766cp-5},
-    {-0x1.024e9d08ce301p-45, 0x1.fafa6d398p-5},
-    {0x1.77c3e779b9fcfp-48, 0x1.0585283764p-4},
-    {0x1.a9a57734f2038p-48, 0x1.15b11a094ap-4},
-    {-0x1.d227d61f9e88dp-45, 0x1.1dd5460c8cp-4},
-    {0x1.d288560689912p-53, 0x1.2e3a740b78p-4},
-    {-0x1.df5de49ddb16p-46, 0x1.367ba3aaa2p-4},
-    {0x1.5b873a39e56dcp-47, 0x1.471ba8a7dep-4},
-    {0x1.75da8a5871b9ap-45, 0x1.4f7aad9bbcp-4},
-    {0x1.ef5f3c10990f4p-45, 0x1.57e3d47c3ap-4},
-    {-0x1.02359c584ac3cp-45, 0x1.68d4eaf26ep-4},
-    {-0x1.41149840eaa65p-46, 0x1.715d0ce368p-4},
-    {-0x1.ff281b9601ce6p-46, 0x1.79efb57b1p-4},
-    {0x1.2b9da13d5c8cbp-47, 0x1.8b350364c6p-4},
-    {-0x1.80743406505e6p-48, 0x1.93e7de0fc4p-4},
-    {-0x1.76b169f6b4949p-49, 0x1.9ca5aa172ap-4},
-    {-0x1.bc8889d0fe1ap-47, 0x1.a56e8325f6p-4},
-    {-0x1.03ad4133e8c4cp-45, 0x1.b721cd1716p-4},
-    {0x1.72a4e1d198491p-46, 0x1.c00c776722p-4},
-    {-0x1.ddd18dedb6656p-45, 0x1.c902a19e66p-4},
-    {0x1.5e533080ecf32p-47, 0x1.d204698cb4p-4},
-    {0x1.7e865b8783768p-45, 0x1.db11ed766ap-4},
-    {0x1.5f7ef576ada0cp-45, 0x1.ed50a4a26ep-4},
-    {0x1.c9a3bd0891bccp-46, 0x1.f68216c9ccp-4},
-    {-0x1.ff229f20ed3d2p-46, 0x1.ffbfc2bbc8p-4},
-    {-0x1.6f30673aae7efp-45, 0x1.0484e4942bp-3},
-    {0x1.dae5ed5e3f34cp-45, 0x1.093025a199p-3},
-    {-0x1.3eea49e637bb3p-45, 0x1.0de1b56357p-3},
-    {0x1.82c6326f70b35p-46, 0x1.1299a4fb3ep-3},
-    {0x1.f04d633b79054p-45, 0x1.175805d158p-3},
-    {0x1.8b891b6d05a73p-48, 0x1.1c1ce9955cp-3},
-    {-0x1.35ca658049a0ap-51, 0x1.20e8624039p-3},
-    {0x1.ff081a4e81f0bp-45, 0x1.25ba8215afp-3},
-    {0x1.1e3f04f63ee01p-45, 0x1.2a935ba5f1p-3},
-    {-0x1.e1471e5cb397ep-45, 0x1.2f7301cf4fp-3},
-    {-0x1.5bd54fd6eb7d9p-45, 0x1.345987bfefp-3},
-    {0x1.fe93f791a7264p-46, 0x1.394700f795p-3},
-    {-0x1.8aebce3ec8738p-45, 0x1.3e3b814974p-3},
-    {0x1.b0722aa2559f2p-45, 0x1.43371cde07p-3},
-    {-0x1.bcb784188a058p-46, 0x1.4839e83507p-3},
-    {0x1.20ca9a2bcc728p-45, 0x1.4d43f8275ap-3},
-    {0x1.bb95ec8fd8f68p-45, 0x1.525561e925p-3},
-    {0x1.4e036062e2e73p-48, 0x1.576e3b0bdep-3},
-    {-0x1.e560b5e7a02b4p-51, 0x1.5c8e998073p-3},
-    {0x1.1e472c751a4cp-49, 0x1.61b6939983p-3},
-    {-0x1.1116ad28239bp-48, 0x1.66e6400da4p-3},
-    {0x1.9ad1d9e405fb9p-46, 0x1.6c1db5f9bbp-3},
-    {-0x1.41149840eaa65p-45, 0x1.715d0ce368p-3},
-    {0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
-    {0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
-    {-0x1.9fc01708d86e7p-48, 0x1.7bf3bde09ap-3},
-    {0x1.4adaf7fe992b6p-45, 0x1.814b4921bdp-3},
-    {-0x1.c0b434f5d2b7ap-46, 0x1.86ab17c10cp-3},
-    {0x1.4c7acd659652fp-45, 0x1.8c13437695p-3},
-    {0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
-    {0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
-    {-0x1.fb7d8e74e02ap-46, 0x1.96fd1b63ap-3},
-    {0x1.7c9690248757ep-46, 0x1.9c7efd734ap-3},
-    {-0x1.0cee0ed4ca7e9p-52, 0x1.a209a84fbdp-3},
-    {0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
-    {0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
-    {0x1.fe6eb5a4df1dfp-49, 0x1.ad39c9c2c6p-3},
-    {0x1.4c9cce6dd603bp-46, 0x1.b2df7a5c5p-3},
-    {-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
-    {-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
-    {0x1.2ee896e06dbe8p-45, 0x1.be46b08735p-3},
-    {-0x1.ff2381071d23ep-49, 0x1.c4087384f5p-3},
-    {-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
-    {-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
-    {-0x1.2386ad8b819b8p-45, 0x1.cfa8e765ccp-3},
-    {0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
-    {0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
-    {0x1.fcc16f3ba09cbp-45, 0x1.db70c7e96ep-3},
-    {-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
-    {-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
-    {0x1.f97877007c127p-46, 0x1.e76124046bp-3},
-    {0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
-    {0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
-    {-0x1.cbc0789055864p-45, 0x1.f37b15bab1p-3},
-    {0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
-    {0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
-    {-0x1.ff229f20ed3d2p-45, 0x1.ffbfc2bbc8p-3},
-    {0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
-    {0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
-    {-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
-    {-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
-    {-0x1.f171b2c5f2274p-48, 0x1.093cc32c91p-2},
-    {-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
-    {-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
-    {0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
-    {0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
-    {0x1.feed957c16a44p-46, 0x1.12cd31b9c98p-2},
-    {0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
-    {0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
-    {0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
-    {0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
-    {-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
-    {-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
-    {0x1.40eb9b53054c3p-46, 0x1.1fe1e5af2cp-2},
-    {0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
-    {0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
-    {0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
-    {0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
-    {0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
-    {0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
-    {-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
-    {-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
-    {-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
-    {-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
-    {0x1.fef311f12b358p-46, 0x1.34413509f78p-2},
-};
+// A simple upper bound for the error of e_x * log(2) - log(r).
+constexpr double HI_ERR = 0x1.0p-85;
 
-constexpr double LOG10_R_ULP[2] = {0x1.0p-96, 0.0};
-
-// Generated with Sollya:
-// > P = fpminimax(log10(1 + x)/x, 6, [|D...|], [-2^-7; 2^-7], absolute);
-// > dirtyinfnorm(log10(1 + x)/x - P, [-2^-7, 2^-7]);
-// 0x1.9535684fb3064623001de9b13e6adf06355b5d75bp-57
-constexpr double COEFFS[7] = {0x1.bcb7b1526e50ep-2, -0x1.bcb7b1526e53fp-3,
-                              0x1.287a763700e4p-3,  -0x1.bcb7b14641063p-4,
-                              0x1.63c61abdf033fp-4, -0x1.28808b8a217fcp-4,
-                              0x1.ffe99fc1908c6p-5};
-
-constexpr double P_ERR = 0x1.0p-52;
-
-// Number of extra range reduction steps.
-constexpr size_t R_STEPS = 5;
-constexpr size_t R_BITS = 4;
-constexpr size_t R_SIZES = 1 << (R_BITS + 1);
-
-// Generated by Sollya with:
-// for i from 0 to 4 do {
-//   N = 11 + 4*i;
-//   print ("{");
-//   for j from -2^4 to 2^4 - 1 do {
-//     r = 2^(-N) * nearestint(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1)) /
-//                             ((1 + j * 2^(-N)) * (1 + (j + 1)*2^(-N))));
-//     print(r, ",");
-//   };
-//   print("},");
-// };
-constexpr double RR[R_STEPS][R_SIZES] = {
-    {
-        0x1.02p0,   0x1.01ep0,  0x1.01cp0,  0x1.01ap0,  0x1.018p0,  0x1.016p0,
-        0x1.014p0,  0x1.012p0,  0x1.01p0,   0x1.00ep0,  0x1.00cp0,  0x1.00ap0,
-        0x1.008p0,  0x1.006p0,  0x1.004p0,  0x1p0,      0x1p0,      0x1.ffcp-1,
-        0x1.ff8p-1, 0x1.ff4p-1, 0x1.ffp-1,  0x1.fecp-1, 0x1.fe8p-1, 0x1.fe4p-1,
-        0x1.fep-1,  0x1.fdcp-1, 0x1.fd8p-1, 0x1.fd4p-1, 0x1.fdp-1,  0x1.fccp-1,
-        0x1.fc8p-1, 0x1.fc4p-1,
-    },
-    {
-        0x1.002p0,   0x1.001ep0,  0x1.001cp0,  0x1.001ap0,  0x1.0018p0,
-        0x1.0016p0,  0x1.0014p0,  0x1.0012p0,  0x1.001p0,   0x1.000ep0,
-        0x1.000cp0,  0x1.000ap0,  0x1.0008p0,  0x1.0006p0,  0x1.0004p0,
-        0x1p0,       0x1p0,       0x1.fffcp-1, 0x1.fff8p-1, 0x1.fff4p-1,
-        0x1.fffp-1,  0x1.ffecp-1, 0x1.ffe8p-1, 0x1.ffe4p-1, 0x1.ffep-1,
-        0x1.ffdcp-1, 0x1.ffd8p-1, 0x1.ffd4p-1, 0x1.ffdp-1,  0x1.ffccp-1,
-        0x1.ffc8p-1, 0x1.ffc4p-1,
-    },
-    {
-        0x1.0002p0,   0x1.0001ep0,  0x1.0001cp0,  0x1.0001ap0,  0x1.00018p0,
-        0x1.00016p0,  0x1.00014p0,  0x1.00012p0,  0x1.0001p0,   0x1.0000ep0,
-        0x1.0000cp0,  0x1.0000ap0,  0x1.00008p0,  0x1.00006p0,  0x1.00004p0,
-        0x1p0,        0x1p0,        0x1.ffffcp-1, 0x1.ffff8p-1, 0x1.ffff4p-1,
-        0x1.ffffp-1,  0x1.fffecp-1, 0x1.fffe8p-1, 0x1.fffe4p-1, 0x1.fffep-1,
-        0x1.fffdcp-1, 0x1.fffd8p-1, 0x1.fffd4p-1, 0x1.fffdp-1,  0x1.fffccp-1,
-        0x1.fffc8p-1, 0x1.fffc4p-1,
-    },
-    {
-        0x1.00002p0,  0x1.00001ep0,  0x1.00001cp0,  0x1.00001ap0,
-        0x1.000018p0, 0x1.000016p0,  0x1.000014p0,  0x1.000012p0,
-        0x1.00001p0,  0x1.00000ep0,  0x1.00000cp0,  0x1.00000ap0,
-        0x1.000008p0, 0x1.000006p0,  0x1.000004p0,  0x1p0,
-        0x1p0,        0x1.fffffcp-1, 0x1.fffff8p-1, 0x1.fffff4p-1,
-        0x1.fffffp-1, 0x1.ffffecp-1, 0x1.ffffe8p-1, 0x1.ffffe4p-1,
-        0x1.ffffep-1, 0x1.ffffdcp-1, 0x1.ffffd8p-1, 0x1.ffffd4p-1,
-        0x1.ffffdp-1, 0x1.ffffccp-1, 0x1.ffffc8p-1, 0x1.ffffc4p-1,
-    },
-    {
-        0x1.000002p0,  0x1.000001ep0,  0x1.000001cp0,  0x1.000001ap0,
-        0x1.0000018p0, 0x1.0000016p0,  0x1.0000014p0,  0x1.0000012p0,
-        0x1.000001p0,  0x1.000000ep0,  0x1.000000cp0,  0x1.000000ap0,
-        0x1.0000008p0, 0x1.0000006p0,  0x1.0000004p0,  0x1p0,
-        0x1p0,         0x1.ffffffcp-1, 0x1.ffffff8p-1, 0x1.ffffff4p-1,
-        0x1.ffffffp-1, 0x1.fffffecp-1, 0x1.fffffe8p-1, 0x1.fffffe4p-1,
-        0x1.fffffep-1, 0x1.fffffdcp-1, 0x1.fffffd8p-1, 0x1.fffffd4p-1,
-        0x1.fffffdp-1, 0x1.fffffccp-1, 0x1.fffffc8p-1, 0x1.fffffc4p-1,
-    },
-};
+// Extra errors from P is from using x^2 to reduce evaluation latency.
+constexpr double P_ERR = 0x1.0p-51;
 
-// log10(2) with 192-bit prepcision generated by SageMath with:
-//   sage: (s, m, e) = RealField(192)(2).log10().sign_exponent_mantissa();
-//   sage: print("MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), ",",
-//   hex((m >> 128) % 2^64), "})");
-const Float192 LOG10_2(/*sign=*/false, /*exponent=*/-193, /*mantissa=*/
-                       MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178,
-                              0x9a209a84fbcff798}));
+// log10(2) with 128-bit prepcision generated by SageMath with:
+//   sage: (s, m, e) = RealField(128)(2).log10().sign_exponent_mantissa();
+//   sage: print("MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})");
+const Float128 LOG10_2(/*sign=*/false, /*exponent=*/-129, /*mantissa=*/
+                       MType({0x8f8959ac0b7c9178, 0x9a209a84fbcff798}));
 
-// -log10(r) with 192-bit precision generated by SageMath with:
+// -log10(r) with 128-bit precision generated by SageMath with:
 //
 // for i in range(128):
-//   r = 2^-8 * round( 2^8 * (1 + (i + 1/2)*2^(-7) - 2^(-15)) / ((1 + i*2^-7)*(1
-//   + (i + 1)*2^-7)) ); s, m, e = RR(r).log10().sign_mantissa_exponent();
+//   r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^(-7)) );
+//   s, m, e = RealField(128)(r).log10().sign_mantissa_exponent();
 //   print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
-//   ",", hex((m >> 128) % 2^64), "})},");
-const Float192 LOG10_R_F192[128] = {
+//         "})},");
+const Float128 LOG10_R1[128] = {
+    {false, 0, MType(0)},
+    {false, -136, MType({0x65af394fe05eafd3, 0xdf3b5ebbda7e186b})},
+    {false, -135, MType({0xa8fb8d87b30163b5, 0xe01d40572f029c16})},
+    {false, -134, MType({0x6bb0170e5bb5d630, 0xa8c1263ac3f57eb3})},
+    {false, -134, MType({0xfc2ea6eb0ea1370e, 0xe1e841bbc26204e5})},
+    {false, -133, MType({0xdc8a199a4bb63382, 0x8dc2eb02274d6ff4})},
+    {false, -133, MType({0x86b57ea610c7db33, 0xaacde920361dd054})},
+    {false, -133, MType({0x5f034a40e6a2f09d, 0xc81618eb15421bab})},
+    {false, -133, MType({0x594a31b2c5cc891c, 0xe59c7e66c5fedb4b})},
+    {false, -133, MType({0x221efda58221904b, 0xf477584f97b654de})},
+    {false, -132, MType({0x68a0dc47567691c9, 0x892e821975106e09})},
+    {false, -132, MType({0x10bc94f44d216b49, 0x9841c66e17dfe7da})},
+    {false, -132, MType({0xe303ea7e23c9d6fb, 0x9fd7be3318306cc5})},
+    {false, -132, MType({0xce697dbaa00d4c7d, 0xaf1cb35bf494a8dd})},
+    {false, -132, MType({0x9c216079dcf0ea96, 0xbe8380a2fa7eba5a})},
+    {false, -132, MType({0x75278940eecfc3a9, 0xc643c7758283a271})},
+    {false, -132, MType({0x2d3467d253e2d1fc, 0xd5de75ec27e4fe68})},
+    {false, -132, MType({0xead4055dcdec7b22, 0xddb904e8f1272a95})},
+    {false, -132, MType({0xe1e0dda0b3d375a4, 0xed88f6bb355fa196})},
+    {false, -132, MType({0x38dc40c4fe11e608, 0xf57e8281ade9d92d})},
+    {false, -131, MType({0x3bcdcfe7b23976cd, 0x82c2941bb20bbe1f})},
+    {false, -131, MType({0x456350b0bda452a6, 0x86cb36632807cdcd})},
+    {false, -131, MType({0x78185dcc37fda01a, 0x8eeaa306458b760a})},
+    {false, -131, MType({0x307643adbbbde1b3, 0x9301839512fc1168})},
+    {false, -131, MType({0x6c449d409f883fe3, 0x9b3dd1d550c41443})},
+    {false, -131, MType({0x8ea7b30c8b4ad886, 0x9f6356aa03c34389})},
+    {false, -131, MType({0x961c6e690d8879b4, 0xa7bd56cdde5d76a2})},
+    {false, -131, MType({0x42643ced81ec14a, 0xabf1ea3e1d7bd7cf})},
+    {false, -131, MType({0x4742fb3d0b5cdd19, 0xb02b9af74c2f879e})},
+    {false, -131, MType({0xf7e2ab36f09e9014, 0xb8ae8671b3d7dd6c})},
+    {false, -131, MType({0x8d3fc63485e7ff13, 0xbcf7dabd87c01afc})},
+    {false, -131, MType({0xf3edc49375fbc5a5, 0xc1467f694d10a581})},
+    {false, -131, MType({0x5fcd7d0ce937375f, 0xc9f3ef07e1f3fc5e})},
+    {false, -131, MType({0x58252dada9f06111, 0xce52d50b94fa253a})},
+    {false, -131, MType({0x62f01e5ff43708ab, 0xd2b74192fae43777})},
+    {false, -131, MType({0x481d9b3131f52639, 0xd72142a84ca85abd})},
+    {false, -131, MType({0xb305ced1419fe924, 0xdb90e68b8abf14af})},
+    {false, -131, MType({0x849266a85513dc6d, 0xe48150cf32888b9c})},
+    {false, -131, MType({0x80ecf3266b4dcf4, 0xe90234c65a15e533})},
+    {false, -131, MType({0xe1e0dda0b3d375a4, 0xed88f6bb355fa196})},
+    {false, -131, MType({0xce3537a3a211b25b, 0xf215a60b6557943f})},
+    {false, -131, MType({0x5dab68307fedefcd, 0xf6a852513757dfbd})},
+    {false, -131, MType({0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
+    {false, -130, MType({0x18aa302171017dcb, 0x8242724a155219f3})},
+    {false, -130, MType({0xabc7e698502d43c0, 0x849812d0ccbb5cbd})},
+    {false, -130, MType({0xc339089a51663370, 0x86f0dab1ab5822b6})},
+    {false, -130, MType({0x26f70b34ce5cf201, 0x894cd27d9f182c63})},
+    {false, -130, MType({0x676f20a87ab433df, 0x8bac02e8ac3e09ac})},
+    {false, -130, MType({0x6db4169cc4b83bc3, 0x8e0e74caae062e24})},
+    {false, -130, MType({0xcd3fdb2fad0d1fd6, 0x907431201c7f651a})},
+    {false, -130, MType({0x49d03e163250d1d4, 0x92dd410ad7bfe103})},
+    {false, -130, MType({0x9ec7dc02d5e723b9, 0x9549add2f8a3c7e0})},
+    {false, -130, MType({0x34698d03a5442573, 0x97b980e7a743d71c})},
+    {false, -130, MType({0x522904d1e47f3de, 0x9a2cc3dff7548556})},
+    {false, -130, MType({0x791a72646c87b976, 0x9ca3807bca9fe93f})},
+    {false, -130, MType({0x3826f190d655d736, 0x9f1dc0a4b9cea286})},
+    {false, -130, MType({0x544ab3e48199b299, 0xa19b8e6f03b60e45})},
+    {false, -130, MType({0xbe775fa82961114e, 0xa41cf41a83643487})},
+    {false, -130, MType({0x45798e5019e6c082, 0xa6a1fc13ad241953})},
+    {false, -130, MType({0x91fb1ed0cdc4d1fb, 0xa92ab0f492b772bd})},
+    {false, -130, MType({0x818b8b9cbbd17b72, 0xabb71d85ef05380d})},
+    {false, -130, MType({0xa50c2fea60c5b3b2, 0xae474cc0397f0d4f})},
+    {false, -130, MType({0x58ea34980ad8b720, 0xb0db49ccc1823c8e})},
+    {false, -130, MType({0x4b5f71941be508a4, 0xb3732006d1fbbba5})},
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-     MType({0xeba2ae5f7d1c7168, 0xe0b571f5c91b0445, 0x817c9fa643880404})},
-    {false, -193,
-     MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
-    {false, -193,
-     MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
-    {false, -193,
-     MType({0xf3cb58bd1bbe04f0, 0x9a741bb171158d29, 0x849e6196487c1d1c})},
-    {false, -193,
-     MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
-    {false, -193,
-     MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
-    {false, -193,
-     MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
-    {false, -193,
-     MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
-    {false, -193,
-     MType({0x84a2c0cd81dbcf53, 0xabe0b522230f7d13, 0x896698dce4cff76c})},
-    {false, -193,
-     MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
-    {false, -193,
-     MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
-    {false, -193,
-     MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
-    {false, -193,
-     MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
-    {false, -193,
-     MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
-    {false, -193,
-     MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
-    {false, -193,
-     MType({0x8a1754a1ee7c990, 0xda982a614e12c6dd, 0x8ff0f2d7960a075c})},
-    {false, -193,
-     MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
-    {false, -193,
-     MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
-    {false, -193,
-     MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
-    {false, -193,
-     MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
-    {false, -193,
-     MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
-    {false, -193,
-     MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
-    {false, -193,
-     MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
-    {false, -193,
-     MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
-    {false, -193,
-     MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
-    {false, -193,
-     MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
-    {false, -193,
-     MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178, 0x9a209a84fbcff798})},
 };
 
-// -log10(r) for further range reduction steps, generated by SageMath with:
+// Range reduction - Step 2:
+//   r(k) = 2^-16 round(2^16 / (1 + k*2^-14)) for k = -2^6 .. 2^7.
+// Output range:
+//   [-0x1.3ffcp-15, 0x1.3e3dp-15]
+
+// We store S2[i] = 2^16 (r(i - 2^6) - 1).
+constexpr int64_t 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,
+    0x94,   0x90,   0x8c,   0x88,   0x84,   0x80,   0x7c,   0x78,   0x74,
+    0x70,   0x6c,   0x68,   0x64,   0x60,   0x5c,   0x58,   0x54,   0x50,
+    0x4c,   0x48,   0x44,   0x40,   0x3c,   0x38,   0x34,   0x30,   0x2c,
+    0x28,   0x24,   0x20,   0x1c,   0x18,   0x14,   0x10,   0xc,    0x8,
+    0x4,    0x0,    -0x4,   -0x8,   -0xc,   -0x10,  -0x14,  -0x18,  -0x1c,
+    -0x20,  -0x24,  -0x28,  -0x2c,  -0x30,  -0x34,  -0x38,  -0x3c,  -0x40,
+    -0x44,  -0x48,  -0x4c,  -0x50,  -0x54,  -0x58,  -0x5c,  -0x60,  -0x64,
+    -0x68,  -0x6c,  -0x70,  -0x74,  -0x78,  -0x7c,  -0x80,  -0x84,  -0x88,
+    -0x8c,  -0x90,  -0x94,  -0x98,  -0x9c,  -0xa0,  -0xa4,  -0xa8,  -0xac,
+    -0xb0,  -0xb4,  -0xb7,  -0xbb,  -0xbf,  -0xc3,  -0xc7,  -0xcb,  -0xcf,
+    -0xd3,  -0xd7,  -0xdb,  -0xdf,  -0xe3,  -0xe7,  -0xeb,  -0xef,  -0xf3,
+    -0xf7,  -0xfb,  -0xff,  -0x103, -0x107, -0x10b, -0x10f, -0x113, -0x117,
+    -0x11b, -0x11f, -0x123, -0x127, -0x12b, -0x12f, -0x133, -0x137, -0x13a,
+    -0x13e, -0x142, -0x146, -0x14a, -0x14e, -0x152, -0x156, -0x15a, -0x15e,
+    -0x162, -0x166, -0x16a, -0x16e, -0x172, -0x176, -0x17a, -0x17e, -0x182,
+    -0x186, -0x18a, -0x18e, -0x192, -0x195, -0x199, -0x19d, -0x1a1, -0x1a5,
+    -0x1a9, -0x1ad, -0x1b1, -0x1b5, -0x1b9, -0x1bd, -0x1c1, -0x1c5, -0x1c9,
+    -0x1cd, -0x1d1, -0x1d5, -0x1d9, -0x1dd, -0x1e0, -0x1e4, -0x1e8, -0x1ec,
+    -0x1f0, -0x1f4, -0x1f8, -0x1fc};
+
+// -log10(r) for the second step, generated by SageMath with:
 //
-// RR = RealField(192);
-// for i in range(5):
-//   N = 11 + 4*i;
-//   print("{");
-//   for j in range(-2^4, 2^4):
-//     r = 2^(-N) * round(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1) ) / ((1 +
-//     j * 2^(-N)) * (1 + (j + 1)*2^(-N)))); a = -RR(r).log10(); if j in [0,
-//     -1]:
-//       r = 1; a = RR(0);
-//     s, m, e = a.sign_mantissa_exponent()
-//     sgn = "{false," if s == 1 else "{true,";
-//     print(sgn, e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
-//     ",", hex((m >> 128) % 2^64), "})},");
-//   print("},");
-const Float192 LOG10_RR[R_STEPS][R_SIZES] = {
-    {
-        {true, -200,
-         MType({0xf52b7aea9ca0c476, 0xdd4a47e1490df56, 0xdd7ea3910f69332e})},
-        {true, -200,
-         MType({0x11c54af8b7b5ac0, 0xbe14368ead9df21c, 0xcfb39f5a164f371c})},
-        {true, -200,
-         MType({0xdc8b8e1f46c98b22, 0xb46a6050fcd513ca, 0xc1e6e4dcf45e0ee0})},
-        {true, -200,
-         MType({0x10c91fa8440cf6a8, 0x799e2aecea385841, 0xb41873accfbaba29})},
-        {true, -200,
-         MType({0xefff2920f96d3591, 0x3440eb8005ad171b, 0xa6484b5ca5f7eaea})},
-        {true, -200,
-         MType({0xaa51f382c80587cb, 0x21e85464267c47a4, 0x98766b7f4c01d932})},
-        {true, -200,
-         MType({0xaf1454629688e40a, 0xee383c5f131898c4, 0x8aa2d3a76e0a0a79})},
-        {true, -201,
-         MType({0xf128ae4671df433, 0x15022fc12f1747d4, 0xf99b06cf1ee618b9})},
-        {true, -201,
-         MType({0x511d8d64285691ec, 0x411e70646c13ae93, 0xddecf4a415784564})},
-        {true, -201,
-         MType({0xd9e8330f453fac1c, 0x662bb5e078c532bf, 0xc23b6ff222d9d207})},
-        {true, -201,
-         MType({0xa9d37fde93b3c989, 0xebcfc062f09deb23, 0xa68677dd5801ce9b})},
-        {true, -201,
-         MType({0xe9eadfa1556bcaec, 0xbfdd2c7f388fc013, 0x8ace0b8973a63413})},
-        {true, -202,
-         MType({0x743a88400b3b04bc, 0xc21595e745f1fa15, 0xde245433c425b5c5})},
-        {true, -202,
-         MType({0xf50e04c0bd73e444, 0x663cf2b27e8f1ffa, 0xa6a5a5637a00afdc})},
-        {true, -203,
-         MType({0xf860837f66c35a6a, 0xba0324530edaa03e, 0xde4011cf2daaff31})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, -204,
-         MType({0x8319f4e45b099e86, 0x20fa8a73c585f633, 0xde69bf8f58005dfc})},
-        {false, -203,
-         MType({0x58e9d6dee00a8fc3, 0x35e3298f4bb2aa0a, 0xde77a8c714b08d28})},
-        {false, -202,
-         MType({0xc6b4c904be2cbf01, 0xa1e66c6203725d4f, 0xa6e42f3ccf49959d})},
-        {false, -202,
-         MType({0xebc8b4278d3f93e9, 0x5ad3480ccfbe988f, 0xde93822dfe812587})},
-        {false, -201,
-         MType({0x9e52f5c983d9caa4, 0xe5f7c52cdf119d4, 0x8b24e77b0cb60a84})},
-        {false, -201,
-         MType({0x9a29985625bf9936, 0x84e42dd54c2ff44e, 0xa7038baa64c5cd39})},
-        {false, -201,
-         MType({0x15c143896f774901, 0xde013ef4ba5abcfa, 0xc2e5ae853064a921})},
-        {false, -201,
-         MType({0x8c6b1043496971e, 0x8f6c475cc382401c, 0xdecb50ebece5e146})},
-        {false, -201,
-         MType({0x95457befdc15d5a7, 0x64a67faf3f1c74c2, 0xfab473bf6c2589b9})},
-        {false, -200,
-         MType({0x825120e3dbb4f086, 0x341aa1df5e8fab33, 0x8b508bf06a597f35})},
-        {false, -200,
-         MType({0xc69ade976589376a, 0x8b19159a7036ac6c, 0x99489f18d0fdba55})},
-        {false, -200,
-         MType({0x132b76e2a5ed5ce1, 0x7b8b2852580ae3e2, 0xa74273c9d23a53b9})},
-        {false, -200,
-         MType({0x6f0c9495185af64, 0xe9ad89ebfeaf5a02, 0xb53e0a7480e3cf3c})},
-        {false, -200,
-         MType({0x3dc11a3e41e693ee, 0xe12996e107931f4b, 0xc33b638a1a7dc81d})},
-        {false, -200,
-         MType({0x2ce27c62f7c5d30e, 0xb43caf8e7b891066, 0xd13a7f7c07506f7d})},
-    },
-    {
-        {true, -204,
-         MType({0xea57bded8e15fa61, 0x634cea6750617a91, 0xde4df4140b42822f})},
-        {true, -204,
-         MType({0x8a2e8bb53b241b41, 0x9185547f24195fab, 0xd069e52741db93b6})},
-        {true, -204,
-         MType({0x265b41c72630e265, 0xfbd7e970aef9dbb8, 0xc285ba757feb2781})},
-        {true, -204,
-         MType({0xd2e480f950aea396, 0xe820a1a6f319285a, 0xb4a173fe56691147})},
-        {true, -204,
-         MType({0xb3300685a98f7031, 0x9aedc1c1ba7d0694, 0xa6bd11c1564a8ace})},
-        {true, -204,
-         MType({0x407aba2445f75697, 0x89de4989587519ca, 0x98d893be108233d7})},
-        {true, -204,
-         MType({0xbe4f7e3cbefc71ce, 0x8d306ba207233c43, 0x8af3f9f41600120a})},
-        {true, -205,
-         MType({0x5449f3ae334e8294, 0x2100088006aeff92, 0xfa1e88c5ef6321c2})},
-        {true, -205,
-         MType({0x2245be0033c6dafe, 0x856a0a3a00fcf3c1, 0xde54e6148d030322})},
-        {true, -205,
-         MType({0xf8b7f31304662b13, 0x8a4399de0be1c455, 0xc28b0bd326b035f2})},
-        {true, -205,
-         MType({0x5abacdedb1dc0b2c, 0xb3a2c1407cf6d38d, 0xa6c0fa00de35f314})},
-        {true, -205,
-         MType({0x874695793b5a312e, 0x220af90d74ea17c7, 0x8af6b09cd55a3e68})},
-        {true, -206,
-         MType({0x70a9c05700a3f2c5, 0xd791cf6a70c3a504, 0xde585f4c5bbbcd3d})},
-        {true, -206,
-         MType({0x1f465fbffe379fdf, 0xe95b52a2781c25e3, 0xa6c2ee3812f90a28})},
-        {true, -207,
-         MType({0x35af2ed67a089e4, 0x10a633f2c4a8ea22, 0xde5a1bf627b1f68f})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, -208,
-         MType({0xe0faf0a837063878, 0x140a2bff40e0d66f, 0xde5cb706384ddbaf})},
-        {false, -207,
-         MType({0xd660ff616f334614, 0xed4a68e5e6e83dde, 0xde5d95658a729eab})},
-        {false, -206,
-         MType({0x542eeb05597ecdea, 0x7c9dc000df79edde, 0xa6c6d6d56238dd6e})},
-        {false, -206,
-         MType({0x5f8e2902ab25d093, 0x3281f1872cdbee94, 0xde5f522b21e3e25a})},
-        {false, -205,
-         MType({0xba7cb3452f07c347, 0x292ef5f6fbe96c00, 0x8afc1e5ae08b4643})},
-        {false, -205,
-         MType({0xde1cc00ce5b7249c, 0xf1466edaa96e356d, 0xa6c8cb3b7e5bbbfd})},
-        {false, -205,
-         MType({0xa521ef720e08d8fe, 0x936a112cb6265f7e, 0xc295afb848dbd759})},
-        {false, -205,
-         MType({0x1d1317f34a2fba9e, 0x8a607fd695dfc3d9, 0xde62cbd21e895473})},
-        {false, -205,
-         MType({0x9c74fa60d5502c53, 0x65a49326981bdf3f, 0xfa301f89dde726b4})},
-        {false, -204,
-         MType({0x4295ea162700a15d, 0xc36b8713ceefe2de, 0x8afed57032bebc7c})},
-        {false, -204,
-         MType({0x10e84795ee07f1c, 0x7068ee550f5ffb37, 0x98e5b6eb49ecd6df})},
-        {false, -204,
-         MType({0xf71013dc630dfafa, 0x5c2e76c953e3e3e5, 0xa6ccb436a3c72fa4})},
-        {false, -204,
-         MType({0x61fc137c9dce9644, 0xb3425e818888a098, 0xb4b3cd52af99afe6})},
-        {false, -204,
-         MType({0xe923dcabcfbadd8e, 0x8e4950fa5c943bbe, 0xc29b023fdcb2dccf})},
-        {false, -204,
-         MType({0xd941d748e2a23bcc, 0xd4d3796d93d6750b, 0xd08252fe9a63d7aa})},
-    },
-    {
-        {true, -208,
-         MType({0x22058fdb723bb37, 0xce31a0d27359396f, 0xde5afa4e86f7f3ee})},
-        {true, -208,
-         MType({0x2e22d8a1e7a512a1, 0x9f63aaa563e9a399, 0xd07557b0de924142})},
-        {true, -208,
-         MType({0xd1fffbb5f7c92f10, 0x67e63bbe1405c20c, 0xc28fb35684fedaab})},
-        {true, -208,
-         MType({0x2473a71714b9ef53, 0x5440d7a131392da8, 0xb4aa0d3f79ce9499})},
-        {true, -208,
-         MType({0xdec055ae33e59ad5, 0xe0e635a681d259e1, 0xa6c4656bbc924352})},
-        {true, -208,
-         MType({0x953365af70272280, 0xda1f690c752fdefe, 0x98debbdb4cdabaf4})},
-        {true, -208,
-         MType({0xc6c161d92abe9aa4, 0x5bf708fead2b177f, 0x8af9108e2a38cf72})},
-        {true, -209,
-         MType({0x78732843e6506315, 0xa448b11f012c975c, 0xfa26c708a87aa929})},
-        {true, -209,
-         MType({0x863d53d84705d0ae, 0xefecdd48ed894c31, 0xde5b697b94f23bf7})},
-        {true, -209,
-         MType({0x87704029104c21a9, 0xb07ebbab782457af, 0xc290087518f9fe3b})},
-        {true, -209,
-         MType({0x4adb1df9649c50c1, 0x9a7eb8811ec3e6bb, 0xa6c4a3f533b3968d})},
-        {true, -209,
-         MType({0xcfe3880042e86013, 0x11fd6995111a926, 0x8af93bfbe440ab33})},
-        {true, -210,
-         MType({0x2cbf7a2e7b9cbcf0, 0xac3bfd925e6b33e1, 0xde5ba1125385c43b})},
-        {true, -210,
-         MType({0x4d418d19762cbd66, 0x53289e84744549cb, 0xa6c4c33a06b7c1d9})},
-        {true, -211,
-         MType({0x4c3b3ad870a5dd12, 0xa74dab3bd6067bc7, 0xde5bbcddc0b533aa})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, -212,
-         MType({0x29423084899fc9b9, 0xbee710a5ace7c8d4, 0xde5be68ef5db7f99})},
-        {false, -211,
-         MType({0x1c57cf04091e59cc, 0x7b644b13993cf4ef, 0xde5bf474b6df8331})},
-        {false, -210,
-         MType({0xd259a65cb34a4a8a, 0x64136e97019d0a3a, 0xa6c501c3dba75dc2})},
-        {false, -210,
-         MType({0x6f497708cc18c4d8, 0xa2912e03afc8cc28, 0xde5c10403fda6db5})},
-        {false, -209,
-         MType({0x865e042dd50e1d3a, 0xbf5dccd967504856, 0x8af992d7c4e2d5b5})},
-        {false, -209,
-         MType({0xe1702b0f17301f13, 0xe22978efa7a9629f, 0xa6c52108dd92e8c1})},
-        {false, -209,
-         MType({0x74a88aedf1375fb4, 0x82cd723bf1524680, 0xc290b2b36adbcda2})},
-        {false, -209,
-         MType({0xa923a00c40053158, 0x4dd6f8576e9188b7, 0xde5c47d76d9be24e})},
-        {false, -209,
-         MType({0x923113e7e6eec47e, 0x5368655f1ce3110a, 0xfa27e074e6b1850f})},
-        {false, -208,
-         MType({0xee64cf0bae6dd4c9, 0x83b16ff7eecace8b, 0x8af9be45eb7d8a41})},
-        {false, -208,
-         MType({0x782dbb6881f2d3d, 0x20c8069e4ae400de, 0x98df8e0e1fab77cd})},
-        {false, -208,
-         MType({0xff9e92ff9157f7ff, 0x9ee5e19f2eecdb54, 0xa6c55f931051bacc})},
-        {false, -208,
-         MType({0x4260c63c04224e80, 0xce16dd7f60311bf6, 0xb4ab32d4bddf830b})},
-        {false, -208,
-         MType({0xbf40b2a151280765, 0x3099a17e0461648b, 0xc29107d328c40080})},
-        {false, -208,
-         MType({0xb32695665b8f1329, 0xfaf478d3d0b312ff, 0xd076de8e516e6348})},
-    },
-    {
-        {true, -212,
-         MType({0xdaf98ea8088ccd55, 0x19be3fabd93832c4, 0xde5bcac37ac6587d})},
-        {true, -212,
-         MType({0x92832d76e51450a5, 0x88a86b30027dc8a5, 0xd0760ee7b9136936})},
-        {true, -212,
-         MType({0xdd67de1487eb3913, 0xc61c028b8f230f14, 0xc29052f02bebe878})},
-        {true, -212,
-         MType({0x8b5c1ba51e27536f, 0xb8cfce8fa982ea7, 0xb4aa96dcd34f6716})},
-        {true, -212,
-         MType({0xe1a1b61dba0d85b7, 0x8fd43f0c9ce444d2, 0xa6c4daadaf3d75e0})},
-        {true, -212,
-         MType({0x60e206e644f9e560, 0x872f9b3fd2141246, 0x98df1e62bfb5a5aa})},
-        {true, -212,
-         MType({0xeef14d02362d0513, 0x2341d13c21a7d8cf, 0x8af961fc04b78746})},
-        {true, -213,
-         MType({0xb56b5c514a13e254, 0x26251c2ccc00d194, 0xfa274af2fc85570b})},
-        {true, -213,
-         MType({0x60d8b357006a3fed, 0x61cd853e796bc2c, 0xde5bd1b658ad4676})},
-        {true, -213,
-         MType({0xa90e9309cbc1b814, 0x3a0de60782dd1939, 0xc29058421de5fe71})},
-        {true, -213,
-         MType({0xb6bc3ab198cec5a2, 0x10652e9b1ce538bb, 0xa6c4de964c2ea0a1})},
-        {true, -213,
-         MType({0x27d91560ba3c3289, 0xd259757215d9aa0d, 0x8af964b2e3864ea9})},
-        {true, -214,
-         MType({0xef6374ae583f4c18, 0x87d6afabfba0644e, 0xde5bd52fc7d8545f})},
-        {true, -214,
-         MType({0x4bf40373e818d7f5, 0x47ca9999c73441f5, 0xa6c4e08a9abea9ae})},
-        {true, -215,
-         MType({0x256483d247de2b54, 0xaf6e93be8e4c1764, 0xde5bd6ec7f7bc110})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, -216,
-         MType({0x5f6593112c03bf0b, 0xd5c0ebcf0fa0221e, 0xde5bd98793024346})},
-        {false, -215,
-         MType({0x11b7b6679d212c54, 0x6d0063a923dd0cc6, 0xde5bda65eede65ed})},
-        {false, -214,
-         MType({0x6079e21db1a5b651, 0xc424c8976e2ebcfa, 0xa6c4e473380da326})},
-        {false, -214,
-         MType({0x8a7f3f059746be25, 0xa9bf32001043629c, 0xde5bdc22a69d9e19})},
-        {false, -213,
-         MType({0xc8dbe969fc9bb1cd, 0x9e8e77fa22e00248, 0x8af96a20a1907043})},
-        {false, -213,
-         MType({0xe129bb635500e8db, 0xaf3be9e7aa56d682, 0xa6c4e66786cc9393})},
-        {false, -213,
-         MType({0xea8e87193d810fa0, 0xfc3aff5aaf056566, 0xc29062e603041758})},
-        {false, -213,
-         MType({0x254d7de51cd5637d, 0x8014f0f360272d82, 0xde5bdf9c1637d9ef})},
-        {false, -213,
-         MType({0xe9795a136f5856c, 0x3a891f89bcf72a40, 0xfa275c89c068b9b3})},
-        {false, -212,
-         MType({0x3c5338931e9d2409, 0x18468a2ba2fa5549, 0x8af96cd780cbca80})},
-        {false, -212,
-         MType({0x3ab6a381bda506d5, 0x362640905714e443, 0x98df2b85ece2a519})},
-        {false, -212,
-         MType({0x13b57109141a8764, 0xfe94a02fc639c0e3, 0xa6c4ea5024795bd2})},
-        {false, -212,
-         MType({0xc613ec4aff62b164, 0x7bddaab60665b883, 0xb4aaa93627905ddb})},
-        {false, -212,
-         MType({0x29f60bd76dda9de7, 0xbae8765350c99434, 0xc2906837f6281a60})},
-        {false, -212,
-         MType({0x6d1b70f85e1ebdd6, 0xcb372dd0da70965b, 0xd076275590410090})},
-    },
-    {
-        {true, -216,
-         MType({0xcf260846db91386b, 0x81645201b36e17b9, 0xde5bd7cadb50f0d8})},
-        {true, -216,
-         MType({0x8f96908bc0be4701, 0xcdea7b1a0dc7ad41, 0xd0761a5b34fd720c})},
-        {true, -216,
-         MType({0x71e2e36ca8d985f3, 0x5ce6604b0911c5ed, 0xc2905ce9d1f24872})},
-        {true, -216,
-         MType({0xdaf377e6a6fc292b, 0x6e0982bd8d96ee, 0xb4aa9f76b22f739a})},
-        {true, -216,
-         MType({0xf97df5dd10eab18e, 0x8a9754fe0c0073a6, 0xa6c4e201d5b4f314})},
-        {true, -216,
-         MType({0x260badb439a3192f, 0xcd77f7489d9ef50b, 0x98df248b3c82c672})},
-        {true, -216,
-         MType({0x35f8aeb949a552f2, 0x9b257b3ce3f82109, 0x8af96712e698ed45})},
-        {true, -217,
-         MType({0x82d6f89adeda85fe, 0x8b6a840831c123d7, 0xfa275331a7eece3b})},
-        {true, -217,
-         MType({0xb88ac395452a99d1, 0x3e79062c7cbb3b7c, 0xde5bd83a093c6718})},
-        {true, -217,
-         MType({0x8416ca30683117d1, 0xf3a098743d20fb63, 0xc2905d3ef11aa442})},
-        {true, -217,
-         MType({0xe98fafcacc5781ed, 0x4f0b030a9742eaff, 0xa6c4e2405f8984dd})},
-        {true, -217,
-         MType({0x9dc90f008c47f3e0, 0xf4e1bab83f55f5a0, 0x8af9673e54890808})},
-        {true, -218,
-         MType({0x1ff62cfc0b87b23a, 0x129bc1c6f293726e, 0xde5bd871a03259cf})},
-        {true, -218,
-         MType({0xb2b2c5c7e8e3dd1b, 0x60f08720313daa3e, 0xa6c4e25fa473e535})},
-        {true, -219,
-         MType({0x4a6c56a5be458d59, 0x3a25757e00f4e3a0, 0xde5bd88d6bad6110})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, 0, MType({0x0, 0x0, 0x0})},
-        {false, -220,
-         MType({0x3f16581bde8d9c4a, 0x225916c2b3f33c90, 0xde5bd8b71ce5fd51})},
-        {false, -219,
-         MType({0x8d1575503bb9c7fd, 0x44397830931fddc, 0xde5bd8c502a38b5e})},
-        {false, -218,
-         MType({0x91d4a95df896a79f, 0xe454dec82bde52e4, 0xa6c4e29e2e48d4cc})},
-        {false, -218,
-         MType({0x242c88441091cf56, 0xa6e2721f2afc3cce, 0xde5bd8e0ce1eae6a})},
-        {false, -217,
-         MType({0x9b6ef45b7cbebc03, 0x80bf0ff2f6cd9f92, 0x8af967953069aa22})},
-        {false, -217,
-         MType({0x7fe96798a2b597a7, 0x55ee1480619827c4, 0xa6c4e2bd7333640c})},
-        {false, -217,
-         MType({0xedc6a1463ce32849, 0x2ed8ba8c6fa81c97, 0xc2905de92f6c85d1})},
-        {false, -217,
-         MType({0xd59f31668df9feb7, 0x6759c94e2d017fac, 0xde5bd9186515104f})},
-        {false, -217,
-         MType({0x337d83d075e0c1ab, 0x5b4c5b5f180b9fe1, 0xfa27544b142d0465})},
-        {false, -216,
-         MType({0x390379ad9871c8a7, 0xb345ef5d90dd6545, 0x8af967c09e5a3178})},
-        {false, -216,
-         MType({0x73d9d08521f9533d, 0xf27a0a6056dcfe57, 0x98df255d6f559668})},
-        {false, -216,
-         MType({0xb70a87a69080a500, 0x99308918494a4a1f, 0xa6c4e2fbfd08b172})},
-        {false, -216,
-         MType({0x8995c0b3074c289d, 0xd5579f970cf000a8, 0xb4aaa09c47738304})},
-        {false, -216,
-         MType({0x72650f85eed7af38, 0xd4ddab9f8032bbab, 0xc2905e3e4e960b8e})},
-        {false, -216,
-         MType({0xeb04ee2cfc701a37, 0xc5b134a5bb25cf2d, 0xd0761be212704b7f})},
-    },
+// for i in range(-2^6, 2^7 + 1):
+//   r = 2^-16 * round( 2^16 / (1 + i*2^(-14)) );
+//   s, m, e = RealField(128)(r).log10().sign_mantissa_exponent();
+//   print("{false," if s == -1 else "{true,", e, ",
+//         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+const Float128 LOG10_R2[193] = {
+    {true, -137, MType({0x7f1ce002fa34131b, 0xdeca729013cd7c31})},
+    {true, -137, MType({0x639afa085dd8b4c7, 0xdb5475b44946d986})},
+    {true, -137, MType({0x5512632fe9a58cb, 0xd7de6b0e10cab7d2})},
+    {true, -137, MType({0xb5380a9953117d07, 0xd468529cfc6fb395})},
+    {true, -137, MType({0x70af2d7d53be1f31, 0xd0f22c609e474741})},
+    {true, -137, MType({0xccd499c49b74cc2, 0xcd7bf858885dcae2})},
+    {true, -137, MType({0x5b51ddc3987ebfb8, 0xca05b6844cba73cf})},
+    {true, -137, MType({0x49375f5189b3782b, 0xc68f66e37d5f545a})},
+    {true, -137, MType({0xf6e57738865c712f, 0xc3190975ac495b7a})},
+    {true, -137, MType({0xca02b10a8c712acd, 0xbfa29e3a6b70547e})},
+    {true, -137, MType({0x78e5038210208151, 0xbc2c25314cc6e6b6})},
+    {true, -137, MType({0xfa099ecd71ee0ea, 0xb8b59e59e23a9524})},
+    {true, -137, MType({0xeeb445ccb8fb09ed, 0xb53f09b3bdb3be28})},
+    {true, -137, MType({0xc352fff18a1c02fb, 0xb1c8673e71159b33})},
+    {true, -137, MType({0x7949e03ecf9b390b, 0xae51b6f98e3e406e})},
+    {true, -137, MType({0x2681f33f30aadedc, 0xaadaf8e4a7069c6c})},
+    {true, -137, MType({0xf01d5496eea213b3, 0xa7642cff4d4277d6})},
+    {true, -137, MType({0xe92ef555ff1de975, 0xa3ed534912c0751d})},
+    {true, -137, MType({0xeb0c7519b3e7c1e0, 0xa0766bc1894a1022})},
+    {true, -137, MType({0xf60d204ff0fe5296, 0x9c21b6e91e7f03a3})},
+    {true, -137, MType({0x125c19a4f057c18b, 0x98aab0491050bea8})},
+    {true, -137, MType({0x7e9383ce1bdf9575, 0x95339bd64cd953e7})},
+    {true, -137, MType({0xbf274f4d8f770253, 0x91bc799065cc57d6})},
+    {true, -137, MType({0x656bd9b758fe44ba, 0x8e454976ecd836ad})},
+    {true, -137, MType({0xbfdd2c7f388fc014, 0x8ace0b8973a63413})},
+    {true, -137, MType({0x83fbf6ed936c493a, 0x8756bfc78bda6ad0})},
+    {true, -137, MType({0x71bfa9a18bec01cc, 0x83df6630c713cc76})},
+    {true, -137, MType({0xf09d19f56dbfef72, 0x8067fec4b6ec2111})},
+    {true, -138, MType({0x4c422713b1642228, 0xf9e11305d9f00dad})},
+    {true, -138, MType({0xc3c7c5699b7a0a4, 0xf2f20cd5f58de39a})},
+    {true, -138, MType({0xb8db7c69e3fa0797, 0xec02eaf8e3c656ff})},
+    {true, -138, MType({0xa083eb05506ff7ed, 0xe513ad6dc7a3a553})},
+    {true, -138, MType({0xc21595e745f1fa15, 0xde245433c425b5c5})},
+    {true, -138, MType({0xb9d5bcdbfe719389, 0xd734df49fc42189b})},
+    {true, -138, MType({0xa17a1e85e93461f4, 0xd0454eaf92e4068b})},
+    {true, -138, MType({0xe3537584da333fda, 0xc955a263aaec6016})},
+    {true, -138, MType({0x963177f24682c2, 0xc265da656731ace5})},
+    {true, -138, MType({0x4ac037347bcfc50e, 0xbb75f6b3ea801b1e})},
+    {true, -138, MType({0x901a736a4364cdfd, 0xb485f74e57997ec6})},
+    {true, -138, MType({0xbb550acc3b9d7247, 0xad95dc33d1355117})},
+    {true, -138, MType({0x663cf2b27e8f1ffb, 0xa6a5a5637a00afdc})},
+    {true, -138, MType({0x5f89bd08feb39952, 0x9fb552dc749e5cca})},
+    {true, -138, MType({0x23c2623c73f494db, 0x98c4e49de3a6bcdd})},
+    {true, -138, MType({0x4937d3b5485af61e, 0x91d45aa6e9a7d7b0})},
+    {true, -138, MType({0xdf14214e7a6d8111, 0x8ae3b4f6a92556d9})},
+    {true, -138, MType({0xbf7cfc14999fb4bc, 0x83f2f38c44988544})},
+    {true, -139, MType({0xa990c0ee569a8d51, 0xfa042ccdbce09d15})},
+    {true, -139, MType({0xa38463e9d941e1c2, 0xec223b0b32227c9e})},
+    {true, -139, MType({0xba0324530edaa03f, 0xde4011cf2daaff31})},
+    {true, -139, MType({0x5e997a02dad7ace7, 0xd05db117f419b857})},
+    {true, -139, MType({0x4a14676d4d0f817e, 0xc27b18e3c9f977c7})},
+    {true, -139, MType({0x857c002ee7a1e473, 0xb4984930f3c0481c})},
+    {true, -139, MType({0x5923b2eb72d8012a, 0xa6b541fdb5cf6d89})},
+    {true, -139, MType({0x21cde8f85ca1f9fd, 0x98d203485473648b})},
+    {true, -139, MType({0xbe08e08b1d212d4, 0x8aee8d0f13e3e09e})},
+    {true, -140, MType({0x695023998e6bd7b0, 0xfa15bea0708795e1})},
+    {true, -140, MType({0x634cea6750617a92, 0xde4df4140b42822f})},
+    {true, -140, MType({0xfbd7e970aef9dbb8, 0xc285ba757feb2781})},
+    {true, -140, MType({0x9aedc1c1ba7d0695, 0xa6bd11c1564a8ace})},
+    {true, -140, MType({0x8d306ba207233c44, 0x8af3f9f41600120a})},
+    {true, -141, MType({0x856a0a3a00fcf3c1, 0xde54e6148d030322})},
+    {true, -141, MType({0xb3a2c1407cf6d38d, 0xa6c0fa00de35f314})},
+    {true, -142, MType({0xd791cf6a70c3a504, 0xde585f4c5bbbcd3d})},
+    {true, -143, MType({0x10a633f2c4a8ea22, 0xde5a1bf627b1f68f})},
+    {true, 0, MType({0x0, 0x0})},
+    {false, -143, MType({0xed4a68e5e6e83ddf, 0xde5d95658a729eab})},
+    {false, -142, MType({0x3281f1872cdbee94, 0xde5f522b21e3e25a})},
+    {false, -141, MType({0xf1466edaa96e356e, 0xa6c8cb3b7e5bbbfd})},
+    {false, -141, MType({0x8a607fd695dfc3d9, 0xde62cbd21e895473})},
+    {false, -140, MType({0xc36b8713ceefe2de, 0x8afed57032bebc7c})},
+    {false, -140, MType({0x5c2e76c953e3e3e6, 0xa6ccb436a3c72fa4})},
+    {false, -140, MType({0x8e4950fa5c943bbf, 0xc29b023fdcb2dccf})},
+    {false, -140, MType({0x20fa8a73c585f634, 0xde69bf8f58005dfc})},
+    {false, -140, MType({0xaa106d9b0a9717a, 0xfa38ec28905810a3})},
+    {false, -139, MType({0x85d70e032de41aec, 0x8b04440780460c2a})},
+    {false, -139, MType({0xbeee21cbb82a9a78, 0x98ec49a311cc30ab})},
+    {false, -139, MType({0xabd7b0fdd8efe6f6, 0xa6d486e8ba5151a0})},
+    {false, -139, MType({0x3221c56e2c1aa912, 0xb4bcfbda377d31cc})},
+    {false, -139, MType({0x57b795a36d9c5f19, 0xc2a5a879470c7c37})},
+    {false, -139, MType({0x131ec142c053ac3b, 0xd08e8cc7a6d0c580})},
+    {false, -139, MType({0x35e3298f4bb2aa0a, 0xde77a8c714b08d28})},
+    {false, -139, MType({0x7133dafdfc44f160, 0xec60fc794ea73ee4})},
+    {false, -139, MType({0x74b37d23121c59d5, 0xfa4a87e012c533eb})},
+    {false, -138, MType({0x93bf5f4207da8a4c, 0x841a257e8f97da22})},
+    {false, -138, MType({0xfdb5990ec6057f4e, 0x8b0f22e919107c0c})},
+    {false, -138, MType({0x2d408a58b1b202fe, 0x92043c3084f41481})},
+    {false, -138, MType({0x1759381b61dfbf01, 0x98f97155b274b1ab})},
+    {false, -138, MType({0x41e90a054df4b9f1, 0x9feec25980cedbbe})},
+    {false, -138, MType({0xa1e66c6203725d50, 0xa6e42f3ccf49959d})},
+    {false, -138, MType({0x8693d36ab45bd7ce, 0xadd9b8007d365d83})},
+    {false, -138, MType({0x91e25bb40ad3f098, 0xb4cf5ca569f12da9})},
+    {false, -138, MType({0xbdf94392c4cc7f6c, 0xbbc51d2c74e07cf0})},
+    {false, -138, MType({0x6fe37973354a82f9, 0xc2baf9967d753f89})},
+    {false, -138, MType({0x97647b4267bfd801, 0xc9b0f1e4632ae79b})},
+    {false, -138, MType({0xdbf5c32a454f7bdf, 0xd0a70617058765ee})},
+    {false, -138, MType({0xd6edfe04c37ba916, 0xd79d362f441b2a92})},
+    {false, -138, MType({0x5ad3480ccfbe9890, 0xde93822dfe812587})},
+    {false, -138, MType({0xc7d9ac765be7e325, 0xe589ea14145ec764})},
+    {false, -138, MType({0x6d8f24b9a3ca011b, 0xec806de265640204})},
+    {false, -138, MType({0xf9b654807dcdd5b2, 0xf3770d99d14b4928})},
+    {false, -138, MType({0xf4513f4745663028, 0xfa6dc93b37d99326})},
+    {false, -137, MType({0xa46e9a72d80da75f, 0x80b25063bc6f2cc6})},
+    {false, -137, MType({0xee60992b51ffac4b, 0x842dca1fba19cce6})},
+    {false, -137, MType({0x1977fa1c786886b3, 0x87a951d204deeaf3})},
+    {false, -137, MType({0xe5f7c52cdf119d5, 0x8b24e77b0cb60a84})},
+    {false, -137, MType({0x3bf9d70da1021a10, 0x8ea08b1b419bf221})},
+    {false, -137, MType({0xfd0406b07523b8e6, 0x921c3cb31392ab94})},
+    {false, -137, MType({0x453ee32c020f2a8, 0x9597fc42f2a18441})},
+    {false, -137, MType({0xcfb3ec22066bf7f6, 0x9913c9cb4ed50d72})},
+    {false, -137, MType({0x215c025bd493ecf9, 0x9c8fa54c983f1cb8})},
+    {false, -137, MType({0x39c116b7ee3a83ec, 0x9f2c93192e68232b})},
+    {false, -137, MType({0xf41f4b3ede2782f0, 0xa2a8870f24ac5f66})},
+    {false, -137, MType({0x61196927723eb75c, 0xa62488ff3c735799})},
+    {false, -137, MType({0xe615e836cb1edab, 0xa9a098e9e5e2a432})},
+    {false, -137, MType({0x6981331c5fc71cfc, 0xad1cb6cf91252372})},
+    {false, -137, MType({0x5f6a4faa054f11fa, 0xb098e2b0ae6af9c2})},
+    {false, -137, MType({0x2a68bc681a74c28, 0xb4151c8dade99205})},
+    {false, -137, MType({0x382ba24d90566403, 0xb7916466ffdb9ded})},
+    {false, -137, MType({0x6ad1abe51dd22e00, 0xbb0dba3d14811652})},
+    {false, -137, MType({0x456d3f7f59b13960, 0xbe8a1e105c1f3b85})},
+    {false, -137, MType({0x738dd8b7d66e9058, 0xc2068fe1470095a4})},
+    {false, -137, MType({0x68e123fed7ff11c6, 0xc5830fb04574f4f1})},
+    {false, -137, MType({0x2f3bd09780c3aa11, 0xc8ff9d7dc7d17225})},
+    {false, -137, MType({0x3b48887f1ce36935, 0xcc7c394a3e706ec5})},
+    {false, -137, MType({0x47ddae655ecc4633, 0xcff8e31619b19578})},
+    {false, -137, MType({0x37fa81eef4819c88, 0xd3759ae1c9f9da5b})},
+    {false, -137, MType({0xff6c4a8d747c65ed, 0xd6f260adbfb37b55})},
+    {false, -137, MType({0x921c29493a33318c, 0xda6f347a6b4e0070})},
+    {false, -137, MType({0xda0631eb65e731d8, 0xddec16483d3e3c27})},
+    {false, -137, MType({0xb3da6c07d110babc, 0xe1690617a5fe4bc2})},
+    {false, -137, MType({0xf2485c7868b8835a, 0xe4e603e9160d97a6})},
+    {false, -137, MType({0x67f5b7ed01344055, 0xe8630fbcfdf0d3ae})},
+    {false, -137, MType({0xf820df445b1d0622, 0xebe02993ce31ff7b})},
+    {false, -137, MType({0xadefc674b7eca5cd, 0xef5d516df76066d0})},
+    {false, -137, MType({0xda6be6dc057d3235, 0xf2da874bea10a1e0})},
+    {false, -137, MType({0x392bdde152ab5ff5, 0xf657cb2e16dc95a9})},
+    {false, -137, MType({0x1bab58e2ec99cf73, 0xf9d51d14ee637444})},
+    {false, -137, MType({0x9b51ef7e3388d692, 0xfd527d00e149bd3e})},
+    {false, -136, MType({0xe914c6a7f3f22fa2, 0x8067f579301c9ef6})},
+    {false, -136, MType({0xd22862eb2081c94, 0x8226b374edf088e2})},
+    {false, -136, MType({0x29ebd0b476cd8fd8, 0x83e57873e27ad153})},
+    {false, -136, MType({0x98feddc2806d01ed, 0x85a44476461854a0})},
+    {false, -136, MType({0x471bfc261a401854, 0x8763177c512896af})},
+    {false, -136, MType({0xb6f89c19b4cd1acd, 0x88b23a5b61430a16})},
+    {false, -136, MType({0xb39aaf34163fb099, 0x8a7119a85909ebe9})},
+    {false, -136, MType({0x1665f0f821541c36, 0x8c2ffff99357e887})},
+    {false, -136, MType({0xa5051754e049c1cb, 0x8deeed4f489679a6})},
+    {false, -136, MType({0x8c5a9a1c57b2e986, 0x8fade1a9b131c159})},
+    {false, -136, MType({0x1d8448438a26a9ae, 0x916cdd0905988a35})},
+    {false, -136, MType({0x8e3a0913ecd2fd02, 0x932bdf6d7e3c477d})},
+    {false, -136, MType({0xbc881a45f47f1d36, 0x94eae8d753911550})},
+    {false, -136, MType({0xf5e51c05499b06d0, 0x96a9f946be0db8d0})},
+    {false, -136, MType({0xc1a43be81a243fde, 0x986910bbf62ba04f})},
+    {false, -136, MType({0xaec3cfebe971beb7, 0x9a282f373466e378})},
+    {false, -136, MType({0x2518b29328614989, 0x9be754b8b13e437c})},
+    {false, -136, MType({0x39d6b147cbe803a4, 0x9da68140a5332b3a})},
+    {false, -136, MType({0x87765e3004ae428d, 0x9f65b4cf48c9af6d})},
+    {false, -136, MType({0x8f896ab28245bac, 0xa124ef64d4888ed6})},
+    {false, -136, MType({0xf8880fb5ca630c87, 0xa2e4310180f93263})},
+    {false, -136, MType({0xb179397cf82e935c, 0xa4a379a586a7ad62})},
+    {false, -136, MType({0x95a8cb717197ad81, 0xa662c9511e22bda3})},
+    {false, -136, MType({0xf6394a34b7f9a4a4, 0xa82220047ffbcba8})},
+    {false, -136, MType({0xffafd8c2b57884e8, 0xa9e17dbfe4c6ead0})},
+    {false, -136, MType({0xa970a643b8a6ac2b, 0xaba0e283851ad980})},
+    {false, -136, MType({0xa89b49fb749d47e0, 0xad604e4f9991014e})},
+    {false, -136, MType({0x66475ed2ac983305, 0xaf1fc1245ac5772e})},
+    {false, -136, MType({0xb4fd6209364bb36f, 0xb06f5be1bf1918e7})},
+    {false, -136, MType({0x8b5ce79b0965962a, 0xb22edb0636da31d6})},
+    {false, -136, MType({0x6724232b07396427, 0xb3ee6133f7149769})},
+    {false, -136, MType({0x2f02b14dcad8a49c, 0xb5adee6b386e62ae})},
+    {false, -136, MType({0xbd6443a81f792e07, 0xb76d82ac339058db})},
+    {false, -136, MType({0xea1cd9625749939a, 0xb92d1df72125eb7c})},
+    {false, -136, MType({0x97775e3142198913, 0xbaecc04c39dd389b})},
+    {false, -136, MType({0xc2a701b809a2bc39, 0xbcac69abb6670aeb})},
+    {false, -136, MType({0x979b990f39e662e3, 0xbe6c1a15cf76d9f6})},
+    {false, -136, MType({0x88395c463ddd82b2, 0xc02bd18abdc2ca45})},
+    {false, -136, MType({0x66f451bd9ba5ed05, 0xc1eb900aba03ad8d})},
+    {false, -136, MType({0x84cfb9413f6437a6, 0xc3ab5595fcf502d9})},
+    {false, -136, MType({0xd2c1c8d32943ca42, 0xc56b222cbf54f6b6})},
+    {false, -136, MType({0x67c0d1fd95192e6, 0xc72af5cf39e4635f})},
+    {false, -136, MType({0xc298bf9edb6441f2, 0xc8ead07da566d0e3})},
+    {false, -136, MType({0xc22d646addde3910, 0xcaaab2383aa27559})},
+    {false, -136, MType({0x7c301e5c7d1ca40, 0xcc6a9aff32603504})},
+    {false, -136, MType({0xfb444464df02505, 0xce2a8ad2c56ba27f})},
+    {false, -136, MType({0x5f1df3591ae898f, 0xcfea81b32c92feec})},
+    {false, -136, MType({0xb43caf8e7b891066, 0xd13a7f7c07506f7d})},
+    {false, -136, MType({0x597fb13f0d0fdf19, 0xd2fa82b36a610c4f})},
+    {false, -136, MType({0x3c21f1c60a60b0d6, 0xd4ba8cf83dd2a06b})},
+    {false, -136, MType({0x2b7455909a0428a4, 0xd67a9e4aba7d7ce5})},
+    {false, -136, MType({0x1438b60573d2da10, 0xd83ab6ab193ca223})},
+    {false, -136, MType({0x49f86400c5ab2b11, 0xd9fad61992edc008})},
+    {false, -136, MType({0xd3c313d148a23c35, 0xdbbafc9660713620})},
+    {false, -136, MType({0xbc56852355e0f0d5, 0xdd7b2a21baaa13cc})},
 };
 
-// > P = fpminimax(log10(1 + x)/x, 4, [|192...|], [-2^-27, 2^-27]);
+// Range reduction - Step 3:
+//   r(k) = 2^-21 round(2^21 / (1 + k*2^-21)) for k = -80 .. 80.
+// Output range:
+//   [-0x1.01928p-22 , 0x1p-22]
+
+// We store S[i] = 2^21 (r(i - 80) - 1).
+constexpr int64_t 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,
+    0x2f,  0x2e,  0x2d,  0x2c,  0x2b,  0x2a,  0x29,  0x28,  0x27,  0x26,  0x25,
+    0x24,  0x23,  0x22,  0x21,  0x20,  0x1f,  0x1e,  0x1d,  0x1c,  0x1b,  0x1a,
+    0x19,  0x18,  0x17,  0x16,  0x15,  0x14,  0x13,  0x12,  0x11,  0x10,  0xf,
+    0xe,   0xd,   0xc,   0xb,   0xa,   0x9,   0x8,   0x7,   0x6,   0x5,   0x4,
+    0x3,   0x2,   0x1,   0x0,   -0x1,  -0x2,  -0x3,  -0x4,  -0x5,  -0x6,  -0x7,
+    -0x8,  -0x9,  -0xa,  -0xb,  -0xc,  -0xd,  -0xe,  -0xf,  -0x10, -0x11, -0x12,
+    -0x13, -0x14, -0x15, -0x16, -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c, -0x1d,
+    -0x1e, -0x1f, -0x20, -0x21, -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28,
+    -0x29, -0x2a, -0x2b, -0x2c, -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33,
+    -0x34, -0x35, -0x36, -0x37, -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e,
+    -0x3f, -0x40, -0x41, -0x42, -0x43, -0x44, -0x45, -0x46, -0x47, -0x48, -0x49,
+    -0x4a, -0x4b, -0x4c, -0x4d, -0x4e, -0x4f, -0x50,
+};
+
+// -log10(r) for the third step, generated by SageMath with:
+//
+// for i in range(-80, 81):
+//   r = 2^-21 * round( 2^21 / (1 + i*2^(-21)) );
+//   s, m, e = RealField(128)(r).log10().sign_mantissa_exponent();
+//   print("{false," if (s == -1) else "{true,", e, ",
+//         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+const Float128 LOG10_R3[161] = {
+    {true, -143, MType({0x54d7e49898ca0093, 0x8af8b9b322ba8c7d})},
+    {true, -143, MType({0xc321bbf16665f29c, 0x893c06529deffc3d})},
+    {true, -143, MType({0x8246df7140c3e4ae, 0x877f52e433ac7ec4})},
+    {true, -143, MType({0x1deaa9e85780e4c1, 0x85c29f67e3ef35bc})},
+    {true, -143, MType({0xcd8a5121a9162d0, 0x8405ebddaeb742cf})},
+    {true, -143, MType({0xb10486fa4644308d, 0x824938459403c7a7})},
+    {true, -143, MType({0x578a2f61eedd4be8, 0x808c849f93d3e5f0})},
+    {true, -144, MType({0x715b4a491790e8a7, 0xfd9fa1d75c4d7ea6})},
+    {true, -144, MType({0xefb6273a04c71573, 0xfa263a53c5f6eaf4})},
+    {true, -144, MType({0x474d901560c17807, 0xf6acd2b464a25420})},
+    {true, -144, MType({0x6b9a5deceb80ec57, 0xf3336af9384dfd7c})},
+    {true, -144, MType({0x2665a32f7cc64f79, 0xefba032240f82a5d})},
+    {true, -144, MType({0x17c8a67316659363, 0xec409b2f7e9f1e16})},
+    {true, -144, MType({0xb62cdd3ef5c8673d, 0xe8c73320f1411bfa})},
+    {true, -144, MType({0x4e4be6d5a4a07422, 0xe54dcaf698dc675e})},
+    {true, -144, MType({0x32f86ff08c92e22, 0xe1d462b0756f4394})},
+    {true, -144, MType({0xce31a0d27359396f, 0xde5afa4e86f7f3ee})},
+    {true, -144, MType({0x7efc3180aee36373, 0xdae191d0cd74bbc1})},
+    {true, -144, MType({0xbb894b1e0ce72fc4, 0xd768293748e3de5e})},
+    {true, -144, MType({0x230f6c7270f8be, 0xd3eec081f9439f19})},
+    {true, -144, MType({0x9f63aaa563e9a399, 0xd07557b0de924142})},
+    {true, -144, MType({0xc2354e441015e7eb, 0xccfbeec3f8ce082d})},
+    {true, -144, MType({0x67d22bcf5a452a4c, 0xc98285bb47f5372c})},
+    {true, -144, MType({0x65c46fa3e3afea18, 0xc6091c96cc061190})},
+    {true, -144, MType({0x67e63bbe1405c20d, 0xc28fb35684fedaab})},
+    {true, -144, MType({0xf061a284212afbad, 0xbf1649fa72ddd5ce})},
+    {true, -144, MType({0x57b0a1901625b539, 0xbb9ce08295a1464c})},
+    {true, -144, MType({0xcc9d1c79d93a9a1e, 0xb82376eeed476f74})},
+    {true, -144, MType({0x5440d7a131392da8, 0xb4aa0d3f79ce9499})},
+    {true, -144, MType({0xca0572f7c9f7a7de, 0xb130a3743b34f90a})},
+    {true, -144, MType({0xdfa464cb37fe6455, 0xadb7398d3178e019})},
+    {true, -144, MType({0x1d26f48efb62e2e0, 0xaa3dcf8a5c988d17})},
+    {true, -144, MType({0xe0e635a681d259e2, 0xa6c4656bbc924352})},
+    {true, -144, MType({0x5f8b022f27cbda35, 0xa34afb315164461d})},
+    {true, -144, MType({0xa40df5ca390a0465, 0x9fd190db1b0cd8c6})},
+    {true, -144, MType({0x8fb76866f01c4f2d, 0x9c582669198a3e9e})},
+    {true, -144, MType({0xda1f690c752fdeff, 0x98debbdb4cdabaf4})},
+    {true, -144, MType({0x112db8a3dc07ee78, 0x95655131b4fc9119})},
+    {true, -144, MType({0x9919c4c22125c79e, 0x91ebe66c51ee045a})},
+    {true, -144, MType({0xac6aa27226204db3, 0x8e727b8b23ad5808})},
+    {true, -144, MType({0x5bf708fead2b1780, 0x8af9108e2a38cf72})},
+    {true, -144, MType({0x8ee54cbc53cd19ed, 0x877fa575658eade6})},
+    {true, -144, MType({0x2ab59d38cc6e2c5, 0x84063a40d5ad36b4})},
+    {true, -144, MType({0x4b0eaf0a99286378, 0x808ccef07a92ad29})},
+    {true, -145, MType({0xa448b11f012c975c, 0xfa26c708a87aa929})},
+    {true, -145, MType({0xb0a1d584117de73b, 0xf333eff8c556e089})},
+    {true, -145, MType({0xe890f9fb57fdabb6, 0xec4118b14bb6870e})},
+    {true, -145, MType({0x261d48c71e693130, 0xe54e41323b962355})},
+    {true, -145, MType({0xefecdd48ed894c32, 0xde5b697b94f23bf7})},
+    {true, -145, MType({0x7944b9957598a88a, 0xd768918d57c75792})},
+    {true, -145, MType({0xa208bc0875093645, 0xd075b9678411fcbf})},
+    {true, -145, MType({0xf6bb94d89da8b432, 0xc982e10a19ceb219})},
+    {true, -145, MType({0xb07ebbab782457b0, 0xc290087518f9fe3b})},
+    {true, -145, MType({0xb512652945eb9165, 0xbb9d2fa8819067be})},
+    {true, -145, MType({0x96d57890e171eea5, 0xb4aa56a4538e753c})},
+    {true, -145, MType({0x94c5854b9cd01726, 0xadb77d688ef0ad4e})},
+    {true, -145, MType({0x9a7eb8811ec3e6bb, 0xa6c4a3f533b3968d})},
+    {true, -145, MType({0x403bd2ab3e0fa2d7, 0x9fd1ca4a41d3b792})},
+    {true, -145, MType({0xcad61d29db384b6b, 0x98def067b94d96f4})},
+    {true, -145, MType({0x2bc55fd6b8a306ec, 0x91ec164d9a1dbb4d})},
+    {true, -145, MType({0x11fd6995111a927, 0x8af93bfbe440ab33})},
+    {true, -145, MType({0x959a26faac7e5494, 0x8406617297b2ed3d})},
+    {true, -146, MType({0xc10eab7266ac6bc0, 0xfa270d6368e21007})},
+    {true, -146, MType({0xbb178b90026b2b2, 0xec41577274ef0439})},
+    {true, -146, MType({0xac3bfd925e6b33e1, 0xde5ba1125385c43b})},
+    {true, -146, MType({0x9d0a01a95b355319, 0xd075ea43049f5d3b})},
+    {true, -146, MType({0x31b3b7b20a6a6496, 0xc29033048834dc64})},
+    {true, -146, MType({0x170da891504620f4, 0xb4aa7b56de3f4ee0})},
+    {true, -146, MType({0x53289e84744549cb, 0xa6c4c33a06b7c1d9})},
+    {true, -146, MType({0x45519048b0ce7e7f, 0x98df0aae01974279})},
+    {true, -146, MType({0xa6118c42bf99407e, 0x8af951b2ced6dde8})},
+    {true, -147, MType({0xe5b474cc5a64cf6, 0xfa273090dcdf429f})},
+    {true, -147, MType({0xa74dab3bd6067bc7, 0xde5bbcddc0b533aa})},
+    {true, -147, MType({0x9f73f4e37357341b, 0xc290484c4921a941})},
+    {true, -147, MType({0x31bf5d5f815220e7, 0xa6c4d2dc7616bdb0})},
+    {true, -147, MType({0x4b987ca5fca242d7, 0x8af95c8e47868b41})},
+    {true, -148, MType({0x19be3fabd93832c5, 0xde5bcac37ac6587d})},
+    {true, -148, MType({0x8fd43f0c9ce444d3, 0xa6c4daadaf3d75e0})},
+    {true, -149, MType({0x61cd853e796bc2c, 0xde5bd1b658ad4676})},
+    {true, -150, MType({0x87d6afabfba0644f, 0xde5bd52fc7d8545f})},
+    {false, 0, MType({0x0, 0x0})},
+    {false, -150, MType({0xa9bf32001043629d, 0xde5bdc22a69d9e19})},
+    {false, -149, MType({0x8014f0f360272d82, 0xde5bdf9c1637d9ef})},
+    {false, -148, MType({0xfe94a02fc639c0e3, 0xa6c4ea5024795bd2})},
+    {false, -148, MType({0xbee710a5ace7c8d4, 0xde5be68ef5db7f99})},
+    {false, -147, MType({0x1a778d8100437e4f, 0x8af972453faf11e8})},
+    {false, -147, MType({0x97d773f8992f7051, 0xa6c4f221608e89fe})},
+    {false, -147, MType({0xc9ee5841a3afa95, 0xc29072dbdd9a0dd5})},
+    {false, -147, MType({0x7b644b13993cf4ef, 0xde5bf474b6df8331})},
+    {false, -147, MType({0x3448f66e2bd7a0ca, 0xfa2776ebec6ccfdb})},
+    {false, -146, MType({0x6a7ca5f1a87a1a3c, 0x8af97d20bf27eccd})},
+    {false, -146, MType({0x245675fe3061108f, 0x98df3f3ab64b431d})},
+    {false, -146, MType({0x64136e97019d0a3b, 0xa6c501c3dba75dc2})},
+    {false, -146, MType({0x6cdadac4d6925bd4, 0xb4aac4bc2f432fa3})},
+    {false, -146, MType({0x2899e23791d29632, 0xc2908823b125aba7})},
+    {false, -146, MType({0x28039e1f0323a4c1, 0xd0764bfa6155c4b5})},
+    {false, -146, MType({0xa2912e03afc8cc28, 0xde5c10403fda6db5})},
+    {false, -146, MType({0x7681cc9f9e0d89f9, 0xec41d4f54cba9991})},
+    {false, -146, MType({0x28dae4b7241255e1, 0xfa279a1987fd3b32})},
+    {false, -145, MType({0xf2b412f8dceda28e, 0x8406afd678d4a2c0})},
+    {false, -145, MType({0xbf5dccd967504857, 0x8af992d7c4e2d5b5})},
+    {false, -145, MType({0x3716dbf950b07f85, 0x91ec7610a82cafed})},
+    {false, -145, MType({0x69eebe0b8e5b18e1, 0x98df598122b5aadd})},
+    {false, -145, MType({0xbb583ce65af56beb, 0x9fd23d2934813ffc})},
+    {false, -145, MType({0xe22978efa7a962a0, 0xa6c52108dd92e8c1})},
+    {false, -145, MType({0xe89bf3898ef27836, 0xadb805201dee1ea3})},
+    {false, -145, MType({0x2c4c997ec90bab0b, 0xb4aae96ef5965b1a})},
+    {false, -145, MType({0x5e3bcd6f21fe6224, 0xbb9dcdf5648f179c})},
+    {false, -145, MType({0x82cd723bf1524680, 0xc290b2b36adbcda2})},
+    {false, -145, MType({0xf1c8f574935e109b, 0xc98397a9087ff6a4})},
+    {false, -145, MType({0x565959c2e4394a59, 0xd0767cd63d7f0c1c})},
+    {false, -145, MType({0xaf0d4157bc4f05be, 0xd769623b09dc8781})},
+    {false, -145, MType({0x4dd6f8576e9188b8, 0xde5c47d76d9be24e})},
+    {false, -145, MType({0xd80c7f46484eee3d, 0xe54f2dab68c095fb})},
+    {false, -145, MType({0x4667957512a6bd26, 0xec4213b6fb4e1c04})},
+    {false, -145, MType({0xe505c36d95a074fa, 0xf334f9fa2547ede1})},
+    {false, -145, MType({0x5368655f1ce3110b, 0xfa27e074e6b1850f})},
+    {false, -144, MType({0xc23a5ac57f06c112, 0x808d63939fc72d83})},
+    {false, -144, MType({0xdf39eb5890580f93, 0x8406d70897f0f4a2})},
+    {false, -144, MType({0xcd896f3e43f38669, 0x87804a995bd7d4a2})},
+    {false, -144, MType({0x83b16ff7eecace8c, 0x8af9be45eb7d8a41})},
+    {false, -144, MType({0x21ec7ae8ffa1531d, 0x8e73320e46e3d23d})},
+    {false, -144, MType({0xf227268d464ae907, 0x91eca5f26e0c6953})},
+    {false, -144, MType({0x680017af3bbaf2d3, 0x956619f260f90c44})},
+    {false, -144, MType({0x20c8069e4ae400de, 0x98df8e0e1fab77cd})},
+    {false, -144, MType({0xe381c4651a67ee13, 0x9c590245aa2568ac})},
+    {false, -144, MType({0xa0e23fffd718794e, 0x9fd2769900689ba2})},
+    {false, -144, MType({0x73508b927f485b97, 0xa34beb082276cd6d})},
+    {false, -144, MType({0x9ee5e19f2eecdb55, 0xa6c55f931051bacc})},
+    {false, -144, MType({0x916daa3c6c8fdc9d, 0xaa3ed439c9fb207f})},
+    {false, -144, MType({0xe265804b77126ed3, 0xadb848fc4f74bb45})},
+    {false, -144, MType({0x52fd36ae943fd7b4, 0xb131bddaa0c047df})},
+    {false, -144, MType({0xce16dd7f60311bf6, 0xb4ab32d4bddf830b})},
+    {false, -144, MType({0x6846c7451d8105ac, 0xb824a7eaa6d4298b})},
+    {false, -144, MType({0x5fd38e2b0650a884, 0xbb9e1d1c5b9ff81e})},
+    {false, -144, MType({0x1cb619369e1c641f, 0xbf179269dc44ab85})},
+    {false, -144, MType({0x3099a17e0461648c, 0xc29107d328c40080})},
+    {false, -144, MType({0x56dbb75e4813a12b, 0xc60a7d58411fb3d0})},
+    {false, -144, MType({0x748c47b1bbe45a07, 0xc983f2f925598236})},
+    {false, -144, MType({0x986da1064b5913e1, 0xccfd68b5d5732873})},
+    {false, -144, MType({0xfaf478d3d0b31300, 0xd076de8e516e6348})},
+    {false, -144, MType({0xfe47f0b26ba754ff, 0xd3f05482994cef77})},
+    {false, -144, MType({0x2e419b90d8e709b7, 0xd769ca92ad1089c2})},
+    {false, -144, MType({0x406d82eaca788b6f, 0xdae340be8cbaeee9})},
+    {false, -144, MType({0x140a2bff40e0d670, 0xde5cb706384ddbaf})},
+    {false, -144, MType({0xb2089d06e51d8034, 0xe1d62d69afcb0cd5})},
+    {false, -144, MType({0x4d0c626a636f2e4f, 0xe54fa3e8f3343f1f})},
+    {false, -144, MType({0x416b93f8c6f48d30, 0xe8c91a84028b2f4e})},
+    {false, -144, MType({0x152eda1dd615c6f5, 0xec42913addd19a25})},
+    {false, -144, MType({0x781173186fc07a66, 0xefbc080d85093c66})},
+    {false, -144, MType({0x43813830e974324d, 0xf3357efbf833d2d5})},
+    {false, -144, MType({0x7a9ea2ef6e1f5d41, 0xf6aef60637531a34})},
+    {false, -144, MType({0x4a3cd2525dccc623, 0xfa286d2c4268cf47})},
+    {false, -144, MType({0x8e19004ae218d5d, 0xfda1e46e1976aed1})},
+    {false, -143, MType({0x9b62aaca25d5d18a, 0x808dade5de3f3aca})},
+    {false, -143, MType({0xbee9a8d43e00613c, 0x824a69a295c0f02b})},
+    {false, -143, MType({0xd8d4b69c2056f729, 0x8407256d334155ed})},
+    {false, -143, MType({0xe7cc28605d7bb77e, 0x85c3e145b6c14a72})},
+    {false, -143, MType({0xff51b4bdc834a8f1, 0x87809d2c2041ac1c})},
+    {false, -143, MType({0x47c0774aa81c3561, 0x893d59206fc3594e})},
+    {false, -143, MType({0xfe4cf331ecb9eb62, 0x8afa1522a5473068})},
+};
+
+// Range reduction - Step 4
+//   r(k) = 2^-28 round(2^28 / (1 + k*2^-28)) for k = -65 .. 64.
+// Output range:
+//   [-0x1.0002143p-29 , 0x1p-29]
+
+// We store S[i] = 2^28 (r(i - 65) - 1).
+constexpr int64_t 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,
+    0x20,  0x1f,  0x1e,  0x1d,  0x1c,  0x1b,  0x1a,  0x19,  0x18,  0x17,  0x16,
+    0x15,  0x14,  0x13,  0x12,  0x11,  0x10,  0xf,   0xe,   0xd,   0xc,   0xb,
+    0xa,   0x9,   0x8,   0x7,   0x6,   0x5,   0x4,   0x3,   0x2,   0x1,   0x0,
+    -0x1,  -0x2,  -0x3,  -0x4,  -0x5,  -0x6,  -0x7,  -0x8,  -0x9,  -0xa,  -0xb,
+    -0xc,  -0xd,  -0xe,  -0xf,  -0x10, -0x11, -0x12, -0x13, -0x14, -0x15, -0x16,
+    -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c, -0x1d, -0x1e, -0x1f, -0x20, -0x21,
+    -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28, -0x29, -0x2a, -0x2b, -0x2c,
+    -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33, -0x34, -0x35, -0x36, -0x37,
+    -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e, -0x3f, -0x40,
+};
+
+// -log10(r) for the fourth step, generated by SageMath with:
+//
+// for i in range(-65, 65):
+//   r = 2^-28 * round( 2^28 / (1 + i*2^(-28)) );
+//   s, m, e = RealField(128)(r).log10().sign_mantissa_exponent();
+//   print("{false," if (s == -1) else "{true,", e, ",
+//         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+const Float128 LOG10_R4[130] = {
+    {true, -151, MType({0xe471a82bbedbe0ae, 0xe1d5464122cf95a4})},
+    {true, -151, MType({0xaf6e93be8e4c1764, 0xde5bd6ec7f7bc110})},
+    {true, -151, MType({0xe44848f0a5779499, 0xdae26797a490f80e})},
+    {true, -151, MType({0x90205533f4e70566, 0xd768f842920f3a98})},
+    {true, -151, MType({0xc01844ace3729e48, 0xd3ef88ed47f688a6})},
+    {true, -151, MType({0x8151a2324e41c7c4, 0xd0761997c646e232})},
+    {true, -151, MType({0xe0edf74d88cacafd, 0xccfcaa420d004734})},
+    {true, -151, MType({0xec0ecc3a5cd27e58, 0xc9833aec1c22b7a6})},
+    {true, -151, MType({0xafd5a7e70a6bf214, 0xc609cb95f3ae3381})},
+    {true, -151, MType({0x39640ff447f81ceb, 0xc2905c3f93a2babe})},
+    {true, -151, MType({0x95db88b5422588b1, 0xbf16ece8fc004d55})},
+    {true, -151, MType({0xd25d952f9beffeec, 0xbb9d7d922cc6eb40})},
+    {true, -151, MType({0xfc0bb71b6ea03578, 0xb8240e3b25f69478})},
+    {true, -151, MType({0x20076ee349cb7b20, 0xb4aa9ee3e78f48f7})},
+    {true, -151, MType({0x4b723ba43353643d, 0xb1312f8c719108b4})},
+    {true, -151, MType({0x8b6d9b2da7657754, 0xadb7c034c3fbd3a9})},
+    {true, -151, MType({0xed1b0a01987ad9b4, 0xaa3e50dcdecfa9cf})},
+    {true, -151, MType({0x7d9c03546f57fc11, 0xa6c4e184c20c8b20})},
+    {true, -151, MType({0x4a12010d0b0c4727, 0xa34b722c6db27794})},
+    {true, -151, MType({0x5f9e7bc4c0f1c851, 0x9fd202d3e1c16f24})},
+    {true, -151, MType({0xcb62eac75cacde29, 0x9c58937b1e3971c9})},
+    {true, -151, MType({0x9a80c413202be52a, 0x98df2422231a7f7d})},
+    {true, -151, MType({0xda197c58c3a6e445, 0x9565b4c8f0649838})},
+    {true, -151, MType({0x974e86fb759f3988, 0x91ec456f8617bbf4})},
+    {true, -151, MType({0xdf415610dadf46b3, 0x8e72d615e433eaa9})},
+    {true, -151, MType({0xbf135a610e7a1ddc, 0x8af966bc0ab92451})},
+    {true, -151, MType({0x43e60366a1cb2e09, 0x877ff761f9a768e5})},
+    {true, -151, MType({0x7adabf4e9c75efce, 0x84068807b0feb85d})},
+    {true, -151, MType({0x7112faf87c6591ee, 0x808d18ad30bf12b3})},
+    {true, -152, MType({0x676043ec6b994be5, 0xfa2752a4f1d0efc0})},
+    {true, -152, MType({0x9fa73d186649999d, 0xf33473ef12f5cfb9})},
+    {true, -152, MType({0xa53db362aa5cc6f0, 0xec419538c4ecc544})},
+    {true, -152, MType({0x9266761de5e05f13, 0xe54eb68207b5d053})},
+    {true, -152, MType({0x81645201b36e17ba, 0xde5bd7cadb50f0d8})},
+    {true, -152, MType({0x8c7a112a9a2b2a52, 0xd768f9133fbe26c5})},
+    {true, -152, MType({0xcdea7b1a0dc7ad42, 0xd0761a5b34fd720c})},
+    {true, -152, MType({0x5ff854b66e7ded1f, 0xc9833ba2bb0ed2a0})},
+    {true, -152, MType({0x5ce6604b0911c5ed, 0xc2905ce9d1f24872})},
+    {true, -152, MType({0xdef75d8816cffc59, 0xbb9d7e3079a7d374})},
+    {true, -152, MType({0x6e0982bd8d96ef, 0xb4aa9f76b22f739a})},
+    {true, -152, MType({0xdb8d1eb50fa7375c, 0xadb7c0bc7b8928d3})},
+    {true, -152, MType({0x8a9754fe0c0073a7, 0xa6c4e201d5b4f314})},
+    {true, -152, MType({0x27cf61a19e032f69, 0x9fd20346c0b2d24e})},
+    {true, -152, MType({0xcd77f7489d9ef50b, 0x98df248b3c82c672})},
+    {true, -152, MType({0x95d3c600cf484f03, 0x91ec45cf4924cf74})},
+    {true, -152, MType({0x9b257b3ce3f82109, 0x8af96712e698ed45})},
+    {true, -152, MType({0xf7afc1d4792b015a, 0x8406885614df1fd7})},
+    {true, -153, MType({0x8b6a840831c123d8, 0xfa275331a7eece3b})},
+    {true, -153, MType({0x3ef142da7335b35a, 0xec4195b647c38612})},
+    {true, -153, MType({0x3e79062c7cbb3b7d, 0xde5bd83a093c6718})},
+    {true, -153, MType({0xbe870ed4ed5b755b, 0xd0761abcec597131})},
+    {true, -153, MType({0xf3a098743d20fb64, 0xc2905d3ef11aa442})},
+    {true, -153, MType({0x124ad974bd15fbca, 0xb4aa9fc017800030})},
+    {true, -153, MType({0x4f0b030a9742eb00, 0xa6c4e2405f8984dd})},
+    {true, -153, MType({0xde664133cead362d, 0x98df24bfc937322e})},
+    {true, -153, MType({0xf4e1bab83f55f5a1, 0x8af9673e54890808})},
+    {true, -154, MType({0x8e0522533c713e98, 0xfa27537802fe0c9f})},
+    {true, -154, MType({0x129bc1c6f293726e, 0xde5bd871a03259cf})},
+    {true, -154, MType({0xe09182166eeb17eb, 0xc2905d6980aef768})},
+    {true, -154, MType({0x60f08720313daa3f, 0xa6c4e25fa473e535})},
+    {true, -154, MType({0xfcc2ea566b3af38b, 0x8af967540b8122fc})},
+    {true, -155, MType({0x3a25757e00f4e3a0, 0xde5bd88d6bad6110})},
+    {true, -155, MType({0x55d3f9e70cf177b8, 0xa6c4e26f46e91b3e})},
+    {true, -156, MType({0x3d4aac85125398d0, 0xde5bd89b516ae82a})},
+    {true, -157, MType({0x9ab5a849a06f400d, 0xde5bd8a24449ac95})},
+    {false, 0, MType({0x0, 0x0})},
+    {false, -157, MType({0xd3cc88fd4ef34c2, 0xde5bd8b02a073729})},
+    {false, -156, MType({0x225916c2b3f33c90, 0xde5bd8b71ce5fd51})},
+    {false, -155, MType({0x17847f98acf08d54, 0xa6c4e28e8bd3930a})},
+    {false, -155, MType({0x44397830931fddd, 0xde5bd8c502a38b5e})},
+    {false, -154, MType({0xc2ab385913176984, 0x8af9677f79717409})},
+    {false, -154, MType({0xe454dec82bde52e5, 0xa6c4e29e2e48d4cc})},
+    {false, -154, MType({0xfe1522b0470d7d7f, 0xc2905dbe9fd7e82f})},
+    {false, -154, MType({0xa6e2721f2afc3cce, 0xde5bd8e0ce1eae6a})},
+    {false, -154, MType({0x75b3458eec3c106c, 0xfa275404b91d27b4})},
+    {false, -153, MType({0x80bf0ff2f6cd9f93, 0x8af967953069aa22})},
+    {false, -153, MType({0xf09cc73b7013b906, 0x98df2528e2a09a29})},
+    {false, -153, MType({0x55ee1480619827c4, 0xa6c4e2bd7333640c})},
+    {false, -153, MType({0x7c2e48d772250b3c, 0xb4aaa052e22207e5})},
+    {false, -153, MType({0x2ed8ba8c6fa81c98, 0xc2905de92f6c85d1})},
+    {false, -153, MType({0x3968c5214f33fc4f, 0xd0761b805b12ddeb})},
+    {false, -153, MType({0x6759c94e2d017fad, 0xde5bd9186515104f})},
+    {false, -153, MType({0x84272d014c70fe58, 0xec4196b14d731d19})},
+    {false, -153, MType({0x5b4c5b5f180b9fe1, 0xfa27544b142d0465})},
+    {false, -152, MType({0x5c22626110c254a4, 0x840688f2dca16327})},
+    {false, -152, MType({0xb345ef5d90dd6545, 0x8af967c09e5a3178})},
+    {false, -152, MType({0x98ce92087c5cb614, 0x91ec468ecf40ed34})},
+    {false, -152, MType({0xf27a0a6056dcfe57, 0x98df255d6f559668})},
+    {false, -152, MType({0xa6061afeb7929f24, 0x9fd2042c7e982d23})},
+    {false, -152, MType({0x99308918494a4a20, 0xa6c4e2fbfd08b172})},
+    {false, -152, MType({0xb1b71c7cca69a844, 0xadb7c1cbeaa72363})},
+    {false, -152, MType({0xd5579f970cf000a9, 0xb4aaa09c47738304})},
+    {false, -152, MType({0xe9cfdf6cf676df42, 0xbb9d7f6d136dd063})},
+    {false, -152, MType({0xd4ddab9f8032bbab, 0xc2905e3e4e960b8e})},
+    {false, -152, MType({0x7c3ed66ab6f39fe9, 0xc9833d0ff8ec3493})},
+    {false, -152, MType({0xc5b134a5bb25cf2e, 0xd0761be212704b7f})},
+    {false, -152, MType({0x96f29dc2c0d26ca0, 0xd768fab49b225061})},
+    {false, -152, MType({0xd5c0ebcf0fa0221e, 0xde5bd98793024346})},
+    {false, -152, MType({0x67d9fb7302d3c705, 0xe54eb85afa10243d})},
+    {false, -152, MType({0x32fbabf2095106f1, 0xec41972ed04bf353})},
+    {false, -152, MType({0x1ce3df2aa59b0889, 0xf334760315b5b096})},
+    {false, -152, MType({0xb5079966dd5143e, 0xfa2754d7ca4d5c14})},
+    {false, -151, MType({0x71ffb12505e19d89, 0x808d19d677097aed})},
+    {false, -151, MType({0x4657417a9e657eae, 0x8406894140833efc})},
+    {false, -151, MType({0x758de3f168f9f8c9, 0x877ff8ac4193fa3d})},
+    {false, -151, MType({0xf2828ffc57f43581, 0x8af968177a3bacb7})},
+    {false, -151, MType({0xb0143e5be77b1053, 0x8e72d782ea7a5672})},
+    {false, -151, MType({0xa121e91e1d8769ef, 0x91ec46ee924ff774})},
+    {false, -151, MType({0xb88a8b9e89e47b9c, 0x9565b65a71bc8fc4})},
+    {false, -151, MType({0xe92d228646302a9c, 0x98df25c688c01f69})},
+    {false, -151, MType({0x25e8abcbf5db5b8c, 0x9c589532d75aa66b})},
+    {false, -151, MType({0x619c26b3c62a45c8, 0x9fd2049f5d8c24cf})},
+    {false, -151, MType({0x8f2693cf6e34c6cc, 0xa34b740c1b549a9d})},
+    {false, -151, MType({0xa166f4fe2ee6b59a, 0xa6c4e37910b407dc})},
+    {false, -151, MType({0x8b3c4d6cd3003616, 0xaa3e52e63daa6c93})},
+    {false, -151, MType({0x3f85a195af160c71, 0xadb7c253a237c8c9})},
+    {false, -151, MType({0xb121f740a191f084, 0xb13131c13e5c1c84})},
+    {false, -151, MType({0xd2f0558312b2e136, 0xb4aaa12f121767cc})},
+    {false, -151, MType({0x97cfc4bff48d77de, 0xb824109d1d69aaa8})},
+    {false, -151, MType({0xf29f4ea7c30c3ba5, 0xbb9d800b6052e51e})},
+    {false, -151, MType({0xd63dfe3883eff4e9, 0xbf16ef79dad31736})},
+    {false, -151, MType({0x358adfbdc6d0009f, 0xc2905ee88cea40f7})},
+    {false, -151, MType({0x36500d0a51aa3b6, 0xc609ce5776986267})},
+    {false, -151, MType({0x32ab7057c2155e78, 0xc9833dc697dd7b8d})},
+    {false, -151, MType({0xb63d3e874add3ff0, 0xccfcad35f0b98c70})},
+    {false, -151, MType({0x80f97ce0f6673948, 0xd0761ca5812c9518})},
+    {false, -151, MType({0x85bf3e340580712d, 0xd3ef8c154936958b})},
+    {false, -151, MType({0xb76d969d42ce9734, 0xd768fb8548d78dd0})},
+    {false, -151, MType({0x8e39b8702d0373a, 0xdae26af5800f7def})},
+    {false, -151, MType({0x6d0063a923dd0cc6, 0xde5bda65eede65ed})},
+};
+
+// > P = fpminimax(log10(1 + x)/x, 3, [|128...|], [-0x1.0002143p-29 , 0x1p-29]);
 // > P;
-// > dirtyinfnorm(log10(1 + x)/x - P, [-2^-27, 2^-27]);
-// 0x1.287a7...p-143
-const Float192 BIG_COEFFS[5]{
-    {false, -194,
-     MType({0x1fc14ee0c0158d73, 0x762ec7601912bf70, 0x58f189dd49436234})},
-    {true, -195,
-     MType({0x8f42a80e947f6357, 0x67836140b941fe04, 0xde5bd8a93728747a})},
-    {false, -194,
-     MType({0xf6f00690b1fa8ba9, 0x78e7c71fc8ca2d3a, 0x943d3b1b7a1af663})},
-    {true, -191,
-     MType({0x255a69358264a2d1, 0xa6ab7555f5a64d33, 0x1bcb7b1526e50e32})},
-    {false, -193,
-     MType({0x3ee3460246fdf301, 0x355baaafad33dc32, 0xde5bd8a937287195})},
+// > dirtyinfnorm(log10(1 + x)/x - P, [-0x1.0002143p-29 , 0x1p-29]);
+// 0x1.64fb8...p-123
+const Float128 BIG_COEFFS[4]{
+    {true, -131, MType({0x6903c4ce1582517d, 0xde5bd8a9373f89a7})},
+    {false, -130, MType({0xb8a21791624e2e8a, 0x943d3b1b7a1af679})},
+    {true, -130, MType({0x355baaafabc25990, 0xde5bd8a937287195})},
+    {false, -129, MType({0x355baaafad33dbd9, 0xde5bd8a937287195})},
 };
 
 // Reuse the output of the fast pass range reduction.
-// |m_x| < 2^-7
+// -2^-8 <= m_x < 2^-7
 double log10_accurate(int e_x, int index, double m_x) {
-  Float192 e_x_f192(static_cast<float>(e_x));
-  Float192 sum = fputil::quick_add(LOG10_R_F192[index],
-                                   fputil::quick_mul(LOG10_2, e_x_f192));
-
-  fputil::DoubleDouble mx{/*lo*/ 0.0, /*hi*/ m_x};
-
-  // Further range reductions.
-  double scale = 0x1.0p+7;
-  for (size_t i = 0; i < R_STEPS; ++i) {
-    scale *= 0x1.0p+4;
-    int id = static_cast<int>(fputil::multiply_add(mx.hi, scale, 0x1.0p+4));
-    double r = RR[i][id];
-    fputil::DoubleDouble rm = fputil::exact_mult(r, mx.hi);
-    rm.hi += r - 1.0;
-    rm.lo = fputil::multiply_add(r, mx.lo, rm.lo);
-    mx = fputil::exact_add(rm.hi, rm.lo);
-    sum = fputil::quick_add(sum, LOG10_RR[i][id]);
-  }
-  // Now |m_x| <= 2^-27
-  Float192 m_hi(mx.hi);
-  Float192 m_lo(mx.lo);
-  Float192 m = fputil::quick_add(m_hi, m_lo);
-  Float192 p = fputil::quick_mul(m, BIG_COEFFS[0]);
-
-  for (size_t i = 1; i < 5; ++i) {
-    auto aa = fputil::quick_add(p, BIG_COEFFS[i]);
-    p = fputil::quick_mul(m, aa);
-  };
 
-  return static_cast<double>(fputil::quick_add(sum, p));
+  Float128 e_x_f128(static_cast<float>(e_x));
+  Float128 sum = fputil::quick_mul(LOG10_2, e_x_f128);
+
+  sum = fputil::quick_add(sum, LOG10_R1[index]);
+
+  int64_t v = static_cast<int64_t>(m_x * 0x1.0p60); // ulp = 2^-60
+
+  // Range reduction - Step 2
+  // Output range: vv2 in [-0x1.3ffcp-15, 0x1.3e3dp-15].
+  // idx2 = trunc(2^14 * (v + 2^-8 + 2^-15))
+  size_t idx2 = static_cast<size_t>((v + 0x10'2000'0000'0000) >> 46);
+  sum = fputil::quick_add(sum, LOG10_R2[idx2]);
+
+  int64_t s2 = S2[idx2];            // |s| <= 2^-7, ulp = 2^-16
+  int64_t sv2 = s2 * v;             // |s*v| < 2^-14, ulp = 2^(-60-16) = 2^-76
+  int64_t spv2 = (s2 << 44) + v;    // |s + v| < 2^-14, ulp = 2^-60
+  int64_t vv2 = (spv2 << 16) + sv2; // |vv2| < 2^-14, ulp = 2^-76
+
+  // Range reduction - Step 3
+  // Output range: vv3 in [-0x1.01928p-22 , 0x1p-22]
+  // idx3 = trunc(2^21 * (v + 80*2^-21 + 2^-22))
+  size_t idx3 = static_cast<size_t>((vv2 + 0x2840'0000'0000'0000) >> 55);
+  sum = fputil::quick_add(sum, LOG10_R3[idx3]);
+
+  int64_t s3 = S3[idx3];           // |s| < 2^-13, ulp = 2^-21
+  int64_t spv3 = (s3 << 55) + vv2; // |s + v| < 2^-21, ulp = 2^-76
+  // |s*v| < 2^-27, ulp = 2^(-76-21) = 2^-97
+  __int128_t sv3 = static_cast<__int128_t>(s3) * static_cast<__int128_t>(vv2);
+  // |vv3| < 2^-21, ulp = 2^-97
+  __int128_t vv3 = (static_cast<__int128_t>(spv3) << 21) + sv3;
+
+  // Range reduction - Step 4
+  // Output range: vv4 in [-0x1.0002143p-29 , 0x1p-29]
+  // idx4 = trunc(2^21 * (v + 65*2^-28 + 2^-29))
+  size_t idx4 = static_cast<size_t>((static_cast<int>(vv3 >> 68) + 131) >> 1);
+
+  sum = fputil::quick_add(sum, LOG10_R4[idx4]);
+
+  __int128_t s4 = static_cast<__int128_t>(S4[idx4]); // |s| < 2^-21, ulp = 2^-28
+  // |s + v| < 2^-28, ulp = 2^-97
+  __int128_t spv4 = (s4 << 69) + vv3;
+  // |s*v| < 2^-42, ulp = 2^(-97-28) = 2^-125
+  __int128_t sv4 = s4 * vv3;
+  // |vv4| < 2^-28, ulp = 2^-125
+  __int128_t vv4 = (spv4 << 28) + sv4;
+
+  Float128 v_f128 = vv4 < 0
+                        ? Float128(true, -125,
+                                   MType({static_cast<uint64_t>(-vv4),
+                                          static_cast<uint64_t>((-vv4) >> 64)}))
+                        : Float128(false, -125,
+                                   MType({static_cast<uint64_t>(vv4),
+                                          static_cast<uint64_t>(vv4 >> 64)}));
+
+  // Polynomial approximation
+  Float128 p = fputil::quick_mul(v_f128, BIG_COEFFS[0]);
+  p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[1]));
+  p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[2]));
+  p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[3]));
+
+  Float128 r = fputil::quick_add(sum, p);
+
+  return static_cast<double>(r);
 }
 
 } // namespace
 
-// TODO(lntue):  Make the implementation correctly rounded for non-FMA targets.
 LLVM_LIBC_FUNCTION(double, log10, (double x)) {
   using FPBits_t = typename fputil::FPBits<double>;
   FPBits_t xbits(x);
-  int x_e = -1023;
+  uint64_t x_u = xbits.uintval();
+
+  int x_e = -FPBits_t::EXPONENT_BIAS;
+
+  if (LIBC_UNLIKELY(x_u == 0x3FF0'0000'0000'0000ULL)) {
+    // log10(1.0) = +0.0
+    return 0.0;
+  }
 
   if (LIBC_UNLIKELY(xbits.uintval() < FPBits_t::MIN_NORMAL ||
                     xbits.uintval() > FPBits_t::MAX_NORMAL)) {
     if (xbits.is_zero()) {
       // return -Inf and raise FE_DIVBYZERO.
       fputil::set_errno_if_required(ERANGE);
       fputil::raise_except_if_required(FE_DIVBYZERO);
       return static_cast<double>(FPBits_t::neg_inf());
     }
     if (xbits.get_sign() && !xbits.is_nan()) {
       fputil::set_errno_if_required(EDOM);
       fputil::raise_except_if_required(FE_INVALID);
       return FPBits_t::build_quiet_nan(0);
     }
     if (xbits.is_inf_or_nan()) {
       return x;
     }
     // Normalize denormal inputs.
     xbits.set_val(x * 0x1.0p52);
     x_e -= 52;
+    x_u = xbits.uintval();
   }
 
   // log10(x) = log10(2^x_e * x_m)
   //          = x_e * log10(2) + log10(x_m)
 
   // Range reduction for log10(x_m):
-  // For each x_m, we would like to find R such that:
-  //   |R * x_m - 1| < C
-  uint64_t x_u = xbits.uintval();
+  // For each x_m, we would like to find r such that:
+  //   -2^-8 <= r * x_m - 1 < 2^-7
   int shifted = x_u >> 45;
   int index = shifted & 0x7F;
-  double r = R[index];
+  double r = RD[index];
 
-  x_e += (x_u >> 52) & 0x7FF;
+  // Add unbiased exponent. Add an extra 1 if the 8 leading fractional bits are
+  // all 1's.
+  x_e += static_cast<int>((x_u + (1ULL << 45)) >> 52);
   double e_x = static_cast<double>(x_e);
 
-  int e_err = (e_x == -1) && (index == 0x7F);
-  int logr_err = (index == 0);
-
-  double err =
-      fputil::multiply_add(e_x, LOG10_2_ULP[e_err], LOG10_R_ULP[logr_err]);
-
   // hi is exact
-  double hi = fputil::multiply_add(e_x, LOG10_2_HI, LOG10_R[index].hi);
-  // lo errors ~ e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo) + rounding err
-  //           <= 2 * (e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo))
-  double lo = fputil::multiply_add(e_x, LOG10_2_LO, LOG10_R[index].lo);
-  // A bound on the error is given
-  //  in "Note on FastTwoSum with Directed Roundings"
-  //  by Paul Zimmermann, https://hal.inria.fr/hal-03798376, 2022.
-  //  Theorem 1 says that
-  //  the difference between a+b and hi+lo is bounded by 2u^2|a+b|
-  //  and also by 2u^2|hi|. Here u=2^-53, thus we get:
-  //  |(a+b)-(hi+lo)| <= 2^-105 min(|a+b|,|hi|)
-  // So the overall errors <= 2^-105 min(|a+b|, |hi|) + 2*(...)
-  fputil::DoubleDouble rr = fputil::exact_add(hi, lo);
+  double hi = fputil::multiply_add(e_x, LOG_2_HI, LOG_R_DD[index].hi);
+  // lo errors ~ e_x * LSB(LOG_2_LO) + LSB(LOG_R[index].lo) + rounding err
+  //           <= 2 * (e_x * LSB(LOG_2_LO) + LSB(LOG_R[index].lo))
+  double lo = fputil::multiply_add(e_x, LOG_2_LO, LOG_R_DD[index].lo);
 
+  // Set m = 1.mantissa.
   uint64_t x_m = (x_u & 0x000F'FFFF'FFFF'FFFFULL) | 0x3FF0'0000'0000'0000ULL;
   double m = FPBits_t(x_m).get_val();
 
-  double u = fputil::multiply_add(r, m, -1.0); // exact
-  err = fputil::multiply_add(u, P_ERR, err);
+  double u, u_sq, err;
+  fputil::DoubleDouble r1;
 
+  // Perform exact range reduction
+#ifdef LIBC_TARGET_CPU_HAS_FMA
+  u = fputil::multiply_add(r, m, -1.0); // exact
+#else
+  uint64_t c_m = x_m & 0x3FFF'E000'0000'0000ULL;
+  double c = FPBits_t(c_m).get_val();
+  u = fputil::multiply_add(r, m - c, CD[index]); // exact
+#endif // LIBC_TARGET_CPU_HAS_FMA
+
+  // Error of u_sq = ulp(u^2);
+  u_sq = u * u;
   // Degree-7 minimax polynomial
-  double u_sq = u * u;
-  double p0 = u * COEFFS[0];
-  double p1 = fputil::multiply_add(u, COEFFS[2], COEFFS[1]);
-  double p2 = fputil::multiply_add(u, COEFFS[4], COEFFS[3]);
-  double p3 = fputil::multiply_add(u, COEFFS[6], COEFFS[5]);
-  double p01 = fputil::multiply_add(u_sq, p1, p0);
-  double p23 = fputil::multiply_add(u_sq, p3, p2);
-  double u4 = u_sq * u_sq;
-  fputil::DoubleDouble re = fputil::exact_add(rr.hi, p01);
-  double ll = fputil::multiply_add(u4, p23, re.lo + rr.lo);
+  double p0 = fputil::multiply_add(u, LOG_COEFFS[1], LOG_COEFFS[0]);
+  double p1 = fputil::multiply_add(u, LOG_COEFFS[3], LOG_COEFFS[2]);
+  double p2 = fputil::multiply_add(u, LOG_COEFFS[5], LOG_COEFFS[4]);
+  double p = fputil::polyeval(u_sq, lo, p0, p1, p2);
+
+  // Exact sum:
+  //   r1.hi + r1.lo = e_x * log(2)_hi - log(r)_hi + u
+  r1 = fputil::exact_add(hi, u);
+  r1.lo += p;
+
+  // Quick double-double multiplication:
+  //   r2.hi + r2.lo ~ r1 * log10(e),
+  // with error bounded by:
+  //   4*ulp( ulp(r2.hi) )
+  fputil::DoubleDouble r2 = fputil::quick_mult(r1, LOG10_E);
+
+  // Technicallly error of r1.lo is bounded by:
+  //    |hi|*ulp(log(2)_lo) + C*ulp(u^2)
+  // To simplify the error computation a bit, we replace |hi|*ulp(log(2)_lo)
+  // with the upper bound: 2^11 * ulp(log(2)_lo) = 2^-85.
+  // Total error is bounded by ~ C * ulp(u^2).
+  err = fputil::multiply_add(u_sq, P_ERR, HI_ERR);
+
   // Lower bound from the result
-  double left = re.hi + (ll - err);
+  double left = r2.hi + (r2.lo - err);
   // Upper bound from the result
-  double right = re.hi + (ll + err);
+  double right = r2.hi + (r2.lo + err);
 
   // Ziv's test if fast pass is accurate enough.
   if (left == right)
     return left;
 
   // Exact cases:
+  if (LIBC_UNLIKELY((x_u & 0x3FFFFF) == 0)) {
     switch (x_u) {
-  case 0x3ff0000000000000: // x = 1.0
-    return 0.0;
     case 0x4024000000000000: // x = 10.0
       return 1.0;
     case 0x4059000000000000: // x = 10^2
       return 2.0;
     case 0x408f400000000000: // x = 10^3
       return 3.0;
     case 0x40c3880000000000: // x = 10^4
       return 4.0;
     case 0x40f86a0000000000: // x = 10^5
       return 5.0;
     case 0x412e848000000000: // x = 10^6
       return 6.0;
     case 0x416312d000000000: // x = 10^7
       return 7.0;
     case 0x4197d78400000000: // x = 10^8
       return 8.0;
     case 0x41cdcd6500000000: // x = 10^9
       return 9.0;
     case 0x4202a05f20000000: // x = 10^10
       return 10.0;
     case 0x42374876e8000000: // x = 10^11
       return 11.0;
     case 0x426d1a94a2000000: // x = 10^12
       return 12.0;
     case 0x42a2309ce5400000: // x = 10^13
       return 13.0;
+    }
+  } else {
+    switch (x_u) {
     case 0x42d6bcc41e900000: // x = 10^14
       return 14.0;
     case 0x430c6bf526340000: // x = 10^15
       return 15.0;
     case 0x4341c37937e08000: // x = 10^16
       return 16.0;
     case 0x4376345785d8a000: // x = 10^17
       return 17.0;
     case 0x43abc16d674ec800: // x = 10^18
       return 18.0;
     case 0x43e158e460913d00: // x = 10^19
       return 19.0;
     case 0x4415af1d78b58c40: // x = 10^20
       return 20.0;
     case 0x444b1ae4d6e2ef50: // x = 10^21
       return 21.0;
     case 0x4480f0cf064dd592: // x = 10^22
       return 22.0;
     }
+  }
 
   return log10_accurate(x_e, index, u);
 }
 
 } // namespace __llvm_libc

libc/test/src/math/CMakeLists.txt

SUITE
    libc_math_unittests
  SRCS
    log10_test.cpp
  DEPENDS
    libc.src.errno.errno
    libc.include.math
    libc.src.math.log10
    libc.src.__support.FPUtil.fp_bits
- FLAGS
-   FMA_OPT__ONLY
 )
 
 add_fp_unittest(
   log10f_test
   NEED_MPFR
   SUITE
     libc_math_unittests
   SRCS

libc/test/src/math/log10_test.cpp

 #include "test/UnitTest/Test.h"
 #include "utils/MPFRWrapper/MPFRUtils.h"
 #include <math.h>
 
 #include <errno.h>
 #include <stdint.h>
 
 namespace mpfr = __llvm_libc::testing::mpfr;
+using __llvm_libc::testing::tlog;
 
 DECLARE_SPECIAL_CONSTANTS(double)
 
 TEST(LlvmLibcLog10Test, SpecialNumbers) {
   EXPECT_FP_EQ(aNaN, __llvm_libc::log10(aNaN));
   EXPECT_FP_EQ(inf, __llvm_libc::log10(inf));
-  EXPECT_TRUE(FPBits(__llvm_libc::log10(neg_inf)).is_nan());
-  EXPECT_FP_EQ(neg_inf, __llvm_libc::log10(0.0));
-  EXPECT_FP_EQ(neg_inf, __llvm_libc::log10(-0.0));
-  EXPECT_TRUE(FPBits(__llvm_libc::log10(-1.0)).is_nan());
-  EXPECT_FP_EQ(zero, __llvm_libc::log10(1.0));
+  EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log10(neg_inf), FE_INVALID);
+  EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log10(0.0), FE_DIVBYZERO);
+  EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log10(-0.0), FE_DIVBYZERO);
+  EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log10(-1.0), FE_INVALID);
+  EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::log10(1.0));
 }
 
 TEST(LlvmLibcLog10Test, TrickyInputs) {
-  constexpr int N = 27;
+  constexpr int N = 35;
   constexpr uint64_t INPUTS[N] = {
       0x3ff0000000000000, // x = 1.0
       0x4024000000000000, // x = 10.0
       0x4059000000000000, // x = 10^2
       0x408f400000000000, // x = 10^3
       0x40c3880000000000, // x = 10^4
       0x40f86a0000000000, // x = 10^5
       0x412e848000000000, // x = 10^6
       0x416312d000000000, // x = 10^7
       0x4197d78400000000, // x = 10^8
       0x41cdcd6500000000, // x = 10^9
       0x4202a05f20000000, // x = 10^10
       0x42374876e8000000, // x = 10^11
       0x426d1a94a2000000, // x = 10^12
       0x42a2309ce5400000, // x = 10^13
       0x42d6bcc41e900000, // x = 10^14
       0x430c6bf526340000, // x = 10^15
       0x4341c37937e08000, // x = 10^16
       0x4376345785d8a000, // x = 10^17
       0x43abc16d674ec800, // x = 10^18
       0x43e158e460913d00, // x = 10^19
       0x4415af1d78b58c40, // x = 10^20
       0x444b1ae4d6e2ef50, // x = 10^21
       0x4480f0cf064dd592, // x = 10^22
-      0x3fefffffffef06ad, 0x3fefde0f22c7d0eb,
-      0x225e7812faadb32f, 0x3fee1076964c2903,
-  };
+      0x3fefffffffef06ad, 0x3fefde0f22c7d0eb, 0x225e7812faadb32f,
+      0x3fee1076964c2903, 0x3fdfe93fff7fceb0, 0x3ff012631ad8df10,
+      0x3fefbfdaa448ed98, 0x44b0c9705a25ce02, 0x2c88d301065c7f9b,
+      0x30160580e7268a99, 0x5ca04103b7eaa345, 0x19ad77dc4a40093f};
   for (int i = 0; i < N; ++i) {
     double x = double(FPBits(INPUTS[i]));
     EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log10, x,
                                    __llvm_libc::log10(x), 0.5);
   }
 }
 
+TEST(LlvmLibcLog10Test, AllExponents) {
+  double x = 0x1.0p-1074;
+  for (int i = -1074; i < 1024; ++i, x *= 2.0) {
+    ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log10, x,
+                                   __llvm_libc::log10(x), 0.5);
+  }
+}
+
 TEST(LlvmLibcLog10Test, InDoubleRange) {
   constexpr uint64_t COUNT = 1234561;
-  constexpr uint64_t STEP = 0x3FF0'0000'0000'0000ULL / COUNT;
+  constexpr uint64_t START = 0x3FD0'0000'0000'0000ULL; // 0.25
+  constexpr uint64_t STOP = 0x4010'0000'0000'0000ULL;  // 4.0
+  // constexpr uint64_t START = 0x3FF0'0000'0000'0000ULL;  // 1.0
+  // constexpr uint64_t STOP = 0x4000'0000'0000'0000ULL;   // 2.0
+  constexpr uint64_t STEP = (STOP - START) / COUNT;
 
   auto test = [&](mpfr::RoundingMode rounding_mode) {
     mpfr::ForceRoundingMode __r(rounding_mode);
     uint64_t fails = 0;
     uint64_t count = 0;
     uint64_t cc = 0;
     double mx, mr = 0.0;
     double tol = 0.5;
 
-    for (uint64_t i = 0, v = 0; i <= COUNT; ++i, v += STEP) {
+    for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
       double x = FPBits(v).get_val();
       if (isnan(x) || isinf(x) || x < 0.0)
         continue;
       libc_errno = 0;
       double result = __llvm_libc::log10(x);
       ++cc;
-      if (isnan(result))
+      if (isnan(result) || isinf(result))
         continue;
 
       ++count;
       // ASSERT_MPFR_MATCH(mpfr::Operation::Log10, x, result, 0.5);
       if (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Log10, x,
                                                result, 0.5, rounding_mode)) {
         ++fails;
         while (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(
             mpfr::Operation::Log10, x, result, tol, rounding_mode)) {
           mx = x;
           mr = result;
           tol *= 2.0;
         }
       }
     }
+    tlog << " Log10 failed: " << fails << "/" << count << "/" << cc
+         << " tests.\n";
+    tlog << "   Max ULPs is at most: " << static_cast<uint64_t>(tol) << ".\n";
     if (fails) {
       EXPECT_MPFR_MATCH(mpfr::Operation::Log10, mx, mr, 0.5, rounding_mode);
     }
   };
 
+  tlog << " Test Rounding To Nearest...\n";
   test(mpfr::RoundingMode::Nearest);
 
+  tlog << " Test Rounding Downward...\n";
   test(mpfr::RoundingMode::Downward);
 
+  tlog << " Test Rounding Upward...\n";
   test(mpfr::RoundingMode::Upward);
 
+  tlog << " Test Rounding Toward Zero...\n";
   test(mpfr::RoundingMode::TowardZero);
 }

utils/bazel/llvm-project-overlay/libc/BUILD.bazel

libc_support_library(
     ],
 )
 
 libc_support_library(
     name = "common_constants",
     srcs = ["src/math/generic/common_constants.cpp"],
     hdrs = ["src/math/generic/common_constants.h"],
     deps = [
+        ":__support_number_pair",
         ":libc_root",
     ],
 )
 
 libc_support_library(
     name = "range_reduction",
     hdrs = [
         "src/math/generic/range_reduction.h",