/* 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