How to Extract the High Part of a 64-bit Integer Multiplication in C ?
Nov 17, 2024 am 02:28 AMExtracting the High Part of 64-bit Integer Multiplication
In C , the multiplication of two 64-bit unsigned integers (uint64_t) results in an uint64_t that only contains the lower part of the product (i.e., (i * j) mod 2^64). If you seek to obtain the higher part of the multiplication, here are some efficient approaches:
Using 128-bit Numbers
If your compiler supports 128-bit numbers (e.g., using __uint128_t in GCC), you can perform a 128-bit multiplication and extract the upper 64 bits. This method is likely the most efficient.
Multiplication Breakdown
If your compiler does not support 128-bit numbers, you can break down each 64-bit integer into the following parts:
- a = (a_hi << 32) a_lo
- b = (b_hi << 32) b_lo
where a_hi, a_lo, b_hi, and b_lo are 32-bit unsigned integers.
Algorithm
To compute the high part of the multiplication (a_hi * b_hi), you can follow these steps:
- Calculate the products a_hi b_hi, a_hi b_lo, b_hi a_lo, and a_lo b_lo.
- Add the products a_hi b_hi and the upper 32 bits of the sum of a_hi b_lo and b_hi * a_lo.
- Add the result from step 2 to the upper 32 bits of a_lo * b_lo.
注意事項(xiàng)
When performing these operations, you must be cautious of integer overflow. The following code illustrates how to handle overflow:
uint64_t a_lo = (uint32_t)a; uint64_t a_hi = a >> 32; uint64_t b_lo = (uint32_t)b; uint64_t b_hi = b >> 32; uint64_t a_x_b_hi = a_hi * b_hi; uint64_t a_x_b_mid = (a_hi * b_lo) >> 32; uint64_t b_x_a_mid = (b_hi * a_lo) >> 32; uint64_t a_x_b_lo = a_lo * b_lo; uint64_t carry_bit = ((uint64_t)(uint32_t)a_x_b_mid + (uint64_t)(uint32_t)b_x_a_mid + (a_x_b_lo >> 32)) >> 32; uint64_t multhi = a_x_b_hi + a_x_b_mid + b_x_a_mid + carry_bit; return multhi;Note that this code may not be perfectly accurate, but it provides a good approximation of the higher part of the multiplication.
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