Homing So

Principle#

To find the absolute value of a number, you simply convert a negative number to a positive one, which can be achieved by calculating its two's complement (one's complement plus one).

Code#

Principle Analysis#

Taking -12 as an example: the two's complement of -12 is 10100 (assuming the machine word length is 5 bits).

The steps are as follows:

• First, obtain the mask mask:

  Copy

  Shifting 10100 left by 4 bits gives 11111. Note that 11111 is a very clever design.

  The reason we choose to right shift to obtain the mask instead of directly right shifting four bits and using & with 0x1 to get the sign bit is that: the mask 11111 is actually entirely composed of the sign bit. If the sign bit is 1, the generated mask can be directly XORed with the original data to get the one's complement, and then adding 1 gives the two's complement.

  If the sign bit is 0, the resulting mask will be all 0s, and XORing 0 with any number yields the number itself.

• Calculate the one's complement:

  Copy

  As shown in the previous step, 10100 XOR 11111 equals 01011, which gives the one's complement.

  If a positive number were here, the mask would be 00000, and XORing would leave it unchanged.

• Add 1:

  Copy

  01011 - 11111 = 01100, resulting in the absolute value of -12, which is 12.

  The reason adding one becomes subtracting the mask is simple: the mask we obtained earlier is 11111, which is actually the two's complement of -1. To add 1, we just need to subtract the mask.

  If a positive number were here, the mask would be 00000, and subtracting 0 would have no effect.

In this way, we have implemented the algorithm for calculating the absolute value. A negative number will be converted to a positive one, while a positive number remains unchanged. Isn't it simple?

Benchmark#

Below are the results obtained using a MacBook Air (M1, 2020) and Apple clang 13.1.6

Below are the results obtained using i5-9500 and gcc 8.5.0 (Red Hat 8.5.0-10)