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#include "heron_k.h"
/**
** Simply works by repeated squaring.
**/
double expk2(double a, unsigned k) [[__unsequenced__]] {
for (;;) {
k /= 2u;
if (!k) break;
a *= a;
}
return a;
}
double expk_rec(double a, unsigned k) [[__unsequenced__]] {
switch (k)
default: {
if (k % 2u)
case 3: {
double ret = expk_rec(a, k/2u);
ret *= ret;
a *= ret;
} else {
/* If k is a power of 2, do something special. */
if (k == (k & -k)) {
[[fallthrough]];
case 4: a = expk2(a, k);
} else {
a = expk_rec(a, k/2u);
[[fallthrough]];
case 2:
a *= a;
}
}
case 1: // only possible with C23
}
return a;
}
double frexp_np(double x, signed exp[static 1]) [[__unsequenced__]] {
if (x < 0.0) return -frexp_np(-x, exp);
signed ret = 0;
if (x > 0.0) {
if (x < 0.5) {
do {
x *= 2.0;
--ret;
} while (x < 0.5);
} else {
while (x >= 1.0) {
x *= 0.5;
++ret;
}
}
}
exp[0] = ret;
return x;
}
double heron1_estimate(double a) [[__unsequenced__]] {
/* A first low quality estimate for the inverse. */
signed e0; frexp_np(a, &e0);
return
(e0 <= 0
? expk_rec(2.0, 1-e0)
: expk_rec(0.5, e0));
}
double heron1_estimate_dir_05(double a) [[__unsequenced__]] {
/* A table of double powers of 2. */
/*Something like (2 << (2 << k)) if only that would be representable
in an integer type. */
double const tab[] = {
0x1P+000,
0x1P+001,
0x1P+002,
0x1P+004,
0x1P+008,
0x1P+016,
0x1P+032,
0x1P+064,
#if (DBL_MAX_EXP*FLT_RDXRDX) > 128
0x1P+128,
#endif
#if (DBL_MAX_EXP*FLT_RDXRDX) > 256
0x1P+256,
#endif
#if (DBL_MAX_EXP*FLT_RDXRDX) > 512
0x1P+512,
#endif
#if (DBL_MAX_EXP*FLT_RDXRDX) > 1024
0x1P+1024,
#endif
#if (DBL_MAX_EXP*FLT_RDXRDX) > 2048
0x1P+2048,
#endif
#if (DBL_MAX_EXP*FLT_RDXRDX) > 4096
0x1P+4096,
#endif
};
/* First estimate log2(log2(a)). This will be the most significant
bit of the exponent that we are looking for. */
constexpr size_t k_max = sizeof tab/sizeof tab[0];
unsigned k = 1u;
while ((k < k_max) && (tab[k]*a < 1.0))
++k;
double x = tab[k-1];
a *= x;
k -= 2u;
/* Now test all the smaller bit positions in the exponent. */
for (;k;--k) {
if (tab[k]*a < 1.0) {
a *= tab[k];
x *= tab[k];
}
}
return x;
}
double heron1_estimate_dir_10(double a) [[__unsequenced__]] {
/* A table of inverses of double powers of 2. */
/* Something like 1.0/(2 << (2 << k)) if only that would be
representable in an integer type. */
double const tab[] = {
0x1P-000,
0x1P-001,
0x1P-002,
0x1P-004,
0x1P-008,
0x1P-016,
0x1P-032,
0x1P-064,
#if (DBL_MIN_EXP*FLT_RDXRDX) < -128
0x1P-128,
#endif
#if (DBL_MIN_EXP*FLT_RDXRDX) < -256
0x1P-256,
#endif
#if (DBL_MIN_EXP*FLT_RDXRDX) < -512
0x1P-512,
#endif
#if (DBL_MIN_EXP*FLT_RDXRDX) < -1024
0x1P-1024,
#endif
#if (DBL_MIN_EXP*FLT_RDXRDX) < -2048
0x1P-2048,
#endif
#if (DBL_MIN_EXP*FLT_RDXRDX) < -4096
0x1P-4096,
#endif
};
/* First estimate -log2(log2(a)). This will be the most significant
bit of the exponent that we are looking for. */
constexpr size_t k_max = sizeof tab/sizeof tab[0];
unsigned k = 1u;
while ((k < k_max) && (tab[k]*a >= 0.5))
++k;
double x = tab[k-1];
a *= x;
k -= 2u;
/* Now test all the smaller bit positions in the exponent. */
for (;k;--k) {
if (tab[k]*a >= 0.5) {
a *= tab[k];
x *= tab[k];
}
}
return x;
}
double heron1_estimate_dir(double a) [[__unsequenced__]] {
if (a < 0.5) return heron1_estimate_dir_05(a);
else if (a >= 1.0) return heron1_estimate_dir_10(a);
else return 1.0;
}
double heron1(double a) [[__unsequenced__]] {
constexpr double epsP1 = 1.0 + 1E-9;
constexpr double epsM1 = 1.0 - 1E-9;
double x = heron1_estimate_dir(a);
/* We are close, correct with the Heron factor. */
for (double prod = a*x;
((prod < epsM1) || (epsP1 < prod));
prod = a*x) {
x *= (2.0 - prod);
}
return x;
}
double expk(double a, signed k) [[__unsequenced__]] {
if (k < 0) return heron1(expk_rec(a, -k));
if (k) return expk_rec(a, k);
else return 1.0;
}
double heron(double a, signed k) [[__unsequenced__]] {
switch (k) {
case 0: return 1.0;
case 1: return a;
}
if (k < 0) return heron1(heron(a, -k));
if (a <= 0.0) return 0.0;
constexpr double eps = 1E-12;
double const k1 = heron1(k);
double const a1 = heron1(a);
double x = (1.0 + a)*k1;
for (;;) {
double xk1 = expk(x, k-1);
double axk1 = a*heron1(xk1);
if (fabs(1.0 - x*xk1*a1) < eps) break;
x = ((k-1)*x + axk1)*k1;
}
return x;
}
/* A normal project would place the `main` in a different compilation
unit. */
int main(int argc, char* argv[argc+1]) {
/* Read the first command line argument for the exponent. */
int k = strtol(argv[1], 0, 0);
for (int i = 2; i < argc; ++i) {
/* Read one command line argument as a double. */
double const a = strtod(argv[i], 0);
double const x = heron(a, k);
signed e0; double m0 = frexp_np(x, &e0);
signed e1; double m1 = frexp(x, &e1);
printf("heron: a=%.5e,\tx=%.5e,\tx**k=%.20e, %gT%+d, %gT%+d\n",
a, x, expk(x, k), m0, e0, m1, e1);
}
return EXIT_SUCCESS;
}
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