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+/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
+/*
+ * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
+ *
+ * Permission to use, copy, modify, and distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
+ * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
+ * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
+ * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
+ */
+/* powl.c
+ *
+ * Power function, long double precision
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, z, powl();
+ *
+ * z = powl( x, y );
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Computes x raised to the yth power. Analytically,
+ *
+ * x**y = exp( y log(x) ).
+ *
+ * Following Cody and Waite, this program uses a lookup table
+ * of 2**-i/32 and pseudo extended precision arithmetic to
+ * obtain several extra bits of accuracy in both the logarithm
+ * and the exponential.
+ *
+ *
+ * ACCURACY:
+ *
+ * The relative error of pow(x,y) can be estimated
+ * by y dl ln(2), where dl is the absolute error of
+ * the internally computed base 2 logarithm. At the ends
+ * of the approximation interval the logarithm equal 1/32
+ * and its relative error is about 1 lsb = 1.1e-19. Hence
+ * the predicted relative error in the result is 2.3e-21 y .
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ *
+ * IEEE +-1000 40000 2.8e-18 3.7e-19
+ * .001 < x < 1000, with log(x) uniformly distributed.
+ * -1000 < y < 1000, y uniformly distributed.
+ *
+ * IEEE 0,8700 60000 6.5e-18 1.0e-18
+ * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
+ *
+ *
+ * ERROR MESSAGES:
+ *
+ * message condition value returned
+ * pow overflow x**y > MAXNUM INFINITY
+ * pow underflow x**y < 1/MAXNUM 0.0
+ * pow domain x<0 and y noninteger 0.0
+ *
+ */
+
+#include "libm.h"
+
+#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
+long double powl(long double x, long double y)
+{
+ return pow(x, y);
+}
+#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
+
+/* Table size */
+#define NXT 32
+
+/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
+ * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
+ */
+static const long double P[] = {
+ 8.3319510773868690346226E-4L,
+ 4.9000050881978028599627E-1L,
+ 1.7500123722550302671919E0L,
+ 1.4000100839971580279335E0L,
+};
+static const long double Q[] = {
+/* 1.0000000000000000000000E0L,*/
+ 5.2500282295834889175431E0L,
+ 8.4000598057587009834666E0L,
+ 4.2000302519914740834728E0L,
+};
+/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
+ * If i is even, A[i] + B[i/2] gives additional accuracy.
+ */
+static const long double A[33] = {
+ 1.0000000000000000000000E0L,
+ 9.7857206208770013448287E-1L,
+ 9.5760328069857364691013E-1L,
+ 9.3708381705514995065011E-1L,
+ 9.1700404320467123175367E-1L,
+ 8.9735453750155359320742E-1L,
+ 8.7812608018664974155474E-1L,
+ 8.5930964906123895780165E-1L,
+ 8.4089641525371454301892E-1L,
+ 8.2287773907698242225554E-1L,
+ 8.0524516597462715409607E-1L,
+ 7.8799042255394324325455E-1L,
+ 7.7110541270397041179298E-1L,
+ 7.5458221379671136985669E-1L,
+ 7.3841307296974965571198E-1L,
+ 7.2259040348852331001267E-1L,
+ 7.0710678118654752438189E-1L,
+ 6.9195494098191597746178E-1L,
+ 6.7712777346844636413344E-1L,
+ 6.6261832157987064729696E-1L,
+ 6.4841977732550483296079E-1L,
+ 6.3452547859586661129850E-1L,
+ 6.2092890603674202431705E-1L,
+ 6.0762367999023443907803E-1L,
+ 5.9460355750136053334378E-1L,
+ 5.8186242938878875689693E-1L,
+ 5.6939431737834582684856E-1L,
+ 5.5719337129794626814472E-1L,
+ 5.4525386633262882960438E-1L,
+ 5.3357020033841180906486E-1L,
+ 5.2213689121370692017331E-1L,
+ 5.1094857432705833910408E-1L,
+ 5.0000000000000000000000E-1L,
+};
+static const long double B[17] = {
+ 0.0000000000000000000000E0L,
+ 2.6176170809902549338711E-20L,
+-1.0126791927256478897086E-20L,
+ 1.3438228172316276937655E-21L,
+ 1.2207982955417546912101E-20L,
+-6.3084814358060867200133E-21L,
+ 1.3164426894366316434230E-20L,
+-1.8527916071632873716786E-20L,
+ 1.8950325588932570796551E-20L,
+ 1.5564775779538780478155E-20L,
+ 6.0859793637556860974380E-21L,
+-2.0208749253662532228949E-20L,
+ 1.4966292219224761844552E-20L,
+ 3.3540909728056476875639E-21L,
+-8.6987564101742849540743E-22L,
+-1.2327176863327626135542E-20L,
+ 0.0000000000000000000000E0L,
+};
+
+/* 2^x = 1 + x P(x),
+ * on the interval -1/32 <= x <= 0
+ */
+static const long double R[] = {
+ 1.5089970579127659901157E-5L,
+ 1.5402715328927013076125E-4L,
+ 1.3333556028915671091390E-3L,
+ 9.6181291046036762031786E-3L,
+ 5.5504108664798463044015E-2L,
+ 2.4022650695910062854352E-1L,
+ 6.9314718055994530931447E-1L,
+};
+
+#define MEXP (NXT*16384.0L)
+/* The following if denormal numbers are supported, else -MEXP: */
+#define MNEXP (-NXT*(16384.0L+64.0L))
+/* log2(e) - 1 */
+#define LOG2EA 0.44269504088896340735992L
+
+#define F W
+#define Fa Wa
+#define Fb Wb
+#define G W
+#define Ga Wa
+#define Gb u
+#define H W
+#define Ha Wb
+#define Hb Wb
+
+static const long double MAXLOGL = 1.1356523406294143949492E4L;
+static const long double MINLOGL = -1.13994985314888605586758E4L;
+static const long double LOGE2L = 6.9314718055994530941723E-1L;
+static const long double huge = 0x1p10000L;
+/* XXX Prevent gcc from erroneously constant folding this. */
+static const volatile long double twom10000 = 0x1p-10000L;
+
+static long double reducl(long double);
+static long double powil(long double, int);
+
+long double powl(long double x, long double y)
+{
+ /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
+ int i, nflg, iyflg, yoddint;
+ long e;
+ volatile long double z=0;
+ long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
+
+ /* make sure no invalid exception is raised by nan comparision */
+ if (isnan(x)) {
+ if (!isnan(y) && y == 0.0)
+ return 1.0;
+ return x;
+ }
+ if (isnan(y)) {
+ if (x == 1.0)
+ return 1.0;
+ return y;
+ }
+ if (x == 1.0)
+ return 1.0; /* 1**y = 1, even if y is nan */
+ if (x == -1.0 && !isfinite(y))
+ return 1.0; /* -1**inf = 1 */
+ if (y == 0.0)
+ return 1.0; /* x**0 = 1, even if x is nan */
+ if (y == 1.0)
+ return x;
+ if (y >= LDBL_MAX) {
+ if (x > 1.0 || x < -1.0)
+ return INFINITY;
+ if (x != 0.0)
+ return 0.0;
+ }
+ if (y <= -LDBL_MAX) {
+ if (x > 1.0 || x < -1.0)
+ return 0.0;
+ if (x != 0.0 || y == -INFINITY)
+ return INFINITY;
+ }
+ if (x >= LDBL_MAX) {
+ if (y > 0.0)
+ return INFINITY;
+ return 0.0;
+ }
+
+ w = floorl(y);
+
+ /* Set iyflg to 1 if y is an integer. */
+ iyflg = 0;
+ if (w == y)
+ iyflg = 1;
+
+ /* Test for odd integer y. */
+ yoddint = 0;
+ if (iyflg) {
+ ya = fabsl(y);
+ ya = floorl(0.5 * ya);
+ yb = 0.5 * fabsl(w);
+ if( ya != yb )
+ yoddint = 1;
+ }
+
+ if (x <= -LDBL_MAX) {
+ if (y > 0.0) {
+ if (yoddint)
+ return -INFINITY;
+ return INFINITY;
+ }
+ if (y < 0.0) {
+ if (yoddint)
+ return -0.0;
+ return 0.0;
+ }
+ }
+ nflg = 0; /* (x<0)**(odd int) */
+ if (x <= 0.0) {
+ if (x == 0.0) {
+ if (y < 0.0) {
+ if (signbit(x) && yoddint)
+ /* (-0.0)**(-odd int) = -inf, divbyzero */
+ return -1.0/0.0;
+ /* (+-0.0)**(negative) = inf, divbyzero */
+ return 1.0/0.0;
+ }
+ if (signbit(x) && yoddint)
+ return -0.0;
+ return 0.0;
+ }
+ if (iyflg == 0)
+ return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
+ /* (x<0)**(integer) */
+ if (yoddint)
+ nflg = 1; /* negate result */
+ x = -x;
+ }
+ /* (+integer)**(integer) */
+ if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
+ w = powil(x, (int)y);
+ return nflg ? -w : w;
+ }
+
+ /* separate significand from exponent */
+ x = frexpl(x, &i);
+ e = i;
+
+ /* find significand in antilog table A[] */
+ i = 1;
+ if (x <= A[17])
+ i = 17;
+ if (x <= A[i+8])
+ i += 8;
+ if (x <= A[i+4])
+ i += 4;
+ if (x <= A[i+2])
+ i += 2;
+ if (x >= A[1])
+ i = -1;
+ i += 1;
+
+ /* Find (x - A[i])/A[i]
+ * in order to compute log(x/A[i]):
+ *
+ * log(x) = log( a x/a ) = log(a) + log(x/a)
+ *
+ * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
+ */
+ x -= A[i];
+ x -= B[i/2];
+ x /= A[i];
+
+ /* rational approximation for log(1+v):
+ *
+ * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
+ */
+ z = x*x;
+ w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
+ w = w - 0.5*z;
+
+ /* Convert to base 2 logarithm:
+ * multiply by log2(e) = 1 + LOG2EA
+ */
+ z = LOG2EA * w;
+ z += w;
+ z += LOG2EA * x;
+ z += x;
+
+ /* Compute exponent term of the base 2 logarithm. */
+ w = -i;
+ w /= NXT;
+ w += e;
+ /* Now base 2 log of x is w + z. */
+
+ /* Multiply base 2 log by y, in extended precision. */
+
+ /* separate y into large part ya
+ * and small part yb less than 1/NXT
+ */
+ ya = reducl(y);
+ yb = y - ya;
+
+ /* (w+z)(ya+yb)
+ * = w*ya + w*yb + z*y
+ */
+ F = z * y + w * yb;
+ Fa = reducl(F);
+ Fb = F - Fa;
+
+ G = Fa + w * ya;
+ Ga = reducl(G);
+ Gb = G - Ga;
+
+ H = Fb + Gb;
+ Ha = reducl(H);
+ w = (Ga + Ha) * NXT;
+
+ /* Test the power of 2 for overflow */
+ if (w > MEXP)
+ return huge * huge; /* overflow */
+ if (w < MNEXP)
+ return twom10000 * twom10000; /* underflow */
+
+ e = w;
+ Hb = H - Ha;
+
+ if (Hb > 0.0) {
+ e += 1;
+ Hb -= 1.0/NXT; /*0.0625L;*/
+ }
+
+ /* Now the product y * log2(x) = Hb + e/NXT.
+ *
+ * Compute base 2 exponential of Hb,
+ * where -0.0625 <= Hb <= 0.
+ */
+ z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
+
+ /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
+ * Find lookup table entry for the fractional power of 2.
+ */
+ if (e < 0)
+ i = 0;
+ else
+ i = 1;
+ i = e/NXT + i;
+ e = NXT*i - e;
+ w = A[e];
+ z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
+ z = z + w;
+ z = scalbnl(z, i); /* multiply by integer power of 2 */
+
+ if (nflg)
+ z = -z;
+ return z;
+}
+
+
+/* Find a multiple of 1/NXT that is within 1/NXT of x. */
+static long double reducl(long double x)
+{
+ long double t;
+
+ t = x * NXT;
+ t = floorl(t);
+ t = t / NXT;
+ return t;
+}
+
+/*
+ * Positive real raised to integer power, long double precision
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, powil();
+ * int n;
+ *
+ * y = powil( x, n );
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns argument x>0 raised to the nth power.
+ * The routine efficiently decomposes n as a sum of powers of
+ * two. The desired power is a product of two-to-the-kth
+ * powers of x. Thus to compute the 32767 power of x requires
+ * 28 multiplications instead of 32767 multiplications.
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic x domain n domain # trials peak rms
+ * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
+ * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
+ * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
+ *
+ * Returns MAXNUM on overflow, zero on underflow.
+ */
+
+static long double powil(long double x, int nn)
+{
+ long double ww, y;
+ long double s;
+ int n, e, sign, lx;
+
+ if (nn == 0)
+ return 1.0;
+
+ if (nn < 0) {
+ sign = -1;
+ n = -nn;
+ } else {
+ sign = 1;
+ n = nn;
+ }
+
+ /* Overflow detection */
+
+ /* Calculate approximate logarithm of answer */
+ s = x;
+ s = frexpl( s, &lx);
+ e = (lx - 1)*n;
+ if ((e == 0) || (e > 64) || (e < -64)) {
+ s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
+ s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
+ } else {
+ s = LOGE2L * e;
+ }
+
+ if (s > MAXLOGL)
+ return huge * huge; /* overflow */
+
+ if (s < MINLOGL)
+ return twom10000 * twom10000; /* underflow */
+ /* Handle tiny denormal answer, but with less accuracy
+ * since roundoff error in 1.0/x will be amplified.
+ * The precise demarcation should be the gradual underflow threshold.
+ */
+ if (s < -MAXLOGL+2.0) {
+ x = 1.0/x;
+ sign = -sign;
+ }
+
+ /* First bit of the power */
+ if (n & 1)
+ y = x;
+ else
+ y = 1.0;
+
+ ww = x;
+ n >>= 1;
+ while (n) {
+ ww = ww * ww; /* arg to the 2-to-the-kth power */
+ if (n & 1) /* if that bit is set, then include in product */
+ y *= ww;
+ n >>= 1;
+ }
+
+ if (sign < 0)
+ y = 1.0/y;
+ return y;
+}
+#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
+// TODO: broken implementation to make things compile
+long double powl(long double x, long double y)
+{
+ return pow(x, y);
+}
+#endif
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