十进制转规格化浮点二进制方法咨询（指数范围[-3,3]，精度4位）

Hey there! Let's walk through exactly how to convert those decimal numbers into the normalized 4-bit precision floating-point format you're working with (exponent range [-3, 3]). I'll break the process down step by step, then apply it to each of your examples to make it totally clear.

Converting a decimal number to this specific floating-point format involves 5 key steps:

Step 1: Split the decimal into integer and fractional parts #

Every decimal number can be split into two parts: the whole number (integer) to the left of the decimal point, and the fraction to the right. For example:

• 3.1416 → integer = 3, fractional = 0.1416
• 0.11 → integer = 0, fractional = 0.11

Step 2: Convert the integer part to binary #

To turn an integer into binary, use the "divide by 2, record the remainder" method:

• Divide the integer by 2, write down the remainder (0 or 1)
• Repeat with the quotient until the quotient is 0
• Reverse the list of remainders to get the binary integer

Example: Convert 3 to binary

• 3 ÷ 2 = 1, remainder 1
• 1 ÷ 2 = 0, remainder 1
• Reverse remainders: 11

Step 3: Convert the fractional part to binary #

For the fractional part, use the "multiply by 2, record the integer part" method:

• Multiply the fraction by 2, write down the integer part (0 or 1)
• Repeat with the new fractional part (after removing the integer you recorded) until you get the precision you need, or until the fraction repeats
• Keep the list of integers in the order you recorded them (no reversing here!)

Example: Convert 0.11 to binary (first few steps)

• 0.11 × 2 = 0.22 → integer part 0
• 0.22 × 2 = 0.44 → integer part 0
• 0.44 × 2 = 0.88 → integer part 0
• 0.88 × 2 = 1.76 → integer part 1
• 0.76 × 2 = 1.52 → integer part 1
• Result so far: 0.00011...

Step 4: Combine and normalize the binary number #

A normalized floating-point number follows the format 1.xxxx * 2^e, where:

• 1.xxxx is the mantissa (with your 4-bit precision, this means 4 total significant bits: the leading 1 plus 3 more bits)
• e is the exponent, which adjusts how far we shift the decimal point to get back to the original number

To normalize:

1. Combine your binary integer and fractional parts (e.g., 3.1416 becomes 11.001001...)
2. Shift the decimal point left or right until there's exactly one 1 to the left of the decimal point
3. The exponent e is equal to the number of positions you shifted the decimal point:
   • Shift left → positive exponent (you're multiplying by 2^e to get back to the original number)
   • Shift right → negative exponent (you're dividing by 2^|e|, which is the same as multiplying by 2^-|e|)

Step 5: Truncate/round to 4-bit precision #

Since your format uses 4-bit precision, we keep the leading 1 plus 3 more bits in the mantissa. If the next bit after those 3 is 1, we round up the last bit (just like decimal rounding).

Let's use these steps for each of your decimal numbers to match the results you provided.

Example 1: 0.11 (decimal) #

1. Split: Integer = 0, Fractional = 0.11
2. Integer binary: 0
3. Fractional binary: 0.00011... (from Step 3 above)
4. Combine & normalize:
   • Combined binary: 0.00011...
   • Shift decimal right 3 positions to get 1.00011...
   • Exponent = -3 (since we shifted right 3 times)
5. 4-bit precision: Keep the leading 1 plus 3 bits → 1.000
6. Final result: 1.000 * 2^-3 (matches your given value)

Example 2: 3.1416 (decimal) #

1. Split: Integer = 3, Fractional = 0.1416
2. Integer binary: 11
3. Fractional binary: 0.001001... (calculated by multiplying 0.1416 by 2 repeatedly)
4. Combine & normalize:
   • Combined binary: 11.001001...
   • Shift decimal left 1 position to get 1.1001001...
   • Exponent = 1 (shifted left 1 time)
5. 4-bit precision: The first 4 significant bits are 1.100, but the next bit is 1 → round up the last bit to get 1.101
6. Final result: 1.101 * 2^1 (matches your given value)

Example 3: 2.718 (decimal) #

1. Split: Integer = 2, Fractional = 0.718
2. Integer binary: 10
3. Fractional binary: 0.1011... (0.718×2=1.436→1; 0.436×2=0.872→0; 0.872×2=1.744→1; 0.744×2=1.488→1)
4. Combine & normalize:
   • Combined binary: 10.1011...
   • Shift decimal left 1 position to get 1.01011...
   • Exponent = 1
5. 4-bit precision: First 4 bits are 1.010, next bit is 1 → round up to 1.011
6. Final result: 1.011 * 2^1 (matches your given value)

Example 4: 7 (decimal) #

1. Split: Integer =7, Fractional=0
2. Integer binary: 111
3. Fractional binary: 0
4. Combine & normalize:
   • Combined binary: 111.0
   • Shift decimal left 2 positions to get 1.110
   • Exponent=2
5. 4-bit precision: 1.110 is already 4 significant bits, no rounding needed
6. Final result: 1.110 * 2^2 (matches your given value)

内容的提问来源于stack exchange，提问作者machinery