关于lua_numbertointeger宏的技术疑问：为何可认定INT_MIN能被浮点数精确表示？

Great question! Let's break this down piece by piece, leaning on IEEE 754 floating-point specs and how they interact with integer representations.

1. 32-bit float (LUA_NUMBER) + 32-bit integer (LUA_INTEGER): Why INT_MIN is exactly representable? #

A 32-bit signed integer’s INT_MIN is -2^31 (-2147483648). For IEEE 754 single-precision (32-bit) floats:

• They use a 1-bit sign flag, 8-bit exponent (with a bias of 127), and 23-bit mantissa (plus an implicit leading 1 for normalized values).
• Pure powers of two are always exactly representable in IEEE floats, as long as their exponent falls within the valid range. For -2^31:
  • The sign bit is set to 1 (marking it negative).
  • The exponent for 2^31 is 31, which adds to the bias gives 31 + 127 = 190—well within the 8-bit exponent’s valid range (-126 to +127, covering values from 2^-126 to 2^127).
  • The mantissa is all zeros, since we’re representing a clean power of two (the implicit leading 1 handles the "1" in 1.0 * 2^31).

No rounding is needed here, so -2^31 fits perfectly into a 32-bit float.

2. 64-bit integer + 64-bit float (double-precision): Does the same hold for INT_MIN? #

Absolutely. A 64-bit signed integer’s INT_MIN is -2^63. IEEE 754 double-precision (64-bit) floats have:

• 1-bit sign, 11-bit exponent (bias of 1023), and 52-bit mantissa (plus implicit leading 1).
• The exponent for 2^63 is 63, which adjusted by the bias is 63 + 1023 = 1086—well within the 11-bit exponent’s valid range (-1022 to +1023, covering 2^-1022 to 2^1023).
• Again, since -2^63 is a pure power of two, the mantissa is all zeros, and the sign bit is set. No rounding occurs, so it’s exactly representable in a 64-bit float.

3. 64-bit integer + 32-bit float: Can INT_MIN be exactly represented? #

Surprisingly, yes—but only for this specific value of INT_MIN (-2^63). Here’s the catch:

• 32-bit floats can’t exactly represent most 64-bit integers (their 23-bit mantissa only guarantees exact integer representations up to ±2^24). However, pure powers of two are an exception.
• -2^63 has an exponent of 63, which falls within the 32-bit float’s exponent range (-126 to +127). The mantissa is all zeros, sign bit is 1, so it fits perfectly without any rounding.
• That said, nearly every other 64-bit integer (like -2^63 + 1 or 2^63 - 1) can’t be exactly represented in a 32-bit float—only powers of two in that exponent range work.

内容的提问来源于stack exchange，提问作者user673679