/*	$OpenBSD: e_powl.c,v 1.5 2013/11/12 20:35:19 martynas Exp $	*/

/*
 * 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
 *
 */

/* Table size */
#define NXT 32
/* log2(Table size) */
#define LNXT 5

/* 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 douba(k) A[k]
#define doubb(k) B[k]
#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

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;
static const 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)
{
    long double w, ya, yb, z;
    long double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb;
    int         i, nflg, iyflg, yoddint;
    long        e;

    if (!isnanl_inline(y) && y == 0.0L) {
        if (issignalingl_inline(x))
            return x + y;
        return (1.0L);
    }

    if (!isnanl_inline(x) && x == 1.0L) {
        if (issignalingl_inline(y))
            return x + y;
        return (1.0L);
    }

    if (isnanl_inline(x))
        return (x + y);
    if (isnanl_inline(y))
        return (x + y);

    w = floorl(y);
    /* Set iyflg to 1 if y is an integer.  */
    iyflg = (w == y);

    /* flag = 1 if x is negative */
    nflg = signbit(x);

    /* Test for odd integer y and negative x (including negative zero)  */
    yoddint = 0;
    if (iyflg && nflg) {
        ya = ldexpl(y, -1);
        yoddint = (ya != floorl(ya));
    }

    if (x == 0.0L) {
        if (y < 0)
            return __math_divzerol(yoddint);
    }

    if (y == 1.0L)
        return (x);

    if (!isfinite(y)) {
        if (x == -1.0L)
            return (1.0L);
        if (y < 0) {
            if (fabsl(x) < 1)
                return ((long double)INFINITY);
            return (0.0L);
        } else {
            if (fabsl(x) < 1)
                return (0.0L);
            return ((long double)INFINITY);
        }
    }

    /* y >= 2**80, infinity (x > 1) or zero (x < 1) */
    if (y >= 0x1p80L) {
        if (x > 1.0L)
            return __math_oflowl(0);
        if (x > 0.0L && x < 1.0L)
            return (0.0L);
        if (x < -1.0L)
            return __math_oflowl(1);
        if (x > -1.0L && x < 0.0L)
            return (-0.0L);
    }
    /* y <= -2**80, zero (x > 1) or infinity (x < 1) */
    if (y <= -0x1p80L) {
        if (x > 1.0L)
            return __math_uflowl(0);
        if (x > 0.0L && x < 1.0L)
            return ((long double)INFINITY);
        if (x < -1.0L)
            return (0.0L);
        if (x > -1.0L && x < 0.0L)
            return ((long double)INFINITY);
    }
    if (x >= LDBL_MAX) {
        if (y > 0.0L)
            return ((long double)INFINITY);
        return (0.0L);
    }

    if (x <= -LDBL_MAX) {
        if (y > 0.0L) {
            if (yoddint)
                return (-(long double)INFINITY);
            return ((long double)INFINITY);
        }
        if (y < 0.0L) {
            if (yoddint)
                return (-0.0L);
            return (0.0L);
        }
    }

    if (x <= 0.0L) {
        if (x == 0.0L) {
            if (y < 0.0L) {
                if (yoddint)
                    return (-(long double)INFINITY);
                return ((long double)INFINITY);
            }
            if (y > 0.0L) {
                if (yoddint)
                    return (-0.0L);
                return (0.0L);
            }
            if (y == 0.0L)
                return (1.0L); /*   0**0   */
            else
                return (0.0L); /*   0**y   */
        } else {
            if (iyflg == 0)
                return __math_invalidl(x); /* (x<0)**(non-int) is NaN */
        }
    }

    /* Integer power of an integer.  */

    if (iyflg) {
        w = floorl(x);
        if ((w == x) && (fabsl(y) < 32768.0L)) {
            w = powil(x, (int)y);
            return (w);
        }
    }

    if (nflg)
        x = fabsl(x);

    /* separate significand from exponent */
    x = frexpl(x, &i);
    e = i;

    /* find significand in antilog table A[] */
    i = 1;
    if (x <= douba(17))
        i = 17;
    if (x <= douba(i + 8))
        i += 8;
    if (x <= douba(i + 4))
        i += 4;
    if (x <= douba(i + 2))
        i += 2;
    if (x >= douba(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 -= douba(i);
    x -= doubb(i / 2);
    x /= douba(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 - ldexpl(z, -1); /*  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 = ldexpl(w, -LNXT); /* divide by 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 = ldexpl(Ga + Ha, LNXT);

    /* Test the power of 2 for overflow */
    if (w > MEXP)
        return __math_oflowl(yoddint); /* overflow */

    if (w < MNEXP)
        return __math_uflowl(yoddint); /* underflow */

    e = w;
    Hb = H - Ha;

    if (Hb > 0.0L) {
        e += 1;
        Hb -= (1.0L / 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 = douba(e);
    z = w * z; /*    2**-e * ( 1 + (2**Hb-1) )    */
    z = z + w;
    z = ldexpl(z, i); /* multiply by integer power of 2 */

    if (nflg) {
        /* For negative x,
         * find out if the integer exponent
         * is odd or even.
         */
        w = ldexpl(y, -1);
        w = floorl(w);
        w = ldexpl(w, 1);
        if (w != y)
            z = -z; /* odd exponent */
    }

    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 = x;

    if (t < LDBL_MAX / NXT) {
        t = ldexpl(t, LNXT);
        t = floorl(t);
        t = ldexpl(t, -LNXT);
    }
    return (t);
}

/*							powil.c
 *
 *	Real raised to integer power, long double precision
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, powil();
 * int n;
 *
 * y = powil( x, n );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns argument x 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, asign, lx;

    if (x == 0.0L) {
        if (nn == 0)
            return (1.0L);
        else if (nn < 0)
            return (LDBL_MAX);
        else
            return (0.0L);
    }

    if (nn == 0)
        return (1.0L);

    if (x < 0.0L) {
        asign = -1;
        x = -x;
    } else
        asign = 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 - 0.5L + lx) * nn * LOGE2L;
    } else {
        s = LOGE2L * e;
    }

    if (s > MAXLOGL)
        return __math_oflowl(asign < 0 && (nn & 1)); /* overflow */

    if (s < MINLOGL)
        return __math_uflowl(asign < 0 && (nn & 1)); /* 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 + 2.0L)) {
        x = 1.0L / x;
        sign = -sign;
    }

    /* First bit of the power */
    if (n & 1)
        y = x;

    else {
        y = 1.0L;
        asign = 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 (asign)
        y = -y; /* odd power of negative number */
    if (sign < 0)
        y = 1.0L / y;
    return (y);
}

#ifdef __strong_reference
#if defined(__GNUCLIKE_PRAGMA_DIAGNOSTIC) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Wmissing-attributes"
#endif
__strong_reference(powl, _powl);
#endif
