Add a "fork" of musl as //fusl.

fusl/src/math/powl.c

+/* 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