// cl_UDS_recip().
// General includes.
#include "cl_sysdep.h"
// Specification.
#include "cl_DS.h"
// Implementation.
#include "cln/abort.h"
namespace cln {
// Compute the reciprocal value of a digit sequence.
// Input: UDS a_MSDptr/a_len/.. of length a_len,
// with 1/2*beta^a_len <= a < beta^a_len.
// Output: UDS b_MSDptr/b_len+2/.. of length b_len+1 (b_len>1), plus 1 more bit
// in the last limb, such that
// beta^b_len <= b <= 2*beta^b_len and
// | beta^(a_len+b_len)/a - b | < 1.
// If a_len > b_len, only the most significant b_len limbs + 3 bits of a
// are used.
extern void cl_UDS_recip (const uintD* a_MSDptr, uintC a_len,
uintD* b_MSDptr, uintC b_len);
// Method:
// Using Newton/Heron iteration.
// Write x = a/beta^a_len and y = b/beta^b_len.
// So we start out with 1/2 <= x < 1 and search an y with 1 <= y <= 2
// and | 1/x - y | < beta^(-b_len).
// For n = 1,2,...,b_len we compute approximations y with 1 <= yn <= 2
// and | 1/x - yn | < beta^(-n). The first n limbs of x, plus the
// next 3 bits (of the (n+1)st limb) enter the computation of yn. Apart
// from that, yn remains valid for any x which shares the same n+1
// most significant limbs.
// Step n = 1:
// Write x = x1/beta + x2/beta^2 + xr with 0 <= xr < 1/(8*beta).
// Divide (beta^2-beta*x1-x2) by x1, gives beta^2-x1*beta-x2 = q*x1+r.
// If this division overflows, i.e. q >= beta, then x1 = beta/2, x2 = 0,
// and we just return y1 = 2.
// Else set qd := ceiling(max(q*x2/beta - r, 0) / (x1+1)) and return
// y1 = (beta+q-qd)/beta.
// Rationale: Obviously 0 <= qd <= q and 0 <= qd <= 2. We have
// beta^2 - beta*(beta+q-qd)*x
// <= beta^2 - (beta+q-qd)*(x1 + x2/beta)
// = beta^2 - beta*x1 - x2 - q*(x1 + x2/beta) + qd*(x1 + x2/beta)
// = q*x1 + r - q*(x1 + x2/beta) + qd*(x1 + x2/beta)
// = r - q*x2/beta + qd*(x1 + x2/beta)
// if qd=0: <= r <= x1-1 < x1
// if qd>0: < r - q*x2/beta + qd*(x1+1) <= x1
// hence always < x1 <= beta*x, hence
// 1 - x*y1 <= x/beta, hence 1/x - y1 <= 1/beta.
// And on the other hand
// beta^2 - beta*(beta+q-qd)*x
// = beta^2 - (beta+q-qd)*(x1 + x2/beta) - beta*(beta+q-qd)*xr
// where the third term is
// <= 2*beta^2*xr < beta/4 <= x1/2 <= beta*x/2.
// Hence
// beta^2 - beta*(beta+q-qd)*x >
// > beta^2 - (beta+q-qd)*(x1 + x2/beta) - beta*x/2
// = r - q*x2/beta + qd*(x1 + x2/beta) - beta*x/2
// >= - qd - beta*x/2 > - beta*x, hence
// 1 - x*y1 >= -x/beta, hence 1/x - y1 >= -1/beta.
// Step n -> m with n < m <= 2*n:
// Write x = xm + xr with 0 <= xr < 1/(8*beta^m).
// Set ym' = 2*yn - xm*yn*yn,
// ym = ym' rounded up to be a multiple of 1/(2*beta^m).
// Rationale:
// 1/x - ym <= 1/x - ym' = 1/x - 2*yn + (x-xr)*yn*yn
// <= 1/x - 2*yn + x*yn*yn = x * (1/x - yn)^2 < x*beta^(-2n)
// < beta^(-2n) <= beta^(-m), and
// 1/x - ym' = 1/x - 2*yn + (x-xr)*yn*yn
// > 1/x - 2*yn + x*yn*yn - 1/(2*beta^m)
// = x * (1/x - yn)^2 - 1/(2*beta^m) >= - 1/(2*beta^m), hence
// 1/x - ym > 1/x - ym' - 1/(2*beta^m) >= -1/beta^m.
// Since it is needed to compute ym as a multiple of 1/(2*beta^m),
// not only as a multiple of 1/beta^m, we compute with zn = 2*yn.
// The iteration now reads zm = round_up(2*zn - xm*zn*zn/2).
// Choice of n:
// So that the computation is minimal, e.g. in the case b_len=10:
// 1 -> 2 -> 3 -> 5 -> 10 and not 1 -> 2 -> 4 -> 8 -> 10.
void cl_UDS_recip (const uintD* a_MSDptr, uintC a_len,
uintD* b_MSDptr, uintC b_len)
{
var uintC y_len = b_len+1;
var uintC x_len = (a_len <= b_len ? a_len+1 : y_len);
var uintD* x_MSDptr;
var uintD* y_MSDptr;
var uintD* y2_MSDptr;
var uintD* y3_MSDptr;
CL_ALLOCA_STACK;
num_stack_alloc(x_len,x_MSDptr=,);
num_stack_alloc(y_len,y_MSDptr=,);
num_stack_alloc(2*y_len,y2_MSDptr=,);
num_stack_alloc(x_len+2*y_len,y3_MSDptr=,);
// Prepare x/2 at x_MSDptr by shifting a right by 1 bit.
if (a_len <= b_len)
{ mspref(x_MSDptr,a_len) =
shiftrightcopy_loop_msp(a_MSDptr,x_MSDptr,a_len,1,0);
}
else
{ mspref(x_MSDptr,b_len) =
shiftrightcopy_loop_msp(a_MSDptr,x_MSDptr,b_len,1,0)
| ((mspref(a_MSDptr,b_len) & -bit(intDsize-3)) >> 1);
}
// Step n = 1.
{ var uintD x1 = mspref(a_MSDptr,0);
var uintD x2 = (a_len > 1 ? (mspref(a_MSDptr,1) & -bit(intDsize-3)) : 0);
if ((x1 == (uintD)bit(intDsize-1)) && (x2 == 0))
{ mspref(y_MSDptr,0) = 4; mspref(y_MSDptr,1) = 0; }
else
{ var uintD q;
var uintD r;
var uintD chi;
var uintD clo;
#if HAVE_DD
divuD((uintDD)(-highlowDD(x1,x2)),x1, q=,r=);
var uintDD c = muluD(q,x2);
chi = highD(c); clo = lowD(c);
#else
divuD((uintD)(-x1 - (x2>0 ? 1 : 0)),(uintD)(-x2),x1, q=,r=);
muluD(q,x2,chi=,clo=);
#endif
if (clo > 0)
chi++;
// qd := ceiling(max(chi-r,0)/(x1+1))
if (chi > r)
{ chi -= r;
if (chi > x1)
{ q--; }
q--;
}
mspref(y_MSDptr,0) = 2 + (q>>(intDsize-1));
mspref(y_MSDptr,1) = q<<1;
}
}
// Other steps.
var int k;
integerlength32((uint32)b_len-1,k=);
// 2^(k-1) < b_len <= 2^k, so we need k steps.
var uintC n = 1;
for (; k>0; k--)
{ // n = ceiling(b_len/2^k) limbs of y have already been computed.
var uintC m = ((b_len-1)>>(k-1))+1; // = ceiling(b_len/2^(k-1))
// Compute zm := 2*zn - round_down(xm/2*zn*zn).
cl_UDS_mul_square(y_MSDptr mspop (n+1),n+1,y2_MSDptr mspop 2*(n+1));
var uintC xm_len = (m < x_len ? m+1 : x_len);
cl_UDS_mul(x_MSDptr mspop xm_len,xm_len,
y2_MSDptr mspop 2*(n+1),2*n+1,
y3_MSDptr mspop (xm_len+2*n+1));
// Round down by just taking the first m+1 limbs at y3_MSDptr.
shift1left_loop_lsp(y_MSDptr mspop (n+1),n+1);
clear_loop_msp(y_MSDptr mspop (n+1),m-n);
subfrom_loop_lsp(y3_MSDptr mspop (m+1),y_MSDptr mspop (m+1),m+1);
n = m;
}
// All n = b_len limbs of y have been computed. Divide by 2.
mspref(b_MSDptr,b_len+1) =
shiftrightcopy_loop_msp(y_MSDptr,b_MSDptr,b_len+1,1,0);
}
// Bit complexity (N := b_len): O(M(N)).
} // namespace cln
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