void log(RR& res, const RR& x)
{
if (x <= 0) Error("argument to log must be positive");
long p = RR::precision();
RR::SetPrecision(p + NumBits(p) + 10);
RR y;
long n;
// re-write x = 2^n * (1 - y), where -1/2 < y < 1/4 (so 3/4 < 1-y < 3/2)
if (x > 0.75 && x < 1.5) {
n = 0;
sub(y, 1, x);
}
else {
n = Lg2(x) - 1;
RR t;
power2(t, -n);
mul(t, t, x);
while (t > 1.5) {
mul(t, t, 0.5);
n++;
}
sub(y, 1, t);
}
// compute s = - ln(1-y) by power series expansion
RR s, s1, t, t1;
s = 0;
xcopy(t, y);
xcopy(t1, y);
long i;
for (i = 2; ; i++) {
add(s1, s, t);
if (s == s1) break;
xcopy(s, s1);
mul(t1, t1, y);
div(t, t1, i);
}
if (n == 0)
t = 0;
else {
ComputeLn2(t);
mul(t, t, n);
}
RR::SetPrecision(p);
sub(res, t, s);
}
long ComputeMax10Power()
{
RRPush push;
RR::SetPrecision(NTL_BITS_PER_LONG);
RR ln2, ln10;
ComputeLn2(ln2);
ComputeLn10(ln10);
long k = to_long( to_RR(NTL_OVFBND/2) * ln2 / ln10 );
return k;
}
ostream& operator<<(ostream& s, const xdouble& a)
{
if (a == 0) {
s << "0";
return s;
}
RRPush push;
long temp_p = long(log(fabs(log(fabs(a))) + 1.0)/log(2.0)) + 10;
RR::SetPrecision(temp_p);
RR ln2, ln10, log_2_10;
ComputeLn2(ln2);
ComputeLn10(ln10);
log_2_10 = ln10/ln2;
ZZ log_10_a = to_ZZ(
(to_RR(a.e)*to_RR(2*NTL_XD_HBOUND_LOG) + log(fabs(a.x))/log(2.0))/log_2_10);
xdouble b;
long neg;
if (a < 0) {
b = -a;
neg = 1;
}
else {
b = a;
neg = 0;
}
ZZ k = xdouble::OutputPrecision() - log_10_a;
xdouble c, d;
c = PowerOf10(to_ZZ(xdouble::OutputPrecision()));
d = PowerOf10(log_10_a);
b = b / d;
b = b * c;
while (b < c) {
b = b * 10.0;
k++;
}
while (b >= c) {
b = b / 10.0;
k--;
}
b = b + 0.5;
k = -k;
ZZ B;
conv(B, b);
long bp_len = xdouble::OutputPrecision()+10;
UniqueArray<char> bp_store;
bp_store.SetLength(bp_len);
char *bp = bp_store.get();
long len, i;
len = 0;
do {
if (len >= bp_len) LogicError("xdouble output: buffer overflow");
bp[len] = IntValToChar(DivRem(B, B, 10));
len++;
} while (B > 0);
for (i = 0; i < len/2; i++) {
char tmp;
tmp = bp[i];
bp[i] = bp[len-1-i];
bp[len-1-i] = tmp;
}
i = len-1;
while (bp[i] == '0') i--;
k += (len-1-i);
len = i+1;
bp[len] = '\0';
if (k > 3 || k < -len - 3) {
// use scientific notation
if (neg) s << "-";
s << "0." << bp << "e" << (k + len);
}
else {
long kk = to_long(k);
if (kk >= 0) {
if (neg) s << "-";
s << bp;
for (i = 0; i < kk; i++)
s << "0";
}
else if (kk <= -len) {
if (neg) s << "-";
s << "0.";
for (i = 0; i < -len-kk; i++)
s << "0";
s << bp;
}
else {
if (neg) s << "-";
for (i = 0; i < len+kk; i++)
s << bp[i];
s << ".";
for (i = len+kk; i < len; i++)
s << bp[i];
}
}
return s;
}