longlong.h – support functions for multi-word arithmetic

Auxiliary asm macros

umul_ppmm(high_prod, low_prod, multipler, multiplicand)

Multiplies two single limb integers MULTIPLER and MULTIPLICAND, and generates a two limb product in HIGH_PROD and LOW_PROD.

smul_ppmm(high_prod, low_prod, multipler, multiplicand)

As for umul_ppmm() but the numbers are signed.

udiv_qrnnd(quotient, remainder, high_numerator, low_numerator, denominator)

Divides an unsigned integer, composed by the limb integers HIGH_NUMERATOR and\ LOW_NUMERATOR, by DENOMINATOR and places the quotient in QUOTIENT and the remainder in REMAINDER. HIGH_NUMERATOR must be less than DENOMINATOR for correct operation.

sdiv_qrnnd(quotient, remainder, high_numerator, low_numerator, denominator)

As for udiv_qrnnd() but the numbers are signed. The quotient is rounded towards \(0\). Note that as the quotient is signed it must lie in the range \([-2^63, 2^63)\).

count_leading_zeros(count, x)

Counts the number of zero-bits from the msb to the first non-zero bit in the limb x. This is the number of steps x needs to be shifted left to set the msb. If x is \(0\) then count is undefined.

count_trailing_zeros(count, x)

As for count_leading_zeros(), but counts from the least significant end. If x is zero then count is undefined.

add_ssaaaa(high_sum, low_sum, high_addend_1, low_addend_1, high_addend_2, low_addend_2)

Adds two limb integers, composed by HIGH_ADDEND_1 and LOW_ADDEND_1, and\ HIGH_ADDEND_2 and LOW_ADDEND_2, respectively. The result is placed in HIGH_SUM and LOW_SUM. Overflow, i.e.carry out, is not stored anywhere, and is lost.

add_sssaaaaaa(high_sum, mid_sum, low_sum, high_addend_1, mid_addend_1, low_addend_1, high_addend_2, mid_addend_2, low_addend_2)

Adds two three limb integers. Carry out is lost.

sub_ddmmss(high_difference, low_difference, high_minuend, low_minuend, high_subtrahend, low_subtrahend)

Subtracts two limb integers, composed by HIGH_MINUEND_1 and LOW_MINUEND_1, and HIGH_SUBTRAHEND_2 and LOW_SUBTRAHEND_2, respectively. The result is placed in\ HIGH_DIFFERENCE and LOW_DIFFERENCE. Overflow, i.e.carry out is not stored anywhere, and is lost.

sub_dddmmmsss(high_diff, mid_diff, low_diff, high_minuend_1, mid_minuend_1, low_minuend_1, high_subtrahend_2, mid_subtrahend_2, low_subtrahend_2)

Subtracts two three limb integers. Borrow out is lost.

byte_swap(x)

Swap the order of the bytes in the word \(x\), i.e. most significant byte becomes least significant byte, etc.

invert_limb(invxl, xl)

Computes an approximate inverse invxl of the limb xl, with an implicit leading~`1`. More formally it computes:

invxl = (B^2 - B*x - 1)/x = (B^2 - 1)/x - B

Note that \(x\) must be normalised, i.e.with msb set. This inverse makes use of the following theorem of Torbjorn Granlund and Peter Montgomery~[Lemma~8.1][GraMon1994]_:

Let \(d\) be normalised, \(d < B\), i.e.it fits in a word, and suppose that \(m d < B^2 \leq (m+1) d\). Let \(0 \leq n \leq B d - 1\). Write \(n = n_2 B + n_1 B/2 + n_0\) with \(n_1 = 0\) or \(1\) and \(n_0 < B/2\). Suppose \(q_1 B + q_0 = n_2 B + (n_2 + n_1) (m - B) + n_1 (d-B/2) + n_0\) and \(0 \leq q_0 < B\). Then \(0 \leq q_1 < B\) and \(0 \leq n - q_1 d < 2 d\).

In the theorem, \(m\) is the inverse of \(d\). If we let m = invxl + B and \(d = x\) we have \(m d = B^2 - 1 < B^2\) and \((m+1) x = B^2 + d - 1 \geq B^2\).

The theorem is often applied as follows: note that \(n_0\) and \(n_1 (d-B/2)\) are both less than \(B/2\). Also note that \(n_1 (m-B) < B\). Thus the sum of all these terms contributes at most \(1\) to \(q_1\). We are left with \(n_2 B + n_2 (m-B)\). But note that \((m-B)\) is precisely our precomputed inverse invxl. If we write \(q_1 B + q_0 = n_2 B + n_2 (m-B)\), then from the theorem, we have \(0 \leq n - q_1 d < 3 d\), i.e.the quotient is out by at most \(2\) and is always either correct or too small.

udiv_qrnnd_preinv(q, r, nh, nl, d, di)

As for udiv_qrnnd() but takes a precomputed inverse di as computed by invert_limb(). The algorithm, in terms of the theorem above, is:

nadj = n1*(d-B/2) + n0
xh, xl = (n2+n1)*(m-B)
xh, xl += nadj + n2*B ( xh, xl = n2*B + (n2+n1)*(m-B) + n1*(d-B/2) + n0 )
_q1 = B - xh - 1
xh, xl = _q1*d + nh, nl - B*d = nh, nl - q1*d - d so that xh = 0 or -1
r = xl + xh*d where xh is 0 if q1 is off by 1, otherwise -1
q = xh - _q1 = xh + 1 + n2