math: new software sqrtl

same approach as in sqrt.

sqrtl was broken on aarch64, riscv64 and s390x targets because
of missing quad precision support and on m68k-sf because of
missing ld80 sqrtl.

this implementation is written for quad precision and then
edited to make it work for both m68k and x86 style ld80 formats
too, but it is not expected to be optimal for them.

note: using fp instructions for the initial estimate when such
instructions are available (e.g. double prec sqrt or rsqrt) is
avoided because of fenv correctness.

src/math/sqrtl.c

@@ -1,7 +1,259 @@
+#include <stdint.h>
 #include <math.h>
+#include <float.h>
+#include "libm.h"
 
+#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 long double sqrtl(long double x)
 {
-	/* FIXME: implement in C, this is for LDBL_MANT_DIG == 64 only */
 	return sqrt(x);
 }
+#elif (LDBL_MANT_DIG == 113 || LDBL_MANT_DIG == 64) && LDBL_MAX_EXP == 16384
+#include "sqrt_data.h"
+
+#define FENV_SUPPORT 1
+
+typedef struct {
+	uint64_t hi;
+	uint64_t lo;
+} u128;
+
+/* top: 16 bit sign+exponent, x: significand.  */
+static inline long double mkldbl(uint64_t top, u128 x)
+{
+	union ldshape u;
+#if LDBL_MANT_DIG == 113
+	u.i2.hi = x.hi;
+	u.i2.lo = x.lo;
+	u.i2.hi &= 0x0000ffffffffffff;
+	u.i2.hi |= top << 48;
+#elif LDBL_MANT_DIG == 64
+	u.i.se = top;
+	u.i.m = x.lo;
+	/* force the top bit on non-zero (and non-subnormal) results.  */
+	if (top & 0x7fff)
+		u.i.m |= 0x8000000000000000;
+#endif
+	return u.f;
+}
+
+/* return: top 16 bit is sign+exp and following bits are the significand.  */
+static inline u128 asu128(long double x)
+{
+	union ldshape u = {.f=x};
+	u128 r;
+#if LDBL_MANT_DIG == 113
+	r.hi = u.i2.hi;
+	r.lo = u.i2.lo;
+#elif LDBL_MANT_DIG == 64
+	r.lo = u.i.m<<49;
+	/* ignore the top bit: pseudo numbers are not handled. */
+	r.hi = u.i.m>>15;
+	r.hi &= 0x0000ffffffffffff;
+	r.hi |= (uint64_t)u.i.se << 48;
+#endif
+	return r;
+}
+
+/* returns a*b*2^-32 - e, with error 0 <= e < 1.  */
+static inline uint32_t mul32(uint32_t a, uint32_t b)
+{
+	return (uint64_t)a*b >> 32;
+}
+
+/* returns a*b*2^-64 - e, with error 0 <= e < 3.  */
+static inline uint64_t mul64(uint64_t a, uint64_t b)
+{
+	uint64_t ahi = a>>32;
+	uint64_t alo = a&0xffffffff;
+	uint64_t bhi = b>>32;
+	uint64_t blo = b&0xffffffff;
+	return ahi*bhi + (ahi*blo >> 32) + (alo*bhi >> 32);
+}
+
+static inline u128 add64(u128 a, uint64_t b)
+{
+	u128 r;
+	r.lo = a.lo + b;
+	r.hi = a.hi;
+	if (r.lo < a.lo)
+		r.hi++;
+	return r;
+}
+
+static inline u128 add128(u128 a, u128 b)
+{
+	u128 r;
+	r.lo = a.lo + b.lo;
+	r.hi = a.hi + b.hi;
+	if (r.lo < a.lo)
+		r.hi++;
+	return r;
+}
+
+static inline u128 sub64(u128 a, uint64_t b)
+{
+	u128 r;
+	r.lo = a.lo - b;
+	r.hi = a.hi;
+	if (a.lo < b)
+		r.hi--;
+	return r;
+}
+
+static inline u128 sub128(u128 a, u128 b)
+{
+	u128 r;
+	r.lo = a.lo - b.lo;
+	r.hi = a.hi - b.hi;
+	if (a.lo < b.lo)
+		r.hi--;
+	return r;
+}
+
+/* a<<n, 0 <= n <= 127 */
+static inline u128 lsh(u128 a, int n)
+{
+	if (n == 0)
+		return a;
+	if (n >= 64) {
+		a.hi = a.lo<<(n-64);
+		a.lo = 0;
+	} else {
+		a.hi = (a.hi<<n) | (a.lo>>(64-n));
+		a.lo = a.lo<<n;
+	}
+	return a;
+}
+
+/* a>>n, 0 <= n <= 127 */
+static inline u128 rsh(u128 a, int n)
+{
+	if (n == 0)
+		return a;
+	if (n >= 64) {
+		a.lo = a.hi>>(n-64);
+		a.hi = 0;
+	} else {
+		a.lo = (a.lo>>n) | (a.hi<<(64-n));
+		a.hi = a.hi>>n;
+	}
+	return a;
+}
+
+/* returns a*b exactly.  */
+static inline u128 mul64_128(uint64_t a, uint64_t b)
+{
+	u128 r;
+	uint64_t ahi = a>>32;
+	uint64_t alo = a&0xffffffff;
+	uint64_t bhi = b>>32;
+	uint64_t blo = b&0xffffffff;
+	uint64_t lo1 = ((ahi*blo)&0xffffffff) + ((alo*bhi)&0xffffffff) + (alo*blo>>32);
+	uint64_t lo2 = (alo*blo)&0xffffffff;
+	r.hi = ahi*bhi + (ahi*blo>>32) + (alo*bhi>>32) + (lo1>>32);
+	r.lo = (lo1<<32) + lo2;
+	return r;
+}
+
+/* returns a*b*2^-128 - e, with error 0 <= e < 7.  */
+static inline u128 mul128(u128 a, u128 b)
+{
+	u128 hi = mul64_128(a.hi, b.hi);
+	uint64_t m1 = mul64(a.hi, b.lo);
+	uint64_t m2 = mul64(a.lo, b.hi);
+	return add64(add64(hi, m1), m2);
+}
+
+/* returns a*b % 2^128.  */
+static inline u128 mul128_tail(u128 a, u128 b)
+{
+	u128 lo = mul64_128(a.lo, b.lo);
+	lo.hi += a.hi*b.lo + a.lo*b.hi;
+	return lo;
+}
+
+
+/* see sqrt.c for detailed comments.  */
+
+long double sqrtl(long double x)
+{
+	u128 ix, ml;
+	uint64_t top;
+
+	ix = asu128(x);
+	top = ix.hi >> 48;
+	if (predict_false(top - 0x0001 >= 0x7fff - 0x0001)) {
+		/* x < 0x1p-16382 or inf or nan.  */
+		if (2*ix.hi == 0 && ix.lo == 0)
+			return x;
+		if (ix.hi == 0x7fff000000000000 && ix.lo == 0)
+			return x;
+		if (top >= 0x7fff)
+			return __math_invalidl(x);
+		/* x is subnormal, normalize it.  */
+		ix = asu128(x * 0x1p112);
+		top = ix.hi >> 48;
+		top -= 112;
+	}
+
+	/* x = 4^e m; with int e and m in [1, 4) */
+	int even = top & 1;
+	ml = lsh(ix, 15);
+	ml.hi |= 0x8000000000000000;
+	if (even) ml = rsh(ml, 1);
+	top = (top + 0x3fff) >> 1;
+
+	/* r ~ 1/sqrt(m) */
+	static const uint64_t three = 0xc0000000;
+	uint64_t r, s, d, u, i;
+	i = (ix.hi >> 42) % 128;
+	r = (uint32_t)__rsqrt_tab[i] << 16;
+	/* |r sqrt(m) - 1| < 0x1p-8 */
+	s = mul32(ml.hi>>32, r);
+	d = mul32(s, r);
+	u = three - d;
+	r = mul32(u, r) << 1;
+	/* |r sqrt(m) - 1| < 0x1.7bp-16, switch to 64bit */
+	r = r<<32;
+	s = mul64(ml.hi, r);
+	d = mul64(s, r);
+	u = (three<<32) - d;
+	r = mul64(u, r) << 1;
+	/* |r sqrt(m) - 1| < 0x1.a5p-31 */
+	s = mul64(u, s) << 1;
+	d = mul64(s, r);
+	u = (three<<32) - d;
+	r = mul64(u, r) << 1;
+	/* |r sqrt(m) - 1| < 0x1.c001p-59, switch to 128bit */
+
+	static const u128 threel = {.hi=three<<32, .lo=0};
+	u128 rl, sl, dl, ul;
+	rl.hi = r;
+	rl.lo = 0;
+	sl = mul128(ml, rl);
+	dl = mul128(sl, rl);
+	ul = sub128(threel, dl);
+	sl = mul128(ul, sl); /* repr: 3.125 */
+	/* -0x1p-116 < s - sqrt(m) < 0x3.8001p-125 */
+	sl = rsh(sub64(sl, 4), 125-(LDBL_MANT_DIG-1));
+	/* s < sqrt(m) < s + 1 ULP + tiny */
+
+	long double y;
+	u128 d2, d1, d0;
+	d0 = sub128(lsh(ml, 2*(LDBL_MANT_DIG-1)-126), mul128_tail(sl,sl));
+	d1 = sub128(sl, d0);
+	d2 = add128(add64(sl, 1), d1);
+	sl = add64(sl, d1.hi >> 63);
+	y = mkldbl(top, sl);
+	if (FENV_SUPPORT) {
+		/* handle rounding modes and inexact exception.  */
+		top = predict_false((d2.hi|d2.lo)==0) ? 0 : 1;
+		top |= ((d1.hi^d2.hi)&0x8000000000000000) >> 48;
+		y += mkldbl(top, (u128){0});
+	}
+	return y;
+}
+#else
+#error unsupported long double format
+#endif