math/pow.c

/*
 * Double-precision x^y function.
 *
 * Copyright (c) 2018, Arm Limited.
 * SPDX-License-Identifier: MIT
 */

#include <float.h>
#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.  IX is the bit representation of x, but
   normalized in the subnormal range using the sign bit 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;

  /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
     |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible.  */
#if HAVE_FAST_FMA
  r = fma (z, invc, -1.0);
#else
  /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|.  */
  double_t zhi = asdouble ((iz + (1ULL << 31)) & (-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 (eval_as_double (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 (eval_as_double (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_t x, double_t 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 eval_as_double (scale + scale * tmp);
}

/* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
   the bit representation of a non-zero finite floating-point value.  */
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.  */
	  /* Without the barrier some versions of clang evalutate the mul
	     unconditionally causing spurious overflow exceptions.  */
	  ix = asuint64 (opt_barrier_double (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);
}
#if USE_GLIBC_ABI
strong_alias (pow, __pow_finite)
hidden_alias (pow, __ieee754_pow)
# if LDBL_MANT_DIG == 53
long double powl (long double x, long double y) { return pow (x, y); }
# endif
#endif