Improve performance of integer formatting (#76726)

* Improve performance of integer formatting

1. The DivRem(..., 10) for each digit in the number ends up being the most expensive aspect of formatting.  This employs a trick other formatting libraries use, which is to have a table for all the formatted values between 00 and 99, and to then DivRem(..., 100) to cut the number of operations in half, which for longer values is meaningful.
2. Avoids going through the digit counting path when we know at the call site it won't be needed.
3. Employs a branch-free, table-based lookup for CountDigits(ulong) to go with a similar approach added for uint.

* Address PR feedback (move license notice)

THIRD-PARTY-NOTICES.TXT

@@ -1143,3 +1143,35 @@ OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABIL
 WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 POSSIBILITY OF SUCH DAMAGE.
+
+License notice for fmtlib/fmt
+-------------------------------
+
+Formatting library for C++
+
+Copyright (c) 2012 - present, Victor Zverovich
+
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of this software and associated documentation files (the
+"Software"), to deal in the Software without restriction, including
+without limitation the rights to use, copy, modify, merge, publish,
+distribute, sublicense, and/or sell copies of the Software, and to
+permit persons to whom the Software is furnished to do so, subject to
+the following conditions:
+
+The above copyright notice and this permission notice shall be
+included in all copies or substantial portions of the Software.
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
+LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
+WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+--- Optional exception to the license ---
+
+As an exception, if, as a result of your compiling your source code, portions
+of this Software are embedded into a machine-executable object form of such
+source code, you may redistribute such embedded portions in such object form
+without including the above copyright and permission notices.

src/libraries/System.Private.CoreLib/src/System/Buffers/Text/FormattingHelpers.CountDigits.cs

@@ -52,60 +52,58 @@ public static int CountDigits(UInt128 value)
             return digits;
         }
 
+        // Based on do_count_digits from https://github.com/fmtlib/fmt/blob/662adf4f33346ba9aba8b072194e319869ede54a/include/fmt/format.h#L1124
         [MethodImpl(MethodImplOptions.AggressiveInlining)]
         public static int CountDigits(ulong value)
         {
-            int digits = 1;
-            uint part;
-            if (value >= 10000000)
+            // Map the log2(value) to a power of 10.
+            ReadOnlySpan<byte> log2ToPow10 = new byte[]
             {
-                if (value >= 100000000000000)
-                {
-                    part = (uint)(value / 100000000000000);
-                    digits += 14;
-                }
-                else
-                {
-                    part = (uint)(value / 10000000);
-                    digits += 7;
-                }
-            }
-            else
-            {
-                part = (uint)value;
-            }
+                1,  1,  1,  2,  2,  2,  3,  3,  3,  4,  4,  4,  4,  5,  5,  5,
+                6,  6,  6,  7,  7,  7,  7,  8,  8,  8,  9,  9,  9,  10, 10, 10,
+                10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15,
+                15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20
+            };
+            Debug.Assert(log2ToPow10.Length == 64);
+            uint index = Unsafe.Add(ref MemoryMarshal.GetReference(log2ToPow10), BitOperations.Log2(value));
 
-            if (part < 10)
-            {
-                // no-op
-            }
-            else if (part < 100)
+            // TODO https://github.com/dotnet/runtime/issues/60948: Use ReadOnlySpan<ulong> instead of ReadOnlySpan<byte>.
+            // Read the associated power of 10.
+            ReadOnlySpan<byte> powersOf10 = new byte[]
             {
-                digits++;
-            }
-            else if (part < 1000)
-            {
-                digits += 2;
-            }
-            else if (part < 10000)
-            {
-                digits += 3;
-            }
-            else if (part < 100000)
-            {
-                digits += 4;
-            }
-            else if (part < 1000000)
-            {
-                digits += 5;
-            }
-            else
+                0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // unused entry to avoid needing to subtract
+                0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // 0
+                0x0A, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // 10
+                0x64, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // 100
+                0xE8, 0x03, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // 1000
+                0x10, 0x27, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, // 10000
+                0xA0, 0x86, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, // 100000
+                0x40, 0x42, 0x0F, 0x00, 0x00, 0x00, 0x00, 0x00, // 1000000
+                0x80, 0x96, 0x98, 0x00, 0x00, 0x00, 0x00, 0x00, // 10000000
+                0x00, 0xE1, 0xF5, 0x05, 0x00, 0x00, 0x00, 0x00, // 100000000
+                0x00, 0xCA, 0x9A, 0x3B, 0x00, 0x00, 0x00, 0x00, // 1000000000
+                0x00, 0xE4, 0x0B, 0x54, 0x02, 0x00, 0x00, 0x00, // 10000000000
+                0x00, 0xE8, 0x76, 0x48, 0x17, 0x00, 0x00, 0x00, // 100000000000
+                0x00, 0x10, 0xA5, 0xD4, 0xE8, 0x00, 0x00, 0x00, // 1000000000000
+                0x00, 0xA0, 0x72, 0x4E, 0x18, 0x09, 0x00, 0x00, // 10000000000000
+                0x00, 0x40, 0x7A, 0x10, 0xF3, 0x5A, 0x00, 0x00, // 100000000000000
+                0x00, 0x80, 0xC6, 0xA4, 0x7E, 0x8D, 0x03, 0x00, // 1000000000000000
+                0x00, 0x00, 0xC1, 0x6F, 0xF2, 0x86, 0x23, 0x00, // 10000000000000000
+                0x00, 0x00, 0x8A, 0x5D, 0x78, 0x45, 0x63, 0x01, // 100000000000000000
+                0x00, 0x00, 0x64, 0xA7, 0xB3, 0xB6, 0xE0, 0x0D, // 1000000000000000000
+                0x00, 0x00, 0xE8, 0x89, 0x04, 0x23, 0xC7, 0x8A, // 10000000000000000000
+            };
+            Debug.Assert((index + 1) * sizeof(ulong) <= powersOf10.Length);
+            ulong powerOf10 = Unsafe.ReadUnaligned<ulong>(ref Unsafe.Add(ref MemoryMarshal.GetReference(powersOf10), index * sizeof(ulong)));
+            if (!BitConverter.IsLittleEndian)
             {
-                Debug.Assert(part < 10000000);
-                digits += 6;
+                powerOf10 = BinaryPrimitives.ReverseEndianness(powerOf10);
             }
 
-            return digits;
+            // Return the number of digits based on the power of 10, shifted by 1
+            // if it falls below the threshold.
+            bool lessThan = value < powerOf10;
+            return (int)(index - Unsafe.As<bool, byte>(ref lessThan)); // while arbitrary bools may be non-0/1, comparison operators are expected to return 0/1
         }
 
         [MethodImpl(MethodImplOptions.AggressiveInlining)]