libm: sync with upstream.
There's potential here to maybe lose some/all of builtins.cpp, but I'll
look at that separately later.
Test: treehugger
Change-Id: I2c2bc1d0753affdd214daeb09fa1ac7cd73db347
diff --git a/libm/upstream-freebsd/lib/msun/bsdsrc/b_tgamma.c b/libm/upstream-freebsd/lib/msun/bsdsrc/b_tgamma.c
index 5cb1f93..493ced3 100644
--- a/libm/upstream-freebsd/lib/msun/bsdsrc/b_tgamma.c
+++ b/libm/upstream-freebsd/lib/msun/bsdsrc/b_tgamma.c
@@ -29,37 +29,46 @@
* SUCH DAMAGE.
*/
+/*
+ * The original code, FreeBSD's old svn r93211, contained the following
+ * attribution:
+ *
+ * This code by P. McIlroy, Oct 1992;
+ *
+ * The financial support of UUNET Communications Services is greatfully
+ * acknowledged.
+ *
+ * The algorithm remains, but the code has been re-arranged to facilitate
+ * porting to other precisions.
+ */
+
/* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
+#include <float.h>
+
+#include "math.h"
+#include "math_private.h"
+
+/* Used in b_log.c and below. */
+struct Double {
+ double a;
+ double b;
+};
+
+#include "b_log.c"
+#include "b_exp.c"
+
/*
- * This code by P. McIlroy, Oct 1992;
+ * The range is broken into several subranges. Each is handled by its
+ * helper functions.
*
- * The financial support of UUNET Communications Services is greatfully
- * acknowledged.
- */
-
-#include <math.h>
-#include "mathimpl.h"
-
-/* METHOD:
- * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
- * At negative integers, return NaN and raise invalid.
- *
- * x < 6.5:
- * Use argument reduction G(x+1) = xG(x) to reach the
- * range [1.066124,2.066124]. Use a rational
- * approximation centered at the minimum (x0+1) to
- * ensure monotonicity.
- *
- * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
- * adjusted for equal-ripples:
- *
- * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
- *
- * Keep extra precision in multiplying (x-.5)(log(x)-1), to
- * avoid premature round-off.
+ * x >= 6.0: large_gam(x)
+ * 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0.
+ * xleft > x > iota: smaller_gam(x) where iota = 1e-17.
+ * iota > x > -itoa: Handle x near 0.
+ * -iota > x : neg_gam
*
* Special values:
* -Inf: return NaN and raise invalid;
@@ -77,201 +86,224 @@
* Maximum observed error < 4ulp in 1,000,000 trials.
*/
-static double neg_gam(double);
-static double small_gam(double);
-static double smaller_gam(double);
-static struct Double large_gam(double);
-static struct Double ratfun_gam(double, double);
-
-/*
- * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
- * [1.066.., 2.066..] accurate to 4.25e-19.
- */
-#define LEFT -.3955078125 /* left boundary for rat. approx */
-#define x0 .461632144968362356785 /* xmin - 1 */
-
-#define a0_hi 0.88560319441088874992
-#define a0_lo -.00000000000000004996427036469019695
-#define P0 6.21389571821820863029017800727e-01
-#define P1 2.65757198651533466104979197553e-01
-#define P2 5.53859446429917461063308081748e-03
-#define P3 1.38456698304096573887145282811e-03
-#define P4 2.40659950032711365819348969808e-03
-#define Q0 1.45019531250000000000000000000e+00
-#define Q1 1.06258521948016171343454061571e+00
-#define Q2 -2.07474561943859936441469926649e-01
-#define Q3 -1.46734131782005422506287573015e-01
-#define Q4 3.07878176156175520361557573779e-02
-#define Q5 5.12449347980666221336054633184e-03
-#define Q6 -1.76012741431666995019222898833e-03
-#define Q7 9.35021023573788935372153030556e-05
-#define Q8 6.13275507472443958924745652239e-06
/*
* Constants for large x approximation (x in [6, Inf])
* (Accurate to 2.8*10^-19 absolute)
*/
-#define lns2pi_hi 0.418945312500000
-#define lns2pi_lo -.000006779295327258219670263595
-#define Pa0 8.33333333333333148296162562474e-02
-#define Pa1 -2.77777777774548123579378966497e-03
-#define Pa2 7.93650778754435631476282786423e-04
-#define Pa3 -5.95235082566672847950717262222e-04
-#define Pa4 8.41428560346653702135821806252e-04
-#define Pa5 -1.89773526463879200348872089421e-03
-#define Pa6 5.69394463439411649408050664078e-03
-#define Pa7 -1.44705562421428915453880392761e-02
-static const double zero = 0., one = 1.0, tiny = 1e-300;
-
-double
-tgamma(x)
- double x;
-{
- struct Double u;
-
- if (x >= 6) {
- if(x > 171.63)
- return (x / zero);
- u = large_gam(x);
- return(__exp__D(u.a, u.b));
- } else if (x >= 1.0 + LEFT + x0)
- return (small_gam(x));
- else if (x > 1.e-17)
- return (smaller_gam(x));
- else if (x > -1.e-17) {
- if (x != 0.0)
- u.a = one - tiny; /* raise inexact */
- return (one/x);
- } else if (!finite(x))
- return (x - x); /* x is NaN or -Inf */
- else
- return (neg_gam(x));
-}
+static const double zero = 0.;
+static const volatile double tiny = 1e-300;
/*
+ * x >= 6
+ *
+ * Use the asymptotic approximation (Stirling's formula) adjusted fof
+ * equal-ripples:
+ *
+ * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
+ *
+ * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
+ * premature round-off.
+ *
* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
*/
-static struct Double
-large_gam(x)
- double x;
-{
- double z, p;
- struct Double t, u, v;
+static const double
+ ln2pi_hi = 0.41894531250000000,
+ ln2pi_lo = -6.7792953272582197e-6,
+ Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
+ Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */
+ Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */
+ Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */
+ Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */
+ Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */
+ Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */
+ Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */
- z = one/(x*x);
- p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
- p = p/x;
+static struct Double
+large_gam(double x)
+{
+ double p, z, thi, tlo, xhi, xlo;
+ struct Double u;
+
+ z = 1 / (x * x);
+ p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
+ z * (Pa6 + z * Pa7))))));
+ p = p / x;
u = __log__D(x);
- u.a -= one;
- v.a = (x -= .5);
- TRUNC(v.a);
- v.b = x - v.a;
- t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
- t.b = v.b*u.a + x*u.b;
- /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
- t.b += lns2pi_lo; t.b += p;
- u.a = lns2pi_hi + t.b; u.a += t.a;
- u.b = t.a - u.a;
- u.b += lns2pi_hi; u.b += t.b;
+ u.a -= 1;
+
+ /* Split (x - 0.5) in high and low parts. */
+ x -= 0.5;
+ xhi = (float)x;
+ xlo = x - xhi;
+
+ /* Compute t = (x-.5)*(log(x)-1) in extra precision. */
+ thi = xhi * u.a;
+ tlo = xlo * u.a + x * u.b;
+
+ /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
+ tlo += ln2pi_lo;
+ tlo += p;
+ u.a = ln2pi_hi + tlo;
+ u.a += thi;
+ u.b = thi - u.a;
+ u.b += ln2pi_hi;
+ u.b += tlo;
return (u);
}
/*
+ * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
+ * [1.066.., 2.066..] accurate to 4.25e-19.
+ *
+ * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
+ */
+static const double
+#if 0
+ a0_hi = 8.8560319441088875e-1,
+ a0_lo = -4.9964270364690197e-17,
+#else
+ a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */
+ a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */
+#endif
+ P0 = 6.2138957182182086e-1,
+ P1 = 2.6575719865153347e-1,
+ P2 = 5.5385944642991746e-3,
+ P3 = 1.3845669830409657e-3,
+ P4 = 2.4065995003271137e-3,
+ Q0 = 1.4501953125000000e+0,
+ Q1 = 1.0625852194801617e+0,
+ Q2 = -2.0747456194385994e-1,
+ Q3 = -1.4673413178200542e-1,
+ Q4 = 3.0787817615617552e-2,
+ Q5 = 5.1244934798066622e-3,
+ Q6 = -1.7601274143166700e-3,
+ Q7 = 9.3502102357378894e-5,
+ Q8 = 6.1327550747244396e-6;
+
+static struct Double
+ratfun_gam(double z, double c)
+{
+ double p, q, thi, tlo;
+ struct Double r;
+
+ q = Q0 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
+ z * (Q6 + z * (Q7 + z * Q8)))))));
+ p = P0 + z * (P1 + z * (P2 + z * (P3 + z * P4)));
+ p = p / q;
+
+ /* Split z into high and low parts. */
+ thi = (float)z;
+ tlo = (z - thi) + c;
+ tlo *= (thi + z);
+
+ /* Split (z+c)^2 into high and low parts. */
+ thi *= thi;
+ q = thi;
+ thi = (float)thi;
+ tlo += (q - thi);
+
+ /* Split p/q into high and low parts. */
+ r.a = (float)p;
+ r.b = p - r.a;
+
+ tlo = tlo * p + thi * r.b + a0_lo;
+ thi *= r.a; /* t = (z+c)^2*(P/Q) */
+ r.a = (float)(thi + a0_hi);
+ r.b = ((a0_hi - r.a) + thi) + tlo;
+ return (r); /* r = a0 + t */
+}
+/*
+ * x < 6
+ *
+ * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
+ * 2.066124]. Use a rational approximation centered at the minimum
+ * (x0+1) to ensure monotonicity.
+ *
* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
* It also has correct monotonicity.
*/
+static const double
+ left = -0.3955078125, /* left boundary for rat. approx */
+ x0 = 4.6163214496836236e-1; /* xmin - 1 */
+
static double
-small_gam(x)
- double x;
+small_gam(double x)
{
- double y, ym1, t;
+ double t, y, ym1;
struct Double yy, r;
- y = x - one;
- ym1 = y - one;
- if (y <= 1.0 + (LEFT + x0)) {
+
+ y = x - 1;
+ if (y <= 1 + (left + x0)) {
yy = ratfun_gam(y - x0, 0);
return (yy.a + yy.b);
}
- r.a = y;
- TRUNC(r.a);
- yy.a = r.a - one;
- y = ym1;
- yy.b = r.b = y - yy.a;
+
+ r.a = (float)y;
+ yy.a = r.a - 1;
+ y = y - 1 ;
+ r.b = yy.b = y - yy.a;
+
/* Argument reduction: G(x+1) = x*G(x) */
- for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
- t = r.a*yy.a;
- r.b = r.a*yy.b + y*r.b;
- r.a = t;
- TRUNC(r.a);
+ for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
+ t = r.a * yy.a;
+ r.b = r.a * yy.b + y * r.b;
+ r.a = (float)t;
r.b += (t - r.a);
}
+
/* Return r*tgamma(y). */
yy = ratfun_gam(y - x0, 0);
- y = r.b*(yy.a + yy.b) + r.a*yy.b;
- y += yy.a*r.a;
+ y = r.b * (yy.a + yy.b) + r.a * yy.b;
+ y += yy.a * r.a;
return (y);
}
/*
- * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
+ * Good on (0, 1+x0+left]. Accurate to 1 ulp.
*/
static double
-smaller_gam(x)
- double x;
+smaller_gam(double x)
{
- double t, d;
- struct Double r, xx;
- if (x < x0 + LEFT) {
- t = x, TRUNC(t);
- d = (t+x)*(x-t);
+ double d, rhi, rlo, t, xhi, xlo;
+ struct Double r;
+
+ if (x < x0 + left) {
+ t = (float)x;
+ d = (t + x) * (x - t);
t *= t;
- xx.a = (t + x), TRUNC(xx.a);
- xx.b = x - xx.a; xx.b += t; xx.b += d;
- t = (one-x0); t += x;
- d = (one-x0); d -= t; d += x;
- x = xx.a + xx.b;
+ xhi = (float)(t + x);
+ xlo = x - xhi;
+ xlo += t;
+ xlo += d;
+ t = 1 - x0;
+ t += x;
+ d = 1 - x0;
+ d -= t;
+ d += x;
+ x = xhi + xlo;
} else {
- xx.a = x, TRUNC(xx.a);
- xx.b = x - xx.a;
+ xhi = (float)x;
+ xlo = x - xhi;
t = x - x0;
- d = (-x0 -t); d += x;
+ d = - x0 - t;
+ d += x;
}
+
r = ratfun_gam(t, d);
- d = r.a/x, TRUNC(d);
- r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
- return (d + r.a/x);
+ d = (float)(r.a / x);
+ r.a -= d * xhi;
+ r.a -= d * xlo;
+ r.a += r.b;
+
+ return (d + r.a / x);
}
/*
- * returns (z+c)^2 * P(z)/Q(z) + a0
+ * x < 0
+ *
+ * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
+ * At negative integers, return NaN and raise invalid.
*/
-static struct Double
-ratfun_gam(z, c)
- double z, c;
-{
- double p, q;
- struct Double r, t;
-
- q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
- p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
-
- /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
- p = p/q;
- t.a = z, TRUNC(t.a); /* t ~= z + c */
- t.b = (z - t.a) + c;
- t.b *= (t.a + z);
- q = (t.a *= t.a); /* t = (z+c)^2 */
- TRUNC(t.a);
- t.b += (q - t.a);
- r.a = p, TRUNC(r.a); /* r = P/Q */
- r.b = p - r.a;
- t.b = t.b*p + t.a*r.b + a0_lo;
- t.a *= r.a; /* t = (z+c)^2*(P/Q) */
- r.a = t.a + a0_hi, TRUNC(r.a);
- r.b = ((a0_hi-r.a) + t.a) + t.b;
- return (r); /* r = a0 + t */
-}
-
static double
-neg_gam(x)
- double x;
+neg_gam(double x)
{
int sgn = 1;
struct Double lg, lsine;
@@ -280,23 +312,29 @@
y = ceil(x);
if (y == x) /* Negative integer. */
return ((x - x) / zero);
+
z = y - x;
if (z > 0.5)
- z = one - z;
- y = 0.5 * y;
+ z = 1 - z;
+
+ y = y / 2;
if (y == ceil(y))
sgn = -1;
- if (z < .25)
- z = sin(M_PI*z);
+
+ if (z < 0.25)
+ z = sinpi(z);
else
- z = cos(M_PI*(0.5-z));
+ z = cospi(0.5 - z);
+
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
if (x < -170) {
+
if (x < -190)
- return ((double)sgn*tiny*tiny);
- y = one - x; /* exact: 128 < |x| < 255 */
+ return (sgn * tiny * tiny);
+
+ y = 1 - x; /* exact: 128 < |x| < 255 */
lg = large_gam(y);
- lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
+ lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */
lg.a -= lsine.a; /* exact (opposite signs) */
lg.b -= lsine.b;
y = -(lg.a + lg.b);
@@ -305,11 +343,58 @@
if (sgn < 0) y = -y;
return (y);
}
- y = one-x;
- if (one-y == x)
+
+ y = 1 - x;
+ if (1 - y == x)
y = tgamma(y);
else /* 1-x is inexact */
- y = -x*tgamma(-x);
+ y = - x * tgamma(-x);
+
if (sgn < 0) y = -y;
- return (M_PI / (y*z));
+ return (M_PI / (y * z));
}
+/*
+ * xmax comes from lgamma(xmax) - emax * log(2) = 0.
+ * static const float xmax = 35.040095f
+ * static const double xmax = 171.624376956302725;
+ * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
+ * ld128: 1.75554834290446291700388921607020320e+03L,
+ *
+ * iota is a sloppy threshold to isolate x = 0.
+ */
+static const double xmax = 171.624376956302725;
+static const double iota = 0x1p-56;
+
+double
+tgamma(double x)
+{
+ struct Double u;
+
+ if (x >= 6) {
+ if (x > xmax)
+ return (x / zero);
+ u = large_gam(x);
+ return (__exp__D(u.a, u.b));
+ }
+
+ if (x >= 1 + left + x0)
+ return (small_gam(x));
+
+ if (x > iota)
+ return (smaller_gam(x));
+
+ if (x > -iota) {
+ if (x != 0.)
+ u.a = 1 - tiny; /* raise inexact */
+ return (1 / x);
+ }
+
+ if (!isfinite(x))
+ return (x - x); /* x is NaN or -Inf */
+
+ return (neg_gam(x));
+}
+
+#if (LDBL_MANT_DIG == 53)
+__weak_reference(tgamma, tgammal);
+#endif