newlib-cygwin/newlib/libm/common/pow.c

/* Double-precision x^y function.
   Copyright (c) 2018 Arm Ltd.  All rights reserved.

   SPDX-License-Identifier: BSD-3-Clause

   Redistribution and use in source and binary forms, with or without
   modification, are permitted provided that the following conditions
   are met:
   1. Redistributions of source code must retain the above copyright
      notice, this list of conditions and the following disclaimer.
   2. Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.
   3. The name of the company may not be used to endorse or promote
      products derived from this software without specific prior written
      permission.

   THIS SOFTWARE IS PROVIDED BY Arm LTD ``AS IS AND ANY EXPRESS OR IMPLIED
   WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
   MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
   IN NO EVENT SHALL Arm LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
   SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
   TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
   PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
   LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
   NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
   SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */

#include "fdlibm.h"
#if !__OBSOLETE_MATH

#include <math.h>
#include <stdint.h>
#include "math_config.h"

/*
Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
*/

#define T __pow_log_data.tab
#define A __pow_log_data.poly
#define Ln2hi __pow_log_data.ln2hi
#define Ln2lo __pow_log_data.ln2lo
#define N (1 << POW_LOG_TABLE_BITS)
#define OFF 0x3fe6955500000000

/* Top 12 bits of a double (sign and exponent bits).  */
static inline uint32_t
top12 (double x)
{
  return asuint64 (x) >> 52;
}

/* Compute y+tail = log(x) where the rounded result is y and tail has about
   additional 15 bits precision.  The bit representation of x if in ix, but
   normalized in the subnormal range using sign bit too for the exponent.  */
static inline double_t
log_inline (uint64_t ix, double_t *tail)
{
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
  uint64_t iz, tmp;
  int k, i;

  /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
     The range is split into N subintervals.
     The ith subinterval contains z and c is near its center.  */
  tmp = ix - OFF;
  i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
  k = (int64_t) tmp >> 52; /* arithmetic shift */
  iz = ix - (tmp & 0xfffULL << 52);
  z = asdouble (iz);
  kd = (double_t) k;

  /* log(x) = k*Ln2 + log(c) + log1p(z/c-1).  */
  invc = T[i].invc;
  logc = T[i].logc;
  logctail = T[i].logctail;

  /* r = z/c - 1, arranged to be exact.  */
#if __HAVE_FAST_FMA
  r = fma (z, invc, -1.0);
#else
  double_t zhi = asdouble (iz & (-1ULL << 32));
  double_t zlo = z - zhi;
  double_t rhi = zhi * invc - 1.0;
  double_t rlo = zlo * invc;
  r = rhi + rlo;
#endif

  /* k*Ln2 + log(c) + r.  */
  t1 = kd * Ln2hi + logc;
  t2 = t1 + r;
  lo1 = kd * Ln2lo + logctail;
  lo2 = t1 - t2 + r;

  /* Evaluation is optimized assuming superscalar pipelined execution.  */
  double_t ar, ar2, ar3, lo3, lo4;
  ar = A[0] * r; /* A[0] = -0.5.  */
  ar2 = r * ar;
  ar3 = r * ar2;
  /* k*Ln2 + log(c) + r + A[0]*r*r.  */
#if __HAVE_FAST_FMA
  hi = t2 + ar2;
  lo3 = fma (ar, r, -ar2);
  lo4 = t2 - hi + ar2;
#else
  double_t arhi = A[0] * rhi;
  double_t arhi2 = rhi * arhi;
  hi = t2 + arhi2;
  lo3 = rlo * (ar + arhi);
  lo4 = t2 - hi + arhi2;
#endif
  /* p = log1p(r) - r - A[0]*r*r.  */
#if POW_LOG_POLY_ORDER == 8
  p = (ar3
       * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
#endif
  lo = lo1 + lo2 + lo3 + lo4 + p;
  y = hi + lo;
  *tail = hi - y + lo;
  return y;
}

#undef N
#undef T
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]

/* Handle cases that may overflow or underflow when computing the result that
   is scale*(1+TMP) without intermediate rounding.  The bit representation of
   scale is in SBITS, however it has a computed exponent that may have
   overflown into the sign bit so that needs to be adjusted before using it as
   a double.  (int32_t)KI is the k used in the argument reduction and exponent
   adjustment of scale, positive k here means the result may overflow and
   negative k means the result may underflow.  */
static inline double
specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
{
  double_t scale, y;

  if ((ki & 0x80000000) == 0)
    {
      /* k > 0, the exponent of scale might have overflowed by <= 460.  */
      sbits -= 1009ull << 52;
      scale = asdouble (sbits);
      y = 0x1p1009 * (scale + scale * tmp);
      return check_oflow (y);
    }
  /* k < 0, need special care in the subnormal range.  */
  sbits += 1022ull << 52;
  /* Note: sbits is signed scale.  */
  scale = asdouble (sbits);
  y = scale + scale * tmp;
  if (fabs (y) < 1.0)
    {
      /* Round y to the right precision before scaling it into the subnormal
	 range to avoid double rounding that can cause 0.5+E/2 ulp error where
	 E is the worst-case ulp error outside the subnormal range.  So this
	 is only useful if the goal is better than 1 ulp worst-case error.  */
      double_t hi, lo, one = 1.0;
      if (y < 0.0)
	one = -1.0;
      lo = scale - y + scale * tmp;
      hi = one + y;
      lo = one - hi + y + lo;
      y = eval_as_double (hi + lo) - one;
      /* Fix the sign of 0.  */
      if (y == 0.0)
	y = asdouble (sbits & 0x8000000000000000);
      /* The underflow exception needs to be signaled explicitly.  */
      force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
    }
  y = 0x1p-1022 * y;
  return check_uflow (y);
}

#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)

/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
   The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1.  */
static inline double
exp_inline (double x, double xtail, uint32_t sign_bias)
{
  uint32_t abstop;
  uint64_t ki, idx, top, sbits;
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t kd, z, r, r2, scale, tail, tmp;

  abstop = top12 (x) & 0x7ff;
  if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
    {
      if (abstop - top12 (0x1p-54) >= 0x80000000)
	{
	  /* Avoid spurious underflow for tiny x.  */
	  /* Note: 0 is common input.  */
	  double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
	  return sign_bias ? -one : one;
	}
      if (abstop >= top12 (1024.0))
	{
	  /* Note: inf and nan are already handled.  */
	  if (asuint64 (x) >> 63)
	    return __math_uflow (sign_bias);
	  else
	    return __math_oflow (sign_bias);
	}
      /* Large x is special cased below.  */
      abstop = 0;
    }

  /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
  /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
  z = InvLn2N * x;
#if TOINT_INTRINSICS
  kd = roundtoint (z);
  ki = converttoint (z);
#elif EXP_USE_TOINT_NARROW
  /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
  kd = eval_as_double (z + Shift);
  ki = asuint64 (kd) >> 16;
  kd = (double_t) (int32_t) ki;
#else
  /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
  kd = eval_as_double (z + Shift);
  ki = asuint64 (kd);
  kd -= Shift;
#endif
  r = x + kd * NegLn2hiN + kd * NegLn2loN;
  /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
  r += xtail;
  /* 2^(k/N) ~= scale * (1 + tail).  */
  idx = 2 * (ki % N);
  top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
  tail = asdouble (T[idx]);
  /* This is only a valid scale when -1023*N < k < 1024*N.  */
  sbits = T[idx + 1] + top;
  /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
  /* Evaluation is optimized assuming superscalar pipelined execution.  */
  r2 = r * r;
  /* Without fma the worst case error is 0.25/N ulp larger.  */
  /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
#if EXP_POLY_ORDER == 4
  tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
#elif EXP_POLY_ORDER == 5
  tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
#elif EXP_POLY_ORDER == 6
  tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
#endif
  if (unlikely (abstop == 0))
    return specialcase (tmp, sbits, ki);
  scale = asdouble (sbits);
  /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
     is no spurious underflow here even without fma.  */
  return scale + scale * tmp;
}

/* Returns 0 if not int, 1 if odd int, 2 if even int.  */
static inline int
checkint (uint64_t iy)
{
  int e = iy >> 52 & 0x7ff;
  if (e < 0x3ff)
    return 0;
  if (e > 0x3ff + 52)
    return 2;
  if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
    return 0;
  if (iy & (1ULL << (0x3ff + 52 - e)))
    return 1;
  return 2;
}

/* Returns 1 if input is the bit representation of 0, infinity or nan.  */
static inline int
zeroinfnan (uint64_t i)
{
  return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
}

double
pow (double x, double y)
{
  uint32_t sign_bias = 0;
  uint64_t ix, iy;
  uint32_t topx, topy;

  ix = asuint64 (x);
  iy = asuint64 (y);
  topx = top12 (x);
  topy = top12 (y);
  if (unlikely (topx - 0x001 >= 0x7ff - 0x001
		|| (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
    {
      /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
	 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1.  */
      /* Special cases: (x < 0x1p-126 or inf or nan) or
	 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan).  */
      if (unlikely (zeroinfnan (iy)))
	{
	  if (2 * iy == 0)
	    return issignaling_inline (x) ? x + y : 1.0;
	  if (ix == asuint64 (1.0))
	    return issignaling_inline (y) ? x + y : 1.0;
	  if (2 * ix > 2 * asuint64 (INFINITY)
	      || 2 * iy > 2 * asuint64 (INFINITY))
	    return x + y;
	  if (2 * ix == 2 * asuint64 (1.0))
	    return 1.0;
	  if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
	    return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
	  return y * y;
	}
      if (unlikely (zeroinfnan (ix)))
	{
	  double_t x2 = x * x;
	  if (ix >> 63 && checkint (iy) == 1)
	    {
	      x2 = -x2;
	      sign_bias = 1;
	    }
	  if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
	    return __math_divzero (sign_bias);
	  /* Without the barrier some versions of clang hoist the 1/x2 and
	     thus division by zero exception can be signaled spuriously.  */
	  return iy >> 63 ? opt_barrier_double (1 / x2) : x2;
	}
      /* Here x and y are non-zero finite.  */
      if (ix >> 63)
	{
	  /* Finite x < 0.  */
	  int yint = checkint (iy);
	  if (yint == 0)
	    return __math_invalid (x);
	  if (yint == 1)
	    sign_bias = SIGN_BIAS;
	  ix &= 0x7fffffffffffffff;
	  topx &= 0x7ff;
	}
      if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
	{
	  /* Note: sign_bias == 0 here because y is not odd.  */
	  if (ix == asuint64 (1.0))
	    return 1.0;
	  if ((topy & 0x7ff) < 0x3be)
	    {
	      /* |y| < 2^-65, x^y ~= 1 + y*log(x).  */
	      if (WANT_ROUNDING)
		return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
	      else
		return 1.0;
	    }
	  return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
							 : __math_uflow (0);
	}
      if (topx == 0)
	{
	  /* Normalize subnormal x so exponent becomes negative.  */
	  ix = asuint64 (x * 0x1p52);
	  ix &= 0x7fffffffffffffff;
	  ix -= 52ULL << 52;
	}
    }

  double_t lo;
  double_t hi = log_inline (ix, &lo);
  double_t ehi, elo;
#if __HAVE_FAST_FMA
  ehi = y * hi;
  elo = y * lo + fma (y, hi, -ehi);
#else
  double_t yhi = asdouble (iy & -1ULL << 27);
  double_t ylo = y - yhi;
  double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
  double_t llo = hi - lhi + lo;
  ehi = yhi * lhi;
  elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25.  */
#endif
  return exp_inline (ehi, elo, sign_bias);
}
#endif