# Unsigned 64-bit integer operations

Module std::math::u64 contains a set of procedures which can be used to perform unsigned 64-bit integer operations. These operations fall into the following categories:

• Arithmetic operations - addition, multiplication, division etc.
• Comparison operations - equality, less than, greater than etc.
• Bitwise operations - binary AND, OR, XOR, bit shifts etc.

All procedures assume that an unsigned 64-bit integer (u64) is encoded using two elements, each containing an unsigned 32-bit integer (u32). When placed on the stack, the least-significant limb is assumed to be deeper in the stack. For example, a u64 value a consisting of limbs a_hi and a_lo would be position on the stack like so:

[a_hi, a_lo, ... ]


Many of the procedures listed below (e.g., overflowing_add, wrapping_add, lt) do not check whether the inputs are encoded using valid u32 values. These procedures do not fail when the inputs are encoded incorrectly, but rather produce undefined results. Thus, it is important to be certain that limbs of input values are valid u32 values prior to calling such procedures.

## Arithmetic operations

ProcedureDescription
overflowing_addPerforms addition of two unsigned 64-bit integers preserving the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [overflow_flag, c_hi, c_lo, ...], where c = (a + b) % 2^64
This takes 6 cycles.
wrapping_addPerforms addition of two unsigned 64-bit integers discarding the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a + b) % 2^64
This takes 7 cycles.
overflowing_subPerforms subtraction of two unsigned 64-bit integers preserving the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [underflow_flag, c_hi, c_lo, ...], where c = (a - b) % 2^64
This takes 11 cycles.
wrapping_subPerforms subtraction of two unsigned 64-bit integers discarding the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a - b) % 2^64
This takes 10 cycles.
overflowing_mulPerforms multiplication of two unsigned 64-bit integers preserving the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi_hi, c_hi_lo, c_lo_hi, c_lo_lo, ...], where c = (a * b) % 2^64
This takes 18 cycles.
wrapping_mulPerforms multiplication of two unsigned 64-bit integers discarding the overflow.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a * b) % 2^64
This takes 11 cycles.
divPerforms division of two unsigned 64-bit integers discarding the remainder.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a // b
This takes 54 cycles.
modPerforms modulo operation of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a % b
This takes 54 cycles.
divmodPerforms divmod operation of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [r_hi, r_lo, q_hi, q_lo ...], where r = a % b, q = a // b
This takes 54 cycles.

## Comparison operations

ProcedureDescription
ltPerforms less-than comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a < b, and 0 otherwise.
This takes 11 cycles.
gtPerforms greater-than comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a > b, and 0 otherwise.
This takes 11 cycles.
ltePerforms less-than-or-equal comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a <= b, and 0 otherwise.
This takes 12 cycles.
gtePerforms greater-than-or-equal comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a >= b, and 0 otherwise.
This takes 12 cycles.
eqPerforms equality comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == b, and 0 otherwise.
This takes 6 cycles.
neqPerforms inequality comparison of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a != b, and 0 otherwise.
This takes 6 cycles.
eqzPerforms comparison to zero of an unsigned 64-bit integer.
The input value is assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == 0, and 0 otherwise.
This takes 4 cycles.
minCompares two unsigned 64-bit integers and drop the larger one from the stack.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a < b, and b otherwise.
This takes 23 cycles.
maxCompares two unsigned 64-bit integers and drop the smaller one from the stack.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a > b, and b otherwise.
This takes 23 cycles.

## Bitwise operations

ProcedureDescription
andPerforms bitwise AND of two unsigned 64-bit integers.
The input values are assumed to be represented using 32-bit limbs, but this is not checked.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a AND b.
This takes 6 cycles.
orPerforms bitwise OR of two unsigned 64-bit integers.
The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a OR b.
This takes 16 cycles.
xorPerforms bitwise XOR of two unsigned 64-bit integers.
The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not.
The stack transition looks as follows:
[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a XOR b.
This takes 6 cycles.
shlPerforms left shift of one unsigned 64-bit integer using the pow2 operation.
The input value to be shifted is assumed to be represented using 32-bit limbs.
The shift value should be in the range [0, 64), otherwise it will result in an error.
The stack transition looks as follows:
[b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64.
This takes 28 cycles.
shrPerforms right shift of one unsigned 64-bit integer using the pow2 operation.
The input value to be shifted is assumed to be represented using 32-bit limbs.
The shift value should be in the range [0, 64), otherwise it will result in an error.
The stack transition looks as follows:
[b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a >> b.
This takes 44 cycles.
rotlPerforms left rotation of one unsigned 64-bit integer using the pow2 operation.
The input value to be shifted is assumed to be represented using 32-bit limbs.
The shift value should be in the range [0, 64), otherwise it will result in an error.
The stack transition looks as follows:
[b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64.
This takes 35 cycles.
rotrPerforms right rotation of one unsigned 64-bit integer using the pow2 operation.
The input value to be shifted is assumed to be represented using 32-bit limbs.
The shift value should be in the range [0, 64), otherwise it will result in an error.
The stack transition looks as follows:
[b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64.
This takes 40 cycles.
clzCounts the number of leading zeros of one unsigned 64-bit integer.
The input value is assumed to be represented using 32 bit limbs, but this is not checked.
The stack transition looks as follows: [n_hi, n_lo, ...] -> [clz, ...], where clz is a number of leading zeros of value n.
This takes 43 cycles.
ctzCounts the number of trailing zeros of one unsigned 64-bit integer.
The input value is assumed to be represented using 32 bit limbs, but this is not checked.
The stack transition looks as follows: [n_hi, n_lo, ...] -> [ctz, ...], where ctz is a number of trailing zeros of value n.
This takes 41 cycles.
cloCounts the number of leading ones of one unsigned 64-bit integer.
The input value is assumed to be represented using 32 bit limbs, but this is not checked.
The stack transition looks as follows: [n_hi, n_lo, ...] -> [clo, ...], where clo is a number of leading ones of value n.
This takes 42 cycles.
ctoCounts the number of trailing ones of one unsigned 64-bit integer.
The input value is assumed to be represented using 32 bit limbs, but this is not checked.
The stack transition looks as follows: [n_hi, n_lo, ...] -> [cto, ...], where cto is a number of trailing ones of value n.
This takes 40 cycles.