# Hacker's Delight (2nd Edition)

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It didn't "pop-up" in a genius' head. Right shifting binary numbers would divide a number by 2 and left shifting the numbers would multiply it by 2. This is because `10` is 2 in binary. Multiplying a number by `10`(be it binary or decimal or hexadecimal) appends a `0` to the number(which is effectively left shifting). Similarly, dividing by `10`(or 2) removes a binary digit from the number(effectively right shifting). This is how the logic really works.

There are plenty of such `bit-twiddlery`(a word I invented a minute ago) in computer world.

http://graphics.stanford.edu/~seander/bithacks.html Here is for the starters.

This is my favorite book: http://www.amazon.com/Hackers-Delight-Edition-Henry-Warren/dp/0321842685/ref=dp_ob_image_bk on bit-twiddlery.

If bit manipulation is something you enjoy, there is an excellent book called Hacker's Delight (2e) by Henry Warren that you will find to be delightful. https://www.amazon.com/dp/0321842685
If you enjoy these bit twiddling hacks, I recommend "Hacker's Delight," by Henry Warren.

Author's website: https://www.amazon.com/Hackers-Delight-2nd-Henry-Warren/dp/0...

You can just use any of the standard bit twiddling hacks to reverse bytes and then shuffle the bytes (see e.g. [Hacker's Delight](https://www.amazon.com/Hackers-Delight-2nd-Henry-Warren/dp/0321842685) and [this list](http://graphics.stanford.edu/%7Eseander/bithacks.html#BitReverseObvious)).

This is mostly to address another answer here, but here's a faster way to get the bit count:

``````#define BC_OffBitCount(in, out)                                \
{                                                            \
out = ~in;                                                 \
switch (sizeof(in)) {                                      \
case 4:                                                  \
out = (out & 0x55555555) + ((out >> 1) & 0x55555555);  \
out = (out & 0x33333333) + ((out >> 2) & 0x33333333);  \
out = (out & 0x0F0F0F0F) + ((out >> 4) & 0x0F0F0F0F);  \
out = (out & 0x00FF00FF) + ((out >> 8) & 0x00FF00FF);  \
out = (out & 0x0000FFFF) + ((out >> 16) & 0x0000FFFF); \
break;                                                 \
case 2:                                                  \
out = (out & 0x5555) + ((out >> 1) & 0x5555);          \
out = (out & 0x3333) + ((out >> 2) & 0x3333);          \
out = (out & 0x0F0F) + ((out >> 4) & 0x0F0F);          \
out = (out & 0x00FF) + ((out >> 8) & 0x00FF);          \
break;                                                 \
case 1:                                                  \
out = (out & 0x55) + ((out >> 1) & 0x55);              \
out = (out & 0x33) + ((out >> 2) & 0x33);              \
out = (out & 0x0F) + ((out >> 4) & 0x0F);              \
break;                                                 \
}                                                          \
}

#define BC_OffCompareGreater(a, b, out)  // Left as an exercise to the reader
``````

And the tests:

``````void test_uint8_t_of_0x00_should_return_8(void) {
uint8_t value = 0;
uint8_t result;
BC_OffBitCount(value, result);
TEST_ASSERT_EQUAL(8, result);
}

void test_uint8_t_of_0x55_should_return_4(void) {
uint8_t value = 0x55;
uint8_t result;
BC_OffBitCount(value, result);
TEST_ASSERT_EQUAL(4, result);
}

void test_uint16_t_of_0xAA_should_return_12(void) {
uint16_t value = 0xAA;
uint16_t result;
BC_OffBitCount(value, result);
TEST_ASSERT_EQUAL(12, result);
}

uint32_t result;
BC_OffBitCount(value, result);
TEST_ASSERT_EQUAL(8, result);
}
``````

I'm sure there's a clever way to remove the repetition, but that can also be left as an exercise to the reader.

Edit: I copied and modified the code from hacker's delight.

This seems a generalization of the question Bitwise transpose of 8 bytes. That question was just about 8x8 transposition, so what you are asking is a bit different. But your question is answered just as well in section 7.3 of the book Hacker's Delight (you might be able to see the relevant pages on Google books). The code that is presented there apparently originates with Guy Steele.

The Hacker's Delight website only contains the source code from the book for the 8x8 and 32x32 cases, but the latter generalizes trivially to your 64x64 case:

``````#include <stdint.h>

void
transpose64(uint64_t a[64]) {
int j, k;
uint64_t m, t;

for (j = 32, m = 0x00000000FFFFFFFF; j; j >>= 1, m ^= m << j) {
for (k = 0; k < 64; k = ((k | j) + 1) & ~j) {
t = (a[k] ^ (a[k | j] >> j)) & m;
a[k] ^= t;
a[k | j] ^= (t << j);
}
}
}
``````

The way that this works is that the function swaps successively smaller blocks of bits, starting with 32x32 blocks (without transposing the bit within those blocks), after that within those 32x32 blocks it swaps the appropriate 16x16 blocks, etc. The variable that holds the block size is `j`. Therefore, the outer loop has `j` succcessively take the values 32, 16, 8, 4, 2 and 1, which means that the outer loop runs six times. The inner loop runs over half the lines of your of bits, the lines where a given bit in the variable `k` is equal to zero. When `j` is 32 those are the lines 0-31, when `j` is 16 those are the lines 0-15 and 32-47, etc. Together the inner part of the loop runs 6*32 = 192 times. What happens inside this inner part is that the mask `m` determines what are the bits that should be swapped, in `t` the xor or those bits are calculated, and that xor-ed lists of bits is used to update the bits in both places appropriately.

The book (and the website) also has a version of this code in which these loops have both been unrolled, and where the mask `m` is not calculated, but just assigned. I guess it depends on things like the number of registers and the size of your instruction cache whether that is an improvement?

To test that this works, suppose we define some bit pattern, say:

``````uint64_t logo[] = {
0b0000000000000000000000000000000000000000000100000000000000000000,
0b0000000000000000000000000000000000000000011100000000000000000000,
0b0000000000000000000000000000000000000000111110000000000000000000,
0b0000000000000000000000000000000000000001111111000000000000000000,
0b0000000000000000000000000000000000000000111111100000000000000000,
0b0000000000000000000000000000000000000000111111100000000000000000,
0b0000000000000000000000000000000000000000011111110000000000000000,
0b0000000000000000000000000000000000000000001111111000000000000000,
0b0000000000000000000000000000000000000000001111111100000000000000,
0b0000000000000000000000000000000010000000000111111100000000000000,
0b0000000000000000000000000000000011100000000011111110000000000000,
0b0000000000000000000000000000000111110000000001111111000000000000,
0b0000000000000000000000000000001111111000000001111111100000000000,
0b0000000000000000000000000000011111111100000000111111100000000000,
0b0000000000000000000000000000001111111110000000011111110000000000,
0b0000000000000000000000000000000011111111100000001111111000000000,
0b0000000000000000000000000000000001111111110000001111111100000000,
0b0000000000000000000000000000000000111111111000000111111100000000,
0b0000000000000000000000000000000000011111111100000011111110000000,
0b0000000000000000000000000000000000001111111110000001111111000000,
0b0000000000000000000000000000000000000011111111100001111111100000,
0b0000000000000000000000001100000000000001111111110000111111100000,
0b0000000000000000000000001111000000000000111111111000011111110000,
0b0000000000000000000000011111110000000000011111111100001111100000,
0b0000000000000000000000011111111100000000001111111110001111000000,
0b0000000000000000000000111111111111000000000011111111100110000000,
0b0000000000000000000000011111111111110000000001111111110000000000,
0b0000000000000000000000000111111111111100000000111111111000000000,
0b0000000000000000000000000001111111111111100000011111110000000000,
0b0000000000000000000000000000011111111111111000001111100000000000,
0b0000000000000000000000000000000111111111111110000011000000000000,
0b0000000000000000000000000000000001111111111111100000000000000000,
0b0000000000000000000000000000000000001111111111111000000000000000,
0b0000000000000000000000000000000000000011111111111100000000000000,
0b0000000000000000000111000000000000000000111111111100000000000000,
0b0000000000000000000111111110000000000000001111111000000000000000,
0b0000000000000000000111111111111100000000000011111000000000000000,
0b0000000000000000000111111111111111110000000000110000000000000000,
0b0000000000000000001111111111111111111111100000000000000000000000,
0b0000000000000000001111111111111111111111111111000000000000000000,
0b0000000000000000000000011111111111111111111111100000000000000000,
0b0000001111110000000000000001111111111111111111100000111111000000,
0b0000001111110000000000000000000011111111111111100000111111000000,
0b0000001111110000000000000000000000000111111111100000111111000000,
0b0000001111110000000000000000000000000000001111000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000001111111111111111111111111111000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111110000000000000000000000000000000000000000111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
0b0000001111111111111111111111111111111111111111111111111111000000,
};
``````

We then call the `transpose32` function and print the resulting bit pattern:

``````#include <stdio.h>

void
printbits(uint64_t a[64]) {
int i, j;

for (i = 0; i < 64; i++) {
for (j = 63; j >= 0; j--)
printf("%c", (a[i] >> j) & 1 ? '1' : '0');
printf("\n");
}
}

int
main() {
transpose64(logo);
printbits(logo);
return 0;
}
``````

And this then gives as output:

``````0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000011111111111111111111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000000000000000000000000111111
0000000000000000000000000000000000000011000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000000000000111111000000011111100000111111
0000000000000000000000000100000000011111000000011111100000111111
0000000000000000000000011110000000011111100000011111100000111111
0000000000000000000001111110000000011111100000011111100000111111
0000000000000000000001111111000000011111100000011111100000111111
0000000000000000000000111111000000011111100000011111100000111111
0000000000000000000000111111100000001111110000011111100000111111
0000000000000000000000011111100000001111110000011111100000111111
0000000000000100000000011111110000001111110000011111100000111111
0000000000001110000000001111110000001111110000011111100000111111
0000000000011110000000001111111000001111110000011111100000111111
0000000001111111000000000111111000000111111000011111100000111111
0000000000111111100000000111111100000111111000011111100000111111
0000000000111111110000000011111100000111111000011111100000111111
0000000000011111111000000011111100000111111000011111100000111111
0000000000001111111100000001111110000011111000011111100000111111
0000000000000111111100000001111110000011111100011111100000111111
0000000000000011111110000000111111000011111100011111100000111111
0001000000000001111111000000111111000011111100011111100000111111
0011110000000001111111100000111111100011111100011111100000111111
0111111000000000111111110000011111100001111100011111100000111111
0111111110000000011111111000011111110001111110011111100000111111
1111111111000000001111111000001111110001111110011111100000111111
0011111111100000000111111100001111111001111110011111100000111111
0001111111111000000011111110000111111001111110011111100000111111
0000111111111100000011111111000111111100111100000000000000111111
0000001111111110000001111111100011111100000000000000000000111111
0000000111111111100000111111110011111000000000000000000000111111
0000000011111111110000011111110001100000000000000000000000111111
0000000000111111111000001111111000000000000000000000000000111111
0000000000011111111110000111111000000000000000000000000000111111
0000000000001111111111000111110000000000011111111111111111111111
0000000000000011111111100011100000000000011111111111111111111111
0000000000000001111111111001000000000000011111111111111111111111
0000000000000000111111111100000000000000011111111111111111111111
0000000000000000001111111100000000000000011111111111111111111111
0000000000000000000111111000000000000000011111111111111111111111
0000000000000000000011110000000000000000000000000000000000000000
0000000000000000000000100000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
``````

Which is nicely flipped, as we hoped for.

### Edit:

This is actually not really what you asked for, as you asked for a non-destructive version of this code. You can get this by having the first swap of the 32x32 blocks go from `x` to `y`. For instance, you might do something like:

``````void
non_destructive_transpose64(uint64_t x[64], uint64_t y[64]) {
int j, k;
uint64_t m, t;

for (k = 0; k < 64; k += 2) {
((uint32_t *) y)[k] = ((uint32_t *) x)[k ^ 64 + 1];
((uint32_t *) y)[k + 1] = ((uint32_t *) x)[k + 1];
}
for (; k < 128; k += 2) {
((uint32_t *) y)[k] = ((uint32_t *) x)[k];
((uint32_t *) y)[k + 1] = ((uint32_t *) x)[k ^ 64];
}
for (j = 16, m = 0x0000FFFF0000FFFF; j; j >>= 1, m ^= m << j) {
for (k = 0; k < 64; k = ((k | j) + 1) & ~j) {
t = (y[k] ^ (y[k | j] >> j)) & m;
y[k] ^= t;
y[k | j] ^= (t << j);
}
}
}
``````

Unlike the other version of the code this does not work regardless of endianness of the architecture. Also, I know that the C standard does not allow you to access an array of `uint64_t` as an array of `uint32_t`. However, I like it that no shifts or xors are needed for the first iteration of the move-the-blocks-around loop when you do it like this.

Hacker's Delight - http://www.amazon.com/Hackers-Delight-Edition-Henry-Warren/d...

EDIT: .. and a good coffeemaker. :)

This seems to have some overlap with Henry Warren's Hacker's Delight: https://www.amazon.com/Hackers-Delight-2nd-Henry-Warren/dp/0...
There is, and I can't recommend it highly enough.

http://www.amazon.com/Hackers-Delight-2nd-Henry-Warren/dp/03...