[libc][math] Implement double precision log function correctly rounded to all rounding modes.

[lntue]

Summary

Implement double precision log function correctly rounded to all
rounding modes.

See https://reviews.llvm.org/D150014 for a more detail description of the algorithm.

Performance

• For 0.5 <= x <= 2, the fast pass hitting rate is about 99.93%.
• Reciprocal throughput from CORE-MATH's perf tool on Ryzen 5900X:

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

-- CORE-MATH reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 17.465 + 0.596 clc/call; Median-Min = 0.602 clc/call; Max = 18.389 clc/call;

-- CORE-MATH reciprocal throughput -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 54.961 + 2.606 clc/call; Median-Min = 2.180 clc/call; Max = 59.583 clc/call;

-- System LIBC reciprocal throughput --
[####################] 100 %
Ntrial = 20 ; Min = 12.608 + 0.276 clc/call; Median-Min = 0.359 clc/call; Max = 13.147 clc/call;

-- LIBC reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 20.952 + 0.468 clc/call; Median-Min = 0.602 clc/call; Max = 21.881 clc/call;

-- LIBC reciprocal throughput -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 18.569 + 0.552 clc/call; Median-Min = 0.601 clc/call; Max = 19.259 clc/call;

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

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

-- CORE-MATH latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 48.431 + 0.699 clc/call; Median-Min = 0.073 clc/call; Max = 51.269 clc/call;

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 64.865 + 3.235 clc/call; Median-Min = 3.475 clc/call; Max = 71.788 clc/call;

-- System LIBC latency --
[####################] 100 %
Ntrial = 20 ; Min = 42.151 + 2.090 clc/call; Median-Min = 2.270 clc/call; Max = 44.773 clc/call;

-- LIBC latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 35.266 + 0.479 clc/call; Median-Min = 0.373 clc/call; Max = 36.798 clc/call;

-- LIBC latency -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 48.518 + 0.484 clc/call; Median-Min = 0.500 clc/call; Max = 49.896 clc/call;

• Accurate pass latency:

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

-- CORE-MATH latency -- with FMA
598.306

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
632.925

-- LIBC latency -- with FMA
455.632

-- LIBC latency -- without FMA
488.564

[lntue]

Update error bounds for fast pass and sync to HEAD.

[lntue]

Sync to HEAD.

[zimmermann6]

I get failures when I try to apply this patch to head (revision 7489301):

patching file libc/config/darwin/arm/entrypoints.txt
patching file libc/config/linux/aarch64/entrypoints.txt
patching file libc/config/linux/x86_64/entrypoints.txt
patching file libc/config/windows/entrypoints.txt
patching file libc/spec/stdc.td
patching file libc/src/math/CMakeLists.txt
patching file libc/src/math/generic/CMakeLists.txt
Hunk #1 succeeded at 784 with fuzz 2 (offset 16 lines).
Hunk #2 FAILED at 785.
Hunk #3 succeeded at 846 (offset -3 lines).
1 out of 3 hunks FAILED -- saving rejects to file libc/src/math/generic/CMakeLists.txt.rej
patching file libc/src/math/generic/common_constants.h
Hunk #1 FAILED at 39.
1 out of 1 hunk FAILED -- saving rejects to file libc/src/math/generic/common_constants.h.rej
patching file libc/src/math/generic/common_constants.cpp
Hunk #1 succeeded at 196 with fuzz 2 (offset -169 lines).
patching file libc/src/math/generic/log.cpp
patching file libc/src/math/generic/log10.cpp
Hunk #1 FAILED at 17.
Hunk #2 FAILED at 41.
Hunk #3 FAILED at 785.
Hunk #4 FAILED at 944.
4 out of 4 hunks FAILED -- saving rejects to file libc/src/math/generic/log10.cpp.rej
patching file libc/src/math/generic/log_range_reduction.h
patching file libc/src/math/log.h
patching file libc/test/src/math/CMakeLists.txt
patching file libc/test/src/math/log_test.cpp

[lntue]

Sync to HEAD.

[zimmermann6]

it works now, thanks. I get for the reciprocal throughput on a AMD EPYC 7282:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 21.404 + 0.252 clc/call; Median-Min = 0.295 clc/call; Max = 23.841 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 13.188 + 0.181 clc/call; Median-Min = 0.036 clc/call; Max = 13.582 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 22.978 + 0.294 clc/call; Median-Min = 0.304 clc/call; Max = 23.516 clc/call;

and for the latency:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 57.524 + 0.353 clc/call; Median-Min = 0.298 clc/call; Max = 58.199 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 49.889 + 0.110 clc/call; Median-Min = 0.058 clc/call; Max = 50.229 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 51.938 + 0.328 clc/call; Median-Min = 0.307 clc/call; Max = 52.466 clc/call;

[mikhail.ramalho]

hey @lntue,

libc/src/math/generic/log_range_reduction.h

62

Can you replace __int128_t with MType? It would fix rv32 compilation.

libc/config/darwin/arm/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.lroundl
     libc.src.math.ldexp
     libc.src.math.ldexpf
     libc.src.math.ldexpl
     libc.src.math.log10
     libc.src.math.log10f
     libc.src.math.log1pf
     libc.src.math.log2f
+    libc.src.math.log
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf
     libc.src.math.logbl
     libc.src.math.modf
     libc.src.math.modff
     libc.src.math.modfl
     libc.src.math.nearbyint

libc/config/linux/aarch64/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.lroundl
     libc.src.math.ldexp
     libc.src.math.ldexpf
     libc.src.math.ldexpl
     libc.src.math.log10
     libc.src.math.log10f
     libc.src.math.log1pf
     libc.src.math.log2f
+    libc.src.math.log
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf
     libc.src.math.logbl
     libc.src.math.modf
     libc.src.math.modff
     libc.src.math.modfl
     libc.src.math.nearbyint

libc/config/linux/x86_64/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.llrintl
     libc.src.math.llround
     libc.src.math.llroundf
     libc.src.math.llroundl
     libc.src.math.log10
     libc.src.math.log10f
     libc.src.math.log1pf
     libc.src.math.log2f
+    libc.src.math.log
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf
     libc.src.math.logbl
     libc.src.math.lrint
     libc.src.math.lrintf
     libc.src.math.lrintl
     libc.src.math.lround

libc/config/windows/entrypoints.txt

set(TARGET_LIBM_ENTRYPOINTS
     libc.src.math.llrintl
     libc.src.math.llround
     libc.src.math.llroundf
     libc.src.math.llroundl
     libc.src.math.log10
     libc.src.math.log10f
     libc.src.math.log1pf
     libc.src.math.log2f
+    libc.src.math.log
     libc.src.math.logf
     libc.src.math.logb
     libc.src.math.logbf
     libc.src.math.logbl
     libc.src.math.lrint
     libc.src.math.lrintf
     libc.src.math.lrintl
     libc.src.math.lround

libc/spec/stdc.td

HeaderSpec Math = HeaderSpec<
 
           FunctionSpec<"log10", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"log10f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"log1pf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"log2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
+          FunctionSpec<"log", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"logf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
 
           FunctionSpec<"logb", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
           FunctionSpec<"logbf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
           FunctionSpec<"logbl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>]>,
 
           FunctionSpec<"modf", RetValSpec<DoubleType>, [ArgSpec<DoubleType>, ArgSpec<DoublePtr>]>,
           FunctionSpec<"modff", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<FloatPtr>]>,

libc/src/math/CMakeLists.txt

 
 add_math_entrypoint_object(log10)
 add_math_entrypoint_object(log10f)
 
 add_math_entrypoint_object(log1pf)
 
 add_math_entrypoint_object(log2f)
 
+add_math_entrypoint_object(log)
 add_math_entrypoint_object(logf)
 
 add_math_entrypoint_object(logb)
 add_math_entrypoint_object(logbf)
 add_math_entrypoint_object(logbl)
 
 add_math_entrypoint_object(llrint)
 add_math_entrypoint_object(llrintf)

libc/src/math/generic/CMakeLists.txt

add_object_library(
   HDRS
     common_constants.h
   SRCS
     common_constants.cpp
   DEPENDS
     libc.src.__support.number_pair
 )
 
+add_header_library(
+  log_range_reduction
+  HDRS
+    log_range_reduction.h
+  DEPENDS
+    .common_constants
+    libc.src.__support.FPUtil.dyadic_float
+)
+
 add_entrypoint_object(
   log10
   SRCS
     log10.cpp
   HDRS
     ../log10.h
   DEPENDS
     .common_constants
+    .log_range_reduction
     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
DEPENDS
     libc.src.__support.FPUtil.fma
     libc.src.__support.FPUtil.polyeval
     libc.src.__support.macros.optimization
     COMPILE_OPTIONS
     -O3
 )
 
 add_entrypoint_object(
+  log
+  SRCS
+    log.cpp
+  HDRS
+    ../log.h
+  DEPENDS
+    .common_constants
+    .log_range_reduction
+    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(
   logf
   SRCS
     logf.cpp
   HDRS
     ../logf.h
   DEPENDS
     .common_constants
     libc.src.__support.FPUtil.except_value_utils

libc/src/math/generic/common_constants.h

 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};
 
+// Logarithm Range Reduction - Step 2, 3, and 4.
+extern const int S2[193];
+extern const int S3[161];
+extern const int S4[130];
+
 // 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.

libc/src/math/generic/common_constants.cpp

alignas(64) const NumberPair<double> LOG_R_DD[128] = {
     {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},
 };
 
+// Logarithm 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).
+alignas(64) const int 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};
+
+// Logarithm 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).
+alignas(64) const int 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,
+};
+
+// Logarithm 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).
+alignas(64) const int 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,
+};
+
 // 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/log.cpp

+//===-- Double-precision log(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/log.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"
+#include "log_range_reduction.h"
+
+namespace __llvm_libc {
+
+// 128-bit precision dyadic floating point numbers.
+using Float128 = typename fputil::DyadicFloat<128>;
+using MType = typename Float128::MantissaType;
+
+namespace {
+
+// A simple upper bound for the error of e_x * log(2) - log(r).
+constexpr double HI_ERR = 0x1.0p-85;
+
+// Extra errors from P is from using x^2 to reduce evaluation latency.
+constexpr double P_ERR = 0x1.0p-50;
+
+// log(2) with 128-bit prepcision generated by SageMath with:
+//   sage: (s, m, e) = RealField(128)(2).log().sign_mantissa_exponent();
+//   sage: print("MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})");
+const Float128 LOG_2(/*sign=*/false, /*exponent=*/-128, /*mantissa=*/
+                     MType({0xc9e3b39803f2f6af, 0xb17217f7d1cf79ab}));
+
+alignas(64) const LogRR LOG_TABLE = {
+    // -log(r) with 128-bit precision generated by SageMath with:
+    //
+    // for i in range(128):
+    //   r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^(-7)) );
+    //   s, m, e = RealField(128)(r).log().sign_mantissa_exponent();
+    //   print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) %
+    //   2^64),
+    //         "})},");
+    /* .step_1= */ {
+        {false, 0, MType(0)},
+        {false, -134, MType({0x662d417ced007a46, 0x8080abac46f38946})},
+        {false, -133, MType({0x91d082dce3ddcd38, 0x8102b2c49ac23a4f})},
+        {false, -133, MType({0xda5f3cc0b3251dbd, 0xc24929464655f45c})},
+        {false, -132, MType({0xb9e3aea6c444ef07, 0x820aec4f3a222380})},
+        {false, -132, MType({0x521016bd904dc968, 0xa33576a16f1f4c64})},
+        {false, -132, MType({0xbe97660a23cc540d, 0xc4a550a4fd9a19a8})},
+        {false, -132, MType({0xe09f5fe2058d6006, 0xe65b9e6eed965c36})},
+        {false, -131, MType({0x1fecdfa819b96098, 0x842cc5acf1d03445})},
+        {false, -131, MType({0xa7c9859530a45153, 0x8cb9de8a32ab368a})},
+        {false, -131, MType({0x976d3b5b45f6ca0b, 0x9defad3e8f73217a})},
+        {false, -131, MType({0xe8b8b88a14ff0ce, 0xaf4ad26cbc8e5be7})},
+        {false, -131, MType({0x6a677b4c8bec22e1, 0xb8069857560707a3})},
+        {false, -131, MType({0xeaf51f66692844ba, 0xc99af2eaca4c4570})},
+        {false, -131, MType({0xa8112e35a60e6375, 0xdb56446d6ad8deff})},
+        {false, -131, MType({0x196ab34ce0bccd12, 0xe442c00de2591b47})},
+        {false, -131, MType({0x4066e87f2c0f7340, 0xf639cc185088fe5d})},
+        {false, -131, MType({0xc17bd40d8d9291ec, 0xff4489cedeab2ca6})},
+        {false, -130, MType({0x9c5a0fe396f40f1e, 0x88bc74113f23def1})},
+        {false, -130, MType({0x88713268840cbcc0, 0x8d515bf11fb94f1c})},
+        {false, -130, MType({0x65c0da506a088484, 0x968b08643409ceb6})},
+        {false, -130, MType({0x411a5b944aca8708, 0x9b2fe580ac80b17d})},
+        {false, -130, MType({0xa9fb6cf0ecb411b7, 0xa489ec199dab06f2})},
+        {false, -130, MType({0xcad2fb8d48054ae0, 0xa93f2f250dac67d1})},
+        {false, -130, MType({0x2c3c2e77904afa78, 0xb2ba75f46099cf8b})},
+        {false, -130, MType({0x34c7bc3d32750fde, 0xb780945bab55dce4})},
+        {false, -130, MType({0x9a631e830fd30904, 0xc11e0b2a8d1e0ddb})},
+        {false, -130, MType({0xaa8b6997a402bf30, 0xc5f57f59c7f46155})},
+        {false, -130, MType({0x2c507fb7a3d0bf6a, 0xcad2d6e7b80bf914})},
+        {false, -130, MType({0x5f53bd2e406e66e7, 0xd49f69e456cf1b79})},
+        {false, -130, MType({0x58a98f2ad65bee9b, 0xd98ec2bade71e539})},
+        {false, -130, MType({0x4d57da945b5d0aaa, 0xde8439c1dec56877})},
+        {false, -130, MType({0xc524848e3443e040, 0xe881bf932af3dac0})},
+        {false, -130, MType({0x11d49f96cb88317b, 0xed89ed86a44a01aa})},
+        {false, -130, MType({0x3b020fa1820c9492, 0xf29877ff38809091})},
+        {false, -130, MType({0x54d2238f75f969b1, 0xf7ad6f26e7ff2ef7})},
+        {false, -130, MType({0xca0cdf301431b60f, 0xfcc8e3659d9bcbec})},
+        {false, -129, MType({0x62dda9d2270fa1f4, 0x8389c3026ac3139b})},
+        {false, -129, MType({0x163ceae88f720f1e, 0x86216b3b0b17188b})},
+        {false, -129, MType({0x9c5a0fe396f40f1e, 0x88bc74113f23def1})},
+        {false, -129, MType({0xf7a5168126a58b9a, 0x8b5ae65d67db9acd})},
+        {false, -129, MType({0x5147bdb6ddcaf59c, 0x8dfccb1ad35ca6ed})},
+        {false, -129, MType({0xdf5bb3b60554e152, 0x934b1089a6dc93c1})},
+        {false, -129, MType({0x4a5004f3ef063313, 0x95f783e6e49a9cfa})},
+        {false, -129, MType({0x2cdec34784707839, 0x98a78f0e9ae71d85})},
+        {false, -129, MType({0xd878bbe3d392be25, 0x9b5b3bb5f088b766})},
+        {false, -129, MType({0x5b035eae273a855f, 0x9e1293b9998c1daa})},
+        {false, -129, MType({0xbb2438273918db7e, 0xa0cda11eaf46390d})},
+        {false, -129, MType({0xf698298adddd7f32, 0xa38c6e138e20d831})},
+        {false, -129, MType({0xe4f5275c2d15c21f, 0xa64f04f0b961df76})},
+        {false, -129, MType({0x8164c759686a2209, 0xa9157039c51ebe70})},
+        {false, -129, MType({0xf72ea07749ce6bd3, 0xabdfba9e468fd6f6})},
+        {false, -129, MType({0x7dd6e688ebb13b03, 0xaeadeefacaf97d35})},
+        {false, -129, MType({0x18ce51fff99479cd, 0xb1801859d56249dc})},
+        {false, -129, MType({0x2756eba00bc33978, 0xb45641f4e350a0d3})},
+        {false, -129, MType({0xbe1116c3466beb6d, 0xb730773578cb90b2})},
+        {false, -129, MType({0x49dc60b2b059a60b, 0xba0ec3b633dd8b09})},
+        {false, -129, MType({0x2efd17781bb3afec, 0xbcf13343e7d9ec7d})},
+        {false, -129, MType({0x37eda996244bccb0, 0xbfd7d1dec0a8df6f})},
+        {false, -129, MType({0x33337789d592e296, 0xc2c2abbb6e5fd56f})},
+        {false, -129, MType({0x1a18fb8f9f9ef280, 0xc5b1cd44596fa51e})},
+        {false, -129, MType({0x688ce7c1a75e341a, 0xc8a5431adfb44ca5})},
+        {false, -129, MType({0x2d7e9307c70c0668, 0xcb9d1a189ab56e76})},
+        {false, -129, MType({0xef2f3f4f861ad6a9, 0xce995f50af69d861})},
+        {false, -129, MType({0x7f9d79f51dcc7301, 0xd19a201127d3c645})},
+        {false, -129, MType({0x7f9d79f51dcc7301, 0xd19a201127d3c645})},
+        {false, -129, MType({0x5f53bd2e406e66e7, 0xd49f69e456cf1b79})},
+        {false, -129, MType({0xad88bba7d0cee8e0, 0xd7a94a92466e833a})},
+        {false, -129, MType({0x96c20cca6efe2ac5, 0xdab7d02231484a92})},
+        {false, -129, MType({0xf40a666c87842843, 0xddcb08dc0717d85b})},
+        {false, -129, MType({0x7fe8e1802aba24d6, 0xe0e30349fd1cec80})},
+        {false, -129, MType({0x7fe8e1802aba24d6, 0xe0e30349fd1cec80})},
+        {false, -129, MType({0x3eadb651b49ac53a, 0xe3ffce3a2aa64922})},
+        {false, -129, MType({0x304e1653e71d9973, 0xe72178c0323a1a0f})},
+        {false, -129, MType({0xe9a767a80d6d97e8, 0xea481236f7d35baf})},
+        {false, -129, MType({0x4f91cf4b33e42998, 0xed73aa4264b0ade9})},
+        {false, -129, MType({0xfc66eb6408ff6433, 0xf0a450d139366ca6})},
+        {false, -129, MType({0xfc66eb6408ff6433, 0xf0a450d139366ca6})},
+        {false, -129, MType({0xac8d42f78d3e65d3, 0xf3da161eed6b9aaf})},
+        {false, -129, MType({0x5a470250d40ebe90, 0xf7150ab5a09f27f4})},
+        {false, -129, MType({0xb780a545a1b54dcf, 0xfa553f7018c966f2})},
+        {false, -129, MType({0xb780a545a1b54dcf, 0xfa553f7018c966f2})},
+        {false, -129, MType({0x8f05924d258c14c5, 0xfd9ac57bd244217e})},
+        {false, -128, MType({0x89d1b09c70c4010a, 0x8072d72d903d588b})},
+        {false, -128, MType({0x30d58c3f7e2ea1f, 0x821b05f3b01d6774})},
+        {false, -128, MType({0x30d58c3f7e2ea1f, 0x821b05f3b01d6774})},
+        {false, -128, MType({0x20f6fafe8fbb68b9, 0x83c5f8299e2b4091})},
+        {false, -128, MType({0xe21f9f89c1ab80b2, 0x8573b71682a7d21a})},
+        {false, -128, MType({0xe21f9f89c1ab80b2, 0x8573b71682a7d21a})},
+        {false, -128, MType({0x1e005d06dbfa8f8, 0x87244c308e670a66})},
+        {false, -128, MType({0x223111a707b6de2c, 0x88d7c11e3ad53cdc})},
+        {false, -128, MType({0x223111a707b6de2c, 0x88d7c11e3ad53cdc})},
+        {false, -128, MType({0x2eb628dba173c82d, 0x8a8e1fb794b09134})},
+        {false, -128, MType({0xbe2ad19415fe25a5, 0x8c47720791e53313})},
+        {false, -128, MType({0xbe2ad19415fe25a5, 0x8c47720791e53313})},
+        {false, -128, MType({0xbddae1ccce247838, 0x8e03c24d73003959})},
+        {false, -128, MType({0x9b00bf167e95da67, 0x8fc31afe30b2c6de})},
+        {false, -128, MType({0x9b00bf167e95da67, 0x8fc31afe30b2c6de})},
+        {false, -128, MType({0x9b92199ed1a4bab1, 0x918586c5f5e4bf01})},
+        {false, -128, MType({0xdf5bb3b60554e152, 0x934b1089a6dc93c1})},
+        {false, -128, MType({0xdf5bb3b60554e152, 0x934b1089a6dc93c1})},
+        {false, -128, MType({0xf3cbc416a2418012, 0x9513c36876083695})},
+        {false, -128, MType({0xbe1188fbc94e2f15, 0x96dfaabd86fa1646})},
+        {false, -128, MType({0xbe1188fbc94e2f15, 0x96dfaabd86fa1646})},
+        {false, -128, MType({0x1d2f89321647b358, 0x98aed221a03458b6})},
+        {false, -128, MType({0x1d2f89321647b358, 0x98aed221a03458b6})},
+        {false, -128, MType({0xe549f9aaea3cb5e1, 0x9a81456cec642e0f})},
+        {false, -128, MType({0xa2554b2dd4619e63, 0x9c5710b8cbb73a42})},
+        {false, -128, MType({0xa2554b2dd4619e63, 0x9c5710b8cbb73a42})},
+        {false, -128, MType({0x30603d87b6df81ad, 0x9e304061b5fda919})},
+        {false, -128, MType({0x30603d87b6df81ad, 0x9e304061b5fda919})},
+        {false, -128, MType({0x67879c5a30cd1242, 0xa00ce1092e5498c3})},
+        {false, -128, MType({0xb7efae08e597e16, 0xa1ecff97c91e267b})},
+        {false, -128, MType({0xb7efae08e597e16, 0xa1ecff97c91e267b})},
+        {false, -128, MType({0x83594fab088c0d65, 0xa3d0a93f45169a4a})},
+        {false, -128, MType({0x83594fab088c0d65, 0xa3d0a93f45169a4a})},
+        {false, -128, MType({0xaf6a62a0dec6e073, 0xa5b7eb7cb860fb88})},
+        {false, -128, MType({0xaf6a62a0dec6e073, 0xa5b7eb7cb860fb88})},
+        {false, -128, MType({0x49362382a768847a, 0xa7a2d41ad270c9d7})},
+        {false, -128, MType({0x49362382a768847a, 0xa7a2d41ad270c9d7})},
+        {false, -128, MType({0x8ba4aea614d05701, 0xa991713433c2b998})},
+        {false, -128, MType({0x8ba4aea614d05701, 0xa991713433c2b998})},
+        {false, -128, MType({0x7fe6607ba902ef3c, 0xab83d135dc633301})},
+        {false, -128, MType({0x7fe6607ba902ef3c, 0xab83d135dc633301})},
+        {false, -128, MType({0xd60864fd949b4bd3, 0xad7a02e1b24efd31})},
+        {false, -128, MType({0xd60864fd949b4bd3, 0xad7a02e1b24efd31})},
+        {false, -128, MType({0x66d235ee63073dd, 0xaf74155120c9011c})},
+        {false, 0, MType(0)},
+    },
+    // -log(r) for the second step, generated by SageMath with:
+    //
+    // 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).log().sign_mantissa_exponent();
+    //   print("{false," if s == -1 else "{true,", e, ",
+    //         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+    /* .step_2 = */
+    {
+        {true, -135, MType({0xa1c6f3fc242ef8d0, 0x803faacac419abf2})},
+        {true, -136, MType({0xa225ebc02e6d9dd4, 0xfc834da16f0d9f57})},
+        {true, -136, MType({0xc33f6ad340ae18a9, 0xf88735ccc7433381})},
+        {true, -136, MType({0x70b2a4d38a242244, 0xf48b0e171249b6bc})},
+        {true, -136, MType({0x1d54819048b811b0, 0xf08ed67fd190e280})},
+        {true, -136, MType({0xaee5983701d2a02b, 0xec928f0686828706})},
+        {true, -136, MType({0x40abb8ab72afa2d2, 0xe89637aab2828aed})},
+        {true, -136, MType({0xdeb547a0d4a26ef9, 0xe499d06bd6eeead5})},
+        {true, -136, MType({0x39c5bdfbcf6087a0, 0xe09d5949751fb909})},
+        {true, -136, MType({0x53ea9bf152de635f, 0xdca0d2430e671d18})},
+        {true, -136, MType({0x25b820436f5f4352, 0xd8a43b582411537e})},
+        {true, -136, MType({0x3c2d13ea1d0be058, 0xd4a794883764ad41})},
+        {true, -136, MType({0x4f3cfa62bcb3ce3a, 0xd0aaddd2c9a18f95})},
+        {true, -136, MType({0xd0fff6cdf14a86c7, 0xccae17375c02737c})},
+        {true, -136, MType({0x7587b5f0453ac3d2, 0xc8b140b56fbbe56a})},
+        {true, -136, MType({0xb358ad16dfd0d085, 0xc4b45a4c85fc84e2})},
+        {true, -136, MType({0x3c86fdce5dbe7314, 0xc0b763fc1fed041d})},
+        {true, -136, MType({0x70764e46ac18a96d, 0xbcba5dc3beb027a6})},
+        {true, -136, MType({0xc63be62b8f285882, 0xb8bd47a2e362c600})},
+        {true, -136, MType({0x72e7b5a386e5e31b, 0xb3c0d59a244325a4})},
+        {true, -136, MType({0xc3ea2cd93f316b34, 0xafc39bac66434f27})},
+        {true, -136, MType({0x1dfb11a7cc892843, 0xabc651d491a7b438})},
+        {true, -136, MType({0xfc679a28e9d9f212, 0xa7c8f8122773f38d})},
+        {true, -136, MType({0xe7bc977eeec42254, 0xa3cb8e64a8a5bbe6})},
+        {true, -136, MType({0xb20f215bd3b58c61, 0x9fce14cb9634cba6})},
+        {true, -136, MType({0xabe2862508d67a98, 0x9bd08b467112f078})},
+        {true, -136, MType({0xd1aacedcefe9d377, 0x97d2f1d4ba2c06f0})},
+        {true, -136, MType({0xf1eb25e77d05f58d, 0x93d54875f265fa2c})},
+        {true, -136, MType({0xcbef6fac33691e95, 0x8fd78f299aa0c375})},
+        {true, -136, MType({0x2720640462a0f8ad, 0x8bd9c5ef33b669e0})},
+        {true, -136, MType({0xe2f1775134c8da75, 0x87dbecc63e7b01ed})},
+        {true, -136, MType({0xff67e201c8c50d67, 0x83de03ae3bbcad2e})},
+        {true, -137, MType({0x3c742a7c76356396, 0xffc0154d588733c5})},
+        {true, -137, MType({0xf90dd6b24aa686ec, 0xf7c4035e21a4052f})},
+        {true, -137, MType({0xca47c52b7d7ffce2, 0xefc7d18dd4485b9e})},
+        {true, -137, MType({0x3703617ad3d8311f, 0xe7cb7fdb71e0db36})},
+        {true, -137, MType({0x7e4cfbd830393b88, 0xdfcf0e45fbce3e80})},
+        {true, -137, MType({0x4f7a29cf0fc2c38e, 0xd7d27ccc736555af})},
+        {true, -137, MType({0x7370ae83f9e72748, 0xcfd5cb6dd9ef05dd})},
+        {true, -137, MType({0x671486eb4cd76f65, 0xc7d8fa2930a84850})},
+        {true, -137, MType({0xe6dbb624f9739782, 0xbfdc08fd78c229b9})},
+        {true, -137, MType({0x6b866e09e57d9079, 0xb7def7e9b361c979})},
+        {true, -137, MType({0x97fa2fd0c9dc723e, 0xafe1c6ece1a058dd})},
+        {true, -137, MType({0x983e80897cf1e60f, 0xa7e47606048b1a65})},
+        {true, -137, MType({0x7199cd06ae5d39b3, 0x9fe705341d236102})},
+        {true, -137, MType({0x43cd18a72a051a96, 0x97e974762c5e8f58})},
+        {true, -137, MType({0x7b6d1248c3e1fd40, 0x8febc3cb332616ff})},
+        {true, -137, MType({0xf5572a8814c703af, 0x87edf332325777c5})},
+        {true, -138, MType({0x26828c92649a3a39, 0xffe0055455887de0})},
+        {true, -138, MType({0x82c550bd1216d82a, 0xefe3e4643a640cf3})},
+        {true, -138, MType({0xda6959f7f0e01bf0, 0xdfe7839214b4e8ae})},
+        {true, -138, MType({0xda93e2fa85a8f214, 0xcfeae2dbe5d6736d})},
+        {true, -138, MType({0xb47505bfa5a03b06, 0xbfee023faf0c2480})},
+        {true, -138, MType({0xb1475a5180a43520, 0xaff0e1bb718186ad})},
+        {true, -138, MType({0xa8740b91c95df537, 0x9ff3814d2e4a36b2})},
+        {true, -138, MType({0x57d895d35921b59c, 0x8ff5e0f2e661e1c6})},
+        {true, -139, MType({0x3c56c598c659c2a3, 0xfff0015535588833})},
+        {true, -139, MType({0x2ef8ec33ed9d782a, 0xdff3c0e497ea4eb1})},
+        {true, -139, MType({0x379eba7e6465ff63, 0xbff7008ff5e0c257})},
+        {true, -139, MType({0x3f972b783fcab757, 0x9ff9c0535073a370})},
+        {true, -140, MType({0xde026e271ee0549d, 0xfff8005551558885})},
+        {true, -140, MType({0xeceb47ea01f6c632, 0xbffb8023febc0c25})},
+        {true, -141, MType({0x7333c57857e1ed52, 0xfffc001554d55888})},
+        {true, -142, MType({0x87dde026fa704374, 0xfffe000555455588})},
+        {true, 0, MType({0x0, 0x0})},
+        {false, -141, MType({0x44999abe2fe2cc65, 0x80010002aab2aac4})},
+        {false, -140, MType({0x4eef381581464ccb, 0x8002000aaaeaac44})},
+        {false, -140, MType({0xdfeb485085f6f454, 0xc004802401440c26})},
+        {false, -139, MType({0x99abe3be3a1c6e93, 0x8004002aacaac445})},
+        {false, -139, MType({0x6bc1e20eac8448b4, 0xa00640535a37a37a})},
+        {false, -139, MType({0x979eedc064c242fd, 0xc00900900a20c275})},
+        {false, -139, MType({0xc72446cc1bf728bd, 0xe00c40e4bd6e4efd})},
+        {false, -138, MType({0xf381b821bbb569e5, 0x800800aabaac446e})},
+        {false, -138, MType({0x569b26aaa485ea5c, 0x900a20f319a3e273})},
+        {false, -138, MType({0x2dcf56c83c80b028, 0xa00c814d7c6a37f8})},
+        {false, -138, MType({0x5f69768284463b9b, 0xb00f21bbe3e388ee})},
+        {false, -138, MType({0xb48ea6c05e2773a1, 0xc0120240510c284c})},
+        {false, -138, MType({0x14d9d76196d8043a, 0xd01522dcc4f87991})},
+        {false, -138, MType({0xe016a611a4415d72, 0xe018839340d4f241})},
+        {false, -138, MType({0x661e135f49a47c40, 0xf01c2465c5e61b6f})},
+        {false, -137, MType({0xbe6bf0fa435e8383, 0x801002ab2ac4499a})},
+        {false, -137, MType({0x9a31ba0cbc030353, 0x881213337898871e})},
+        {false, -137, MType({0x54b57dfe0c4c840f, 0x901443cccd362c9f})},
+        {false, -137, MType({0x7ad1e9c315328f7e, 0x98169478296fad41})},
+        {false, -137, MType({0x1f3f686cf3d6be22, 0xa01905368e2389b3})},
+        {false, -137, MType({0xf105b66ec4703ede, 0xa81b9608fc3c50ec})},
+        {false, -137, MType({0x610848c68df4d233, 0xb01e46f074b0a0f3})},
+        {false, -137, MType({0xd6aef30cd312169a, 0xb82117edf8832797})},
+        {false, -137, MType({0xf3ac379608053d9d, 0xc024090288c2a339})},
+        {false, -137, MType({0xe6e2acf8f4d4c24a, 0xc8271a2f2689e388})},
+        {false, -137, MType({0xce6ae474d860359f, 0xd02a4b74d2ffca44})},
+        {false, -137, MType({0x28bb3cd9f2a65fb5, 0xd82d9cd48f574c00})},
+        {false, -137, MType({0x54f30dbef38a8066, 0xe0310e4f5ccf70e1})},
+        {false, -137, MType({0x224a96f5a7471c46, 0xe8349fe63cb35564})},
+        {false, -137, MType({0x6ea920591aa02e1b, 0xf038519a305a2b1b})},
+        {false, -137, MType({0xd462b63756c87e80, 0xf83c236c39273972})},
+        {false, -136, MType({0x338f77605fe77f2a, 0x80200aaeac44ef38})},
+        {false, -136, MType({0x3ff51287882500ed, 0x842213b747fec7bb})},
+        {false, -136, MType({0xcc394b3ef0ebeb12, 0x88242cd07084ed02})},
+        {false, -136, MType({0x1ab9679b55f78a6b, 0x8c2655faa6a1323f})},
+        {false, -136, MType({0x7025697d10af0436, 0x90288f366b237771})},
+        {false, -136, MType({0x17e4b7ac6c600cb4, 0x942ad8843ee1a9cd})},
+        {false, -136, MType({0x7013925a9a8da7f3, 0x982d31e4a2b7c418})},
+        {false, -136, MType({0xfd1a09c848e3950e, 0x9c2f9b581787cf0d})},
+        {false, -136, MType({0x84dd2de6e3d90a37, 0xa03214df1e39e1bd})},
+        {false, -136, MType({0x318b2ddd9d0a33b4, 0xa4349e7a37bc21ed})},
+        {false, -136, MType({0xbc031e6f5acfd4a8, 0xa8373829e502c47a})},
+        {false, -136, MType({0x9dd91e52c79fd070, 0xac39e1eea7080dbc})},
+        {false, -136, MType({0x4af78fa1cb48a12d, 0xb03c9bc8fecc51e3})},
+        {false, -136, MType({0x72de1d99ce252efd, 0xb43f65b96d55f55a})},
+        {false, -136, MType({0xefb1dbe721934877, 0xb74187bc8ccffa84})},
+        {false, -136, MType({0xb4b080f230c87598, 0xbb446dd4d9bca499})},
+        {false, -136, MType({0xda6a7cd19c7fa4f2, 0xbf476404a05f88f2})},
+        {false, -136, MType({0xdf00e3783b50ecfb, 0xc34a6a4c61d5cc3c})},
+        {false, -136, MType({0xda2e5e02ab4e183c, 0xc74d80ac9f42a52d})},
+        {false, -136, MType({0xea5f6ee99d30c626, 0xcb50a725d9cf5ce6})},
+        {false, -136, MType({0xa96d5956531d7d8b, 0xcf53ddb892ab4f55})},
+        {false, -136, MType({0xa8fc636eb36afa75, 0xd35724654b0beb95})},
+        {false, -136, MType({0xf67e2b827bfc4421, 0xd75a7b2c842cb451})},
+        {false, -136, MType({0xa6d8c817516303e6, 0xdb5de20ebf4f4026})},
+        {false, -136, MType({0x69b36ae5962e85f4, 0xdf61590c7dbb3a02})},
+        {false, -136, MType({0x24693eec2a831cc3, 0xe364e02640be6188})},
+        {false, -136, MType({0x94a339d56a55ab4a, 0xe768775c89ac8b70})},
+        {false, -136, MType({0xfa9998fbf9703bf4, 0xeb6c1eafd9dfa1eb})},
+        {false, -136, MType({0xcafdc27227b71eaa, 0xef6fd620b2b7a503})},
+        {false, -136, MType({0x688d4282f6026aa3, 0xf3739daf959aaafc})},
+        {false, -136, MType({0xe54e9e3804464cdd, 0xf777755d03f4e0b6})},
+        {false, -136, MType({0xcb78b383f4b59dce, 0xfb7b5d297f388a12})},
+        {false, -136, MType({0xee055fc515062c04, 0xff7f551588de024f})},
+        {false, -135, MType({0x207812b43382acdd, 0x81c1ae90d131de38})},
+        {false, -135, MType({0xdc90c4c4b61f3a87, 0x83c3baa726a721cc})},
+        {false, -135, MType({0x1a03f13fb2c978b1, 0x85c5cece05941dbc})},
+        {false, -135, MType({0xb36f282e83a7dc36, 0x87c7eb05aec1304f})},
+        {false, -135, MType({0x6ad14c3dfa414391, 0x89ca0f4e62f9c476})},
+        {false, -135, MType({0xe8dd4ea0d48b88e5, 0x8bcc3ba8630c51f4})},
+        {false, -135, MType({0xc02515afe8caeb90, 0x8dce7013efca5d96})},
+        {false, -135, MType({0x741ceaf3349f3cf1, 0x8fd0ac914a08795f})},
+        {false, -135, MType({0x83f7cd4929d2c28c, 0x91d2f120b29e44bb})},
+        {false, -135, MType({0x795d03ebc2fd03fa, 0x93d53dc26a666cb1})},
+        {false, -135, MType({0xfaf74f1d1ad16acc, 0x95d79276b23eac12})},
+        {false, -135, MType({0xe2de134f72fee429, 0x97d9ef3dcb07cbad})},
+        {false, -135, MType({0x58d8dba6cadac5d5, 0x99dc5417f5a5a27d})},
+        {false, -135, MType({0xf07d90bc5aae40a4, 0x9bdec10572ff15da})},
+        {false, -135, MType({0x1deaf79d9fc40374, 0x9d6098046659ea6b})},
+        {false, -135, MType({0x7ba63e6769b81999, 0x9f63131450b07988})},
+        {false, -135, MType({0x59ebfc9335094e59, 0xa1659638404d5f92})},
+        {false, -135, MType({0x16aae012b5026f71, 0xa36821707622f97a})},
+        {false, -135, MType({0xff5d4f2c0e4b9cae, 0xa56ab4bd3326b378})},
+        {false, -135, MType({0x855838b5119dcb28, 0xa76d501eb8510941})},
+        {false, -135, MType({0x75f70cbbe9cf1603, 0xa96ff395469d8630})},
+        {false, -135, MType({0x36a53ad4d5541cc9, 0xab729f211f0ac57e})},
+        {false, -135, MType({0x4c5934ec32d20d9, 0xad7552c2829a7270})},
+        {false, -135, MType({0x3977e89aec59bfa2, 0xaf780e79b2514889})},
+        {false, -135, MType({0x913d4e3dc55c3e6e, 0xb17ad246ef3713bc})},
+        {false, -135, MType({0x777b52a9e70d8bcc, 0xb37d9e2a7a56b09d})},
+        {false, -135, MType({0x55de916fd30591de, 0xb580722494be0c91})},
+        {false, -135, MType({0xe79cfb37be2861e4, 0xb7834e357f7e2600})},
+        {false, -135, MType({0x90983104d3805389, 0xb986325d7bab0c89})},
+        {false, -135, MType({0xb860504baa6f984d, 0xbb891e9cca5be12e})},
+        {false, -135, MType({0x29178d6ff5712b96, 0xbd8c12f3acaad68b})},
+        {false, -135, MType({0x7236fa47ba19a198, 0xbf8f0f6263b53102})},
+        {false, -135, MType({0x4f34d64cafcc50e3, 0xc19213e9309b46f2})},
+        {false, -135, MType({0x120cc62eb0a8db3e, 0xc3952088548080e4})},
+        {false, -135, MType({0x11aa5084779060e3, 0xc5983540108b59be})},
+        {false, -135, MType({0x1c35fd6236c8dcf1, 0xc79b5210a5e55ef5})},
+        {false, -135, MType({0xed4576a7e4b878fe, 0xc99e76fa55bb30bd})},
+        {false, -135, MType({0x6caf4bb8fd2c1131, 0xcb20d7fa3a336081})},
+        {false, -135, MType({0x3f24a6cbb09c654f, 0xcd240b10753e78de})},
+        {false, -135, MType({0x78bc003bb81e40f3, 0xcf2746407e0ff09f})},
+        {false, -135, MType({0x56647301edfd8e8b, 0xd12a898a95dff002})},
+        {false, -135, MType({0x28fe1c4d04ca4ed9, 0xd32dd4eefde9b2ef})},
+        {false, -135, MType({0xe1ea9ea6cbf57379, 0xd531286df76b892a})},
+        {false, -135, MType({0xa3832028141a5cc2, 0xd7348407c3a6d688})},
+        {false, -135, MType({0x557421dd379d3ead, 0xd937e7bca3e0131b})},
+        {false, -135, MType({0x3cff8e87a99bcaf0, 0xdb3b538cd95ecb67})},
+        {false, -135, MType({0x99255ef34bd0801f, 0xdd3ec778a56da093})},
+        {false, -135, MType({0x42b33220abfa15cd, 0xdf424380495a489c})},
+        {false, -135, MType({0x503b378faa97dbc0, 0xe145c7a406758e83})},
+        {false, -135, MType({0xbdf2ca006f59b544, 0xe34953e41e135282})},
+        {false, -135, MType({0x1979190af37ed16f, 0xe54ce840d18a8a3e})},
+        {false, -135, MType({0x31863ff7cf898c9c, 0xe75084ba623540f4})},
+        {false, -135, MType({0xc983284f60293647, 0xe9542951117097b0})},
+        {false, -135, MType({0x510a969ebe03f804, 0xeb57d605209cc57e})},
+        {false, -135, MType({0x9f53bffc6d23fe30, 0xed5b8ad6d11d1797})},
+        {false, -135, MType({0xb286c6e113337886, 0xef5f47c66457f199})},
+        {false, -135, MType({0xb6ed80852ae6fd63, 0xf0e21acdd6e7d412})},
+        {false, -135, MType({0xdf437fb0f616082d, 0xf2e5e5f25450c5a2})},
+        {false, -135, MType({0xf237cff1acb306b3, 0xf4e9b935685dbe0b})},
+        {false, -135, MType({0x52dbfafb4121a092, 0xf6ed94975480b696})},
+        {false, -135, MType({0xd81648249cece4c, 0xf8f178185a2ebfd9})},
+        {false, -135, MType({0xad95e6b0b96903d3, 0xfaf563b8bae001eb})},
+        {false, -135, MType({0x176cd56887ac7fe9, 0xfcf95778b80fbc98})},
+        {false, -135, MType({0x65f4c7397f1f478d, 0xfefd5358933c478c})},
+    },
+    // -log(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).log().sign_mantissa_exponent();
+    //   print("{false," if (s == -1) else "{true,", e, ",
+    //         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+    /* .step_3 = */
+    {
+        {true, -142, MType({0x374b294076d669c3, 0x9fff38014d52e45a})},
+        {true, -142, MType({0x7f6f05dcdbeb776e, 0x9dff3cf940fad85a})},
+        {true, -142, MType({0x3d55e21d41bbadf9, 0x9bff41e134f1cb36})},
+        {true, -142, MType({0xccdba2d54aadbc5c, 0x99ff46b92936bcf4})},
+        {true, -142, MType({0x71dd16d3073f79b2, 0x97ff4b811dc8ad9d})},
+        {true, -142, MType({0x5837f3df1a58dd48, 0x95ff503912a69d37})},
+        {true, -142, MType({0x93cad3bcdd26fd6d, 0x93ff54e107cf8bc9})},
+        {true, -142, MType({0x2075312a827f14fa, 0x91ff5978fd42795b})},
+        {true, -142, MType({0xe21764e139c98f60, 0x8fff5e00f2fe65f2})},
+        {true, -142, MType({0xa492a29551751b4c, 0x8dff6278e9025197})},
+        {true, -142, MType({0x1bc8f5f658f1c3a2, 0x8bff66e0df4d3c50})},
+        {true, -142, MType({0xe39d3faf42340ed7, 0x89ff6b38d5de2622})},
+        {true, -142, MType({0x7ff3326682c02485, 0x87ff6f80ccb40f16})},
+        {true, -142, MType({0x5caf4fbe343cf928, 0x85ff73b8c3cdf731})},
+        {true, -142, MType({0xcdb6e554348f7fe8, 0x83ff77e0bb2ade79})},
+        {true, -142, MType({0xef009c2457de25d, 0x81ff7bf8b2c9c4f6})},
+        {true, -143, MType({0x8883333c57b57c74, 0xffff000155535558})},
+        {true, -143, MType({0xf32668f39c70d183, 0xfbff07f145931f44})},
+        {true, -143, MType({0x459a73c6a6486fe3, 0xf7ff0fc13650e7bd})},
+        {true, -143, MType({0x37b18cca7dd3a29f, 0xf3ff1771278aaecd})},
+        {true, -143, MType({0x513f610d21bcfc78, 0xefff1f01193e7480})},
+        {true, -143, MType({0xea190b95c0690b7b, 0xebff26710b6a38e1})},
+        {true, -143, MType({0x2a150f64f0ad1743, 0xe7ff2dc0fe0bfbfd})},
+        {true, -143, MType({0x90b5174e995e9d1, 0xe3ff34f0f121bddd})},
+        {true, -143, MType({0x4ed512b9b93ea2bf, 0xdfff3c00e4a97e8c})},
+        {true, -143, MType({0x934cea217ab794a2, 0xdbff42f0d8a13e15})},
+        {true, -143, MType({0x3e4ebe948afd2c76, 0xd7ff49c0cd06fc83})},
+        {true, -143, MType({0x87b7c0f5bcfee2e1, 0xd3ff5070c1d8b9df})},
+        {true, -143, MType({0x776666228cb6371b, 0xcfff5700b7147634})},
+        {true, -143, MType({0xe53a60f3514db358, 0xcbff5d70acb8318b})},
+        {true, -143, MType({0x79149c3b6e57fa86, 0xc7ff63c0a2c1ebef})},
+        {true, -143, MType({0xaad734c98416df2a, 0xc3ff69f0992fa568})},
+        {true, -143, MType({0xc26573679ed28334, 0xbfff70008fff5e00})},
+        {true, -143, MType({0xd7a3c6db6540809f, 0xbbff75f0872f15c0})},
+        {true, -143, MType({0xd277bde645fb1aad, 0xb7ff7bc07ebcccb1})},
+        {true, -143, MType({0x6ac80145a4087793, 0xb3ff817076a682dc})},
+        {true, -143, MType({0x287c4db30271e265, 0xafff87006eea3849})},
+        {true, -143, MType({0x637d6de42eeb151e, 0xabff8c706785ed00})},
+        {true, -143, MType({0x43b5348b6b898a8c, 0xa7ff91c06077a10a})},
+        {true, -143, MType({0xc10e7657978bd7f6, 0xa3ff96f059bd546e})},
+        {true, -143, MType({0xa37503f457310e59, 0x9fff9c0053550735})},
+        {true, -143, MType({0x82d5a40a3aa022ff, 0x9bffa0f04d3cb966})},
+        {true, -143, MType({0xc71e0d3ee3df5f4d, 0x97ffa5c047726b08})},
+        {true, -143, MType({0xa83ce0352bdbd79b, 0x93ffaa7041f41c23})},
+        {true, -143, MType({0x2e21a18d4680e8e4, 0x8fffaf003cbfccbe})},
+        {true, -143, MType({0x30bcb3e4e5dfbd28, 0x8bffb37037d37cdf})},
+        {true, -143, MType({0x57ff51d75c66d64a, 0x87ffb7c0332d2c8d})},
+        {true, -143, MType({0x1bdb87fdbe299f43, 0x83ffbbf02ecadbcf})},
+        {true, -144, MType({0x88885dde02700703, 0xffff800055551555})},
+        {true, -144, MType({0xd259ca803a0c1870, 0xf7ff87e04d94724c})},
+        {true, -144, MType({0xe514130851c7070a, 0xefff8f80464fce8f})},
+        {true, -144, MType({0x30a16898f3073a64, 0xe7ff96e03f832a2a})},
+        {true, -144, MType({0xc4ed64517b2949ce, 0xdfff9e00392a8526})},
+        {true, -144, MType({0x51e4fb4e32cf6350, 0xd7ffa4e03341df90})},
+        {true, -144, MType({0x277672a88350bcce, 0xcfffab802dc53971})},
+        {true, -144, MType({0x359153772a490f06, 0xc7ffb1e028b092d3})},
+        {true, -144, MType({0xc265ece6b481a0e, 0xbfffb80023ffebc0})},
+        {true, -144, MType({0xdb2781c03fa132f6, 0xb7ffbde01faf4440})},
+        {true, -144, MType({0x7287c95c845ada33, 0xafffc3801bba9c5e})},
+        {true, -144, MType({0x423b56b1263e5a77, 0xa7ffc8e0181df421})},
+        {true, -144, MType({0x5a3752ca4c076fa3, 0x9fffce0014d54b91})},
+        {true, -144, MType({0x6a71e2b27eb3f573, 0x97ffd2e011dca2b6})},
+        {true, -144, MType({0xc2e21b72cff39d8f, 0x8fffd7800f2ff997})},
+        {true, -144, MType({0x537ff612feb7ac9e, 0x87ffdbe00ccb503c})},
+        {true, -145, MType({0x5888873333c57c18, 0xffffc00015554d55})},
+        {true, -145, MType({0xfa51421842311c42, 0xefffc7c01193f9d1})},
+        {true, -145, MType({0x2c4ed6de475b942c, 0xdfffcf000e4aa5fa})},
+        {true, -145, MType({0xce77678cbb6fcb88, 0xcfffd5c00b7151d8})},
+        {true, -145, MType({0xc26629a679ed3b, 0xbfffdc0008fffd78})},
+        {true, -145, MType({0x23287cb9d3072728, 0xafffe1c006eea8e1})},
+        {true, -145, MType({0xd5a37540fd057315, 0x9fffe7000535541c})},
+        {true, -145, MType({0xf82e21c1fce36810, 0x8fffebc003cbff32})},
+        {true, -146, MType({0x5588887ddde02702, 0xffffe00005555455})},
+        {true, -146, MType({0x9ac4ed72adf5b295, 0xdfffe7800392aa14})},
+        {true, -146, MType({0xc26648066b482, 0xbfffee00023fffaf})},
+        {true, -146, MType({0x455a3754b292c077, 0x9ffff380014d552e})},
+        {true, -147, MType({0x5558888833333c58, 0xfffff00001555535})},
+        {true, -147, MType({0xe000c2665736679f, 0xbffff700008ffff5})},
+        {true, -148, MType({0x5555888885ddde02, 0xfffff80000555551})},
+        {true, -149, MType({0xd555588888733334, 0xfffffc0000155554})},
+        {false, 0, MType({0x0, 0x0})},
+        {false, -148, MType({0xeaaaac44444eeeef, 0x80000200000aaaaa})},
+        {false, -147, MType({0xaaaac444459999ac, 0x80000400002aaaac})},
+        {false, -147, MType({0x2000c2667596679f, 0xc00009000090000a})},
+        {false, -146, MType({0xaaac44446eeef381, 0x8000080000aaaaba})},
+        {false, -146, MType({0x655a3755f81815cc, 0xa0000c80014d557c})},
+        {false, -146, MType({0xc26684c66b482, 0xc000120002400051})},
+        {false, -146, MType({0xbac4ed7c40fb07eb, 0xe00018800392ab40})},
+        {false, -145, MType({0xaac44449999abe2c, 0x8000100002aaab2a})},
+        {false, -145, MType({0x82e21d79cbb6812, 0x9000144003cc00cd})},
+        {false, -145, MType({0xd5a37569adb01dc3, 0xa00019000535568d})},
+        {false, -145, MType({0x33287d01e8c9d1d9, 0xb0001e4006eeac74})},
+        {false, -145, MType({0xc266a32679ed48, 0xc000240009000288})},
+        {false, -145, MType({0xde77685122b2764b, 0xd0002a400b7158d1})},
+        {false, -145, MType({0x2c4ed810a8063f03, 0xe00031000e4aaf5b})},
+        {false, -145, MType({0xa5143e7be891c8f, 0xf00038401194062e})},
+        {false, -144, MType({0xac4444eeef3813a1, 0x800020000aaaaeaa})},
+        {false, -144, MType({0x5b7ff7fe1339025b, 0x880024200ccb5a6e})},
+        {false, -144, MType({0x42e21e26caf39e33, 0x900028800f300668})},
+        {false, -144, MType({0xf271e66fa5554bc6, 0x98002d2011dcb29e})},
+        {false, -144, MType({0x5a3757e0615cc676, 0xa000320014d55f19})},
+        {false, -144, MType({0xca3b5d8210ca5cab, 0xa8003720181e0bde})},
+        {false, -144, MType({0xf287d25f3cb032bb, 0xb0003c801bbab8f6})},
+        {false, -144, MType({0xe3278d840be28cdb, 0xb80042201faf6669})},
+        {false, -144, MType({0xc266dfe6b482076, 0xc000480024001440})},
+        {false, -144, MType({0x3d9166de380a6d3d, 0xc8004e2028b0c282})},
+        {false, -144, MType({0xa7768b356ba61e4b, 0xd00054802dc57139})},
+        {false, -144, MType({0xd9e51a1849db73c1, 0xd8005b203342206f})},
+        {false, -144, MType({0xc4ed8a9d907eb521, 0xe0006200392ad02e})},
+        {false, -144, MType({0xb8a197dea928acd7, 0xe80069203f838080})},
+        {false, -144, MType({0x65144cf7dcc72d3b, 0xf000708046503170})},
+        {false, -144, MType({0xda5a1108890d9f6a, 0xf80078204d94e308})},
+        {false, -143, MType({0xc4445999abe2ce2c, 0x800040002aaacaaa})},
+        {false, -143, MType({0x1fdbbb4f3bffc832, 0x840044102ecb2431})},
+        {false, -143, MType({0x97ff8f39ec91b4ee, 0x88004840332d7e1d})},
+        {false, -143, MType({0x74bcfcf0b3f0a95d, 0x8c004c9037d3d876})},
+        {false, -143, MType({0x2e21f80ca6813aff, 0x900051003cc03342})},
+        {false, -143, MType({0x6c3d4629170ce87f, 0x9400559041f48e87})},
+        {false, -143, MType({0x71e84e3b80a8881, 0x98005a404772ea4d})},
+        {false, -143, MType({0x6d62fdcbdd6bec3, 0x9c005f104d3d469a})},
+        {false, -143, MType({0xa375a6b701dc77c0, 0xa00064005355a375})},
+        {false, -143, MType({0x450f331826ad6b05, 0xa400691059be00e7})},
+        {false, -143, MType({0x83b60ea8bd0aa459, 0xa8006e4060785ef6})},
+        {false, -143, MType({0x277e691469dd13f5, 0xac0073906786bdab})},
+        {false, -143, MType({0x287d6e0a0d1e25eb, 0xb00079006eeb1d0d})},
+        {false, -143, MType({0xaec94b3be9b060f5, 0xb4007e9076a77d24})},
+        {false, -143, MType({0x1279365fce280cce, 0xb80084407ebdddfa})},
+        {false, -143, MType({0xdba5732f3e83e04a, 0xbc008a1087303f95})},
+        {false, -143, MType({0xc26759679ed5b754, 0xc00090009000a200})},
+        {false, -143, MType({0xaed95aca5edb5109, 0xc400961099310543})},
+        {false, -143, MType({0xb917091d2687160f, 0xc8009c40a2c36967})},
+        {false, -143, MType({0x293d1c2a0378e75d, 0xcc00a290acb9ce76})},
+        {false, -143, MType({0x776977bf9766f5a7, 0xd000a900b7163478})},
+        {false, -143, MType({0x4bbb31b14776a18b, 0xd400af90c1da9b78})},
+        {false, -143, MType({0x7e5297d76c8564ba, 0xd800b640cd09037f})},
+        {false, -143, MType({0x1751360f8461c447, 0xdc00bd10d8a36c98})},
+        {false, -143, MType({0x4ed9dc3c63f44c41, 0xe000c400e4abd6cc})},
+        {false, -143, MType({0x8d10a4466a5894d5, 0xe400cb10f1244226})},
+        {false, -143, MType({0x6a1af81bb4e6510e, 0xe800d240fe0eaeb1})},
+        {false, -143, MType({0xae1f97b0542a677a, 0xec00d9910b6d1c77})},
+        {false, -143, MType({0x51469efe81d014cc, 0xf000e10119418b84})},
+        {false, -143, MType({0x7bb98c06d77a18b4, 0xf400e891278dfbe2})},
+        {false, -143, MType({0x85a344d0868bed17, 0xf800f04136546d9d})},
+        {false, -143, MType({0xf7301d6990e307cc, 0xfc00f8114596e0c0})},
+        {false, -142, MType({0x4446eef38140138f, 0x80008000aaabaaac})},
+        {false, -142, MType({0x10f5e43296105497, 0x82008408b2cbe5b8})},
+        {false, -142, MType({0xedbd4f83ef63f730, 0x84008820bb2d2189})},
+        {false, -142, MType({0xfeb654fd541c638e, 0x86008c48c3d05e27})},
+        {false, -142, MType({0x7ffadeb8882f7674, 0x88009080ccb69b98})},
+        {false, -142, MType({0xc5a59fd36bd44397, 0x8a0094c8d5e0d9e1})},
+        {false, -142, MType({0x3bd217701b27dddb, 0x8c009920df50190a})},
+        {false, -142, MType({0x669c93b50e4a2595, 0x8e009d88e9055918})},
+        {false, -142, MType({0xe22234cd39f29cd4, 0x9000a200f3019a12})},
+        {false, -142, MType({0x6280efe8307d41d9, 0x9200a688fd45dc00})},
+        {false, -142, MType({0xb3d7923a436f6fc4, 0x9400ab2107d31ee7})},
+        {false, -142, MType({0xba45c3fca574c5a0, 0x9600afc912aa62cf})},
+        {false, -142, MType({0x71ec0b6d8cd413d1, 0x9800b4811dcca7bf})},
+        {false, -142, MType({0xeeebcfd0565c5006, 0x9a00b949293aedbd})},
+        {false, -142, MType({0x5d675c6da8c98fc3, 0x9c00be2134f634d2})},
+        {false, -142, MType({0x181e39398a2099a, 0x9e00c30940ff7d04})},
+        {false, -142, MType({0x375f8195cc8b1d29, 0xa000c8014d57c65a})},
+
+    },
+    // -log(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).log().sign_mantissa_exponent();
+    //   print("{false," if (s == -1) else "{true,", e, ",
+    //         MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
+    /* .step_4 = */
+    {
+        {true, -149, MType({0x4cd24d68ff2f11ae, 0x81fffef7f002cb2b})},
+        {true, -150, MType({0x455555888887ddde, 0xfffffe0000055555})},
+        {true, -150, MType({0xf0fa101f52b3971f, 0xfbfffe0fe0051653})},
+        {true, -150, MType({0x9c9329d659ed3734, 0xf7fffe1f8004d94a})},
+        {true, -150, MType({0x4821006d9b58462e, 0xf3fffe2ee0049e31})},
+        {true, -150, MType({0xf3a3f025142f8c21, 0xeffffe3e000464ff})},
+        {true, -150, MType({0x9f1c53bcc1c4b11c, 0xebfffe4ce0042dae})},
+        {true, -150, MType({0x4a8a8474a17fdd30, 0xe7fffe5b8003f835})},
+        {true, -150, MType({0xf5eeda0cb0df586d, 0xe3fffe69e003c48b})},
+        {true, -150, MType({0xa149aac4ed772adf, 0xdffffe78000392aa})},
+        {true, -150, MType({0x4c9b4b5d54f0bc96, 0xdbfffe85e0036289})},
+        {true, -150, MType({0xf7e40f15e50a759f, 0xd7fffe938003341f})},
+        {true, -150, MType({0xa32447ae9b975e05, 0xd3fffea0e0030766})},
+        {true, -150, MType({0x4e5c4567767ebdd5, 0xcffffeae0002dc55})},
+        {true, -150, MType({0xf98c570073bbbd19, 0xcbfffebae002b2e3})},
+        {true, -150, MType({0xa4b4c9b9915d03dd, 0xc7fffec780028b0a})},
+        {true, -150, MType({0x4fd5e952cd845a28, 0xc3fffed3e00264c1})},
+        {true, -150, MType({0xfaf0000c26664806, 0xbffffee000023fff})},
+        {true, -150, MType({0xa60356a59a49b57f, 0xbbfffeebe0021cbe})},
+        {true, -150, MType({0x5110345f27878a9b, 0xb7fffef78001faf5})},
+        {true, -150, MType({0xfc16def8cc8a4f61, 0xb3ffff02e001da9b})},
+        {true, -150, MType({0xa7179ab287cdcbd8, 0xafffff0e0001bbaa})},
+        {true, -150, MType({0x5212aa4c57dea809, 0xabffff18e0019e19})},
+        {true, -150, MType({0xfd084f063b5a0bf8, 0xa7ffff23800181df})},
+        {true, -150, MType({0xa7f8c8a030ed3fab, 0xa3ffff2de00166f6})},
+        {true, -150, MType({0x52e4555a37554b29, 0x9fffff3800014d55})},
+        {true, -150, MType({0xfdcb31f44d5e9676, 0x9bffff41e00134f3})},
+        {true, -150, MType({0xa8ad99ae71e48997, 0x97ffff4b80011dca})},
+        {true, -150, MType({0x538bc648a3d12c90, 0x93ffff54e00107d1})},
+        {true, -150, MType({0xfe65f002e21cc765, 0x8fffff5e0000f2ff})},
+        {true, -150, MType({0xa93c4d9d2bcd821a, 0x8bffff66e000df4e})},
+        {true, -150, MType({0x540f14577ff704b2, 0x87ffff6f8000ccb5})},
+        {true, -150, MType({0xfede77f1ddba1731, 0x83ffff77e000bb2b})},
+        {true, -151, MType({0x5355555888888333, 0xffffff0000015555})},
+        {true, -151, MType({0xa8e7ba8d659ed7dc, 0xf7ffff0fc0013652})},
+        {true, -151, MType({0xfe747e025142fc61, 0xefffff1f0001193f})},
+        {true, -151, MType({0x53fbfb374a1800c7, 0xe7ffff2dc000fe0d})},
+        {true, -151, MType({0xa97e8aac4ed77513, 0xdfffff3c0000e4aa})},
+        {true, -151, MType({0xfefc81e15e50a947, 0xd7ffff49c000cd07})},
+        {true, -151, MType({0x547633567767ed66, 0xcfffff570000b715})},
+        {true, -151, MType({0xa9ebee8b9915d174, 0xc7ffff63c000a2c2})},
+        {true, -151, MType({0xff5e0000c2666573, 0xbfffff7000008fff})},
+        {true, -151, MType({0x54ccb135f2787966, 0xb7ffff7bc0007ebd})},
+        {true, -151, MType({0xaa3848ab287cdd4e, 0xafffff8700006eea})},
+        {true, -151, MType({0xffa109e063b5a12d, 0xa7ffff91c0006077})},
+        {true, -151, MType({0x55073555a3755504, 0x9fffff9c00005355})},
+        {true, -151, MType({0xaa6b088ae71e48d5, 0x97ffffa5c0004772})},
+        {true, -151, MType({0xffccbe002e21cca2, 0x8fffffaf00003cbf})},
+        {true, -151, MType({0x552c8d3577ff706a, 0x87ffffb7c000332d})},
+        {true, -152, MType({0x551555558888885e, 0xffffff8000005555})},
+        {true, -152, MType({0xffce8fc025142fe3, 0xefffff8f8000464f})},
+        {true, -152, MType({0xaa8526aac4ed7764, 0xdfffff9e0000392a})},
+        {true, -152, MType({0x5539711567767ee3, 0xcfffffab80002dc5})},
+        {true, -152, MType({0xffebc0000c26665f, 0xbfffffb8000023ff})},
+        {true, -152, MType({0xaa9c5e6ab287cdd9, 0xafffffc380001bba})},
+        {true, -152, MType({0x554b91555a375553, 0x9fffffce000014d5})},
+        {true, -152, MType({0xfff997c002e21ccb, 0x8fffffd780000f2f})},
+        {true, -153, MType({0x554d555558888887, 0xffffffc000001555})},
+        {true, -153, MType({0xaaa5fa2aac4ed777, 0xdfffffcf00000e4a})},
+        {true, -153, MType({0xfffd780000c26666, 0xbfffffdc000008ff})},
+        {true, -153, MType({0x55541cd555a37555, 0x9fffffe700000535})},
+        {true, -154, MType({0x5554555555888888, 0xffffffe000000555})},
+        {true, -154, MType({0xffffaf00000c2666, 0xbfffffee0000023f})},
+        {true, -155, MType({0x5555355555588889, 0xfffffff000000155})},
+        {true, -156, MType({0x5555515555558889, 0xfffffff800000055})},
+        {false, 0, MType({0x0, 0x0})},
+        {false, -155, MType({0xaaaaacaaaaaac444, 0x800000040000002a})},
+        {false, -154, MType({0xaaaabaaaaaac4444, 0x80000008000000aa})},
+        {false, -154, MType({0x5100000c2666, 0xc000001200000240})},
+        {false, -153, MType({0xaaab2aaaaac44444, 0x80000010000002aa})},
+        {false, -153, MType({0x55568dd555a37555, 0xa000001900000535})},
+        {false, -153, MType({0x2880000c26667, 0xc000002400000900})},
+        {false, -153, MType({0xaaaf5b2aac4ed778, 0xe000003100000e4a})},
+        {false, -152, MType({0xaaaeaaaaac444445, 0x8000002000000aaa})},
+        {false, -152, MType({0x6684002e21cce, 0x9000002880000f30})},
+        {false, -152, MType({0x555f19555a375558, 0xa0000032000014d5})},
+        {false, -152, MType({0xaab8f6eab287cde2, 0xb000003c80001bba})},
+        {false, -152, MType({0x1440000c26666e, 0xc000004800002400})},
+        {false, -152, MType({0x5571399567767efb, 0xd000005480002dc5})},
+        {false, -152, MType({0xaad02eaac4ed778b, 0xe00000620000392a})},
+        {false, -152, MType({0x3170402514301d, 0xf000007080004650})},
+        {false, -151, MType({0xaacaaaaac444445a, 0x8000004000002aaa})},
+        {false, -151, MType({0x557e1d7577ff70a7, 0x880000484000332d})},
+        {false, -151, MType({0x3342002e21ccf8, 0x9000005100003cc0})},
+        {false, -151, MType({0xaaea4ccae71e494d, 0x9800005a40004772})},
+        {false, -151, MType({0x55a37555a37555a7, 0xa000006400005355})},
+        {false, -151, MType({0x5ef62063b5a207, 0xa800006e40006078})},
+        {false, -151, MType({0xab1d0cab287cde6e, 0xb000007900006eea})},
+        {false, -151, MType({0x55ddf975f2787ade, 0xb800008440007ebd})},
+        {false, -151, MType({0xa20000c2666759, 0xc000009000009000})},
+        {false, -151, MType({0xab6966cb9915d3e1, 0xc800009c4000a2c2})},
+        {false, -151, MType({0x563477567767f078, 0xd00000a90000b715})},
+        {false, -151, MType({0x1037e215e50ad20, 0xd80000b64000cd08})},
+        {false, -151, MType({0xabd6caac4ed779dc, 0xe00000c40000e4aa})},
+        {false, -151, MType({0x56aeaf774a1806b0, 0xe80000d24000fe0d})},
+        {false, -151, MType({0x18b82025143039f, 0xf00000e100011940})},
+        {false, -151, MType({0xac6d9acd659ee0ad, 0xf80000f040013652})},
+        {false, -150, MType({0xabaaaaac444446ef, 0x800000800000aaaa})},
+        {false, -150, MType({0x1218811ddba1d9b, 0x840000882000bb2c})},
+        {false, -150, MType({0x569b96577ff70c5f, 0x880000908000ccb5})},
+        {false, -150, MType({0xac1907bd2bcd8b3b, 0x8c0000992000df4e})},
+        {false, -150, MType({0x19a1002e21cd235, 0x900000a20000f300})},
+        {false, -150, MType({0x571ee468a3d1394e, 0x940000ab200107d1})},
+        {false, -150, MType({0xaca7bbae71e4988b, 0x980000b480011dca})},
+        {false, -150, MType({0x234ce144d5ea7f0, 0x9c0000be200134f4})},
+        {false, -150, MType({0x57c6555a37555f82, 0xa00000c800014d55})},
+        {false, -150, MType({0xad5c8cc030ed5744, 0xa40000d2200166f6})},
+        {false, -150, MType({0x2f7b1063b5a273b, 0xa80000dc800181e0})},
+        {false, -150, MType({0x5898006c57dec76f, 0xac0000e720019e19})},
+        {false, -150, MType({0xae3dbab287cdefe3, 0xb00000f20001bbaa})},
+        {false, -150, MType({0x3e92118cc8a789f, 0xb40000fd2001da9c})},
+        {false, -150, MType({0x599a765f2787b9aa, 0xb80001088001faf5})},
+        {false, -150, MType({0xaf51fec59a49eb0a, 0xbc00011420021cbe})},
+        {false, -150, MType({0x510000c266684c6, 0xc000012000024000})},
+        {false, -150, MType({0x5ad4c172cd849ee9, 0xc400012c200264c1})},
+        {false, -150, MType({0xb0a08bb9915d5179, 0xc800013880028b0a})},
+        {false, -150, MType({0x673a92073bc1480, 0xcc0001452002b2e4})},
+        {false, -150, MType({0x5c4e6567767f2009, 0xd00001520002dc55})},
+        {false, -150, MType({0xb2310dce9b97cc1d, 0xd400015f20030766})},
+        {false, -150, MType({0x81bf115e50af0c7, 0xd800016c80033420})},
+        {false, -150, MType({0x5e0f5f7d54f14614, 0xdc00017a20036289})},
+        {false, -150, MType({0xb40baac4ed77c410, 0xe0000188000392aa})},
+        {false, -150, MType({0xa11262cb0e002c7, 0xe40001962003c48c})},
+        {false, -150, MType({0x60202674a1809a47, 0xe80001a48003f835})},
+        {false, -150, MType({0xb63901dcc1c582a0, 0xec0001b320042dae})},
+        {false, -150, MType({0xc5c1025143073df, 0xf00001c200046500})},
+        {false, -150, MType({0x6289aa8d9b594616, 0xf40001d120049e31})},
+        {false, -150, MType({0xb8c22bd659ee5155, 0xf80001e08004d94a})},
+        {false, -150, MType({0xf05f03f52b4cdae, 0xfc0001f020051654})},
+        {false, -149, MType({0xb2aaaac44444999a, 0x800001000002aaaa})},
+
+    }};
+// > P = fpminimax((log(1 + x) - x)/x^2, 2, [|1, 128...|],
+//                 [-0x1.0002143p-29 , 0x1p-29]);
+// > P;
+// > dirtyinfnorm(log(1 + x)/x - x*P, [-0x1.0002143p-29 , 0x1p-29]);
+// 0x1.99a3...p-121
+const Float128 BIG_COEFFS[3]{
+    {true, -129, MType({0xb59c58e5554d581c, 0x800000000006a710})},
+    {false, -129, MType({0xde05c7c94ae9cbae, 0xaaaaaaaaaaaaaabd})},
+    {true, -128, MType({0x0, 0x8000000000000000})},
+};
+
+// Reuse the output of the fast pass range reduction.
+// -2^-8 <= m_x < 2^-7
+double log_accurate(int e_x, int index, double m_x) {
+
+  Float128 e_x_f128(static_cast<float>(e_x));
+  Float128 sum = fputil::quick_mul(LOG_2, e_x_f128);
+  sum = fputil::quick_add(sum, LOG_TABLE.step_1[index]);
+
+  Float128 v_f128 = log_range_reduction(m_x, LOG_TABLE, sum);
+  sum = fputil::quick_add(sum, v_f128);
+
+  // 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, p);
+
+  Float128 r = fputil::quick_add(sum, p);
+
+  return static_cast<double>(r);
+}
+
+} // namespace
+
+LLVM_LIBC_FUNCTION(double, log, (double x)) {
+  using FPBits_t = typename fputil::FPBits<double>;
+  FPBits_t xbits(x);
+  uint64_t x_u = xbits.uintval();
+
+  int x_e = -FPBits_t::EXPONENT_BIAS;
+
+  if (LIBC_UNLIKELY(x_u == 0x3FF0'0000'0000'0000ULL)) {
+    // log(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();
+  }
+
+  // log(x) = log(2^x_e * x_m)
+  //        = x_e * log(2) + log(x_m)
+
+  // Range reduction for log(x_m):
+  // 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 = RD[index];
+
+  // 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);
+
+  // hi is exact
+  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, 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
+
+  // Exact sum:
+  //   r1.hi + r1.lo = e_x * log(2)_hi - log(r)_hi + u
+  r1 = fputil::exact_add(hi, u);
+
+  // Error of u_sq = ulp(u^2);
+  u_sq = u * u;
+  // Total error is bounded by ~ C * ulp(u^2).
+  // Degree-7 minimax polynomial
+  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 + r1.lo, p0, p1, p2);
+
+  // 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) + 2^-85.
+  err = fputil::multiply_add(u_sq, P_ERR, HI_ERR);
+
+  // Lower bound from the result
+  double left = r1.hi + (p - err);
+  // Upper bound from the result
+  double right = r1.hi + (p + err);
+
+  // Ziv's test if fast pass is accurate enough.
+  if (left == right)
+    return left;
+
+  return log_accurate(x_e, index, u);
+}
+
+} // namespace __llvm_libc

libc/src/math/generic/log10.cpp

 #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"
+#include "log_range_reduction.h"
 
 namespace __llvm_libc {
 
 // 128-bit precision dyadic floating point numbers.
 using Float128 = typename fputil::DyadicFloat<128>;
 using MType = typename Float128::MantissaType;
 
 namespace {
 
 constexpr fputil::DoubleDouble LOG10_E = {0x1.95355baaafad3p-57,
                                           0x1.bcb7b1526e50ep-2};
 
 // A simple upper bound for the error of e_x * log(2) - log(r).
 constexpr double HI_ERR = 0x1.0p-85;
 
 // Extra errors from P is from using x^2 to reduce evaluation latency.
 constexpr double P_ERR = 0x1.0p-51;
 
 // 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}));
 
+const LogRR LOG10_TABLE = {
     // -log10(r) with 128-bit precision generated by SageMath with:
     //
     // for i in range(128):
     //   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),
+    //   print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) %
+    //   2^64),
     //         "})},");
-const Float128 LOG10_R1[128] = {
+    /* .step_1 = */ {
         {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})},
         {false, -130, MType({0x9e405fb8bcb1ff1e, 0xb60edafcdd99ad1d})},
         {false, -130, MType({0x9e405fb8bcb1ff1e, 0xb60edafcdd99ad1d})},
         {false, -130, MType({0xf7e2ab36f09e9014, 0xb8ae8671b3d7dd6c})},
         {false, -130, MType({0xc669639640c305bb, 0xbb522e5dbf37f63b})},
         {false, -130, MType({0xa3dc9e464e98764b, 0xbdf9def04cf980ff})},
         {false, -130, MType({0xffd3256b59fa9c59, 0xc0a5a490dea95b5e})},
         {false, -130, MType({0xb0a2d48672a051a5, 0xc3558be085e3f4bc})},
         {false, -130, MType({0xb0a2d48672a051a5, 0xc3558be085e3f4bc})},
         {false, -130, MType({0xacb2ca5d4ca1c10e, 0xc609a1bb4aa98f59})},
         {false, -130, MType({0x43690b9e3cde0d02, 0xc8c1f3399ca7d33b})},
         {false, -130, MType({0x18b1fd60383f7e5a, 0xcb7e8db1cfe04827})},
         {false, -130, MType({0x248757e5f45af3d, 0xce3f7eb9a517c969})},
         {false, -130, MType({0x7c4acd605be48bc1, 0xd104d427de7fbcc4})},
         {false, -130, MType({0x7c4acd605be48bc1, 0xd104d427de7fbcc4})},
         {false, -130, MType({0x58ff63629a92652d, 0xd3ce9c15e10ec927})},
         {false, -130, MType({0x6b49be3bd8c89f10, 0xd69ce4e16303fcdd})},
         {false, -130, MType({0xe6dd603a881e9060, 0xd96fbd2e2814c9cc})},
         {false, -130, MType({0xe6dd603a881e9060, 0xd96fbd2e2814c9cc})},
         {false, -130, MType({0x89e281c98c1d705c, 0xdc4733e7cbcbfc8c})},
         {false, -130, MType({0xdc0db7cf0cce9f32, 0xdf2358439aa5dd12})},
         {false, -130, MType({0xfdf1c5b846db9deb, 0xe20439c27a7c01b8})},
         {false, -130, MType({0xfdf1c5b846db9deb, 0xe20439c27a7c01b8})},
         {false, -130, MType({0x3dd7eab48869c402, 0xe4e9e832e2da0c05})},
         {false, -130, MType({0x4e8fcc900b41daef, 0xe7d473b2e5db8f2a})},
         {false, -130, MType({0x4e8fcc900b41daef, 0xe7d473b2e5db8f2a})},
         {false, -130, MType({0x7593e1a9e917359a, 0xeac3ecb24a3ac7b4})},
         {false, -130, MType({0xe7741396b49e1ce5, 0xedb863f4b73f982d})},
         {false, -130, MType({0xe7741396b49e1ce5, 0xedb863f4b73f982d})},
         {false, -130, MType({0xc8ba4f8f47b85a5c, 0xf0b1ea93f34675a7})},
         {false, -130, MType({0x7007c1276821b705, 0xf3b09202359f9787})},
         {false, -130, MType({0x7007c1276821b705, 0xf3b09202359f9787})},
         {false, -130, MType({0x7ee19afe6db7e324, 0xf6b46c0c8c8fdea1})},
         {false, -130, MType({0xedf54f37f6d40420, 0xf9bd8add584687f0})},
         {false, -130, MType({0xedf54f37f6d40420, 0xf9bd8add584687f0})},
         {false, -130, MType({0xefe52ccf03e7dee1, 0xfccc00fedba4e6fb})},
         {false, -130, MType({0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
         {false, -130, MType({0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
         {false, -129, MType({0xe0b571f5c91b0446, 0x817c9fa643880404})},
         {false, -129, MType({0x7178594bef2def59, 0x830c17427ea55eca})},
         {false, -129, MType({0x7178594bef2def59, 0x830c17427ea55eca})},
         {false, -129, MType({0x9a741bb171158d2a, 0x849e6196487c1d1c})},
         {false, -129, MType({0x9a741bb171158d2a, 0x849e6196487c1d1c})},
         {false, -129, MType({0x1a618264446cb495, 0x863388eb55ebd295})},
         {false, -129, MType({0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
         {false, -129, MType({0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
         {false, -129, MType({0xabe0b522230f7d14, 0x896698dce4cff76c})},
         {false, -129, MType({0xabe0b522230f7d14, 0x896698dce4cff76c})},
         {false, -129, MType({0xd28e8adafea703b4, 0x8b04972e9d4d3011})},
         {false, -129, MType({0x208422d83be34b27, 0x8ca59def7b5cefc5})},
         {false, -129, MType({0x208422d83be34b27, 0x8ca59def7b5cefc5})},
         {false, -129, MType({0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
         {false, -129, MType({0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
         {false, -129, MType({0xda982a614e12c6dd, 0x8ff0f2d7960a075c})},
         {false, -129, MType({0xda982a614e12c6dd, 0x8ff0f2d7960a075c})},
         {false, -129, MType({0x38401fc1c1b5c2c, 0x919b58b0d999bbc8})},
         {false, -129, MType({0x38401fc1c1b5c2c, 0x919b58b0d999bbc8})},
         {false, -129, MType({0xa9b55d3f16da746a, 0x9348f6614f821394})},
         {false, -129, MType({0xa9b55d3f16da746a, 0x9348f6614f821394})},
         {false, -129, MType({0x88d2d1473d4f7f5, 0x94f9d870aac256a5})},
         {false, -129, MType({0x88d2d1473d4f7f5, 0x94f9d870aac256a5})},
         {false, -129, MType({0x7c1e117dea19e9e6, 0x96ae0bb05c35d5bd})},
         {false, -129, MType({0x7c1e117dea19e9e6, 0x96ae0bb05c35d5bd})},
         {false, -129, MType({0x336db0630f536fb9, 0x98659d3dd9b12532})},
         {false, 0, MType(0)},
-};
-
-// 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:
     //
     // 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] = {
+    /* .step_2 = */
+    {
         {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})},
-};
-
-// 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] = {
+    /* .step_3 = */
+    {
         {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] = {
+    /* .step_4 = */
+    {
         {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, [-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.
 // -2^-8 <= m_x < 2^-7
 double log10_accurate(int e_x, int index, double m_x) {
 
   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_TABLE.step_1[index]);
 
-  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)}));
+  Float128 v_f128 = log_range_reduction(m_x, LOG10_TABLE, sum);
 
   // 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);
#endif // LIBC_TARGET_CPU_HAS_FMA
   // 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).
+  // Total error is bounded by ~ C * ulp(u^2) + 2^-85.
   err = fputil::multiply_add(u_sq, P_ERR, HI_ERR);
 
   // Lower bound from the result
   double left = r2.hi + (r2.lo - err);
   // Upper bound from the result
   double right = r2.hi + (r2.lo + err);
 
   // Ziv's test if fast pass is accurate enough.

libc/src/math/generic/log_range_reduction.h

+//===-- Extra range reduction steps for accurate pass of logarithms -------===//
+//
+// 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_LOG_RANGE_REDUCTION_H
+#define LLVM_LIBC_SRC_MATH_GENERIC_LOG_RANGE_REDUCTION_H
+
+#include "common_constants.h"
+#include "src/__support/FPUtil/dyadic_float.h"
+
+namespace __llvm_libc {
+
+// Struct to store -log*(r) for 4 range reduction steps.
+struct LogRR {
+  fputil::DyadicFloat<128> step_1[128];
+  fputil::DyadicFloat<128> step_2[193];
+  fputil::DyadicFloat<128> step_3[161];
+  fputil::DyadicFloat<128> step_4[130];
+};
+
+// Perform logarithm range reduction steps 2-4.
+// Inputs from the first step of range reduction:
+//   m_x : the reduced argument after the first step of range reduction
+//         satisfying  -2^-8 <= m_x < 2^-7  and  ulp(m_x) >= 2^-60.
+//   idx1: index of the -log(r1) table from the first step.
+// Outputs of the extra range reduction steps:
+//   sum: adding -log(r1) - log(r2) - log(r3) - log(r4) to the resulted sum.
+//   return value: the reduced argument v satisfying:
+//                 -0x1.0002143p-29 <= v < 0x1p-29,  and  ulp(v) >= 2^(-125).
+LIBC_INLINE fputil::DyadicFloat<128>
+log_range_reduction(double m_x, const LogRR &log_table,
+                    fputil::DyadicFloat<128> &sum) {
+  using Float128 = typename fputil::DyadicFloat<128>;
+  using MType = typename Float128::MantissaType;
+
+  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, log_table.step_2[idx2]);
+
+  int64_t s2 = static_cast<int64_t>(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, log_table.step_3[idx3]);
+
+  int64_t s3 = static_cast<int64_t>(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, log_table.step_4[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;
+
+  return (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)}));
+}
+
+} // namespace __llvm_libc
+
+#endif // LLVM_LIBC_SRC_MATH_GENERIC_LOG_RANGE_REDUCTION_H

libc/src/math/log.h

+//===-- Implementation header for log ---------------------------*- 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_LOG_H
+#define LLVM_LIBC_SRC_MATH_LOG_H
+
+namespace __llvm_libc {
+
+double log(double x);
+
+} // namespace __llvm_libc
+
+#endif // LLVM_LIBC_SRC_MATH_LOG_H

libc/test/src/math/CMakeLists.txt

add_fp_unittest(
   DEPENDS
     libc.src.errno.errno
     libc.include.math
     libc.src.math.expm1f
     libc.src.__support.FPUtil.fp_bits
 )
 
 add_fp_unittest(
+ log_test
+ NEED_MPFR
+ SUITE
+   libc_math_unittests
+ SRCS
+   log_test.cpp
+ DEPENDS
+   libc.src.errno.errno
+   libc.include.math
+   libc.src.math.log
+   libc.src.__support.FPUtil.fp_bits
+)
+
+add_fp_unittest(
   logf_test
   NEED_MPFR
   SUITE
     libc_math_unittests
   SRCS
     logf_test.cpp
   DEPENDS
     libc.src.errno.errno

libc/test/src/math/log_test.cpp

+//===-- Unittests for log -------------------------------------------------===//
+//
+// 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/__support/FPUtil/FPBits.h"
+#include "src/errno/libc_errno.h"
+#include "src/math/log.h"
+#include "test/UnitTest/FPMatcher.h"
+#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(LlvmLibcLogTest, SpecialNumbers) {
+  EXPECT_FP_EQ(aNaN, __llvm_libc::log(aNaN));
+  EXPECT_FP_EQ(inf, __llvm_libc::log(inf));
+  EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log(neg_inf), FE_INVALID);
+  EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log(0.0), FE_DIVBYZERO);
+  EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log(-0.0), FE_DIVBYZERO);
+  EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log(-1.0), FE_INVALID);
+  EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::log(1.0));
+}
+
+TEST(LlvmLibcLogTest, TrickyInputs) {
+  constexpr int N = 30;
+  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, 0x3fdfe93fff7fceb0, 0x3ff012631ad8df10,
+      0x3fefbfdaa448ed98,
+  };
+  for (int i = 0; i < N; ++i) {
+    double x = double(FPBits(INPUTS[i]));
+    EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log, x, __llvm_libc::log(x),
+                                   0.5);
+  }
+}
+
+TEST(LlvmLibcLogTest, AllExponents) {
+  double x = 0x1.0p-1074;
+  for (int i = -1074; i < 1024; ++i, x *= 2.0) {
+    ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log, x, __llvm_libc::log(x),
+                                   0.5);
+  }
+}
+
+TEST(LlvmLibcLogTest, InDoubleRange) {
+  constexpr uint64_t COUNT = 1234561;
+  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 = 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::log(x);
+      ++cc;
+      if (isnan(result) || isinf(result))
+        continue;
+
+      ++count;
+      // ASSERT_MPFR_MATCH(mpfr::Operation::Log, x, result, 0.5);
+      if (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Log, x, result,
+                                               0.5, rounding_mode)) {
+        ++fails;
+        while (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(
+            mpfr::Operation::Log, x, result, tol, rounding_mode)) {
+          mx = x;
+          mr = result;
+          tol *= 2.0;
+        }
+      }
+    }
+    tlog << " Log 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::Log, 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);
+}