libm/common/exp2.c

/* Double-precision 2^x 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"

#define N (1 << EXP_TABLE_BITS)
#define Shift __exp_data.exp2_shift
#define T __exp_data.tab
#define C1 __exp_data.exp2_poly[0]
#define C2 __exp_data.exp2_poly[1]
#define C3 __exp_data.exp2_poly[2]
#define C4 __exp_data.exp2_poly[3]
#define C5 __exp_data.exp2_poly[4]
#define C6 __exp_data.exp2_poly[5]

/* 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 1.  */
      sbits -= 1ull << 52;
      scale = asdouble (sbits);
      y = 2 * (scale + scale * tmp);
      return check_oflow (y);
    }
  /* k < 0, need special care in the subnormal range.  */
  sbits += 1022ull << 52;
  scale = asdouble (sbits);
  y = scale + scale * tmp;
  if (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;
      lo = scale - y + scale * tmp;
      hi = 1.0 + y;
      lo = 1.0 - hi + y + lo;
      y = eval_as_double (hi + lo) - 1.0;
      /* Avoid -0.0 with downward rounding.  */
      if (WANT_ROUNDING && y == 0.0)
	y = 0.0;
      /* 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);
}

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

double
exp2 (double x)
{
  uint32_t abstop;
  uint64_t ki, idx, top, sbits;
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t kd, 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.  */
	return WANT_ROUNDING ? 1.0 + x : 1.0;
      if (abstop >= top12 (1024.0))
	{
	  if (asuint64 (x) == asuint64 (-INFINITY))
	    return 0.0;
	  if (abstop >= top12 (INFINITY))
	    return 1.0 + x;
	  if (!(asuint64 (x) >> 63))
	    return __math_oflow (0);
	  else if (asuint64 (x) >= asuint64 (-1075.0))
	    return __math_uflow (0);
	}
      if (2 * asuint64 (x) > 2 * asuint64 (928.0))
	/* Large x is special cased below.  */
	abstop = 0;
    }

  /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */
  /* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */
  kd = eval_as_double (x + Shift);
  ki = asuint64 (kd); /* k.  */
  kd -= Shift; /* k/N for int k.  */
  r = x - kd;
  /* 2^(k/N) ~= scale * (1 + tail).  */
  idx = 2 * (ki % N);
  top = ki << (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;
  /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */
  /* Evaluation is optimized assuming superscalar pipelined execution.  */
  r2 = r * r;
  /* Without fma the worst case error is 0.5/N ulp larger.  */
  /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */
#if EXP2_POLY_ORDER == 4
  tmp = tail + r * C1 + r2 * C2 + r * r2 * (C3 + r * C4);
#elif EXP2_POLY_ORDER == 5
  tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
#elif EXP2_POLY_ORDER == 6
  tmp = tail + r * C1 + 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^-65 and scale > 2^-928, so there
     is no spurious underflow here even without fma.  */
  return scale + scale * tmp;
}
#endif